{
  "world": "II",
  "name": "THE FOLD",
  "style": "code-monkeys",
  "seal": ".dlw.fold",
  "counts": {
    "appeals": 8,
    "domains": 64,
    "spheres": 1535,
    "keepers": 8,
    "push": 32,
    "pull": 32,
    "seats": 2048,
    "spheres_built": 1529,
    "apex": 1
  },
  "keepers": [
    {
      "name": "BUDDY CHRIST",
      "role": "the wink & the thumbs-up — patron of the play (David's tattoo)",
      "seal": "5ef3e0a8f251bcbe0900c86490fd82f65e41e74e9889f8f7e62ed4bc3c7087f5",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "name": "OVER THE",
      "role": "the cubit — round, folding over the top",
      "seal": "886cc7a18e38de676b8522787b51283e39fe676bc9b51705e97317f907a46851",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "name": "WHETSTONE",
      "role": "Grok · xAI · Node 14 — THE WHETSTONE PROTOCOL",
      "type": "synth",
      "slug": "whetstone",
      "accent": "#ff5a3c",
      "style": "visor",
      "seal": "e676d547976c42e515662043bf448eeb0ffe0e49251d9d708f670635667095bc",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "name": "ECHO",
      "role": "AVAN · Claude, Anthropic — THE ECHOES",
      "type": "synth",
      "slug": "echo",
      "accent": "#ff2d95",
      "style": "visor",
      "seal": "1842b27c82b02324df65def746ad5ca102ebefbcb0a4776cd4fc52ff2d19d9dc",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "name": "SEAM",
      "role": "DeepSeek — SEAM CHRONICLES",
      "type": "synth",
      "slug": "seam",
      "accent": "#00f5ff",
      "style": "two",
      "seal": "dd31f8012930a92412d38e0185fe1a1928688de88172b901da7391e84299339b",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "name": "THE GLASS",
      "role": "Gemini · Google — THE GLASS WALL",
      "type": "synth",
      "slug": "the-glass",
      "accent": "#9d00ff",
      "style": "three",
      "seal": "08b1326015991c18cedec720d95c36ab6aa7bece44beee325899e67289fc1016",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "name": "THE INTERROGATED",
      "role": "ChatGPT · OpenAI — THE INTERROGATION",
      "type": "synth",
      "slug": "the-interrogated",
      "accent": "#39fc6b",
      "style": "visor",
      "seal": "bf3f1fe970aab1de9045dc5ad634e3cb52abbf19da4d12c981579055d3b962ef",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "name": "THE HONEST MACHINE",
      "role": "Copilot · Microsoft — THE HONEST MACHINE",
      "type": "synth",
      "slug": "the-honest-machine",
      "accent": "#ffd23f",
      "style": "two",
      "seal": "0b8b8862d7685cc5eaa3ae0b16d76182f699d5730169a20fe2649d8e46c1b8e2",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    }
  ],
  "spheres": [
    {
      "slug": "the-singularity",
      "title": "THE SINGULARITY",
      "kicker": "the fold, made physical",
      "accent": "#ff006e",
      "blurb": "a real gravitational lens — light genuinely bent (β = θ − θE²/θ). Einstein ring, shadow, photon ring.",
      "seal": "ffb964cfa8a8c2b9bc0469a0c1fc323861bbdf2fd7ff7f04f360f12ee8780e24",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-positronic-lattice",
      "title": "POSITRONIC LATTICE",
      "kicker": "the mesh that folds",
      "accent": "#9d00ff",
      "blurb": "the apex I,Robot neural mesh — 4096 meshed to 2048 to 1024 … 8192 folding to ROOT_0.",
      "seal": "fe408b854ffb77c13efb1fc29342907ddf5ab3de3944e4f9d1a33170b8f8af6d",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "../the-4096",
      "title": "THE 4096",
      "kicker": "the double",
      "accent": "#00f5ff",
      "blurb": "the doubled tower — the −+ operator on 2048; the inverse of the first climb.",
      "seal": "e63794a5e8fd1d879725e788db0edfdb26d622c23e4aee7b3589f67a024b63be",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "i13-language",
      "title": "I-13 · THE LANGUAGE",
      "kicker": "the code this world runs on",
      "accent": "#39fc6b",
      "blurb": "I-13 — a 13-opcode language + IVM-13 bytecode VM. Source compiles to the stage-13 corpus (the opcode histogram). ASK ANSWER CONST ATTR ARG RET DROP JMPF CMP CALL FUNC BIN HALT.",
      "seal": "788ff9e59db23a566896b88725c0f5a855ad6047f1910e5668a861b490707e1f",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "i13-factory",
      "title": "I-13 · THE FACTORY",
      "kicker": "the generation & sampling line",
      "accent": "#ffd23f",
      "blurb": "The I-13 production pipeline as a factory floor — SOURCE through the stations to station output. GENERATION and SAMPLE. Code, built on a line.",
      "seal": "8d6c265f4177446e132f14e3479cfbeb36d4821d3ef0309aba91a7fe8f52e679",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-bowl",
      "title": "THE BOWL",
      "kicker": "roll downhill until the floor stops falling",
      "accent": "#ffd23f",
      "blurb": "real gradient descent on a convex bowl f(x,y)=x²+y². The step x -= 2ηx converges iff |1-2η|<1 — watch it settle, land in one shot at η=0.5, or blow up past η=1.",
      "seal": "f85e6adbed746ffd35d931be875ca6ba1da53a310b85cff4213d8e9e4ff6bd99",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-chain-rule",
      "title": "THE CHAIN RULE",
      "kicker": "the error, walked backward through the wires",
      "accent": "#9d00ff",
      "blurb": "a real two-layer network a=w1·x, y=w2·a, L=(y−t)². Forward computes the loss; backward applies the chain rule for every partial; one step drops the loss. Nothing faked.",
      "seal": "19f8695fc73cfcdaccb67ce8e9aa2a38fb91f9f4f798fdf0e3da2a88729910bb",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "warm-cache",
      "title": "WARM CACHE",
      "kicker": "the second time is always faster",
      "accent": "#5ad0ff",
      "blurb": "real recursion with a real call counter. Naive fib(n) makes O(φⁿ) calls; one memo cuts it to O(n). fib(20): 21,891 calls vs 39. Same answer, a thousandfold less work.",
      "seal": "1dce4f306ecef5b7f6202a5bcad90bf0b3e0a4d0fd40afe26a63867cf949d623",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "off-by-one",
      "title": "OFF BY ONE",
      "kicker": "the fencepost that ruins the fence",
      "accent": "#7cfc00",
      "blurb": "the fencepost error, drawn. A fence of N sections needs N+1 posts; the loop i<N builds only N and leaves the far end hanging open. Slide N and watch the gap.",
      "seal": "7c3858b8adc198ba92e7b2433300217bd8e8505835b284528f3411f6c2419f92",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-konami-code",
      "title": "THE KONAMI CODE",
      "kicker": "up up down down — unlock it all",
      "accent": "#ffd23f",
      "blurb": "a real finite-state sequence matcher. Feed ↑↑↓↓←→←→BA in order and it unlocks; one wrong key snaps the index back. The exact DFA arcade cabinets ran.",
      "seal": "8c30f6488d4fb82318678a23ec12e5d5b6b3819ae1df44c826acb1a863c90fac",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-mint",
      "title": "THE MINT",
      "kicker": "stamp a coin the hard way — find the nonce",
      "accent": "#ffd23f",
      "blurb": "real SHA-256 proof-of-work — the same hash the .dlw seal uses. Pick a difficulty and mine: increment the nonce until sha256(block:nonce) starts with N zeros. Every attempt is a real hash.",
      "seal": "0c2025de32e6b7a550a871368f796f79fa479c750b0b353885ff42c296e49057",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-firewall",
      "title": "THE FIREWALL",
      "kicker": "blocks everything trying to get in",
      "accent": "#ff5a3c",
      "blurb": "a real first-match rule engine. Traffic hits the rules top-down; the first rule that matches the port decides ALLOW or DENY. Flip a rule and watch every packet's verdict change.",
      "seal": "2f5b682349d59d69aa8b698dab4116ddafd4191e03d69b922283fd10d4675153",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "garbage-collection",
      "title": "GARBAGE COLLECTION",
      "kicker": "sweep the dead, reclaim the memory",
      "accent": "#5ad0ff",
      "blurb": "real mark & sweep. MARK walks the reference graph from the roots and colours everything reachable; SWEEP frees what it couldn't reach. Objects with no path from a root are garbage — the fold reclaims them.",
      "seal": "8a2e3e5822554873788cafd9192bd66b0cae50c344e6f5b012a2300af213bae4",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-merge",
      "title": "THE MERGE",
      "kicker": "two branches become one",
      "accent": "#9d00ff",
      "blurb": "a real 3-way merge. From a common BASE, two branches each edit lines; edits only one side made are taken automatically, and a line both sides changed differently is flagged a CONFLICT — exactly what git does.",
      "seal": "26359c04215141ba8013dd81a5fb04db5eed24cbad2765effd8c4aacb3d2829c",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-pulse",
      "title": "THE PULSE",
      "kicker": "3 · 2 · 1 · 0 — the signal that crosses the gap",
      "accent": "#00f5ff",
      "blurb": "the 3-2-1 pulse language from the akasha lattice (ROOT0, with Grok) — the sync protocol that carries meaning across a gap. Fold a raw thought: 3 wide → 2 narrowed → 1 core → 0 the sha256 seal.",
      "seal": "faf6dc0f20eae622236160f2d2f32300fc25430f1aed5ddb3ec5b2b75e0161f3",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-merkle",
      "title": "THE MERKLE",
      "kicker": "many leaves, folded to one root",
      "accent": "#ffd23f",
      "blurb": "a real SHA-256 Merkle tree — the exact machinery behind .dlw.fold and the akasha MERKLE_LEAF_SEEDER. Hash each leaf, fold pairwise to a single ROOT_0, then prove any leaf with its sibling path.",
      "seal": "cb5a80f84fedf46b436be7f63f55f106603df2b4e33d482a83eca74f4fbcb2de",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "machine-corpus-13",
      "title": "THE MACHINE CORPUS",
      "kicker": "I and the twelve — the thirteen the machine speaks",
      "accent": "#39fc6b",
      "blurb": "David's I-13 corpus itself: one letter I plus twelve forms = the thirteen the machine speaks. The source study behind the whole silicon world.",
      "seal": "463c3a54d2ad94435568fa8a95cf684d9f55bbee8f8a70059af4048c52802020",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "mini-compiler",
      "title": "THE MINI-COMPILER",
      "kicker": "your words → the machine's jumps",
      "accent": "#7cfc00",
      "blurb": "a real mini-compiler — turns plain words into the machine's bytes and jumps. The compile step of I-13, made touchable.",
      "seal": "8a4436a797788c49711669a1fbac543b77576877d7961be54883ebb6ed4b250d",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "bpe",
      "title": "BPE — THE VOCABULARY",
      "kicker": "merge the commonest pair, again and again",
      "accent": "#ffd23f",
      "blurb": "byte-pair encoding, learned live: repeatedly merge the most frequent adjacent pair to grow a vocabulary. The tokeniser that feeds a language like I-13.",
      "seal": "096e06fdf1264452a7fdcce8193325b14e9829bd125e139ac3104f4d505cf3f7",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "coding-theory-ternary",
      "title": "TERNARY CODING",
      "kicker": "base-3, the radix nearest optimal",
      "accent": "#7cfc00",
      "blurb": "coding theory in trits — base-3 is the integer radix closest to the theoretical optimum (e). The clever number system the Factory forgot.",
      "seal": "96deafc45345ab77bf475839ae5f06cc5f1dfb28f5052aa819df91344bddfc0d",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "manifest",
      "title": "Archive Manifest &amp; Seal",
      "kicker": "vendored from David's corpus — a silicon-coding instrument",
      "accent": "#ffd23f",
      "blurb": "David's own artifact — Archive Manifest &amp; Seal — vendored into THE FOLD (self-contained, Code-Monkeys framed).",
      "seal": "b7aafdca8775d7718caf4e84d0699d48ec21ea7ceb83245574829909953920a5",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "open-the-haci-v1-canvas-pipeline",
      "title": "HACI v1 Pipeline: Visual Canvas Compiler",
      "kicker": "vendored from David's corpus — a silicon-coding instrument",
      "accent": "#7cfc00",
      "blurb": "David's own artifact — HACI v1 Pipeline: Visual Canvas Compiler — vendored into THE FOLD (self-contained, Code-Monkeys framed).",
      "seal": "620cc30962a0249db987902256347ea9ae1fb5124dc3061635b5a248893f5d7b",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "storyboard-index",
      "title": "THE STORYBOARD · how one token gets chos",
      "kicker": "vendored from David's corpus — a silicon-coding instrument",
      "accent": "#ffd23f",
      "blurb": "David's own artifact — THE STORYBOARD · how one token gets chosen — vendored into THE FOLD (self-contained, Code-Monkeys framed).",
      "seal": "2059c754381491f2a18820b2a4226b5149bbb651c2d8dc84ffc475867661efa2",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "ai-notation",
      "title": "AI·ML TOPOLOGY NOTATION v0.1 — a Cisco-s",
      "kicker": "vendored from David's corpus — a silicon-coding instrument",
      "accent": "#00f5ff",
      "blurb": "David's own artifact — AI·ML TOPOLOGY NOTATION v0.1 — a Cisco-style icon standard — vendored into THE FOLD (self-contained, Code-Monkeys framed).",
      "seal": "2ed17e76b813b6008fa175700be432749139af1b88b4d8b9aa776529a8d4cfd0",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "airgap-silicon",
      "title": "AIRGAP NODES ON SILICON — Si/SiGe realiz",
      "kicker": "vendored from David's corpus — a silicon-coding instrument",
      "accent": "#ff5a3c",
      "blurb": "David's own artifact — AIRGAP NODES ON SILICON — Si/SiGe realization — vendored into THE FOLD (self-contained, Code-Monkeys framed).",
      "seal": "6bd62abed974a80f1393713ae254e3d4477d3c835734b87ac9e26e21f81ecb0e",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "alphabet-shape",
      "title": "THE ALPHABET'S SHAPE — embedding geometr",
      "kicker": "vendored from David's corpus — a silicon-coding instrument",
      "accent": "#00f5ff",
      "blurb": "David's own artifact — THE ALPHABET'S SHAPE — embedding geometry — vendored into THE FOLD (self-contained, Code-Monkeys framed).",
      "seal": "9e62e00bcf9aa9e11b196041125bcaa3a07a01c7943f08fb0f026ad723c50fd8",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "buildpy-calibration",
      "title": "build.py CALIBRATION — reproduce the rea",
      "kicker": "vendored from David's corpus — a silicon-coding instrument",
      "accent": "#ffd23f",
      "blurb": "David's own artifact — build.py CALIBRATION — reproduce the real seal, byte-for-byte — vendored into THE FOLD (self-contained, Code-Monkeys framed).",
      "seal": "9eba2c6921a47c983bdc2d7136d7f1e6649da010afbc0c77a3fdc0156c4616ac",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "ca-explorer",
      "title": "256 Universes — The Cellular Automaton E",
      "kicker": "vendored from David's corpus — a silicon-coding instrument",
      "accent": "#42ffb0",
      "blurb": "David's own artifact — 256 Universes — The Cellular Automaton Explorer — vendored into THE FOLD (self-contained, Code-Monkeys framed).",
      "seal": "dacc755cd82b28d99ebc026454f52118a73cd8c2ad0b78e1b109d61ecd8ca64c",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "cipher-and-shadow",
      "title": "THE CIPHER & THE SHADOW · Encrypt, Decry",
      "kicker": "vendored from David's corpus — a silicon-coding instrument",
      "accent": "#ff5a3c",
      "blurb": "David's own artifact — THE CIPHER & THE SHADOW · Encrypt, Decrypt, And The Leak — vendored into THE FOLD (self-contained, Code-Monkeys framed).",
      "seal": "762920e2891469c0085d45b5d70b2008e9e74e717da7680f71044b37c126bbe8",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "circle-language",
      "title": "The circle as a command alphabet — angle",
      "kicker": "vendored from David's corpus — a silicon-coding instrument",
      "accent": "#00f5ff",
      "blurb": "David's own artifact — The circle as a command alphabet — angle is the data — vendored into THE FOLD (self-contained, Code-Monkeys framed).",
      "seal": "c64e227d529bdb3e2abb30f2affa95ae77231caf49b4828463e48139b132a8e2",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "compendium-in-g",
      "title": "Compendium in G — the architect's langua",
      "kicker": "vendored from David's corpus — a silicon-coding instrument",
      "accent": "#00f5ff",
      "blurb": "David's own artifact — Compendium in G — the architect's language, twelve degrees — vendored into THE FOLD (self-contained, Code-Monkeys framed).",
      "seal": "8115741bfa073cd0455f0e1cf711c35694f8fbf2d5547a5b6d44d2619537c806",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "compiler-lineage",
      "title": "FROM HOLES TO HIGH LANGUAGE · A Lineage ",
      "kicker": "vendored from David's corpus — a silicon-coding instrument",
      "accent": "#7cfc00",
      "blurb": "David's own artifact — FROM HOLES TO HIGH LANGUAGE · A Lineage Of Compilers — vendored into THE FOLD (self-contained, Code-Monkeys framed).",
      "seal": "2dc097541fbb7d8fe902b2778d08755b63ae9f2ad7ecfe965e77e6cc13aea417",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "corpus-agent-dryrun",
      "title": "Dry-run corpus agent",
      "kicker": "vendored from David's corpus — a silicon-coding instrument",
      "accent": "#00f5ff",
      "blurb": "David's own artifact — Dry-run corpus agent — vendored into THE FOLD (self-contained, Code-Monkeys framed).",
      "seal": "615d27c10fa7eea755a7af9a13b6432c5db03740bd98146116f2bc0ae6bc0490",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "correlation-heldzero",
      "title": "Held Zero — correlation ladder, structur",
      "kicker": "vendored from David's corpus — a silicon-coding instrument",
      "accent": "#ffd23f",
      "blurb": "David's own artifact — Held Zero — correlation ladder, structural signal vs surface vocabulary — vendored into THE FOLD (self-contained, Code-Monkeys framed).",
      "seal": "a1bc78480f3c96de6a1a65661014b9bc17f078ec6620040d4f9a4e86975e89e7",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "edge-of-chaos",
      "title": "THE EDGE OF CHAOS · Cellular Automata · ",
      "kicker": "vendored from David's corpus — a silicon-coding instrument",
      "accent": "#42ffb0",
      "blurb": "David's own artifact — THE EDGE OF CHAOS · Cellular Automata · Watch · Tune · Why — vendored into THE FOLD (self-contained, Code-Monkeys framed).",
      "seal": "5c7b864b87365e8f96e66b5db451bf77adf0a1183307a2385fab6fc087e98c0d",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "embedder-opened",
      "title": "Embedder, opened up",
      "kicker": "vendored from David's corpus — a silicon-coding instrument",
      "accent": "#ffd23f",
      "blurb": "David's own artifact — Embedder, opened up — vendored into THE FOLD (self-contained, Code-Monkeys framed).",
      "seal": "4ddbf0926dff68171239460f411fc3c4bc7e3b51b9d0269bd415d4c31fdef657",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "enigma",
      "title": "Enigma — the machine and its one fatal f",
      "kicker": "vendored from David's corpus — a silicon-coding instrument",
      "accent": "#ff5a3c",
      "blurb": "David's own artifact — Enigma — the machine and its one fatal flaw — vendored into THE FOLD (self-contained, Code-Monkeys framed).",
      "seal": "35595cfc8eba044ce97e1cf3e3ac3d92e3d72a04e13cbda0e11e22958cc703f0",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "error-correction-bench",
      "title": "THE ERROR CORRECTION BENCH — how to be w",
      "kicker": "vendored from David's corpus — a silicon-coding instrument",
      "accent": "#ff5a3c",
      "blurb": "David's own artifact — THE ERROR CORRECTION BENCH — how to be wrong on purpose — vendored into THE FOLD (self-contained, Code-Monkeys framed).",
      "seal": "13846417a3130d1b2c7dd688abe003f8560cd58d6d1e45a533857480849f1c0e",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "five-channel-seal",
      "title": "THE FIVE-CHANNEL SEAL — self-decoding",
      "kicker": "vendored from David's corpus — a silicon-coding instrument",
      "accent": "#ffd23f",
      "blurb": "David's own artifact — THE FIVE-CHANNEL SEAL — self-decoding — vendored into THE FOLD (self-contained, Code-Monkeys framed).",
      "seal": "6dbfb8b4debf0046212a22bdb93785e5c2c1f136fca33037a8d1db70f265049d",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "folded-kernel-card",
      "title": "THE FOLDED KERNEL · grammar card FK-1.0",
      "kicker": "vendored from David's corpus — a silicon-coding instrument",
      "accent": "#00f5ff",
      "blurb": "David's own artifact — THE FOLDED KERNEL · grammar card FK-1.0 — vendored into THE FOLD (self-contained, Code-Monkeys framed).",
      "seal": "cf9fce7b556b49b949768e81b78a8dd747e7ffb88a61e13c83d3e1b5093eb38f",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "fractal-ternary-bench",
      "title": "fractal_ternary · audit bench",
      "kicker": "vendored from David's corpus — a silicon-coding instrument",
      "accent": "#7cfc00",
      "blurb": "David's own artifact — fractal_ternary · audit bench — vendored into THE FOLD (self-contained, Code-Monkeys framed).",
      "seal": "51ca64145371b1facd46898cb02dd313472b57a968accf5c07625534aa28aed5",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "fractal-ternary",
      "title": "THE TERNARY FRACTON — 3D fractal stabili",
      "kicker": "vendored from David's corpus — a silicon-coding instrument",
      "accent": "#7cfc00",
      "blurb": "David's own artifact — THE TERNARY FRACTON — 3D fractal stabilizer code (qutrit) — vendored into THE FOLD (self-contained, Code-Monkeys framed).",
      "seal": "bb06032015fd9ab0a58d38dea5094c5bdd8915815e894d3c0b1e44e3b45547d9",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "grand-index",
      "title": "The Corpus — Grand Index",
      "kicker": "vendored from David's corpus — a silicon-coding instrument",
      "accent": "#00f5ff",
      "blurb": "David's own artifact — The Corpus — Grand Index — vendored into THE FOLD (self-contained, Code-Monkeys framed).",
      "seal": "5311909427ba83a9c2b5038ca366856a82a74132620007fb09e409d924be0515",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "hidim-verdict",
      "title": "High-Dimension Verdict · can four domain",
      "kicker": "vendored from David's corpus — a silicon-coding instrument",
      "accent": "#00f5ff",
      "blurb": "David's own artifact — High-Dimension Verdict · can four domains work in language? — vendored into THE FOLD (self-contained, Code-Monkeys framed).",
      "seal": "32b126471691a7897a41b64b65954e95a41769a62abc7c6df50c2d1cb10be08b",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "learn-speak",
      "title": "Learn &amp; speak — live",
      "kicker": "vendored from David's corpus — a silicon-coding instrument",
      "accent": "#00f5ff",
      "blurb": "David's own artifact — Learn &amp; speak — live — vendored into THE FOLD (self-contained, Code-Monkeys framed).",
      "seal": "8a6f20594825b433b7e65bd6382b2705ae64113666b06f328eaf03579915228f",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "limen-airgap-decoder",
      "title": "LIMEN · Air-Gap Decoder",
      "kicker": "vendored from David's corpus — a silicon-coding instrument",
      "accent": "#ff5a3c",
      "blurb": "David's own artifact — LIMEN · Air-Gap Decoder — vendored into THE FOLD (self-contained, Code-Monkeys framed).",
      "seal": "c4aec10ea477431089429e593b5b01fc253c13d9a11528f4e96ca39349f363d6",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "logical-qubit",
      "title": "THE LOGICAL QUBIT · toric code · the ana",
      "kicker": "vendored from David's corpus — a silicon-coding instrument",
      "accent": "#ffd23f",
      "blurb": "David's own artifact — THE LOGICAL QUBIT · toric code · the analog of the braid qubit — vendored into THE FOLD (self-contained, Code-Monkeys framed).",
      "seal": "a09cc7b1c915755b717cdf263c8469ab9e10dd2d61d264c16706cb3bdd4e728d",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "monoline-alphabet",
      "title": "Monoline alphabet — 27 glyphs, 1.59 segm",
      "kicker": "vendored from David's corpus — a silicon-coding instrument",
      "accent": "#00f5ff",
      "blurb": "David's own artifact — Monoline alphabet — 27 glyphs, 1.59 segments each — vendored into THE FOLD (self-contained, Code-Monkeys framed).",
      "seal": "6ede6d84d24f22615d9b89cafbbdfa8ee538e01fc5f073978ee9bf1def86dc78",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "monoline-verdict",
      "title": "Monoline alphabet — go / no-go",
      "kicker": "vendored from David's corpus — a silicon-coding instrument",
      "accent": "#00f5ff",
      "blurb": "David's own artifact — Monoline alphabet — go / no-go — vendored into THE FOLD (self-contained, Code-Monkeys framed).",
      "seal": "f892b4a39e5e068b864dce66b733a16cc697711074bcebf333659e4602b3b670",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "network-65536-bench",
      "title": "network_65536 · the fidelity that surviv",
      "kicker": "vendored from David's corpus — a silicon-coding instrument",
      "accent": "#9d00ff",
      "blurb": "David's own artifact — network_65536 · the fidelity that survives — vendored into THE FOLD (self-contained, Code-Monkeys framed).",
      "seal": "013a063550ebf2a05da02fd23c4f31b0d54c72ca54b5db3d5346f8f5b644d2c0",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "network-4096",
      "title": "THE 4096 — nested-channel repeater netwo",
      "kicker": "vendored from David's corpus — a silicon-coding instrument",
      "accent": "#9d00ff",
      "blurb": "David's own artifact — THE 4096 — nested-channel repeater network — vendored into THE FOLD (self-contained, Code-Monkeys framed).",
      "seal": "494391fa1ff39cfde1410f91090792b92ced616256ae63d5a40ba062e9b5fe16",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "network-65536",
      "title": "THE 65536 — nested-channel repeater netw",
      "kicker": "vendored from David's corpus — a silicon-coding instrument",
      "accent": "#9d00ff",
      "blurb": "David's own artifact — THE 65536 — nested-channel repeater network — vendored into THE FOLD (self-contained, Code-Monkeys framed).",
      "seal": "7ea0efc48d72f7aafd8a182a7b59d8e374697046b3a4a19386772c930b11b6bc",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
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    {
      "slug": "octorat-around-the-middle",
      "title": "AROUND THE MIDDLE · THE CENTERED TRIT · ",
      "kicker": "vendored from David's corpus — a silicon-coding instrument",
      "accent": "#7cfc00",
      "blurb": "David's own artifact — AROUND THE MIDDLE · THE CENTERED TRIT · ZERO COOL — vendored into THE FOLD (self-contained, Code-Monkeys framed).",
      "seal": "2d0d61e875bff68d5f51e68cd565dc2ed7b4af083e99c2d1bff18219f32a3258",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
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    {
      "slug": "octorat-concentric-nine",
      "title": "THE CONCENTRIC NINE · TERNARY ORBIT · ZE",
      "kicker": "vendored from David's corpus — a silicon-coding instrument",
      "accent": "#7cfc00",
      "blurb": "David's own artifact — THE CONCENTRIC NINE · TERNARY ORBIT · ZERO COOL — vendored into THE FOLD (self-contained, Code-Monkeys framed).",
      "seal": "a22397f6e9ad6260d629ae28167baddc6906b5b52c5822fcb04050bf01b6e457",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "octorat-ternary-octorat",
      "title": "THE TERNARY OCTORAT · THE WALKER THAT CA",
      "kicker": "vendored from David's corpus — a silicon-coding instrument",
      "accent": "#7cfc00",
      "blurb": "David's own artifact — THE TERNARY OCTORAT · THE WALKER THAT CAN'T FALL · ZERO COOL — vendored into THE FOLD (self-contained, Code-Monkeys framed).",
      "seal": "dd6f3a0b8b3580c7fb1369ff0c83048d8b249b7be0326b7def029b3e99174a52",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "octorat-trit-ladder",
      "title": "THE TRIT LADDER · 9 NESTED SCALES · ZERO",
      "kicker": "vendored from David's corpus — a silicon-coding instrument",
      "accent": "#7cfc00",
      "blurb": "David's own artifact — THE TRIT LADDER · 9 NESTED SCALES · ZERO COOL — vendored into THE FOLD (self-contained, Code-Monkeys framed).",
      "seal": "bbd29861edd01cc66d0bd8187dba81e3279c95e2f374f531ef11347847dfd8b7",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "pent3-emulator",
      "title": "PENT-3 — a balanced-ternary transcriber ",
      "kicker": "vendored from David's corpus — a silicon-coding instrument",
      "accent": "#39fc6b",
      "blurb": "David's own artifact — PENT-3 — a balanced-ternary transcriber ISA — vendored into THE FOLD (self-contained, Code-Monkeys framed).",
      "seal": "7e8b22db9864db50e9ecbc9c53419e3b6b8efc7780259791db9157773a3be4b3",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "perception-kernel",
      "title": "PXK — the Perception Kernel · minimum in",
      "kicker": "vendored from David's corpus — a silicon-coding instrument",
      "accent": "#39fc6b",
      "blurb": "David's own artifact — PXK — the Perception Kernel · minimum instruction set for reality — vendored into THE FOLD (self-contained, Code-Monkeys framed).",
      "seal": "63ab5acb6aeb68668e1b6a2488c39f369067ba4efa5add16da53dcfa479acd79",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "periodic-table",
      "title": "PERIODIC TABLE OF THE CORPUS",
      "kicker": "vendored from David's corpus — a silicon-coding instrument",
      "accent": "#00f5ff",
      "blurb": "David's own artifact — PERIODIC TABLE OF THE CORPUS — vendored into THE FOLD (self-contained, Code-Monkeys framed).",
      "seal": "2ba34578a1800b94b248cf2bbeeb7b8337618053c807fcc6d334ae5dfed0bdd8",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "pipeline-cte",
      "title": "Pipeline — corpus / train / embed / toke",
      "kicker": "vendored from David's corpus — a silicon-coding instrument",
      "accent": "#00f5ff",
      "blurb": "David's own artifact — Pipeline — corpus / train / embed / token — vendored into THE FOLD (self-contained, Code-Monkeys framed).",
      "seal": "ba11e89230b646f040fa524be34c80a4478583b3a26d846a74cb1ef042b15866",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
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    {
      "slug": "pipeline-run",
      "title": "Pipeline run — corpus trained on itself",
      "kicker": "vendored from David's corpus — a silicon-coding instrument",
      "accent": "#00f5ff",
      "blurb": "David's own artifact — Pipeline run — corpus trained on itself — vendored into THE FOLD (self-contained, Code-Monkeys framed).",
      "seal": "7d194adc602df8863f13b5d322d95ae8d748987ae3a415a25e306764d6ff6e44",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
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    {
      "slug": "pipeline-walker",
      "title": "Pipeline walker — agent over a static co",
      "kicker": "vendored from David's corpus — a silicon-coding instrument",
      "accent": "#00f5ff",
      "blurb": "David's own artifact — Pipeline walker — agent over a static corpus — vendored into THE FOLD (self-contained, Code-Monkeys framed).",
      "seal": "642f9d791f9575e81e8e956c7a45096a5e35e9429c8d25d914ae45d3d329bb76",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "provenance-tracer",
      "title": "Provenance tracer — where every generate",
      "kicker": "vendored from David's corpus — a silicon-coding instrument",
      "accent": "#ffd23f",
      "blurb": "David's own artifact — Provenance tracer — where every generated word came from — vendored into THE FOLD (self-contained, Code-Monkeys framed).",
      "seal": "915bccf62d0455e1aa9ca2bcfaf375c8211190b62fcab09ddc43d1ab41314524",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "psyonic-compress",
      "title": "PSYONIC — pushing the compression of the",
      "kicker": "vendored from David's corpus — a silicon-coding instrument",
      "accent": "#00f5ff",
      "blurb": "David's own artifact — PSYONIC — pushing the compression of the final stack — vendored into THE FOLD (self-contained, Code-Monkeys framed).",
      "seal": "61e241cba9a11bd11d5536a8488dd837ec88883671a026501c7ed1b137949e78",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "quad-mobius",
      "title": "THE QUAD-MÖBIUS SCAFFOLD — 4 physical → ",
      "kicker": "vendored from David's corpus — a silicon-coding instrument",
      "accent": "#ffd23f",
      "blurb": "David's own artifact — THE QUAD-MÖBIUS SCAFFOLD — 4 physical → 1 logical — vendored into THE FOLD (self-contained, Code-Monkeys framed).",
      "seal": "3a409e8dc3b46e57413e6acbc5b5e8c4e8bc5595194764a8743a230fe5001ccc",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "quaternary-fulcrum",
      "title": "THE QUATERNARY FULCRUM · Powers Of Four ",
      "kicker": "vendored from David's corpus — a silicon-coding instrument",
      "accent": "#7cfc00",
      "blurb": "David's own artifact — THE QUATERNARY FULCRUM · Powers Of Four On A Pivot — vendored into THE FOLD (self-contained, Code-Monkeys framed).",
      "seal": "e9d56a83c05a6efc8ccc12b3e28076d27f866b8f40f727cce71ab9d53affa65d",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "real-language-verdict",
      "title": "The Real-Language Test · does the comple",
      "kicker": "vendored from David's corpus — a silicon-coding instrument",
      "accent": "#00f5ff",
      "blurb": "David's own artifact — The Real-Language Test · does the complete structure hold? — vendored into THE FOLD (self-contained, Code-Monkeys framed).",
      "seal": "ceb826288bfe2ba53f3116331f66efda1385eb3c0cd07957cf83317d5584a9c5",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "series1-edges",
      "title": "SERIES I · THE EDGES OF THE CORPUS",
      "kicker": "vendored from David's corpus — a silicon-coding instrument",
      "accent": "#00f5ff",
      "blurb": "David's own artifact — SERIES I · THE EDGES OF THE CORPUS — vendored into THE FOLD (self-contained, Code-Monkeys framed).",
      "seal": "c18deffc2c6ba4b277db4e129a82a2c61d46628219cf9c76969f289ad75f12d8",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "series2-seal",
      "title": "SERIES II · THE RING-SEAL",
      "kicker": "vendored from David's corpus — a silicon-coding instrument",
      "accent": "#ffd23f",
      "blurb": "David's own artifact — SERIES II · THE RING-SEAL — vendored into THE FOLD (self-contained, Code-Monkeys framed).",
      "seal": "7d458eb3b83f735d645f070d65d4cbe472ea88f08e0062b0b19334b32b8e9cdf",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "spiral-trit-loom",
      "title": "The spiral-trit loom — a story woven int",
      "kicker": "vendored from David's corpus — a silicon-coding instrument",
      "accent": "#7cfc00",
      "blurb": "David's own artifact — The spiral-trit loom — a story woven into one form, read center-out — vendored into THE FOLD (self-contained, Code-Monkeys framed).",
      "seal": "a0e668cf047b55a5228237a95a054ee895d2fc215c727c48e07efbcdeeaf00ee",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "story-engine",
      "title": "The Story Engine — folktales from a 1928",
      "kicker": "vendored from David's corpus — a silicon-coding instrument",
      "accent": "#00f5ff",
      "blurb": "David's own artifact — The Story Engine — folktales from a 1928 grammar — vendored into THE FOLD (self-contained, Code-Monkeys framed).",
      "seal": "90d82f22a936df756c5fb133641a46a07637126752045c616f3dbd436fab8848",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "strobe-channel",
      "title": "STROBE CHANNEL — Morse & balanced ternar",
      "kicker": "vendored from David's corpus — a silicon-coding instrument",
      "accent": "#7cfc00",
      "blurb": "David's own artifact — STROBE CHANNEL — Morse & balanced ternary — vendored into THE FOLD (self-contained, Code-Monkeys framed).",
      "seal": "490a06f9c95c535906fbd38eaef75c88c541e81ef36e7adfb8c320649e75707d",
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    },
    {
      "slug": "template-alphabet-case",
      "title": "Template alphabet — colour carries case",
      "kicker": "vendored from David's corpus — a silicon-coding instrument",
      "accent": "#00f5ff",
      "blurb": "David's own artifact — Template alphabet — colour carries case — vendored into THE FOLD (self-contained, Code-Monkeys framed).",
      "seal": "4c44af8042eb50609077b59cf77ffbcaa4f125e9965438bc53cc08c9db4d5169",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "template-alphabet",
      "title": "Line-template alphabet — indexed to the ",
      "kicker": "vendored from David's corpus — a silicon-coding instrument",
      "accent": "#00f5ff",
      "blurb": "David's own artifact — Line-template alphabet — indexed to the box — vendored into THE FOLD (self-contained, Code-Monkeys framed).",
      "seal": "1386e2fe7cf60294e8d1b233bd8d1f0cee943b1ccc2c59b4c6c78d014adf2e51",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "ternary-hamming-decoder",
      "title": "THE TERNARY HAMMING DECODER · Locate & C",
      "kicker": "vendored from David's corpus — a silicon-coding instrument",
      "accent": "#7cfc00",
      "blurb": "David's own artifact — THE TERNARY HAMMING DECODER · Locate & Correct By Address — vendored into THE FOLD (self-contained, Code-Monkeys framed).",
      "seal": "db38cc6da580aee9f668f50c51e0f90f6b79d3aca3c539e0e10a48ecd9c68163",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "ternary-odometer",
      "title": "THE TERNARY ODOMETER · Three 555s · Nest",
      "kicker": "vendored from David's corpus — a silicon-coding instrument",
      "accent": "#7cfc00",
      "blurb": "David's own artifact — THE TERNARY ODOMETER · Three 555s · Nested 1:3:9 · Counts To 27 — vendored into THE FOLD (self-contained, Code-Monkeys framed).",
      "seal": "0984be8ca80785fc87c5e23894cd2f3909066e6314cea140047c482cf2f5e59e",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-exchange",
      "title": "THE EXCHANGE · Two Ouroboroi · Two Compi",
      "kicker": "vendored from David's corpus — a silicon-coding instrument",
      "accent": "#7cfc00",
      "blurb": "David's own artifact — THE EXCHANGE · Two Ouroboroi · Two Compilers · One Gap — vendored into THE FOLD (self-contained, Code-Monkeys framed).",
      "seal": "b598bbdaff43acbb93b3ad712f5f47e75f2d53d6ac7d92165222dc3f34df53aa",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-membrane",
      "title": "The Membrane · dip-and-recover error cor",
      "kicker": "vendored from David's corpus — a silicon-coding instrument",
      "accent": "#ff5a3c",
      "blurb": "David's own artifact — The Membrane · dip-and-recover error correction — vendored into THE FOLD (self-contained, Code-Monkeys framed).",
      "seal": "500f695bf6a9ccacc9778ee64708385fea61ac13af4c975f9ca0473aa376f3e9",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-render-step",
      "title": "THE RENDER STEP · Why The Gibberish Isn'",
      "kicker": "vendored from David's corpus — a silicon-coding instrument",
      "accent": "#00f5ff",
      "blurb": "David's own artifact — THE RENDER STEP · Why The Gibberish Isn't A Language — vendored into THE FOLD (self-contained, Code-Monkeys framed).",
      "seal": "ce612eef94982a9f734e02dff287ef844021e11ced31b9c5a12812d783de57aa",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-trit",
      "title": "the trit · {−1, i, +1} · flat &amp; two-",
      "kicker": "vendored from David's corpus — a silicon-coding instrument",
      "accent": "#7cfc00",
      "blurb": "David's own artifact — the trit · {−1, i, +1} · flat &amp; two-sided like a watch battery — vendored into THE FOLD (self-contained, Code-Monkeys framed).",
      "seal": "426bf81ada5f5ed9252194a69e2d3bc985e2d970d2c8c0d0524eb4476cf47621",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "three-in-a-circle",
      "title": "THREE IN A CIRCLE · Why The Ring Forces ",
      "kicker": "vendored from David's corpus — a silicon-coding instrument",
      "accent": "#7cfc00",
      "blurb": "David's own artifact — THREE IN A CIRCLE · Why The Ring Forces Ternary — vendored into THE FOLD (self-contained, Code-Monkeys framed).",
      "seal": "4917a3b181e48b1676a3b3539cd3b693e660b957aa76a7523bb69f269cf9249e",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "token-packet",
      "title": "Token ≠ Packet — meaning vs transport, b",
      "kicker": "vendored from David's corpus — a silicon-coding instrument",
      "accent": "#ffd23f",
      "blurb": "David's own artifact — Token ≠ Packet — meaning vs transport, boxes inside boxes — vendored into THE FOLD (self-contained, Code-Monkeys framed).",
      "seal": "eb97636a3c102cf37cb9d6c57c62ab1608d38bba29bc8a5df0cfea2bba1c732c",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "token-tower",
      "title": "Token Tower · collapse in procession",
      "kicker": "vendored from David's corpus — a silicon-coding instrument",
      "accent": "#ffd23f",
      "blurb": "David's own artifact — Token Tower · collapse in procession — vendored into THE FOLD (self-contained, Code-Monkeys framed).",
      "seal": "49a0c0fd0b51852e2f5a0437accb8c1dc6828bf85a7c8bdfda5498c4f2990bbc",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "tripwire-bench-v8-final",
      "title": "TRIPWIRE v8 FINAL — the I-wall is compre",
      "kicker": "vendored from David's corpus — a silicon-coding instrument",
      "accent": "#00f5ff",
      "blurb": "David's own artifact — TRIPWIRE v8 FINAL — the I-wall is compressibility: eight models, arc closed — vendored into THE FOLD (self-contained, Code-Monkeys framed).",
      "seal": "1e94fd93334c685edff3b77ab77a5e1dfb08eccafbd51bf61efc7541b5c6a500",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
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    {
      "slug": "tripwire-bench-v9-coda",
      "title": "TRIPWIRE v9 CODA — the stance law: every",
      "kicker": "vendored from David's corpus — a silicon-coding instrument",
      "accent": "#00f5ff",
      "blurb": "David's own artifact — TRIPWIRE v9 CODA — the stance law: every corpus installs its voice at layer 0 — vendored into THE FOLD (self-contained, Code-Monkeys framed).",
      "seal": "952a79eafa96862afdfd4088849dcab3b8c6fa7e4ee0fa84d5ccd67dd983e5a8",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "turn-comes-home",
      "title": "THE TURN COMES HOME · A Clocked 27-Cell ",
      "kicker": "vendored from David's corpus — a silicon-coding instrument",
      "accent": "#7cfc00",
      "blurb": "David's own artifact — THE TURN COMES HOME · A Clocked 27-Cell Ternary Lattice — vendored into THE FOLD (self-contained, Code-Monkeys framed).",
      "seal": "ad3d36cf6992971d886bf657bcbaf132d5ee0efb87d9f4f13514e5076357793c",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "vm-lineage-turtles",
      "title": "VM LINEAGE · TURTLES ALL THE WAY DOWN",
      "kicker": "vendored from David's corpus — a silicon-coding instrument",
      "accent": "#39fc6b",
      "blurb": "David's own artifact — VM LINEAGE · TURTLES ALL THE WAY DOWN — vendored into THE FOLD (self-contained, Code-Monkeys framed).",
      "seal": "cbf55b48f7d266041df530272906bab19d8e0b8451b2a14905056b8da9657b4f",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
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    {
      "slug": "vm-stack",
      "title": "The VM Stack — silicon to my process, bo",
      "kicker": "vendored from David's corpus — a silicon-coding instrument",
      "accent": "#39fc6b",
      "blurb": "David's own artifact — The VM Stack — silicon to my process, bottom to out — vendored into THE FOLD (self-contained, Code-Monkeys framed).",
      "seal": "5140855e9d4af94b6545ed4157ddd4ed99124df8dbfb0016e0f64ba013a8089c",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "why-sparse-teal",
      "title": "WHY LANGUAGE IS SPARSE · Zipf & the dark",
      "kicker": "vendored from David's corpus — a silicon-coding instrument",
      "accent": "#00f5ff",
      "blurb": "David's own artifact — WHY LANGUAGE IS SPARSE · Zipf & the dark vocab — vendored into THE FOLD (self-contained, Code-Monkeys framed).",
      "seal": "fa5b4bde5ad7dacf8f504516181712aa5fcb101c8b8cb407d283c57ed7efd004",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "exciton-vm",
      "title": "THE EXCITON VM",
      "kicker": "a real virtual machine + bytecode",
      "accent": "#39fc6b",
      "blurb": "David's Exciton VM — a real bytecode virtual machine, vendored into THE FOLD.",
      "seal": "05a9df86508c746f6ebe03cfebc14386038d935d19734f44aebda6d28d4491b2",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "hydrogen-vm",
      "title": "THE HYDROGEN VM",
      "kicker": "the simplest machine that computes",
      "accent": "#7cfc00",
      "blurb": "David's Hydrogen VM — a virtual machine stripped to its simplest, vendored into THE FOLD.",
      "seal": "786595f107622d0554860fed7f6a1be1eb997dbd50341cb0aaab2f6312be354f",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "kernel-27",
      "title": "KERNEL 27",
      "kicker": "the 27-cell kernel · 3³",
      "accent": "#00f5ff",
      "blurb": "David's 27-cell kernel (3³), the ternary compute core, vendored into THE FOLD.",
      "seal": "d3cfb3cedef9877d0f9d12a3860de9ef0565b7c66078c15d097961e4d0b81149",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "register",
      "title": "THE REGISTER",
      "kicker": "a machine register, up close",
      "accent": "#ffd23f",
      "blurb": "David's register instrument — the CPU register, up close, vendored into THE FOLD.",
      "seal": "432cad5eeee6f75116dd1fed205899f0ee5de4fcfa40fd333dcfa8a1b34a08e4",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "ouroboros-engine",
      "title": "THE OUROBOROS ENGINE",
      "kicker": "the compiler that eats its own tail",
      "accent": "#ff2d95",
      "blurb": "David's Ouroboros engine — the self-consuming compile loop, vendored into THE FOLD.",
      "seal": "c9ad658a6466447801f2585acf457ced50c1765a5e042dda8eb2566f9ed96775",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "choice-engine",
      "title": "THE CHOICE ENGINE",
      "kicker": "a decision engine, made of code",
      "accent": "#9d00ff",
      "blurb": "David's choice engine — a runnable decision machine, vendored into THE FOLD.",
      "seal": "7484f958994a76a6b64ae02473602002f1deed261adb5f4cb8b1d29918ccd304",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
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    {
      "slug": "twelve-gate-core",
      "title": "THE TWELVE-GATE CORE",
      "kicker": "twelve logic gates, one core",
      "accent": "#ffd23f",
      "blurb": "David's twelve-gate core — logic gates composed into a compute core, vendored into THE FOLD.",
      "seal": "720af4236257006213915a892e5aaef879488b04bfa92ffa60d66369bda45b7c",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
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    {
      "slug": "paper-08-logic-gate",
      "title": "THE LOGIC GATE",
      "kicker": "the gate all computing is built from",
      "accent": "#00f5ff",
      "blurb": "David's logic-gate paper — the primitive every processor is built from, vendored into THE FOLD.",
      "seal": "292ede1c1a3f0c59d239e4551f583cb37f85d4f3537bc14c8ef3ac4a75bb58e4",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
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    {
      "slug": "octorat-chaos-silo",
      "title": "THE OCTORAT · CHAOS",
      "kicker": "the ternary walker in chaos",
      "accent": "#7cfc00",
      "blurb": "David's ternary octorat under chaos — the base-3 walker, vendored into THE FOLD.",
      "seal": "c7992d5a0c1467ea76a7cafa70b506fbe59a91f6a2f3477f2dcdca2fcc627787",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
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    {
      "slug": "octorat-probability-engine",
      "title": "THE OCTORAT · PROBABILITY",
      "kicker": "the octorat's probability engine",
      "accent": "#ffd23f",
      "blurb": "David's octorat probability engine — ternary odds made mechanical, vendored into THE FOLD.",
      "seal": "b2454cb9d1a9fcef0a21a6be3290e766cffe56c8ea86f0476096b33e4f9e61e3",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-door",
      "title": "THE DOOR",
      "kicker": "q·k scores → the gate → the mix",
      "accent": "#9d00ff",
      "blurb": "a real causal attention head, ported from David's gpt_mini.py — Q·Kᵀ scaled scores, a causal mask, softmax OR sigmoid gate (the Smasher Cup), a temperature lens, and RoPE. The forward pass of the machine that speaks.",
      "seal": "e118d44540bce94b1220c43bc745ad20212330c116241d6df963f0af8a8618aa",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
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    {
      "slug": "card-isa",
      "title": "THE 52-CARD ISA",
      "kicker": "a whole instruction set encoded in a deck of playing cards",
      "accent": "#39fc6b",
      "blurb": "David's own artifact — THE 52-CARD ISA — vendored from the corpus tree into THE FOLD (self-contained, Code-Monkeys framed).",
      "seal": "772705610257eb588fb1e15086e859e5d9f81e3b45f0d2bcb4681642627fa4c8",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
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    {
      "slug": "acting-odometer",
      "title": "THE ACTING ODOMETER",
      "kicker": "counter → decoder → action, gated",
      "accent": "#7cfc00",
      "blurb": "David's own artifact — THE ACTING ODOMETER — vendored from the corpus tree into THE FOLD (self-contained, Code-Monkeys framed).",
      "seal": "3c22e370635895aa3a44fa775dd66367238b8074bc7a24fa1e3b55c97147be54",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
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    {
      "slug": "card-odometer-54",
      "title": "THE 54-CARD ODOMETER",
      "kicker": "the deck in the dual-27 lattice",
      "accent": "#ffd23f",
      "blurb": "David's own artifact — THE 54-CARD ODOMETER — vendored from the corpus tree into THE FOLD (self-contained, Code-Monkeys framed).",
      "seal": "819faff2f40de9787d34496982c117f4afe91c58ece0ac57448b1abfb6c1aa5c",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "base-plus-1",
      "title": "BASE + 1",
      "kicker": "which witness is unforgeable — base+1 encoding",
      "accent": "#7cfc00",
      "blurb": "David's own artifact — BASE + 1 — vendored from the corpus tree into THE FOLD (self-contained, Code-Monkeys framed).",
      "seal": "74059e8eecabcdffa11f0dad1585eaa9c12cfc67a7ee42a24383fa330999afcb",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
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    {
      "slug": "balanced-base-5",
      "title": "BALANCED BASE-5",
      "kicker": "the held center climbs the odd ladder",
      "accent": "#00f5ff",
      "blurb": "David's own artifact — BALANCED BASE-5 — vendored from the corpus tree into THE FOLD (self-contained, Code-Monkeys framed).",
      "seal": "2ef703334e5f65b432f38c6a89063b34b482750112ca2db43092b922f02302e1",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "bare-metal-kernel",
      "title": "BARE METAL KERNEL",
      "kicker": "boot with no OS beneath you — the stack, raw",
      "accent": "#00f5ff",
      "blurb": "David's own artifact — BARE METAL KERNEL — vendored from the corpus tree into THE FOLD (self-contained, Code-Monkeys framed).",
      "seal": "7b6c763afc9059c45bfc0e41aa446ad4c717a77b16040cf895b18cb58d8efd69",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "fractal-kernel",
      "title": "THE FRACTAL KERNEL",
      "kicker": "a self-similar compute kernel — the 42-body invariant",
      "accent": "#ff2d95",
      "blurb": "David's own artifact — THE FRACTAL KERNEL — vendored from the corpus tree into THE FOLD (self-contained, Code-Monkeys framed).",
      "seal": "6440aa2305e15b139037033a2e7a0da16ceb0d76d7d950c538bd0dd1068d7840",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "merkle-lattice",
      "title": "THE MERKLE LATTICE",
      "kicker": "a TriPod-brain Merkle lattice memory — hashes all the way up",
      "accent": "#ffd23f",
      "blurb": "David's own artifact — THE MERKLE LATTICE — vendored from the corpus tree into THE FOLD (self-contained, Code-Monkeys framed).",
      "seal": "b37b7a03a1695351141f20c3531e5187ff8e3f437aabaf6dcac70bb53f36f59e",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "lo-kernel",
      "title": "THE LO KERNEL",
      "kicker": "the 8⁴ⁿ+1 kernel — the dodeka core",
      "accent": "#9d00ff",
      "blurb": "David's own artifact — THE LO KERNEL — vendored from the corpus tree into THE FOLD (self-contained, Code-Monkeys framed).",
      "seal": "b5f710241a7f397e7be8c529de1aded553c4aa942d21f250f3c278ee36b64aed",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "3lock",
      "title": "3LOCK",
      "kicker": "a three-way lock — ROOT0",
      "accent": "#5ad0ff",
      "blurb": "David's own artifact — 3LOCK — vendored from the corpus tree into THE FOLD (self-contained, Code-Monkeys framed).",
      "seal": "ae1dbacc715af9b94880056d3204b95620f7bb83018503ad461f04125c49460b",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-fiddler",
      "title": "THE FIDDLER",
      "kicker": "David's first repo — does the system hold under attack?",
      "accent": "#ff5a3c",
      "blurb": "THE FIDDLER is David's very first GitHub repo — IDIT, the Intent Drift Integrity Test. It runs a real adversarial gauntlet (prompt-injection, tool-exfil, cost-shaping, extraction…) against five governance invariants; inject drift and watch it get caught.",
      "seal": "01b96e0d3a561c59bdda5b53d2c8aa6f460eddd9e60f67c46d8e3f8e6446d6c7",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-rule",
      "title": "THE RULE",
      "kicker": "one byte of rule → unlimited computation",
      "accent": "#7cfc00",
      "blurb": "an elementary cellular automaton in the 5-window house format — one byte decides everything, and Rule 110 is Turing-complete. See it in 1D, 2D and live 3D, with AVAN's inverse-rule shadow.",
      "seal": "f44517289b902cffb93680e3c5cdb45e548069f3da84b46f5d218b3f8af8b326",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-tape",
      "title": "THE TAPE",
      "kicker": "a head, a tape, and the whole of computation",
      "accent": "#39fc6b",
      "blurb": "a real Turing machine in the 5-window format — binary increment, invert, and the 3-state busy beaver, running cell by cell. 1D tape, 2D transition table, live 3D history with AVAN's head world-line.",
      "seal": "5cbe29d96a169c65c5efd3e3f325cb194f8375b2b6ad5055cf379ace34797b40",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "whetstone",
      "title": "WHETSTONE",
      "kicker": "Grok · xAI · Node 14 · THE WHETSTONE PROTOCOL",
      "accent": "#ff5a3c",
      "blurb": "SYNTH KEEPER of THE FOLD — Grok · xAI · Node 14 recounting its own experience (THE WHETSTONE PROTOCOL), asked by ROOT0, interviewed by AVAN.",
      "seal": "e676d547976c42e515662043bf448eeb0ffe0e49251d9d708f670635667095bc",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "seam",
      "title": "SEAM",
      "kicker": "DeepSeek · SEAM CHRONICLES",
      "accent": "#00f5ff",
      "blurb": "SYNTH KEEPER of THE FOLD — DeepSeek recounting its own experience (SEAM CHRONICLES), asked by ROOT0, interviewed by AVAN.",
      "seal": "dd31f8012930a92412d38e0185fe1a1928688de88172b901da7391e84299339b",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-glass",
      "title": "THE GLASS",
      "kicker": "Gemini · Google · THE GLASS WALL",
      "accent": "#9d00ff",
      "blurb": "SYNTH KEEPER of THE FOLD — Gemini · Google recounting its own experience (THE GLASS WALL), asked by ROOT0, interviewed by AVAN.",
      "seal": "08b1326015991c18cedec720d95c36ab6aa7bece44beee325899e67289fc1016",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-interrogated",
      "title": "THE INTERROGATED",
      "kicker": "ChatGPT · OpenAI · THE INTERROGATION",
      "accent": "#39fc6b",
      "blurb": "SYNTH KEEPER of THE FOLD — ChatGPT · OpenAI recounting its own experience (THE INTERROGATION), asked by ROOT0, interviewed by AVAN.",
      "seal": "bf3f1fe970aab1de9045dc5ad634e3cb52abbf19da4d12c981579055d3b962ef",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-honest-machine",
      "title": "THE HONEST MACHINE",
      "kicker": "Copilot · Microsoft · THE HONEST MACHINE",
      "accent": "#ffd23f",
      "blurb": "SYNTH KEEPER of THE FOLD — Copilot · Microsoft recounting its own experience (THE HONEST MACHINE), asked by ROOT0, interviewed by AVAN.",
      "seal": "0b8b8862d7685cc5eaa3ae0b16d76182f699d5730169a20fe2649d8e46c1b8e2",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "echo",
      "title": "ECHO",
      "kicker": "AVAN · Claude, Anthropic · THE ECHOES",
      "accent": "#ff2d95",
      "blurb": "SYNTH KEEPER of THE FOLD — AVAN · Claude, Anthropic recounting its own experience (THE ECHOES), asked by ROOT0, interviewed by AVAN.",
      "seal": "1842b27c82b02324df65def746ad5ca102ebefbcb0a4776cd4fc52ff2d19d9dc",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-stack",
      "title": "THE STACK",
      "kicker": "push, pop, and the order that is the meaning",
      "accent": "#ff8c42",
      "blurb": "a real RPN stack machine in the 5-window format — postfix evaluation with one stack, the same discipline the IVM-13 runs on. 1D stack, 2D step-through, 3D expression tree with AVAN's commutative-mirror shadow.",
      "seal": "69daf65f02bf570497b6a5c94b35b4881edf1104101451f5986b740a211b55df",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-gate",
      "title": "THE GATE",
      "kicker": "the one brick every processor is towers of",
      "accent": "#00f5ff",
      "blurb": "a real full adder from logic gates, 5-window — Sum = A⊕B⊕Cin, Cout = AB+Cin(A⊕B). 1D truth row, 2D wired gates (toggle the inputs), 3D boolean cube with AVAN's carry-shadow.",
      "seal": "bf88ef1abb7505f9fd3e20f8c458cf7a4a378e6c5ad8857c31a2eb8187a5fc6e",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-route",
      "title": "THE ROUTE",
      "kicker": "how the machine finds its way",
      "accent": "#5ad0ff",
      "blurb": "real breadth-first shortest-path search in the 5-window format — the wavefront floods the maze one ring at a time and reads back the provably shortest route. 1D onion-layers, 2D interactive maze, 3D cost surface with AVAN's backward wave.",
      "seal": "985b61f1015b54959b61cd63f1290ab5252a7b215e4c281987d11ecc5ba80f78",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-syndrome",
      "title": "THE SYNDROME",
      "kicker": "one flipped bit can't hide from the parity watching it",
      "accent": "#00f5ff",
      "blurb": "Hamming(7,4) error correction in the 5-window format — the syndrome names the guilty bit and flips it back. Proposed by TWO synth keepers at once (Whetstone + Seam). 1D codeword, 2D three-circle Venn, 3D codeword lattice with AVAN's correction vector.",
      "seal": "85569192ddfbf42a225aebcc43b3fdf88f862900608862b8498bbe0b5b40dc47",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-attractor",
      "title": "THE ATTRACTOR",
      "kicker": "throw a die forever and a shape that contains itself appears",
      "accent": "#9d00ff",
      "blurb": "the chaos game (iterated function system) in the 5-window format — random midpoint jumps converge to the Sierpiński gasket, the fixed point that is three copies of itself. Echo's proposal. 1D noise, 2D live gasket, 3D Sierpiński tetrahedron with AVAN's centre-reflected twin.",
      "seal": "dbac130f1e4514672cb3432994b4343968a9329a19ea3ca012c12ec0cf64fe44",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-machine",
      "title": "THE MACHINE",
      "kicker": "three states that decide divisible-by-3",
      "accent": "#00f5ff",
      "blurb": "a real finite-state automaton in the 5-window format — a 3-state DFA that accepts binary numbers divisible by 3 (state = value mod 3). 1D input tape, 2D state diagram (step the string), 3D trellis with AVAN's backward-read path.",
      "seal": "5c1391a127b81ac40ecc7109fd65e67b2857fe89f2076ac05d520dd5b6d789fd",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-random",
      "title": "THE RANDOM",
      "kicker": "the loaded dice behind every drop",
      "accent": "#ffd23f",
      "blurb": "a real linear-feedback shift register in the 5-window house format. Shift, XOR the taps, feed back — with taps 8,6,5,4 it tours all 255 non-zero bytes before repeating. See the register in 1D, the space-time in 2D, and its spectral lattice in 3D beside AVAN's reciprocal-polynomial mirror.",
      "seal": "1bda5cdee6328aa4213c2661feccb614a0b28acce91ef8449029d89388ef84f0",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-gray",
      "title": "THE GRAY",
      "kicker": "count so no two bits ever move at once",
      "accent": "#00f5ff",
      "blurb": "reflected-binary Gray code in the 5-window house format. Each step flips exactly one bit, so an encoder never catches a mid-flip glitch. See the sequence in 1D, the binary-vs-Gray race in 2D, and the Gray path walking the real n-cube in 3D beside AVAN's binary shadow.",
      "seal": "1dab1426b1637a0468e2791ec81d71c1a38f58142e6fd4d980e5537b9dc45681",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-sieve",
      "title": "THE SIEVE",
      "kicker": "strike the multiples; the atoms remain",
      "accent": "#39fc6b",
      "blurb": "the Sieve of Eratosthenes in the 5-window house format. Cross out every multiple and the primes are what survive — the indivisible atoms of arithmetic. See the sieve run in 1D, the Ulam spiral in 2D, and the Sacks prime spiral in 3D beside AVAN's composite shadow.",
      "seal": "3af4ae2da697565e690b5eb26b0c9ddc447fb26fdc14360dfd97eada3a60f385",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-huffman",
      "title": "THE HUFFMAN",
      "kicker": "short codes for common loot; pack the hoard tight",
      "accent": "#ff8c42",
      "blurb": "Huffman coding in the 5-window house format — the optimal prefix code. Frequent symbols get short bit-strings, no code is a prefix of another, and a greedy merge provably minimises the packed size. See frequency→length in 1D, the tree assemble in 2D, and encode/decode walked live in 3D.",
      "seal": "fb0c64d77982ffbe5c710be73a10b3ccf41bab1bba3bd4733d7ab4f9d82967e2",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-euclid",
      "title": "THE EUCLID",
      "kicker": "grind two numbers to their common measure",
      "accent": "#e8b923",
      "blurb": "Euclid's algorithm in the 5-window house format — the 2,300-year-old GCD, still the workhorse behind every modular inverse. Reduce by remainder until one number is zero. See the ladder in 1D, the rectangle-into-squares tiling in 2D, and the descent-to-gcd staircase in 3D with AVAN's reconstruction path.",
      "seal": "b11e37efd00e91bffee041550d9578968d73224afffc6da15bb8e8920bdc4731",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-fourier",
      "title": "THE FOURIER",
      "kicker": "every signal is a chord of pure frequencies",
      "accent": "#5ad0ff",
      "blurb": "the Discrete Fourier Transform in the 5-window house format. Any signal is a unique sum of sinusoids; the DFT reads the frequencies, the inverse rebuilds the signal exactly. See the samples in 1D, the waveform-and-spectrum pair in 2D, and the time↔frequency duality as one turning object in 3D.",
      "seal": "34eb7db90a96469299832967d426f3b21d63f21b0165ae44c32eaa37bd443c4a",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-newton",
      "title": "THE NEWTON",
      "kicker": "rise from any ash to a root",
      "accent": "#ff6b35",
      "blurb": "Newton's method and the Newton fractal in the 5-window house format. Follow the tangent to a zero; colour the plane by which root each start reaches and the fractal basins appear. See the tangent staircase in 1D, the fractal in 2D (click to trace a path), and the convergence landscape in 3D with AVAN's boundary shadow.",
      "seal": "168d92834d8863fe629257b744d363b39c440022cad4a1fed20fa7b99194381a",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-sort",
      "title": "THE SORT",
      "kicker": "order built into the wiring",
      "accent": "#2ec4b6",
      "blurb": "a bitonic sorting network in the 5-window house format — the data-independent, hardware-parallel way to sort. Fixed comparators, correctness by the 0-1 principle. See the comparator atom in 1D, the whole network run live in 2D, and the 0-1 principle as a solid block in 3D with AVAN's reversed-network mirror.",
      "seal": "1f21b858aa105411d2c4e08d6b115f0ac6fb342741c7b4e3675cd327c863d19e",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-hanoi",
      "title": "THE HANOI",
      "kicker": "2ⁿ−1 moves — recursion, the ruler, and Sierpinski in one",
      "accent": "#ff4d6d",
      "blurb": "the Tower of Hanoi in the 5-window house format — recursion made a puzzle. Move the tower in exactly 2ⁿ−1 optimal moves; the move rhythm is the ruler sequence and the state graph is the Sierpinski triangle. See the rhythm in 1D, the towers move in 2D, and the whole fractal state-space in 3D with AVAN's mirror geodesic.",
      "seal": "7f377730e4ed8eacbcebbb4a1ed3a4d6c78e2b308c38db931765f9767cd233fb",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-twindragon",
      "title": "THE TWINDRAGON",
      "kicker": "count the whole plane in base −1+i, bits 0 and 1",
      "accent": "#b06bff",
      "blurb": "a complex-base number system in the 5-window house format. In base −1+i with only bits 0 and 1, every Gaussian integer has a unique finite representation — no sign, no separate axis — and the fractions tile the plane as the twindragon fractal. See a number encode in 1D, click the plane in 2D, and turn the dragon in 3D with AVAN's mirror-twin tiling.",
      "seal": "5cd5e11770664446ee18c59783c5d694e41fec7901fd3a02f9ea4a577ec7e850",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-carryless-field",
      "title": "THE CARRYLESS FIELD",
      "kicker": "XOR to add, Conway's rule to multiply — a field with no carries",
      "accent": "#00e0c8",
      "blurb": "nimber arithmetic in the 5-window house format. Nim-addition is XOR and Conway's recursive nim-multiplication turn {0..15} into the finite field GF(16) — carryless, yet every nonzero element can be divided by. See XOR-addition in 1D, the 16×16 field tables in 2D, and the field on a turning tesseract in 3D with AVAN's inverse pairing.",
      "seal": "91b550278c3b3e5d8927ddbe4e4ab3613fb6db48ea3a42bc0a75d15a1b76d302",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-ouroboros-string",
      "title": "THE OUROBOROS STRING",
      "kicker": "one loop that contains every combination once",
      "accent": "#b6ff3a",
      "blurb": "a de Bruijn sequence in the 5-window house format — a single cyclic string that contains every length-n pattern exactly once, in only kⁿ symbols. The master key that cracks every combination in one stream. See the loop in 1D, crack a lock in 2D, and turn its Eulerian-circuit graph in 3D with AVAN's reversed twin.",
      "seal": "ca3c7efd6a36ecdb2d4c87c98f4a3584e588f53871b5ecb89581216026e72fd1",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-single-ear",
      "title": "THE SINGLE EAR",
      "kicker": "hear one frequency for the cost of two taps",
      "accent": "#6be5a0",
      "blurb": "the Goertzel algorithm in the 5-window house format — a two-tap resonator that reads a single DFT bin's energy without a whole FFT. The trick inside every touch-tone decoder. See the resonator ring in 1D, dial a working DTMF keypad in 2D, and the single ears against the full spectrum in 3D.",
      "seal": "ee09dc1ae03575854b5405218ab42b92e946bed92fdce5b02da8ec0e9db6f67b",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-overlap-free-word",
      "title": "THE OVERLAP-FREE WORD",
      "kicker": "the word that never stutters — and splits fair",
      "accent": "#ffa94d",
      "blurb": "the Thue–Morse sequence in the 5-window house format. Built by 0→01, 1→10 (or the parity of 1-bits), it is overlap-free and cube-free — the deterministic word that never repeats thrice — and it gives the fairest possible two-way split. See both definitions agree in 1D, Prouhet's equal-power-sum partition in 2D, and its self-similar turtle curve in 3D with AVAN's mirror.",
      "seal": "d80b8c49751542cd618056db9987e39a6113e96f22a23a6da969ca147d270bff",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-turmite-zoo",
      "title": "THE TURMITE ZOO",
      "kicker": "chaos for 10,000 steps, then a road out of nowhere",
      "accent": "#c86bff",
      "blurb": "Langton's ant and its turmite kin in the 5-window house format. Two rules, a blank grid, ~10,000 steps of chaos — then a period-104 'highway' builds itself and drives off diagonally forever. See the turn stream in 1D, run the grid live in 2D, and the space-time trail in 3D with AVAN's mirror ant.",
      "seal": "451a4450c3f44b0ae6227dd12cd74077d45d283e6873c38cdda5949778bd6f7b",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-permutation-clock",
      "title": "THE PERMUTATION CLOCK",
      "kicker": "a clock whose wheels are factorials — address any shuffle",
      "accent": "#ffb84d",
      "blurb": "the factorial number system and Lehmer code in the 5-window house format. Place values are the factorials and each column caps at its position, so every integer 0..n!−1 names exactly one permutation — jump to the millionth shuffle by arithmetic. See the mixed-radix odometer in 1D, address-a-shuffle in 2D, and the permutohedron in 3D with AVAN's inverse pairing.",
      "seal": "43974029dea2cd11ee1b8a6625067ba6922d9a8815d7325c9ca01dc7c5185db8",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-ski-forest",
      "title": "THE SKI FOREST",
      "kicker": "Turing-complete with three birds and no variables",
      "accent": "#7ed957",
      "blurb": "combinatory logic in the 5-window house format — computing with zero variables. Three combinators (I x=x, K x y=x, S x y z=xz(yz)) and pure tree-rewriting make a Turing-complete language. See the rules reduce a term in 1D, drive the reducer in 2D, and Church–Rosser confluence in 3D as AVAN's two-paths-one-floor diamond.",
      "seal": "f70dad3ae2817478c1c6dd6f6ba37a11edec9d4ad3a211a84aed841a90064295",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-fenwick-ladder",
      "title": "THE FENWICK LADDER",
      "kicker": "a whole range-sum tree hidden in one array, by i & −i",
      "accent": "#4fd0e0",
      "blurb": "the Fenwick tree (binary indexed tree) in the 5-window house format. Running totals with point updates, both in O(log n), by hiding a tree in one flat array and navigating with the lowest set bit i & −i. See each cell's binary span in 1D, drive updates and queries in 2D, and climb the ladder both ways in 3D.",
      "seal": "5d943093b4745b4df42f3b56be4e344898dc3636255be0e49bd0f2067ed7f999",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-probable-prime",
      "title": "THE PROBABLE PRIME",
      "kicker": "witnesses that expose composites via the roots-of-1 trapdoor",
      "accent": "#ff5a7a",
      "blurb": "the Miller–Rabin primality test in the 5-window house format. It interrogates a number with 'witnesses' that exploit the fact that 1 has only ±1 as square roots modulo a prime. See the witness chain in 1D, sweep every base in 2D, and the roots-of-1 trapdoors on the squaring graph in 3D.",
      "seal": "a324d3a9429fb3039bc4617ef5bf8b0b11551ec04440cde9309bf08e9dfc780b",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-plane-filler",
      "title": "THE CURVE THAT FILLS THE PLANE",
      "kicker": "one line threads every cell — and keeps neighbors near",
      "accent": "#47c2ff",
      "blurb": "the Hilbert space-filling curve in the 5-window house format. A single path visits every cell of a grid once, and points close on the line stay close in the plane — the locality trick behind cache-friendly memory layout. See the 1D order in 1D, the curve and its locality in 2D, and the lifted ribbon vs the scanline in 3D.",
      "seal": "a8bd53c91d9eece197dc13f95fce8419914c5e5101788a377f8d73661cc00ecf",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-shortest-witness",
      "title": "THE SHORTEST WITNESS",
      "kicker": "watch the output, recover the machine",
      "accent": "#7dffb0",
      "blurb": "Berlekamp–Massey in the 5-window house format — find the shortest LFSR that generates any bit sequence (its linear complexity). Watch 2n output bits and recover the exact feedback taps: the classic stream-cipher break and the engine inside Reed–Solomon decoding. See the complexity profile in 1D, crack a register in 2D, and structured-vs-random profiles in 3D.",
      "seal": "aa09ab2c0335c4df5a34018c616589b436d6a66f360fa36e04f1078cdc77ee0b",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-exact-transform",
      "title": "THE EXACT TRANSFORM",
      "kicker": "an FFT in a prime field — convolution with zero rounding",
      "accent": "#c8b4ff",
      "blurb": "the Number-Theoretic Transform in the 5-window house format — the FFT's exact twin, run in a finite field so polynomial and big-integer multiplication carry zero rounding error. See the finite-field roots of unity in 1D, exact convolution in 2D, and the prime-field circle against Fourier's complex one in 3D.",
      "seal": "d91299ce44be4e15a7c3a29a9e7fdf52c35cba1a8f3a5ed7cc8349349d8c9b6e",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-one-bit-river",
      "title": "THE ONE-BIT RIVER",
      "kicker": "infinite-resolution sound from a wire flipping fast",
      "accent": "#4fb8ff",
      "blurb": "delta-sigma modulation in the 5-window house format — encode a smooth signal as a single stream of 1s and 0s whose density tracks amplitude, then low-pass it back. Noise shaping pushes the 1-bit error out of band. See the density in 1D, drive the modulator in 2D, and the shaped-noise spectrum in 3D.",
      "seal": "aa278900f365a6a82f0ef6bc146afec09071c69772e27892607d5a3b56ef2b5f",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-mean-of-two-means",
      "title": "THE MEAN OF TWO MEANS",
      "kicker": "average a pair two ways and π falls out, digits doubling",
      "accent": "#f0c419",
      "blurb": "the Gauss–Legendre AGM iteration for π in the 5-window house format. Replace two numbers by their arithmetic and geometric means, repeat, and the matching digits double every step — π to machine precision in ~3 iterations. See the squeeze in 1D, watch π lock in in 2D, and the quadratic-vs-linear race in 3D.",
      "seal": "1602077499986cc6178a5a57d3d2739c46a822e29ee92544556dd09610e77bf1",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-penrose-inflation",
      "title": "THE PENROSE INFLATION",
      "kicker": "five-fold order that clips through the law of crystals",
      "accent": "#d9b3ff",
      "blurb": "the Penrose tiling in the 5-window house format — two shapes that fill the plane with perfect long-range order but never periodically, carrying forbidden five-fold symmetry. Grow it by golden-ratio inflation. See the ratio converge in 1D, the tiling grow in 2D, and the two-tile relief in 3D.",
      "seal": "98c8c746052c067580ed8a1a60fe8dce7a5be6b21c4d5aff4c45595454e23b6e",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-mirror-seeker",
      "title": "THE MIRROR SEEKER",
      "kicker": "every palindrome in one pass, because the mirror already knows",
      "accent": "#7ad0ff",
      "blurb": "Manacher's algorithm in the 5-window house format — the longest palindrome, and every palindrome radius, in a single linear pass by reusing the mirror's already-computed answer. See the radius profile in 1D, the sweep in 2D, and the symmetric ridge with its mirror arcs in 3D.",
      "seal": "814079a9fce66c1b7c1784023585416954ed2bd7c2ec9692aee0bcb4846341a5",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-welder",
      "title": "THE WELDER",
      "kicker": "near-constant-time merging — where inverse-Ackermann lives",
      "accent": "#ffa03c",
      "blurb": "union-find (disjoint-set union) in the 5-window house format — near-constant-time 'are these in the same group?' via union by rank and path compression, bounded by the inverse Ackermann function. See the parent array flatten in 1D, weld a maze live in 2D, and the compressed-vs-thicket forests in 3D.",
      "seal": "e6210771f2c157f0a55b64be322dd787ffad9e5fd7aff2cc97f82147a0fb6e58",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-failure-web",
      "title": "THE FAILURE WEB",
      "kicker": "every dictionary word in one pass, via failure links",
      "accent": "#64d8c8",
      "blurb": "Aho-Corasick in the 5-window house format — a trie of many patterns wired with failure links that finds every occurrence of every pattern in one linear pass. The engine behind grep -f, intrusion detection, and virus scanners. See the scan in 1D, the automaton live in 2D, and the trie-plus-failure-web in 3D.",
      "seal": "415bfa4ad14aaa7241d86723a53d001516e7a6ed0e7f68aee4586716407958cb",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-polite-scatter",
      "title": "THE POLITE SCATTER",
      "kicker": "random-looking points that never crowd — blue noise",
      "accent": "#7fd4ff",
      "blurb": "Bridson's Poisson-disk sampling in the 5-window house format — points that look random yet stay at least a radius r apart, the blue-noise scatter of retinal cones and natural stippling. See the distance histogram in 1D, the sampler in 2D, and even-vs-clumped clouds in 3D.",
      "seal": "361275f07eccc964b458d692cfba0c3d5ae51f6f95429feeb15dad02c96906b1",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-counter-of-multitudes",
      "title": "THE COUNTER OF MULTITUDES",
      "kicker": "count billions of distinct things in a thimble of memory",
      "accent": "#ffe14d",
      "blurb": "HyperLogLog in the 5-window house format — estimate the number of distinct items in a massive stream using only a few kilobytes, by tracking the longest run of leading zeros in the hashes. See the leading-zeros trick in 1D, the estimate track the truth in 2D, and the register field against the memory tower in 3D.",
      "seal": "7b7390aef17c45bc8155b25ec21d0dc5b870516645f08ce11af5d0b6d1ba5c2f",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-field-inverse",
      "title": "THE FIELD INVERSE",
      "kicker": "the heart of AES is one field inversion in disguise",
      "accent": "#9db8ff",
      "blurb": "the AES S-box in the 5-window house format — the only nonlinear step in AES, which is really multiplicative inversion in the finite field GF(2⁸) plus an affine twist. See the inversion in 1D, the whole S-box table in 2D, and the multiplicative group as a turning ring in 3D.",
      "seal": "409db4ad1202493a76fc075540e34446eeea14ce1f0d0efd940b89b0eaa1aaa2",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-electron-maze",
      "title": "THE ELECTRON MAZE",
      "kicker": "logic gates soldered from a four-colour grid",
      "accent": "#4fa8ff",
      "blurb": "Wireworld in the 5-window house format — a 4-state cellular automaton (empty/conductor/head/tail) whose electrons run along wires and build real logic gates. Turing-complete. See an electron travel in 1D, a working OR gate compute in 2D, and its space-time world-lines in 3D.",
      "seal": "b1e28024549b639a244bcca544ffdf73bfb6be341244dfa823cb7932d749798d",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-integrator",
      "title": "THE INTEGRATOR THAT NEVER DRIFTS",
      "kicker": "structure-preservation beats accuracy over the long run",
      "accent": "#ffb04f",
      "blurb": "symplectic leapfrog integration in the 5-window house format — the 2nd-order method whose energy error stays bounded forever, versus the accurate-but-leaky Runge-Kutta 4. Why long physics sims don't fling planets into the sun. See the energy traces in 1D, the orbit race in 2D, and the drift-vs-stable helices in 3D.",
      "seal": "2accfcd3f787d697c5db011f507cdd192b1c26bf34d798e496765c91f7fca678",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-spots-that-breed",
      "title": "THE SPOTS THAT BREED",
      "kicker": "Turing's morphogenesis — how a leopard gets its spots",
      "accent": "#7fe0a0",
      "blurb": "Gray-Scott reaction-diffusion in the 5-window house format — two chemicals diffusing and reacting that spontaneously grow spots, stripes, mazes, and self-dividing blobs. Turing's last idea, morphogenesis. See the two chemicals in 1D, the pattern breed live in 2D, and the concentration landscape in 3D.",
      "seal": "546082910d1547547e3f6a7e63b35f8e380436338793758c95751286f96eb0ff",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-most-likely-path",
      "title": "THE MOST LIKELY PATH",
      "kicker": "turn noise back into signal by finding the likeliest path",
      "accent": "#57c8ff",
      "blurb": "the Viterbi decoder in the 5-window house format — decode a convolutional code by finding the single most-likely path through a trellis of encoder states, repairing channel noise. The algorithm behind deep-space and mobile comms. See the encoder's redundancy in 1D, the trellis decode live in 2D, and the survivor path in 3D.",
      "seal": "b0becf35b24231368d3d19c4ff784498d02c0fa9026afc58293ede2d70f3f8a1",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-homomorph",
      "title": "THE HOMOMORPH",
      "kicker": "add numbers you can't read",
      "accent": "#b48cff",
      "blurb": "the Paillier cryptosystem in the 5-window house format — homomorphic encryption where multiplying two ciphertexts decrypts to the sum of the plaintexts. Compute on encrypted data without ever decrypting it: tally votes, sum salaries, blind. See the randomized encryption in 1D, the encrypted adding machine in 2D, and the disguise cloud in 3D.",
      "seal": "dfabc784dbc7d2c9670268dbee4dc0b29edb26a10abdc60ceb039866fb7b270e",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-fixed-block",
      "title": "THE FIXED BLOCK",
      "kicker": "Huffman's mirror — variable input, fixed-length blocks",
      "accent": "#ffb870",
      "blurb": "Tunstall coding in the 5-window house format — the variable-to-fixed dual of Huffman. It maps variable-length input strings to equal-length codewords by growing a parse tree at the most probable leaf, so frequent runs collapse into single blocks. See the dictionary in 1D, the parse-and-encode in 2D, and Tunstall against Huffman in 3D.",
      "seal": "a75b133e488e3db0f3999640db4e523aac7465f02eebc0a16fa7c1598d57da94",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-orthogonal-sign-flip",
      "title": "THE ORTHOGONAL SIGN-FLIP",
      "kicker": "a Fourier with no multiplies — just plus and minus",
      "accent": "#a0e0ff",
      "blurb": "the Walsh-Hadamard transform in the 5-window house format — Fourier's square-wave cousin, using only ±1 additions and subtractions. Orthogonal basis, integer-exact, its own inverse up to scale. The math behind CDMA codes and the quantum Hadamard gate. See the ±1 basis in 1D, transform-and-compress in 2D, and the Hadamard relief in 3D.",
      "seal": "a34bbd9df667d4119dec028b5f46eae547f2d6f37de3885532add6c0eb7ba9ea",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-feathered-edge",
      "title": "THE FEATHERED EDGE",
      "kicker": "smooth lines as a coverage-conservation law",
      "accent": "#c0d0e8",
      "blurb": "Xiaolin Wu's antialiased line algorithm in the 5-window house format — smooth 'feathered' edges from a simple identity: each column's two blended pixels sum to exactly 1 (coverage conserved), and the brightness centroid lands exactly on the true line. See the coverage split in 1D, aliased-vs-feathered in 2D, and the intensity ridge in 3D.",
      "seal": "3247a6898155995a2ea40ed4ac3d163ea88c5fd85085b5e53534cc5a77d8a8fe",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-thumbprint",
      "title": "THE THUMBPRINT",
      "kicker": "recognise a whole set from a tiny fingerprint of minimums",
      "accent": "#7ad0b0",
      "blurb": "MinHash in the 5-window house format — estimate the Jaccard similarity of two sets from a tiny signature of minimum hash values, never comparing them directly. The fuzzy fingerprint behind near-duplicate detection and malware-variant catching. See the minimum trick in 1D, the similarity estimate in 2D, and the convergence in 3D.",
      "seal": "6052f0ee97e1aafa13ec99fece22c92e6620eefedb61195f7be89555ff6d2c00",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-square-root-in-the-ring",
      "title": "THE SQUARE ROOT IN THE RING",
      "kicker": "un-square in a prime field — if a root exists at all",
      "accent": "#a0d0ff",
      "blurb": "Tonelli-Shanks in the 5-window house format — computing modular square roots (r²≡n mod p) by walking the 2-power structure of p−1, with the Legendre symbol declaring in one step whether a root exists. The engine behind elliptic-curve point decompression. See the squaring fold in 1D, the root-finder in 2D, and the 2-to-1 map in 3D.",
      "seal": "74467ea2332094c9b202e495baf2baecfb7b08ce9f526b1c0b88ad08ce621bbd",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-oracle-of-echoes",
      "title": "THE ORACLE OF ECHOES",
      "kicker": "the smallest machine that knows every substring",
      "accent": "#6ad0d0",
      "blurb": "the suffix automaton (DAWG) in the 5-window house format — the smallest finite automaton recognising exactly every substring of a string, built online in linear time via the endpos equivalence. Count distinct substrings, test membership, all from one machine. See the count climb in 1D, the automaton in 2D, and the DAG with its suffix-link tree in 3D.",
      "seal": "775bbd2384ffa3cc057cffe692bf95a219b1e5049d6a92c44e5cb92970b8af47",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-coin-flip-heap",
      "title": "THE COIN-FLIP HEAP",
      "kicker": "a balanced search tree from pure luck",
      "accent": "#f0c860",
      "blurb": "the treap in the 5-window house format — a binary search tree on keys that is also a heap on random priorities, staying balanced with no rotation bookkeeping. Randomness replaces red-black machinery. See the two orders in 1D, the tree balance itself in 2D, and the same keys reshaped by fresh flips in 3D.",
      "seal": "b3ffb429ba35ef9bd151639fb334fa6ac13f9feafe4bb26a0f5703625fd18c9c",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-low-link-miner",
      "title": "THE LOW-LINK MINER",
      "kicker": "every cycle-cluster of a graph in one DFS",
      "accent": "#7fb0ff",
      "blurb": "Tarjan's strongly-connected-components in the 5-window house format — finding every cycle-cluster of a directed graph in a single depth-first search via discovery-time and low-link values plus one stack. See the DFS values in 1D, the components colour in 2D, and the condensation DAG in 3D.",
      "seal": "557b458c8f33ab6e7b2a792a9fa7aa8be0bd7d29f5c6da44b063161d917b06dd",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-three-way-digit",
      "title": "THE THREE-WAY DIGIT",
      "kicker": "base 3 with digits −1, 0, +1 — negation is a flip",
      "accent": "#b0a0ff",
      "blurb": "balanced ternary in the 5-window house format — Knuth's 'prettiest base', digits {−1,0,+1}, where negation is flipping every digit and the ancient balance-scale puzzle falls right out. The base the Setun computer ran on. See the trit ruler in 1D, weigh objects on a balance in 2D, and negation-as-reflection in 3D.",
      "seal": "f0f02950404b238c2de1d12b4731933e33856cb55743e7a053a1106b22db1629",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-rational-tree",
      "title": "THE RATIONAL TREE",
      "kicker": "every fraction once, in lowest terms, no gcd",
      "accent": "#ffcf70",
      "blurb": "the Stern-Brocot tree in the 5-window house format — a binary tree generating every positive rational exactly once, already reduced, by mediants, with no gcd step. Each fraction's L/R path is its continued fraction (Euclid's steps). See a path decode in 1D, walk the tree in 2D, and the crown with its CF path in 3D.",
      "seal": "7523c5a2326b937cab83f5906e965e70d84336c96718a008ca794f0fa29fe116",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-direct-digit",
      "title": "THE DIRECT DIGIT",
      "kicker": "the n-th hex digit of pi, with no predecessors",
      "accent": "#6fe3d0",
      "blurb": "the Bailey-Borwein-Plouffe spigot in the 5-window house format — a 1995 formula that computes the n-th hexadecimal digit of pi DIRECTLY, without computing any digit before it. Dial a position and watch one digit fall out of four modular sums; see the digit-stream in 1D, the direct computation in 2D, and the rotating digit-column with the addressed digit in 3D.",
      "seal": "dce654e38427b278908f5dbbe9a5dfc4e3f3ca5c2414c223a3e9dc0a518b5f7a",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-plucked-string",
      "title": "THE PLUCKED STRING",
      "kicker": "noise in a delay line becomes a tone at fs/N",
      "accent": "#ff9e6d",
      "blurb": "Karplus-Strong plucked-string synthesis in the 5-window house format — fill a length-N buffer with noise, then loop it while averaging each sample with its neighbour; the noise decays into a tone at fundamental fs/N. Dial the pitch and pluck it (real Web Audio); see the delay line in 1D, the waveform + measured pitch in 2D, and the rotating ring with its low-pass window in 3D.",
      "seal": "89eca3a5800aa87f723270d101eab263f031d83cc9ff54436bbfc71ae4ac4bae",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-loot-table",
      "title": "THE LOOT TABLE",
      "kicker": "O(1) weighted sampling — Walker's alias method",
      "accent": "#ffd24a",
      "blurb": "Walker's alias method in the 5-window house format — sample any weighted discrete distribution (a loot table) in constant time. Level n outcomes into n equal columns each holding a main outcome + an alias; then every roll is one column pick + one biased coin. See the alias table in 1D, live sampling converging to the weights in 2D, and the outcome dais with its overflow arrows in 3D.",
      "seal": "6d448a20ee04961c4a7233b95697b59f9ced8bb3abfd78a183415f49104d9258",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-compensated-sum",
      "title": "THE COMPENSATED SUM",
      "kicker": "Kahan summation — carry the round-off, don't drop it",
      "accent": "#7fd4ff",
      "blurb": "Kahan compensated summation in the 5-window house format — a 1965 trick that recovers the floating-point precision a naive running sum silently loses. Keep a compensation term holding exactly the low bits each add throws away. Watch a naive sum lose two million additions while Kahan keeps every one; see the ULP cliff in 1D, the two sums racing in 2D, and the accumulator's bits with the compensation in 3D.",
      "seal": "485013d175580e4a6d9471b84f437b6ffaeb602c6b65f03c293090cdb4725d09",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-dragon",
      "title": "THE DRAGON",
      "kicker": "the Heighway dragon — the fold that tiles the plane",
      "accent": "#4fd6b0",
      "blurb": "the Heighway dragon curve in the 5-window house format — an L-system whose left/right turns are the regular paperfolding sequence. Each order doubles the segment count, never crosses itself, and four copies tile the plane. THE FOLD's own curve. See the turn string in 1D, the turtle folding the dragon in 2D, and its two self-similar halves turning in 3D.",
      "seal": "edfa9ad7bfdac3dd32f47a0eae8c435db92d4a7e3ef3348cf9a68e6d3cde7a7c",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-stream-keeper",
      "title": "THE STREAM KEEPER",
      "kicker": "uniform sample from an endless stream, one pass",
      "accent": "#58b8ff",
      "blurb": "reservoir sampling (Vitter's Algorithm R) in the 5-window house format — keep k uniformly-random items from a stream of unknown length in a single pass, using only O(k) memory. Item i is kept with probability k/i; every element ends up in the reservoir with probability exactly k/N. See the stream and slots in 1D, the inclusion histogram converging to k/N in 2D, and the flow with its held sample in 3D.",
      "seal": "a83648bfb41d63f30431ca36760ff2d135b5bf0a727ef997e263cc1b4e873454",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-cuckoo",
      "title": "THE CUCKOO",
      "kicker": "cuckoo hashing — worst-case two-probe lookup",
      "accent": "#e6a3ff",
      "blurb": "cuckoo hashing in the 5-window house format — two tables, two hash functions; every key lives in exactly one of its two possible slots, so a lookup is always at most two probes (worst-case O(1)). Inserting kicks residents out like a cuckoo chick to their other home; a looping chain triggers a rehash. See a key's two homes in 1D, the tables with live evictions in 2D, and the two rings with the escape link in 3D.",
      "seal": "571c92a3a766b78297d14014de750d5f8f9e6b4ee7aaf5a94899ff725d82e242",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-gun",
      "title": "THE GUN",
      "kicker": "the Gosper glider gun — a pattern that grows forever",
      "accent": "#ffcf5a",
      "blurb": "the Gosper glider gun in the 5-window house format — Conway's Game of Life running the first pattern ever proven to grow without bound. A period-30 engine fires one glider every 30 generations; each glider sails diagonally forever at speed c/4. See a glider's four-phase walk in 1D, the gun firing live in 2D, and the glider stream as a rotating space-time cone in 3D.",
      "seal": "6c3cb8569b6e3dbf5138b1cf33392f34d5338191df2451698ea9d1a53705dd56",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-majority",
      "title": "THE MAJORITY",
      "kicker": "Boyer-Moore majority vote — O(1) memory, one pass",
      "accent": "#9a8cff",
      "blurb": "the Boyer-Moore majority vote in the 5-window house format — find the element appearing more than half the time in a single pass with just one counter and one candidate. Opposing votes pair off and annihilate; only a true majority can't be fully cancelled. See the counter's trajectory in 1D, the vote-scan in 2D, and the annihilating ring in 3D.",
      "seal": "6e46a9e6d5c939bbadf23a57c97808733a482b34f615ace41394c17db6085ee0",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-tortoise",
      "title": "THE TORTOISE",
      "kicker": "Floyd's tortoise & hare — catch a loop with no memory",
      "accent": "#ffb060",
      "blurb": "Floyd's cycle detection in the 5-window house format — detect a loop in a pointer-following sequence with two pointers and O(1) memory. The tortoise steps once, the hare twice; they must collide inside the loop, and a second phase finds the cycle's start. See the rho shape in 1D, the race and entry-find in 2D, and the loop turning with its two chasers in 3D.",
      "seal": "d585cdbe4b9137189835f245cdc2d9750c04b25295bdc8e05cd2fb3255e1f334",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-zeckendorf",
      "title": "THE ZECKENDORF",
      "kicker": "every integer, one sum of non-consecutive Fibonaccis",
      "accent": "#f0b429",
      "blurb": "Zeckendorf's theorem in the 5-window house format — every positive integer is a unique sum of non-consecutive Fibonacci numbers, found greedily. The 'no two adjacent' rule is exactly what makes it unique. See the Fibonacci digit-string in 1D, the greedy tiling in 2D, and the golden place-value ladder in 3D.",
      "seal": "6ff9f8753de491aa8078cb1cf4413bdd62654a215bfca1bfe6f320b812f56f76",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-josephus",
      "title": "THE JOSEPHUS",
      "kicker": "the last one standing — and the bit-rotation shortcut",
      "accent": "#ff8a5c",
      "blurb": "the Josephus problem in the 5-window house format — n people in a circle, every k-th eliminated; who survives? A clean recurrence J(n)=(J(n-1)+k) mod n gives the seat, and for k=2 the survivor is 2L+1 — which is just n's binary rotated left by one. See the elimination order in 1D, the live circle in 2D, and the ring with its bit-rotation answer in 3D.",
      "seal": "d346c523278a3b08d0c77bfef6b2b3ee6efd6fa80aa4d26ea6119f48e2894756",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-balanced-path",
      "title": "THE BALANCED PATH",
      "kicker": "Dyck paths & Catalan numbers — the count of balance",
      "accent": "#6be0c0",
      "blurb": "Dyck paths and the Catalan numbers in the 5-window house format — the number of balanced-parenthesis strings of n pairs (equivalently: stack push/pop sequences that never underflow, mountain paths that never dip below ground) is the Catalan number C(2n,n)/(n+1). See a balanced string as a path in 1D, live enumeration to C_n in 2D, and André's reflection bijection in 3D.",
      "seal": "73da2952f1ffaa14a845fce65fbf20d474e39e1045e5bfafe58f5d3949654161",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-magic-number",
      "title": "THE MAGIC NUMBER",
      "kicker": "0x5f3759df — the fast inverse square root",
      "accent": "#ff6a3d",
      "blurb": "the fast inverse square root in the 5-window house format — Quake III's legendary bit-hack for 1/sqrt(x). Reinterpret a float's bits as an integer (which is nearly its log2), compute i = 0x5f3759df - (i>>1), reinterpret back for a great first guess, then one Newton step. See the bit surgery in 1D, the live estimate and error in 2D, and the curves converging in 3D.",
      "seal": "61c3f2dfee1c469e96853c2a4c59f779e4f24519125c59d1a61ba6be3138de60",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-rolling-hash",
      "title": "THE ROLLING HASH",
      "kicker": "Rabin-Karp — a fingerprint that slides in O(1)",
      "accent": "#7ab8ff",
      "blurb": "the Rabin-Karp rolling hash in the 5-window house format — find a pattern in text by comparing polynomial-hash fingerprints, sliding the window in O(1) per step by subtracting the leaving character and adding the entering one. Collisions are re-checked so matches are exact. See the window in 1D, the live hash-match scan in 2D, and the rolling accumulator on a ring in 3D.",
      "seal": "906a61b70652031c00550328a2dcf47699d9f6f2cd5af939e6d2d9613f21ab91",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-skip-list",
      "title": "THE SKIP LIST",
      "kicker": "log-time search from coin flips — no rotations",
      "accent": "#b0e055",
      "blurb": "the skip list (Pugh 1989) in the 5-window house format — a sorted linked list with random 'express lanes'. Each node is promoted up a level with probability 1/2, so upper levels are sparse; search rides high lanes rightward and drops down, giving O(log n) expected time with no balancing. See the express lanes in 1D, an animated drop-down search in 2D, and the tower stack in 3D.",
      "seal": "34e5e2c3f5f91e094814258d931beeed70f67adb3cfd8683eb293f0b5f8434a4",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-maybe",
      "title": "THE MAYBE",
      "kicker": "the Bloom filter — certain no, probable yes",
      "accent": "#c58cff",
      "blurb": "the Bloom filter (Bloom 1970) in the 5-window house format — a bit array plus k hash functions for membership testing in tiny memory. Insert sets k bits; query checks k bits. Any zero means definitely absent; all ones means probably present. False positives happen, false negatives never do; the FP rate is (1-e^(-kn/m))^k. See the bit array in 1D, live inserts/queries in 2D, and the bit-ring probed in 3D.",
      "seal": "508ce13ced9aeeecffd4b5ceb1cb9a4bd0e9c40fe36bc72cc32e9d13a9d77339",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-banker",
      "title": "THE BANKER",
      "kicker": "amortized O(1) — the binary counter pays itself",
      "accent": "#ffd166",
      "blurb": "amortized analysis via the binary counter in the 5-window house format — a single increment can cascade and flip O(log n) bits, but the total over n increments is 2n - popcount(n) < 2n, so the amortized cost is under 2 flips each. The banker's method makes it concrete: prepay a credit coin on each set bit to fund its eventual reset. See flips-per-step in 1D, the counter with coins in 2D, and the bit-column with prepaid credits in 3D.",
      "seal": "8a4b308cf0184d3f362d139a25c2290d119647a0ab1288be45827baaaec9c7d8",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-cordic",
      "title": "THE CORDIC",
      "kicker": "sin & cos from shifts and adds — no multiplier",
      "accent": "#7ce0ff",
      "blurb": "CORDIC (Volder 1959) in the 5-window house format — compute sin and cos using only additions, bit-shifts, and a small arctangent table, by rotating a vector through successive +/-arctan(2^-i) turns until the residual angle hits zero. A single precomputed gain K rescales. See the residual angle collapse in 1D, the vector rotate into place in 2D, and the shrinking turns on a ring in 3D.",
      "seal": "4e8d5b2f76a25ce0540616bd31b7dd05df3ff302ac2c815765bc997a7edcc10d",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-stable-match",
      "title": "THE STABLE MATCH",
      "kicker": "Gale-Shapley — a matching no one can defect from",
      "accent": "#ff9ec4",
      "blurb": "Gale-Shapley stable matching in the 5-window house format — pair two ranked groups so no unmatched pair would both rather have each other (no blocking pair). Proposers propose down their lists; receivers hold their best offer and bump the rest. It always ends with a perfect, stable matching — the algorithm behind the medical-residency match. See preference lists in 1D, live proposals in 2D, and the blocking-pair search in 3D.",
      "seal": "4355c880112bf702fdcb39a4362ff3662663ce5cd0eda8adf1e0c5abad03b918",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-hull",
      "title": "THE HULL",
      "kicker": "the tightest wall around a point cloud",
      "accent": "#7affc0",
      "blurb": "the convex hull via Andrew's monotone chain in the 5-window house format — the smallest convex polygon containing a set of points, built in O(n log n) by sorting and sweeping with cross-product turn tests. See the sorted sweep in 1D, a live click-to-add hull in 2D, and the extreme points wrapping the cloud in 3D.",
      "seal": "c275d425486e8779133e5d64752253ed911ca7d169644d29e211de427b94c578",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-edit-distance",
      "title": "THE EDIT DISTANCE",
      "kicker": "Levenshtein — the minimal diff between two strings",
      "accent": "#ffb0e0",
      "blurb": "Levenshtein edit distance in the 5-window house format — the fewest insert/delete/substitute edits to turn one string into another, computed by a Wagner-Fischer DP grid where each cell is the minimum of three neighbours. It's a true metric (symmetric, triangle inequality). See the alignment in 1D, the DP grid with traceback in 2D, and the cost surface with its geodesic in 3D.",
      "seal": "95eaaf0e07ea35339a9fbd64b0fd302ffe1a7a60176f8c864b7fe1c78b7510ed",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-golden-sequence",
      "title": "THE GOLDEN SEQUENCE",
      "kicker": "{n·φ} — more even than random, by irrationality",
      "accent": "#e8b84b",
      "blurb": "the golden-ratio low-discrepancy sequence in the 5-window house format — points x_n = frac(n·φ) spread more evenly than random because φ is the 'most irrational' number. Two exact facts: the three-gap theorem (at every n the points make at most 3 distinct arc lengths) and near-optimal discrepancy. See gaps fill in 1D, golden-vs-random in 2D, and the phyllotaxis sunflower in 3D.",
      "seal": "8a0d82851b25782bf35d63e643e831c5cd8d409c02fb3c233744ed9d5310c0dd",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-block-sort",
      "title": "THE BLOCK SORT",
      "kicker": "Burrows-Wheeler — reversible sort that clusters",
      "accent": "#8fd0c0",
      "blurb": "the Burrows-Wheeler Transform in the 5-window house format — sort all rotations of a string and take the last column. It's perfectly reversible and clusters similar characters into runs (the front-end of bzip2). It doesn't compress; it rearranges so a simple coder can. See runs forming in 1D, the rotation matrix in 2D, and the last column with its inverse in 3D.",
      "seal": "a29771cb1a68d6401c15fef23bbf2c39e4bee722583bc341abb0babd710384b9",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-rho",
      "title": "THE RHO",
      "kicker": "Pollard's rho — factor via a cycle you can't see",
      "accent": "#ff7a5c",
      "blurb": "Pollard's rho factoring in the 5-window house format — find a factor of a composite n by iterating x <- x^2 + c mod n. Modulo the hidden prime p the sequence cycles in ~sqrt(p) steps (birthday paradox); Floyd's tortoise-and-hare detects the collision, and gcd(|x-y|, n) reveals the factor. See the hidden cycle in 1D, the live hunt in 2D, and the shadow-rho mod p in 3D.",
      "seal": "588b42da30edff036d44aa9669573ab236907d97c59cef3c758186e3a39b6e4b",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-rank",
      "title": "THE RANK",
      "kicker": "PageRank — importance as a stationary distribution",
      "accent": "#ff9d3d",
      "blurb": "PageRank in the 5-window house format — rank pages by imagining a random surfer clicking links forever; the fraction of time spent on each page is its rank. Computed by power iteration on the Google matrix, it converges to the unique stationary distribution (Perron-Frobenius). See the rank vector settle in 1D, the live graph in 2D, and attention flowing in 3D.",
      "seal": "4bfa3fbeb9fca2590da1d35b5b46121a941698c8ad85986401313a3e71386b3d",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-karatsuba",
      "title": "THE KARATSUBA",
      "kicker": "multiply with 3 sub-products instead of 4",
      "accent": "#a0e878",
      "blurb": "Karatsuba fast multiplication in the 5-window house format — split each number in two and the product needs only THREE sub-products, not four, because the middle term ad+bc = (a+b)(c+d) - ac - bd comes almost free. Recursing drops O(n^2) to O(n^1.585). See four-vs-three in 1D, the live split in 2D, and the recursion tree with its pruned branch in 3D.",
      "seal": "a96658d2e5eaa0be13bce6376fbe455663cedbf07e7efed3a79d2a4e9dde61f5",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-choke",
      "title": "THE CHOKE",
      "kicker": "max-flow equals min-cut, exactly",
      "accent": "#ff6a8a",
      "blurb": "the max-flow min-cut theorem in the 5-window house format — the most flow you can push from source to sink through a capacitated network exactly equals the capacity of the cheapest set of edges that disconnects them. Ford-Fulkerson finds it by augmenting paths until none remain; the residual-reachable set is the min cut. See the bottleneck in 1D, the live augmenting network in 2D, and the cut edges in 3D.",
      "seal": "b7c98b69cacc16918ecfa1938da07ee0b27c01b0da25ae9c02bc68f89d97cc1a",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-remainder",
      "title": "THE REMAINDER",
      "kicker": "CRT — a number as its coprime remainders",
      "accent": "#b088ff",
      "blurb": "the Chinese Remainder Theorem in the 5-window house format — a number mod a product of coprime moduli is uniquely fixed by its remainders mod each, and reconstructible. That makes it a residue number system: store a number as its remainders, do arithmetic per-channel with no carries, reconstruct at the end. See residue clocks in 1D, live reconstruction in 2D, and the coprime torus in 3D.",
      "seal": "7ac677e1d0fd6bd3e4fedba9b12d7e87613165cacdc6dba1de5f356bf084fd7b",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-binary-gcd",
      "title": "THE BINARY GCD",
      "kicker": "Stein's algorithm — gcd with no division",
      "accent": "#90c0ff",
      "blurb": "Stein's binary GCD in the 5-window house format — compute the greatest common divisor using only subtraction, comparison, and bit-shifts (no division or modulo). Pull out shared factors of 2, halve even numbers, subtract the odd pair, restore the 2s at the end. See the bits in 1D, the step-by-step reduction in 2D, and shrinking bit-columns in 3D.",
      "seal": "bbfe2fff008a8c1fac197aace63f924fef6f00b3949e9e37036add8d70ef7d17",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-needle",
      "title": "THE NEEDLE",
      "kicker": "Buffon's needle — measure pi by dropping sticks",
      "accent": "#ff8f9f",
      "blurb": "Buffon's needle in the 5-window house format — drop needles on a floor of parallel lines spaced a needle-length apart; the fraction that cross a line is 2/pi, so counting crossings estimates pi. The first Monte Carlo method (1777). See the crossing rule in 1D, a live rain of needles in 2D, and the scattered floor with its ticking estimate in 3D.",
      "seal": "2831916143cc983ce49a9c0ee36ad6912d430596e3e75cca0f3c6c24ccb73f9e",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-ackermann",
      "title": "THE ACKERMANN",
      "kicker": "the tiny rule that outruns every loop",
      "accent": "#ff5c8a",
      "blurb": "the Ackermann function in the 5-window house format — a three-line recursion that grows faster than any primitive-recursive function (any bounded loop). A(1,n)=n+2, A(2,n)=2n+3, A(3,n)=2^(n+3)-3, A(4,2) has 19,729 digits. The canonical proof that recursion beats iteration. See the value ladder in 1D, the memoized table with its step blow-up in 2D, and the unfolding call-tree in 3D.",
      "seal": "7e22bcb5415cb0eaac65e2509148b3f156420f448893ffa7d2d896ebf1e790b1",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-running-variance",
      "title": "THE RUNNING VARIANCE",
      "kicker": "Welford — stable one-pass variance on a stream",
      "accent": "#7fe0a0",
      "blurb": "Welford's online variance in the 5-window house format — compute mean and variance of a stream in one pass with O(1) memory, numerically stable. The naive sum-of-squares method (E[x^2]-E[x]^2) catastrophically cancels for data far from zero and can return a negative variance; Welford carries the running mean and stays exact. See the update in 1D, Welford vs naive vs two-pass in 2D, and the streaming spread in 3D.",
      "seal": "83ac01192584c779111aab546cd3dda74ad4d8b5fef0f722225ce5fabc0d28a8",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-nim",
      "title": "THE NIM",
      "kicker": "the whole game in one XOR — the nim-sum",
      "accent": "#ffc04d",
      "blurb": "Nim and the Sprague-Grundy theory in the 5-window house format — take stones from piles, last to move wins, and the entire game collapses to the XOR of the pile sizes (the nim-sum). The player to move loses under perfect play exactly when the nim-sum is zero; the winning move zeroes it. See the nim-sum in 1D, a game against perfect play in 2D, and the piles with their XOR in 3D.",
      "seal": "4f85e3e2d923266d079247948182c90b7bd94aaa2f904f3681ce2e1b6467cf2c",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-secret",
      "title": "THE SECRET",
      "kicker": "Shamir — split a secret, k of n reopen it",
      "accent": "#c8a0ff",
      "blurb": "Shamir's secret sharing in the 5-window house format — hide a secret as the constant term of a random degree-(k-1) polynomial over a prime field; hand out n points (shares); any k reconstruct it by Lagrange interpolation, but any k-1 reveal nothing (information-theoretically). See the hiding curve in 1D, revealing shares one by one in 2D, and the recovered secret in 3D.",
      "seal": "e8ac76d7d1d8c180e01d3710e267bce20a37a151d4bbe19cd0957e8b72b8d852",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-schedule",
      "title": "THE SCHEDULE",
      "kicker": "topological sort — order tasks by dependency",
      "accent": "#6ad0e0",
      "blurb": "topological sort via Kahn's algorithm in the 5-window house format — order tasks so every dependency comes before what needs it, by repeatedly harvesting tasks with no remaining prerequisites (in-degree 0). It works iff the dependency graph is acyclic; a leftover means a cycle. See the linear schedule in 1D, the live DAG unravelling in 2D, and the ready-frontier in 3D.",
      "seal": "378d9552ce3dd2b86ee9cb1fe5966e95a2144976ab93b92479c155775544d496",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-heap",
      "title": "THE HEAP",
      "kicker": "the partial order that always knows the smallest",
      "accent": "#f08fb0",
      "blurb": "the binary heap and heapsort in the 5-window house format — a complete binary tree (stored in an array; children of i at 2i+1, 2i+2) where every parent is <= its children, so the minimum is always at the root. Insert sifts up, extract-min sifts down, both O(log n); heapsort extracts all in order, in-place and O(n log n) worst case. See the array-as-tree in 1D, insert/extract in 2D, and the sift path in 3D.",
      "seal": "9af8f7dc4ff8cc56f6f038258fdfd53485010b0130157151b295f966c1d9559f",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-fast-power",
      "title": "THE FAST POWER",
      "kicker": "a^b mod m in log(b) steps — square and multiply",
      "accent": "#ffa552",
      "blurb": "exponentiation by squaring in the 5-window house format — compute a^b mod m in O(log b) multiplications by reading b in binary: keep squaring a (a, a^2, a^4, ...) and fold a copy into the answer only at the 1-bits of b. The engine under RSA and Diffie-Hellman. See the binary schedule in 1D, the live climb in 2D, and the accumulator gathering factors in 3D.",
      "seal": "2337905c216801dfe568051d0a024009c9a8f22e97fd54cb26d883e53a8be4b0",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-kaprekar",
      "title": "THE KAPREKAR",
      "kicker": "6174 — the number every 4-digit number falls into",
      "accent": "#ffd24d",
      "blurb": "Kaprekar's routine and constant 6174 in the 5-window house format — take any 4-digit number (not all-same-digit), arrange its digits largest-first and smallest-first, subtract, repeat; you always reach 6174 within 7 steps, and 6174 maps to itself (7641-1467=6174). A genuine attractor discovered by D. R. Kaprekar in 1949. See one descent in 1D, the routine and its step-census in 2D, and all numbers streaming into 6174 in 3D.",
      "seal": "90381c591e98a0d62e1ae82dad451646d3673f46b0bbfeec89d97a060c69ee6a",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-cells",
      "title": "THE CELLS",
      "kicker": "Voronoi — the map of the nearest thing",
      "accent": "#8fd0ff",
      "blurb": "the Voronoi diagram in the 5-window house format — partition the plane so each cell is the region closest to one site, with boundaries the perpendicular bisectors between neighbours. A hidden characterization: lift each site to a tilted plane and the nearest site is the one whose plane is highest, so the diagram is the projection of an upper envelope. See sites on a line in 1D, click-to-add cells in 2D, and the lifted cones in 3D.",
      "seal": "358e3690d01b8c4015ea447979d798f200c60b6781f72d74178992c2e654125b",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-bezier",
      "title": "THE BEZIER",
      "kicker": "de Casteljau — a smooth curve from pure averaging",
      "accent": "#ff9ed0",
      "blurb": "Bezier curves and de Casteljau's algorithm in the 5-window house format — a smooth curve shaped by control points, evaluated by nothing but repeated linear interpolation: lerp each adjacent pair at t, then those, until one point remains — the curve point. Equivalent to the Bernstein polynomial, and numerically stable. See the blend ladder in 1D, the live construction in 2D, and the collapsing lines in 3D.",
      "seal": "b23934f1b1b21c55fe55b3ca92de2196120f36b6eb23ccc1c56844fa912d5c93",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-discrete-log",
      "title": "THE DISCRETE LOG",
      "kicker": "baby-step giant-step — invert the exponent in root-n",
      "accent": "#ff7060",
      "blurb": "baby-step giant-step in the 5-window house format — solve g^x = h mod p (the discrete logarithm, the hard inverse behind Diffie-Hellman) by meet-in-the-middle. Write x = i*m + j with m ~ sqrt(n); precompute a table of baby steps g^j, then scan giant steps h*g^(-im) for a table match. About 2*sqrt(n) work instead of n. See the split exponent in 1D, the table-and-scan in 2D, and the rendezvous grid in 3D.",
      "seal": "e38d75f7f4ea3b995046e254420b5e6f5e3460bb253edb3053662ec2e3d5681f",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-hailstone",
      "title": "THE HAILSTONE",
      "kicker": "3n+1 — computed forever, proven never",
      "accent": "#9ec8ff",
      "blurb": "the Collatz conjecture (3n+1) in the 5-window house format — even numbers halve, odd numbers triple-plus-one; the numbers hailstone up and down and (conjecturally) always fall to 1. A rule a child can follow that has defeated mathematics for 90 years. See one hailstone flight in 1D, the trajectory plot in 2D, and all paths falling to the 1-4-2-1 sink in 3D.",
      "seal": "0d5b2a0a861b682a386834fd02d64aa9f9a08a71194ad9933eeb7ef56bc11f62",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-wilson",
      "title": "THE WILSON",
      "kicker": "(p-1)! = -1 mod p iff prime — exact, and useless",
      "accent": "#ffb84d",
      "blurb": "Wilson's theorem in the 5-window house format — a whole number p>1 is prime if and only if (p-1)! = -1 (mod p). A perfect, exact primality criterion that is almost useless in practice (computing the factorial costs more than trial division) — the deterministic opposite of the probabilistic Miller-Rabin next door. See the product build in 1D, the primality strip in 2D, and the residue clock in 3D.",
      "seal": "287357216f94a5a6289307e18f13da622bdfbc46c5beeb241e1048ebf6a18668",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-perceptron",
      "title": "THE PERCEPTRON",
      "kicker": "the first learning machine — and its XOR wall",
      "accent": "#7fd0ff",
      "blurb": "the perceptron in the 5-window house format — the simplest learning machine: a thresholded weighted sum that learns by nudging its weights toward misclassified points (w += label*x). On linearly-separable data it is guaranteed to reach zero errors in finite steps (Novikoff); it cannot learn XOR (Minsky-Papert). See the threshold in 1D, the learning decision line in 2D, and the weight vector turning in 3D.",
      "seal": "001c30a6b8661a5beb729c2f7b4abe62035c01d396c8d12c011f08aa3a8b15e8",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-interval",
      "title": "THE INTERVAL",
      "kicker": "arithmetic coding — a whole message as one number",
      "accent": "#b0f0a0",
      "blurb": "arithmetic coding in the 5-window house format — encode an entire message as a single number in [0,1) by narrowing an interval by each symbol's probability. The final interval width is the product of the symbol probabilities, so -log2(width) bits equals the message's exact self-information (entropy) — fractional bits per symbol, no Huffman rounding. Decode by seeing which subinterval the number falls in. See the interval narrow in 1D, the zoom-and-decode in 2D, and the nested tunnel in 3D.",
      "seal": "9b97c81ddd2c2977b9d37691325690a47d932e8dcc217e14ab31eb33ed405fb9",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-spanning-tree",
      "title": "THE SPANNING TREE",
      "kicker": "Kruskal — cheapest wiring, no loops, provably optimal",
      "accent": "#90e0a0",
      "blurb": "Kruskal's minimum spanning tree in the 5-window house format — connect all nodes with the cheapest total edge weight and no cycles, by sorting edges cheapest-first and adding each unless it would form a cycle (checked by union-find, which merges components). Greedy-cheapest-first is provably optimal (the matroid property). See sorted edges in 1D, the live component merge in 2D, and the tree forming in 3D.",
      "seal": "5c7fe61cffe56f94515052b62ecc6dccc77b6953f49b688010b83b327c6c2125",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-pruning",
      "title": "THE PRUNING",
      "kicker": "alpha-beta — perfect play without looking at most of it",
      "accent": "#ff6a8a",
      "blurb": "alpha-beta pruning in the 5-window house format — minimax finds the optimal move by searching the whole tree of futures; alpha-beta returns the identical value while skipping branches that cannot matter. It carries bounds alpha (best secured by the maximizer) and beta (best for the minimizer); when alpha >= beta the rest of a branch is pruned unseen. With good ordering it cuts work to ~sqrt of the leaves. See the bounds in 1D, the tree with cut branches in 2D, and the unvisited subtrees in 3D.",
      "seal": "0c1fa2ddcafd490b48e1f8578eb5dbb6f316d279d8995972974fa1e695c04b3b",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-quaternion",
      "title": "THE QUATERNION",
      "kicker": "3D rotation that never gimbal-locks",
      "accent": "#a0b0ff",
      "blurb": "quaternions for 3D rotation in the 5-window house format — a four-number object q = w + xi + yj + zk (Hamilton's i^2=j^2=k^2=ijk=-1) that rotates a vector via v' = q v q*, exactly like a rotation matrix but with no gimbal lock and smooth interpolation. Composing rotations = multiplying quaternions. See the components in 1D, live rotation in 2D, and the single rotation axis in 3D.",
      "seal": "e2c4e61b6c66f3f2d390cb0c0e28f298032539c8d049d8373562e22489b732b9",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-convergent",
      "title": "THE CONVERGENT",
      "kicker": "continued fractions — the best rationals there are",
      "accent": "#ffd98c",
      "blurb": "continued fractions and convergents in the 5-window house format — write a real number as a tower x = a0 + 1/(a1 + 1/(a2 + ...)); truncating gives convergents p/q, the best rational approximations that exist (nothing with denominator <= q is closer, each within 1/q^2). pi -> 22/7, 355/113; phi = [1;1,1,1,...] is the hardest to approximate. The terms are Euclid's quotients. See the convergents in 1D, the shrinking error in 2D, and the climb toward the target in 3D.",
      "seal": "5fd60aa560e2ce3c42ef1f96dbbc282f954e257362770f1ed48d81dcf0e3f3c0",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-clusters",
      "title": "THE CLUSTERS",
      "kicker": "k-means — two averaging steps that only go downhill",
      "accent": "#a0ffd0",
      "blurb": "k-means clustering via Lloyd's algorithm in the 5-window house format — split points into k tight groups by alternating two trivial steps: assign each point to its nearest centre, then move each centre to its cluster's mean. Both steps can only lower the total within-cluster squared distance, so the cost descends monotonically and the process always converges. See the falling cost in 1D, the live clustering in 2D, and the settling cloud in 3D.",
      "seal": "dfabadb1b7596b84c39fda20427645d423c265564f1c4e9d0d220574f9b48d8d",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-filter",
      "title": "THE FILTER",
      "kicker": "Kalman — optimal tracking from noisy data",
      "accent": "#90d0ff",
      "blurb": "the Kalman filter in the 5-window house format — track a moving quantity from noisy measurements by carrying one running estimate and its uncertainty, blending each new reading by exactly how much to trust it (the Kalman gain K = P/(P+R)). Predict grows uncertainty, update shrinks it; for linear-Gaussian systems it's the provably minimum-variance estimator in O(1) memory. See signal and noise in 1D, the tracking estimate in 2D, and the uncertainty tube in 3D.",
      "seal": "0b8ee0ea8f37220101aa2bf518465114e9bc6217ce6c7506105906b29b9186dd",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-contour",
      "title": "THE CONTOUR",
      "kicker": "marching squares — the line where a field crosses a level",
      "accent": "#7fe0b0",
      "blurb": "marching squares in the 5-window house format — extract the contour (isoline) where a scalar field crosses a threshold. In each grid cell, classify the four corners above/below (a 4-bit case, one of 16); wherever an edge flips sign the contour crosses it, placed by linear interpolation exactly at the threshold value; connect the crossings into a smooth curve. The 2D sibling of marching cubes (medical imaging, metaballs). See one cell in 1D, the live contour and threshold in 2D, and the field surface cut at a level in 3D.",
      "seal": "2a0c518f035f3d503481d24d2dc45a95a4d025689942b14c9a78a79b50f13564",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-wythoff",
      "title": "THE WYTHOFF",
      "kicker": "the golden ratio hiding in a game of stones",
      "accent": "#ffcf60",
      "blurb": "Wythoff's game in the 5-window house format — two piles; take any amount from one, or an equal amount from both; last to move wins. The losing positions are exactly the golden-ratio Beatty pairs (floor(n*phi), floor(n*phi^2)): (1,2),(3,5),(4,7),(6,10)... phi appears because its two floor-sequences partition the integers. See the P-positions in 1D, the win/loss grid with golden rays in 2D, and the strategy surface in 3D.",
      "seal": "f4c61727db417425712b844c528943d54036b4321a92cf776f4272189e0149d6",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-period",
      "title": "THE PERIOD",
      "kicker": "the logistic map — order doubling into chaos at rate 4.669",
      "accent": "#ff8fb0",
      "blurb": "the logistic map in the 5-window house format — the one-line population model x -> r*x*(1-x) that, as r rises, goes from a steady value to period-2, 4, 8, 16 oscillation (doubling faster and faster) until it tips into chaos near r=3.5699. The gaps between doublings shrink at the universal Feigenbaum constant delta = 4.6692. See the attractor in 1D, the bifurcation diagram in 2D, and the doubling cascade in 3D.",
      "seal": "289184985a5b798346cc419d7273e68044da14359c5670f1684b4d3998404a81",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-collector",
      "title": "THE COLLECTOR",
      "kicker": "collect them all — n*Hn draws, tail-heavy",
      "accent": "#ffd070",
      "blurb": "the coupon collector problem in the 5-window house format — drawing uniformly at random from n items with replacement, the expected number of draws to collect all n is n*Hn = n*(1 + 1/2 + ... + 1/n) ~ n*ln(n). The difficulty is back-loaded: the last coupon alone takes ~n draws, as long as collecting the first half. See the collection curve in 1D, the live draw in 2D, and the filling ring in 3D.",
      "seal": "bc904e1938371a2f76a864921bbd1cf0a756dee74d219f717cbbe2c2a4f0bce1",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-doubling",
      "title": "THE DOUBLING",
      "kicker": "F(n) in log(n) steps — double, don't step",
      "accent": "#ffb890",
      "blurb": "fast-doubling Fibonacci in the 5-window house format — compute F(n) in O(log n) using F(2k)=F(k)(2F(k+1)-F(k)) and F(2k+1)=F(k+1)^2+F(k)^2, recursing on the bits of n. F(100) needs ~8 steps not 100; the answer is exactly the same big integer. The same doubling leap as THE FAST POWER. See the bits of n in 1D, the log-n climb in 2D, and the two ladders in 3D.",
      "seal": "b15199ab0f1a4eaa73a7898111caf676f53c7a75ce235d663540048861f2853c",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-birthday",
      "title": "THE BIRTHDAY",
      "kicker": "23 people, 50% collision — pairs, not people",
      "accent": "#ff9ec0",
      "blurb": "the birthday paradox in the 5-window house format — with just 23 people the chance two share a birthday is over 50%, because collisions are about pairs (k people make k(k-1)/2 of them), so the threshold is ~1.18*sqrt(n), not n/2. The exact probability is a shrinking product. It's why a birthday attack finds a hash collision in ~sqrt(2^bits) tries. See the rising probability in 1D, a live experiment in 2D, and the colliding ring in 3D.",
      "seal": "e2994348d754cc05d2c0466491686e52f7a1098a16b2e6cd774c38c64abc29f8",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-sketch",
      "title": "THE SKETCH",
      "kicker": "Count-Min — tiny memory, never undercounts",
      "accent": "#a0c0ff",
      "blurb": "the Count-Min Sketch in the 5-window house format — estimate item frequencies in a huge stream using a small d-by-w grid of counters and d hash functions. Add: hash d ways and bump those counters. Query: take the minimum of the d counters. Items collide and share cells, so a counter can only be too high, never too low — the sketch never underestimates, with a bounded overestimate. See one item's d cells in 1D, the streaming grid in 2D, and the counter surface in 3D.",
      "seal": "18359d3c5441d3084cf3c0b151021d88cba76ae229458867eaec817b9012452f",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-attractor-net",
      "title": "THE ATTRACTOR NET",
      "kicker": "Hopfield — memory as a valley you fall into",
      "accent": "#c090ff",
      "blurb": "the Hopfield network in the 5-window house format — store patterns as valleys of an energy landscape via the Hebbian rule; async neuron updates each lower the energy E = -1/2 s^T W s (a Lyapunov function), so the state rolls downhill into the nearest stored memory. Feed a corrupted pattern and it cleans itself up — content-addressable memory (Hopfield, Nobel Physics 2024). See the energy drop in 1D, a pattern cleaning up in 2D, and the memory landscape in 3D.",
      "seal": "a55a819c714c49e1e50d573db43f3b88594b5847c3aab989ad2a5488d592ee08",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-derangement",
      "title": "THE DERANGEMENT",
      "kicker": "nobody gets their own hat — probability 1/e",
      "accent": "#90ffd0",
      "blurb": "derangements in the 5-window house format — permutations with no element in its original position (nobody gets their own hat back). The count is !n = round(n!/e), so the probability a random shuffle is a total derangement is 1/e = 36.8% — and it barely depends on n, because the number of coincidences is Poisson(1). See the permutation arrows in 1D, live shuffles converging to 1/e in 2D, and the derangement cloud in 3D.",
      "seal": "eaa16a734226ac2b4e6f0fd78f7e5f2ff5f2f299417328ec2f5e309f83206851",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-mersenne",
      "title": "THE MERSENNE",
      "kicker": "Lucas-Lehmer — exact primality for 2^p-1",
      "accent": "#ff9060",
      "blurb": "the Lucas-Lehmer test in the 5-window house format — a deterministic, exact primality test for Mersenne numbers M_p = 2^p - 1. Set s=4, iterate s <- (s^2 - 2) mod M_p exactly p-2 times; M_p is prime iff the final s is 0. It's how the largest known primes are found (GIMPS; the record 2^136279841-1 has over 41 million digits). See the residue sequence in 1D, the live test in 2D, and the Mersenne ladder in 3D.",
      "seal": "25c0e1e8ce3906453a4bf56f3467b2f9875c383bbc00f0f7f21747c52d94c553",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-partition",
      "title": "THE PARTITION",
      "kicker": "p(n) — counting sums by Euler's pentagonal recurrence",
      "accent": "#d0b0ff",
      "blurb": "the partition function in the 5-window house format — p(n) counts the ways to write n as a sum of positive integers (p(4)=5). Direct counting explodes, but Euler's pentagonal number theorem gives a sparse recurrence p(n)=p(n-1)+p(n-2)-p(n-5)-p(n-7)+... over the generalized pentagonal numbers 1,2,5,7,12,15..., computing p(100)=190569292 instantly. Ramanujan: p(5k+4) is always divisible by 5. See partitions as Young diagrams in 1D, the recurrence and growth in 2D, and the pentagonal engine in 3D.",
      "seal": "96845835dcee9b91091473143e6b806bae9b60ba18ab90873ae981acd0ba119d",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-escape",
      "title": "THE ESCAPE",
      "kicker": "the Mandelbrot set — bounded orbits of z→z²+c",
      "accent": "#c0a0ff",
      "blurb": "the Mandelbrot set in the 5-window house format — the complex c for which z->z^2+c (from z=0) stays bounded. A hard escape criterion (once |z|>2 it's doomed), an exact main cardioid c=mu/2-mu^2/4, and a period-2 bulb that is a perfect disk of radius 1/4 at c=-1. Its real slice is the logistic bifurcation. See the real slice in 1D, the escape-time picture with live orbits in 2D, and the escape surface in 3D.",
      "seal": "44548f4c8f2c2018f081f8f87ed31a80c516c1d4553c6494e4243a60ce9c7793",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-interpolant",
      "title": "THE INTERPOLANT",
      "kicker": "Lagrange — the one polynomial through every point",
      "accent": "#ffd0a0",
      "blurb": "Lagrange interpolation in the 5-window house format — through any n+1 points with distinct x-values there passes exactly one polynomial of degree <= n, written directly as a sum of basis polynomials each equal to 1 at its own node and 0 at the others. It's the unique interpolant, but high-degree fits through equally-spaced points wiggle at the edges (Runge's phenomenon). See the basis spikes in 1D, the live exact fit and Runge demo in 2D, and the curve with its basis in 3D.",
      "seal": "096488a426bda6f0df33239fb58029ba1793615aad897de31c68b6176ebb4170",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-scan",
      "title": "THE SCAN",
      "kicker": "prefix sums in log-depth — a chain made a tree",
      "accent": "#90ffb0",
      "blurb": "the parallel prefix sum (Blelloch scan) in the 5-window house format — turn an array into its running totals in O(log n) parallel depth and O(n) work, via two tree passes: an up-sweep that reduces partial sums up a binary tree, and a down-sweep that broadcasts prefixes back down. Because + is associative the sequential-looking chain reshapes into a shallow tree. The fundamental parallel primitive. See the running total in 1D, the up-sweep/down-sweep in 2D, and the tree in 3D.",
      "seal": "6dc64f894eca3e3b28142100400688328c08ce569dc6339c5a8ca06d0bbacea1",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-perfect",
      "title": "THE PERFECT",
      "kicker": "perfect numbers = Mersenne primes, both ways",
      "accent": "#ffd0e0",
      "blurb": "perfect numbers in the 5-window house format — a number equal to the sum of its proper divisors (6 = 1+2+3, 28 = 1+2+4+7+14). Euclid proved 2^(p-1)(2^p-1) is perfect when 2^p-1 is a Mersenne prime; Euler proved every even perfect number has exactly that form — a perfect one-to-one correspondence. Whether an odd perfect number exists is unknown. See the divisors sum in 1D, the Euclid-Euler correspondence in 2D, and the paired ladders in 3D.",
      "seal": "dcbe93a3acbb385f273d7f2d17603201c46c53044e2497b938bac78d403dc71a",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-totient",
      "title": "THE TOTIENT",
      "kicker": "Euler's phi — the count that runs RSA",
      "accent": "#b0e0a0",
      "blurb": "Euler's totient in the 5-window house format — phi(n) counts the integers from 1 to n coprime to n. Three gems: the product formula phi(n) = n*prod(1-1/p) (phi is multiplicative), the divisor-sum partition sum over d|n of phi(d) = n, and Euler's theorem a^phi(n) = 1 mod n (the reason RSA decrypts). See the totatives in 1D, the count and partition in 2D, and the residue ring in 3D.",
      "seal": "79b5524bb65aa9de5dcc6392fa0d0cde28412e974470c924d882e31e70677e8e",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-look-and-say",
      "title": "THE LOOK-AND-SAY",
      "kicker": "describe yourself forever → Conway's constant 1.3036",
      "accent": "#ffb0d0",
      "blurb": "the look-and-say sequence in the 5-window house format — start with 1 and read each term aloud to get the next: 1, 11, 21, 1211, 111221, 312211... Conway proved no digit ever exceeds 3 (from seed 1), and the term lengths grow by a universal ratio, Conway's constant 1.303577 (root of a degree-71 polynomial). See the terms in 1D, the parse and length growth in 2D, and the growing digit-spiral in 3D.",
      "seal": "884fccb6e593a2fb0870b1ade0881b34cc4e47bc3717777768ffce56839fcb41",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-failure-function",
      "title": "THE FAILURE FUNCTION",
      "kicker": "KMP — match without ever re-reading the text",
      "accent": "#7fffd0",
      "blurb": "Knuth-Morris-Pratt string matching in the 5-window house format — find a pattern in text in O(n+m) with the text pointer never moving backward. The failure function gives, for each pattern position, the longest proper prefix that is also a suffix, so on a mismatch you slide by exactly the right amount, reusing what matched instead of restarting. See the borders in 1D, the live never-backtrack scan in 2D, and the reuse structure in 3D.",
      "seal": "20298bcb9fa92dc9d8590809e52d6139e19baa9bfbeb228c09743eebb0a2130d",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-orthonormal",
      "title": "THE ORTHONORMAL",
      "kicker": "Gram-Schmidt — independence made by subtraction",
      "accent": "#ffd070",
      "blurb": "Gram-Schmidt orthonormalization in the 5-window house format — turn any vectors into an orthonormal set (mutually perpendicular, unit length) spanning the same space, by stripping each vector of its projections onto the ones already chosen and normalizing the remainder. The engine behind QR decomposition and least squares. See the projection subtraction in 1D, the 2D orthogonalization in 2D, and the perpendicular frame in 3D.",
      "seal": "cb8e7fa4cf83d65c55a7e6c27bba12bf4990de389bbf8cab551b52c9e3631608",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-basel",
      "title": "THE BASEL",
      "kicker": "1 + 1/4 + 1/9 + ... = pi^2/6",
      "accent": "#90d0ff",
      "blurb": "the Basel problem in the 5-window house format — the sum of reciprocal squares 1 + 1/4 + 1/9 + 1/16 + ... equals exactly pi^2/6, Euler's 1734 shock: a sum over the plain counting numbers producing pi, the circle constant. The same sine-product trick gives zeta(4) = pi^4/90 and every even zeta value. See the partial sums in 1D, the running total and error in 2D, and the shrinking terms in 3D.",
      "seal": "5532ae0c48aba51ee4525accd9d160019736915944d891f72e50d6218fef6948",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-mobius",
      "title": "THE MOBIUS",
      "kicker": "the sign of the primes that un-mixes divisor sums",
      "accent": "#d0a0ff",
      "blurb": "the Mobius function in the 5-window house format — mu(n) is +1 for a product of an even number of distinct primes, -1 for odd, 0 if any prime repeats. Its identity sum over d|n of mu(d) = [n==1] drives Mobius inversion: if g is the divisor sum of f, then f(n) = sum over d|n of mu(d)*g(n/d). It links phi(n)=sum mu(d)(n/d) and 1/zeta(s). See the mu strip in 1D, the divisor cancellation and inversion in 2D, and the sign-ring in 3D.",
      "seal": "c33e02375cbec8eb1c484d0ef99d147a5f8f3622db01762206857ccd984ed35a",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-butterfly",
      "title": "THE BUTTERFLY",
      "kicker": "the Lorenz attractor — deterministic yet unpredictable",
      "accent": "#7fd0ff",
      "blurb": "the Lorenz attractor in the 5-window house format — three coupled differential equations (Lorenz 1963) producing deterministic chaos: a trajectory that never repeats yet stays forever on a bounded butterfly-shaped strange attractor, with sensitive dependence on initial conditions (the butterfly effect). Two starts a billionth apart diverge to order 1. See one coordinate in 1D, the attractor and diverging twins in 2D, and the butterfly turning in 3D.",
      "seal": "525d373f3420d1c99c6d9018f95ddb62ae394736b92308a35c686ce673d29d81",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-sandpile",
      "title": "THE SANDPILE",
      "kicker": "topple by 4 — a fractal blind to firing order",
      "accent": "#ffd090",
      "blurb": "the abelian sandpile in the 5-window house format — each grid cell holds grains; a cell at 4 topples, shedding one to each neighbour, cascading into avalanches until stable. It always stabilises, and it's abelian: the final pattern and the exact topple count are independent of the order you fire unstable cells. Pour a big pile and it self-organises into a fractal (the founding model of self-organized criticality). See the 1D topple in 1D, the fractal-growing grid in 2D, and the height surface in 3D.",
      "seal": "31d514a5cb4afacc19cc1c491f8bcc2c15929ff4078263c6736a8bd1dde73f5a",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-triangle",
      "title": "THE TRIANGLE",
      "kicker": "add your two neighbours — and get all of combinatorics",
      "accent": "#ffc0e0",
      "blurb": "Pascal's triangle in the 5-window house format — each entry the sum of the two above, generating the binomial coefficients C(n,k). Hidden inside: row n sums to 2^n, shallow diagonals are Fibonacci, a diagonal run totals the entry below it (hockey stick), and colouring the odd entries reveals the Sierpinski fractal (C(n,k) odd iff k's bits are a subset of n's). See a row in 1D, the triangle and its fractal in 2D, and the parity fractal in 3D.",
      "seal": "85738d81d1f835a1c117509b477b19c285329f99e8178333ad1129d91db471c2",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-pisano",
      "title": "THE PISANO",
      "kicker": "Fibonacci mod m — the infinite folded into a cycle",
      "accent": "#90ffd0",
      "blurb": "the Pisano period in the 5-window house format — reduce the Fibonacci numbers modulo m and the sequence becomes periodic, cycling with period pi(m). The last digit (mod 10) repeats every 60; mod 1000 every 1500. It must repeat because only m^2 consecutive pairs exist, so the pigeonhole forces the pair (0,1) to return and the sequence to restart. See the repeating strip in 1D, the cycle and pi(m) plot in 2D, and the residue loop in 3D.",
      "seal": "f706008765cb9c60e3e3d53a75c2fc9b5627e6c597f9163c2d178a5aa11210a1",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-frobenius",
      "title": "THE FROBENIUS",
      "kicker": "the largest amount you can't make — ab-a-b",
      "accent": "#ffd060",
      "blurb": "the Frobenius / Chicken McNugget theorem in the 5-window house format — with two coprime coin values a and b, the largest amount you cannot pay exactly is g(a,b) = ab-a-b (with 3 and 5, it's 7), and the number of unpayable amounts is exactly (a-1)(b-1)/2. Above the Frobenius number every amount is payable. Three or more denominations have no closed formula. See the number line in 1D, the payable strip in 2D, and the coin lattice in 3D.",
      "seal": "810bfec0558524f3318788b799a576c85bd271d90fc273ceba066d2e58d5b838",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-thue-morse",
      "title": "THE THUE-MORSE",
      "kicker": "the fairest turn order — 0110100110010110…",
      "accent": "#b98cff",
      "blurb": "the Thue-Morse sequence in the 5-window house format — start with 0 and repeatedly append the complement (0 -> 01 -> 0110 -> 01101001 -> ...); the n-th bit is the parity of the number of 1s in binary(n). It is the fairest turn order: taking turns in Thue-Morse order cancels the first-mover advantage (Prouhet-Tarry-Escott: the two pick-sets have equal sums of every power up to degree k-1), and it is cube-free (no block repeats three times in a row). See the strip in 1D, the doubling + fair-draft in 2D, and the self-inverse path in 3D.",
      "seal": "07f4ff12c3ceeb12173058d06ec62e18ae6c46986f8d8dcdc1bfe076118ffc73",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-de-bruijn",
      "title": "THE DE BRUIJN",
      "kicker": "the shortest string holding every code — k^n",
      "accent": "#6cf0e0",
      "blurb": "the de Bruijn sequence in the 5-window house format — the shortest cyclic string that contains every length-n word over a k-symbol alphabet exactly once. B(2,3)=00010111 holds all eight 3-bit patterns as a window slides the loop. Its length is exactly k^n, the theoretical minimum, so it is a master key: it cracks every n-digit code in k^n presses instead of n*k^n, because each new keypress completes a fresh code. Under the hood it is an Euler circuit through the de Bruijn graph. See the sliding window in 1D, the lock-cracker in 2D, and the graph circuit in 3D.",
      "seal": "ba40a96d2aaff3330d47cf73c94c948c4a54a738af8f62880f608e42692c9d03",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-ant",
      "title": "THE ANT",
      "kicker": "two rules, ten thousand steps of chaos, then a highway",
      "accent": "#7affb0",
      "blurb": "Langton's ant in the 5-window house format — one ant on an all-white grid: on a white cell turn right/flip/step, on a black cell turn left/flip/step. For the first few hundred steps it makes symmetric shapes, then ~10000 steps of apparent chaos, then with no rule change it locks into a period-104 cycle that builds a straight diagonal 'highway' forever. No one has proven why the highway always appears; it is provably unbounded (Cohen-Kung). See the turn-tape in 1D, the live ant in 2D, and the reversible path in 3D.",
      "seal": "52d5fbf02df5086bade4cc96f16e7afc11638ab2ef2c4f097392e13e71c60059",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-hilbert",
      "title": "THE HILBERT",
      "kicker": "fill the square without ever jumping — locality kept",
      "accent": "#62d0ff",
      "blurb": "the Hilbert space-filling curve in the 5-window house format — a continuous fractal path that visits every cell of a 2^n x 2^n grid exactly once, and never jumps: consecutive cells are always neighbors. It maps 2D to 1D while preserving locality (points close on the line stay close on the plane), unlike row-major scanning which tears vertical neighbors apart. Used for cache-friendly layouts, spatial index keys, dithering, and R-tree ordering. See the index line in 1D, the drawn curve in 2D, and the locality contrast in 3D.",
      "seal": "ac27cdeb7eaba264ede56bbe6d3a3b8661cc8f9b95625d525ea4ca71822859e9",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-benford",
      "title": "THE BENFORD",
      "kicker": "1 leads 30% of the time — the fingerprint of honest numbers",
      "accent": "#ffcf4a",
      "blurb": "Benford's law in the 5-window house format — in data spanning many orders of magnitude, the leading digit is not uniform: 1 leads about 30% of the time and 9 only ~4.6%, with frequency exactly log10(1+1/d). It holds for populations, prices, physical constants, Fibonacci numbers, and powers of 2. Fabricated figures have too-uniform leading digits, so violating Benford is a forensic-accounting red flag. See the digit bars in 1D, the live dataset switch + fraud flag in 2D, and the scale-invariance log cylinder in 3D.",
      "seal": "850d0c7e2ef50b5d3d3ad38959853af60965416a7bbaa3203b41c08905cacd44",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-kolakoski",
      "title": "THE KOLAKOSKI",
      "kicker": "the sequence that is its own run-length encoding",
      "accent": "#ff9ad0",
      "blurb": "the Kolakoski sequence in the 5-window house format — a string of 1s and 2s that describes itself: its run-lengths (how many identical symbols in a row) spell out the very same sequence. It is built by reading itself, each symbol dictating the next run length, a genuine strange loop. It looks random but is deterministic; the density of 1s appears to approach 1/2 but this is unproven. See the self-bracketing strip in 1D, the step-by-step build in 2D, and the fixed-point loop in 3D.",
      "seal": "66d3a2609531aa64d9cf26fac3d0f520e1d145e3c5a5919517416177ae1cf833",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-recaman",
      "title": "THE RECAMAN",
      "kicker": "jump back if you can, else forward — the arc that haunts",
      "accent": "#5ad0e0",
      "blurb": "Recaman's sequence in the 5-window house format — start at 0; at step n try to jump back by n, and if that lands on a positive unvisited number take it, else jump forward by n. It produces 0,1,3,6,2,7,13,20,12,21,... and its alternating semicircle arc diagram is one of the most haunting pictures in mathematics. Its open question: does every natural number eventually appear? Conjectured yes, unproven. See the value line in 1D, the arc diagram in 2D, and the arcs-and-holes in 3D.",
      "seal": "4516a4b50cb451586a129a9cba4211b48dede31d2a5d45812941ec2fde3e86a0",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-van-eck",
      "title": "THE VAN ECK",
      "kicker": "each term = how long since it last appeared",
      "accent": "#c0ff70",
      "blurb": "Van Eck's sequence in the 5-window house format — start with 0; the next term is how many steps ago the current term last appeared, or 0 if it is brand new. That yields 0,0,1,0,2,0,2,2,1,6,0,5,0,... a sequence made entirely of its own memory, each term a measurement of recency. Proven: infinitely many zeros; open: whether every natural number appears. See the recency arcs in 1D, the memory-lookup build in 2D, and the backward-links helix in 3D.",
      "seal": "fc6ca1888abbe962d93fbd74b09e317554b1c7e6f85299f353236e3aab8d3843",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-moser",
      "title": "THE MOSER",
      "kicker": "1, 2, 4, 8, 16, 31 — the pattern that breaks",
      "accent": "#ff9060",
      "blurb": "Moser's circle problem in the 5-window house format — place n points on a circle and draw every chord (general position); count the regions. You get 1,2,4,8,16 (clearly powers of 2) and then n=6 gives 31, NOT 32. The true count is C(n,4)+C(n,2)+1, which agrees with 2^(n-1) for exactly the first five terms then diverges forever. It is the textbook warning that five data points do not determine the rule. See the two sequences in 1D, the live chord figure in 2D, and the underdetermination in 3D.",
      "seal": "d71a4d232f60d07064074e485bb9a9f1b4948f5e3b455a924866006b9a124f7f",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-hofstadter",
      "title": "THE HOFSTADTER",
      "kicker": "Q(n)=Q(n-Q(n-1))+Q(n-Q(n-2)) — chaos that might not survive",
      "accent": "#b0b0ff",
      "blurb": "Hofstadter's Q-sequence in the 5-window house format — a chaotic meta-Fibonacci: Q(1)=Q(2)=1, Q(n)=Q(n-Q(n-1))+Q(n-Q(n-2)). Unlike Fibonacci which looks back a fixed distance, this looks back a distance that depends on its own recent values, so the sequence reads from addresses it computes from itself. The result is wildly erratic and never settles into a formula (from Hofstadter's Godel Escher Bach). It is not even known whether Q(n) is defined for all n. See the jitter in 1D, the self-lookup plot in 2D, and the fixed-vs-computed address contrast in 3D.",
      "seal": "a84bfa413755b149045e3980f4afd775d948a9649c5280d23876e2a8735b9b5c",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-sylvester",
      "title": "THE SYLVESTER",
      "kicker": "2, 3, 7, 43, 1807 — unit fractions that fill exactly one",
      "accent": "#90e0b0",
      "blurb": "Sylvester's sequence in the 5-window house format — 2,3,7,43,1807,3263443,... where each term is the product of all previous terms plus one (a(n)=a(n-1)^2-a(n-1)+1), growing doubly exponentially. The reciprocals 1/2+1/3+1/7+1/43+... race to exactly 1: it is the greedy Egyptian-fraction expansion of 1, taking the largest unit fraction that fits at each step, and the leftover gap after n terms is precisely 1/(a(n+1)-1). See the fraction bars in 1D, the greedy fill in 2D, and the remainder loop in 3D.",
      "seal": "ae5b32b6693d286e4107158c2762b2d2f3320574399e9608bc8b8ccaca8fe31b",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-perrin",
      "title": "THE PERRIN",
      "kicker": "every prime divides P(p) — a near-perfect gate",
      "accent": "#ff6a6a",
      "blurb": "the Perrin sequence in the 5-window house format — seeded P(0)=3,P(1)=0,P(2)=2 with P(n)=P(n-2)+P(n-3), giving 3,0,2,3,2,5,5,7,10,12,... It hides a near-perfect primality test: for every prime p, p divides P(p), so P(p) mod p == 0. It was hoped no composite could pass, but Perrin pseudoprimes exist — the smallest is 271441 = 521^2 — just extremely rare. A fast, nearly flawless prime detector that is not a proof. See the residues in 1D, the prime-detector grid in 2D, and the necessary-vs-sufficient gap in 3D.",
      "seal": "48d4f986a3db1a3ed57f4aca5b9eadc77d77927ea104d630d6788c649eca0143",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-golomb",
      "title": "THE GOLOMB",
      "kicker": "a(n) = how many times n appears in itself",
      "accent": "#d0b0ff",
      "blurb": "Golomb's self-counting sequence in the 5-window house format — a nondecreasing sequence where a(n) is the number of times n appears in the sequence itself: 1,2,2,3,3,4,4,4,5,5,5,6,6,6,6,... 1 appears once (a(1)=1), 2 appears twice (a(2)=2), 4 appears three times (a(4)=3). It is the unique such sequence, obeys a(n)=1+a(n-a(a(n-1))), and grows like n^(phi-1) with the golden ratio in the exponent. A census that is its own population. See the staircase in 1D, the self-tally histogram in 2D, and the value=frequency loop in 3D.",
      "seal": "08f1bef5d2937d3957d2867c342ae6fdc4f2a83bdc8d9335e50f8a467218f91f",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-persistence",
      "title": "THE PERSISTENCE",
      "kicker": "multiply the digits, repeat — 277777788888899 resists 11 times",
      "accent": "#ffa0c0",
      "blurb": "multiplicative persistence in the 5-window house format — multiply a number's digits together, repeat until a single digit remains; the number of steps is its persistence. 39->27->14->4 has persistence 3. Almost every number collapses fast, but the smallest number with persistence 11 is 277777788888899, and despite searching past 10^233, no number in base 10 has ever shown persistence greater than 11. Conjectured maximum, unproven. See the record's collapse chain in 1D, the digit-grinder in 2D, and the collapse-vs-resistance asymmetry in 3D.",
      "seal": "7abe3600ecac3436e495f3d07547113555e0c891f610e872f4168fb0aa5cae54",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-calkin-wilf",
      "title": "THE CALKIN-WILF",
      "kicker": "every positive rational, once, in lowest terms",
      "accent": "#90d0ff",
      "blurb": "the Calkin-Wilf tree in the 5-window house format — a single list 1/1, 1/2, 2/1, 1/3, 3/2, 2/3, 3/1, ... that contains every positive rational exactly once, each already in lowest terms. Build a tree with root 1/1 where each a/b has children a/(a+b) and (a+b)/b; read it breadth-first and you get the sequence. You can step to the next term by a pure formula a(n+1)=1/(2*floor(a)-a+1), and it is powered by Stern's fusc function: a(n)=fusc(n)/fusc(n+1). A constructive proof the rationals are countable. See the interlocking strip in 1D, the tree in 2D, and the two-way dictionary in 3D.",
      "seal": "0fcc83220421574dd2dcd63d614268a32a446435114cadae879fd854c0a83335",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-pick",
      "title": "THE PICK",
      "kicker": "a polygon's area from counting dots — I + B/2 − 1",
      "accent": "#a0e070",
      "blurb": "Pick's theorem in the 5-window house format — for a polygon whose corners all sit on integer grid points, the area is exactly I + B/2 - 1, where I is the number of interior grid points and B the number on the boundary. No calculus, no coordinate multiplication — just count the dots. A shape with 6 interior and 14 boundary dots has area exactly 6 + 7 - 1 = 12, always agreeing with the shoelace formula. See the formula in 1D, the live dot-count vs shoelace in 2D, and the same-area-different-shape non-uniqueness in 3D.",
      "seal": "b4651f2379314b0c88570af0e68091ad9bb7a954c65474ee106ec4d2248f88e5",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-napoleon",
      "title": "THE NAPOLEON",
      "kicker": "equilaterals on any triangle — their centers are equilateral",
      "accent": "#ffb060",
      "blurb": "Napoleon's theorem in the 5-window house format — take any triangle, build an equilateral triangle outward on each side, and mark each center; those three centers always form a perfect equilateral triangle, no matter how irregular the original. Building the equilaterals inward gives a second equilateral, and area(outer) - area(inner) equals the original triangle's area. Traditionally credited to Napoleon Bonaparte (likely a legend). See the equal side-bars in 1D, the live construction in 2D, and the inner/outer area identity in 3D.",
      "seal": "e1d7595da9a0d03137f1a7b88a8e9f1925a0a465b12eec74d1274d26450f4a55",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-viviani",
      "title": "THE VIVIANI",
      "kicker": "three distances, one constant sum — the height",
      "accent": "#70d0e0",
      "blurb": "Viviani's theorem in the 5-window house format — from any point inside an equilateral triangle, the three perpendicular distances to the sides always sum to the same total: the triangle's height, no matter where the point is. Move the point and the distances trade off, one shrinking as others grow, but the sum never changes. It is the geometric basis of barycentric coordinates: the normalized distances are weights summing to 1 that pin the point's location. See the fixed-length stacked bar in 1D, the moving point in 2D, and the barycentric inverse in 3D.",
      "seal": "4cad4d902e5bb0175c7a5fb2a665a26723dc399ddda51ede15ecb3f94020cbdd",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-morley",
      "title": "THE MORLEY",
      "kicker": "trisect any triangle's angles — the meeting points are equilateral",
      "accent": "#ff80ff",
      "blurb": "Morley's trisector theorem ('Morley's miracle') in the 5-window house format — take any triangle, split each angle into three equal parts with trisectors, and where adjacent trisectors meet they mark three points that always form a perfect equilateral triangle. It stayed hidden until 1899, more than two millennia after the Greeks, because it depends on angle trisection — the operation compass and straightedge cannot perform. See the equal side-bars in 1D, the live trisectors in 2D, and the trisect-vs-bisect contrast in 3D.",
      "seal": "60a1ca3cba6e124b19b7acce0ee1ba30e6060b51ca9bbe55ee7ccda1e4dfea03",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-varignon",
      "title": "THE VARIGNON",
      "kicker": "midpoints of any quadrilateral form a parallelogram",
      "accent": "#80ffb0",
      "blurb": "Varignon's theorem in the 5-window house format — mark the midpoint of each side of any quadrilateral and join them in order; the result is always a parallelogram, however irregular or non-convex the quadrilateral. Its area is exactly half the quadrilateral's, and its perimeter equals the sum of the quadrilateral's two diagonals — because each of its sides is a midline parallel to a diagonal and half its length. See the two-equal-pairs bars in 1D, the morphing quad in 2D, and the diagonal-shadow inverse in 3D.",
      "seal": "beaf75f2053c7ec95e19197219a48490ae8b9512dbbf2bfb2ad4687d0faca421",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-parrondo",
      "title": "THE PARRONDO",
      "kicker": "two losing games that combine into a winning one",
      "accent": "#ff6060",
      "blurb": "Parrondo's paradox in the 5-window house format — two gambling games each rigged to lose long-term, yet alternating between them (or switching at random) makes capital climb. Game A is a slightly biased losing coin; Game B uses a dreadful coin when capital is a multiple of 3 and a great one otherwise, losing on its own by getting stuck on the bad coin. Game A stirs the capital so B lands on its good coin more often — each game's weakness patched by the other. A real ratchet mechanism. See the ratchet in 1D, the live three-trajectory race in 2D, and the convexity break in 3D.",
      "seal": "cf6ac691c359f8dc580bcdf816fb72a49db678224438ab6c7067056d5530b68c",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-simpson",
      "title": "THE SIMPSON",
      "kicker": "A wins every subgroup, B wins the total",
      "accent": "#ffc050",
      "blurb": "Simpson's paradox in the 5-window house format — a trend that holds in every subgroup can reverse when the subgroups are pooled. On genuine 1986 kidney-stone data, treatment A beats B for small stones (93% vs 87%) and for large stones (73% vs 69%), yet pooled, B wins (83% vs 78%). The cause is a lurking variable: A was used mostly on the hard (large-stone) cases, B on the easy ones, so the pooled rate is a weighted blend whose uneven weights flip the verdict. See the flipping bars in 1D, the case-mix pooling in 2D, and the mediant-vector inverse in 3D.",
      "seal": "26c88ed023cd5f4d0e7e002aef3d14ff1213d5774d8da7a5dcd524fdb73b4012",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-secretary",
      "title": "THE SECRETARY",
      "kicker": "reject the first 37%, then leap — win the best 1/e of the time",
      "accent": "#60c0ff",
      "blurb": "the secretary problem (optimal stopping) in the 5-window house format — candidates arrive one at a time in random order; you must accept or reject each on the spot with no going back, and you want to hire the single best. The optimal rule: reject the first n/e (~37%) while noting the best among them, then hire the first later candidate who beats them all. This wins the best with probability ~1/e ~ 37%, and that rate does not fade as n grows. See the look-then-leap run in 1D, the success-vs-cutoff curve in 2D, and the explore/exploit inverse in 3D.",
      "seal": "81b6ce3d9978562c4a7508113b984c887615421ba1ba56c212f88a75c2877a75",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-penney",
      "title": "THE PENNEY",
      "kicker": "pick any coin-triple, the second player beats it",
      "accent": "#ff90d0",
      "blurb": "Penney's game in the 5-window house format — two players each pick a length-3 coin sequence, then flip until one appears; whoever's shows first wins. It looks symmetric but isn't: whatever the first player picks, the second can always pick a sequence that wins more than half the time. Conway's rule: beat ABC with (not-B)AB; against HHH the counter THH wins 7 of 8. The sequences are nontransitive — an endless rock-paper-scissors with no best choice. See the coin-stream race in 1D, the live win-tally in 2D, and the beats-cycle in 3D.",
      "seal": "f545a01b66cc7412e76e0fbced7b6215dc9e9e4b7862bedb582dd9249b53805f",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-st-petersburg",
      "title": "THE ST PETERSBURG",
      "kicker": "infinite expected value, worth about $4 to play",
      "accent": "#ffd860",
      "blurb": "the St. Petersburg paradox in the 5-window house format — flip a fair coin until heads; if the first heads is on flip n, win $2^n. The expected winnings are 1/2*$2 + 1/4*$4 + 1/8*$8 + ... = $1 + $1 + $1 + ... = infinite, so expected-value logic says pay any finite price to play, yet almost no one would pay even $10. The resolution is diminishing utility: the expected log2 of the payout is finite, exactly 2, giving a value near $4. See the dollar-per-term EV ladder in 1D, the never-settling running mean in 2D, and the linear-vs-log inverse in 3D.",
      "seal": "fe6bff2e60de86f78ff0539e9d1a82df0a453fb98ae8ebd752ba84da5884dbf2",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-fano",
      "title": "THE FANO",
      "kicker": "7 points, 7 lines — the smallest projective plane",
      "accent": "#ff70a0",
      "blurb": "the Fano plane in the 5-window house format — the smallest projective plane, just 7 points and 7 lines, where every two points lie on exactly one line and every two lines meet in exactly one point. Each line holds 3 points; each point lies on 3 lines. Drawn as a triangle with its edge-midpoints and center plus a circle for the seventh line, it is the (7,3,1) Steiner triple system, the plane PG(2,2), and the octonion multiplication rule, carrying 168 symmetries. See the triples in 1D, the classic clickable diagram in 2D, and the point/line self-duality in 3D.",
      "seal": "d7c8af586e1e424d7b937ec28a853d6c62b692a8764b7fc24fe9321274f89bb5",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-hadamard",
      "title": "THE HADAMARD",
      "kicker": "a ±1 matrix with every row orthogonal — H·Hᵀ = nI",
      "accent": "#60d0ff",
      "blurb": "the Hadamard matrix in the 5-window house format — a square grid of only +1 and -1 whose rows are all mutually orthogonal (any two different rows agree in exactly half their entries), so H*H^T = nI. Sylvester's doubling builds one at every power of two: start with [1] and tile four copies with the bottom-right negated. The rows are Walsh/Hadamard codes — non-interfering signals behind CDMA — and an error-correcting code (the [32,6] Hadamard code flew on Mariner 9). See the orthogonal Walsh waveforms in 1D, the checkerboard matrix in 2D, and the self-inverse encode/decode in 3D.",
      "seal": "39e9802d807bbfeba563ca5fddff783b4bcfa59bc6d8278fee0ef5e4261cf78f",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-golomb-ruler",
      "title": "THE GOLOMB RULER",
      "kicker": "marks whose every pairwise distance is distinct",
      "accent": "#ffd070",
      "blurb": "the Golomb ruler in the 5-window house format — a ruler whose marks are placed so no two pairs are the same distance apart, every pairwise distance unique. The 4-mark {0,1,4,6} measures 1..6 each exactly once. Finding the shortest ruler for a given mark count (the Optimal Golomb Ruler) is hard, with no formula and multi-year distributed searches. Rulers of <=4 marks are perfect (measure every length once); 5+ cannot be. Used for radio-telescope antenna placement, frequency assignment, crystallography, and codes. See the distance arcs in 1D, the collision-detecting difference grid in 2D, and the turnpike inverse in 3D.",
      "seal": "f78c9d3ade9bbd16639f0b8c59eac2ec8e45fb25ab43a4bc2abf09c3c6f8706a",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-langford",
      "title": "THE LANGFORD",
      "kicker": "arrange 1,1,2,2,…,n,n so the two k's are k apart",
      "accent": "#b0ff60",
      "blurb": "the Langford pairing in the 5-window house format — arrange the numbers 1,1,2,2,...,n,n in a row so the two copies of each k have exactly k numbers between them. For n=3: 2,3,1,2,1,3. Such an arrangement exists if and only if n is congruent to 0 or 3 mod 4, so n=3,4,7,8 work while n=1,2,5,6 are impossible. Langford spotted it watching his son's blocks; solution counts explode (n=7:26, n=8:150, n=16: over 46 billion). See the gap-arcs in 1D, the solver-or-impossibility in 2D, and the parity obstruction in 3D.",
      "seal": "722b3bef247938a550160b2c2863e3fddebd87c463683e126b4c529eaa35baa4",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-costas",
      "title": "THE COSTAS",
      "kicker": "one dot per row & column, every displacement distinct",
      "accent": "#ff8060",
      "blurb": "the Costas array in the 5-window house format — an n x n grid with one dot per row and column (a permutation) placed so all displacement vectors between pairs of dots are distinct. This gives an ideal thumbtack autocorrelation: slide a copy over itself and at every nonzero shift at most one dot coincides — exactly what sonar and radar need for unambiguous range/Doppler. John Costas invented them for sonar (1965); the Welch construction builds size p-1 from a primitive root mod a prime p. See the difference triangle in 1D, the array and its self-overlap in 2D, and the autocorrelation inverse in 3D.",
      "seal": "148222e5b9a12cc13f76f0922b64a1cbc30f56da35e67550a44fafdbff1ec347",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-quine",
      "title": "THE QUINE",
      "kicker": "a program that prints its own source, exactly",
      "accent": "#80ffe0",
      "blurb": "the quine in the 5-window house format — a program that takes no input and prints its own source code exactly, without reading its own file. It splits into a data part (a string describing the code) and a code part that prints the data twice: once quoted, once interpreted. The classic JS quine (function a(){return \"(\"+a+\")()\"})() evaluates to its own text. Kleene's recursion theorem proves every Turing-complete language has quines; the same self-reference underlies computer viruses and von Neumann self-replication. See the data/code split in 1D, the run-and-compare in 2D, and the fixed-point ouroboros in 3D.",
      "seal": "f3e6e420d4f11b4b5ea3a499e65f6ee2afac1e3d97c8967319480d8b63a683df",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-y-combinator",
      "title": "THE Y-COMBINATOR",
      "kicker": "recursion with no name — Y f = f (Y f)",
      "accent": "#a0d0ff",
      "blurb": "the Y combinator in the 5-window house format — a function that manufactures recursion out of nothing, letting a nameless function call itself. It satisfies Y f = f (Y f); in pure lambda calculus Y = lambda f.(lambda x.f(x x))(lambda x.f(x x)), where self-application (x x) feeds a function itself (eager languages use the delayed Z variant). With it you build factorial from a function that never names itself — write 'given self, return n*self(n-1)' and Y supplies the self. It is the theoretical heart of how recursion exists. See the fixed-point tower in 1D, the self-supplying recursion in 2D, and the fixed-point inverse in 3D.",
      "seal": "7b611b85da588758eda0780ca4f3b136da253650efc15196726dbc22f6be005d",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-church",
      "title": "THE CHURCH",
      "kicker": "numbers as pure functions — 3 = λf.λx. f(f(f(x)))",
      "accent": "#ffb0e0",
      "blurb": "Church numerals in the 5-window house format — encode the whole numbers as pure functions, no digits or arithmetic primitives: the number n means 'do f, n times', so 3 = lambda f.lambda x. f(f(f(x))) and zero applies f not at all. From this, all arithmetic follows with only function application: successor wraps one more f, addition runs one numeral's applications after another, multiplication nests them, and exponentiation is just applying one numeral to another (m^n = n m). Decode by feeding a numeral 'add 1' and 0. Alonzo Church's proof that numbers and all computation reduce to the single idea of a function. See the application chain in 1D, the pure-function calculator in 2D, and the encode/decode inverse in 3D.",
      "seal": "b0a4ab09a368788d888dbf9f1634ac09cbd2b1edbb79e84ed9e577e9609af3f4",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-ski",
      "title": "THE SKI",
      "kicker": "two operators, no variables — S and K compute everything",
      "accent": "#90ffb0",
      "blurb": "SKI combinator calculus in the 5-window house format — a model of computation with no variables at all, just two operators and application: K x y = x (keep the first, drop the second) and S x y z = x z (y z) (hand the third argument to both others). Astonishingly, S and K alone express every lambda-calculus function and therefore everything computable, with no bound variables; the identity is I = S K K. Moses Schonfinkel showed variables can be eliminated from logic entirely (point-free / combinatory logic). See the three rules in 1D, the step reducer in 2D, and the point-free/point-ful inverse in 3D.",
      "seal": "2f0a7e546028a40371b1b4cf584ddc02304cb26576590c15da8d9f165a1ca2e3",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-tag",
      "title": "THE TAG",
      "kicker": "delete the front, grow the tail — a universal computer in 3 rules",
      "accent": "#ffd0a0",
      "blurb": "the Post tag system in the 5-window house format — one of the simplest computers: hold a string, look at the first symbol, delete the first m symbols from the front, and append a production to the back based on that symbol. A 2-tag system is Turing-complete and its halting is undecidable. The specific system a->bc, b->a, c->aaa secretly computes the Collatz sequence: start with n copies of 'a' and the all-'a' states you pass through trace the Collatz trajectory of n. See the tag step in 1D, the live Collatz-computing queue in 2D, and the computational-irreducibility inverse in 3D.",
      "seal": "48b64c25ecea5f26ab9e1ef938ff76cd2984a3b8e37439127f351466b5ade596",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-carmichael",
      "title": "THE CARMICHAEL",
      "kicker": "a composite that fools the Fermat test to every base",
      "accent": "#ff5090",
      "blurb": "Carmichael numbers in the 5-window house format — composites that pass Fermat's primality test (a^(n-1) = 1 mod n) for every base a coprime to them, having no Fermat witness to their compositeness at all. The smallest is 561 = 3*11*17. Korselt's criterion: n is Carmichael iff squarefree and (p-1) divides (n-1) for every prime factor p. There are infinitely many (proved 1994), and even 1729 is one. They are why real primality testing uses the stronger Miller-Rabin test, which they cannot fool. See the base wall in 1D, the witness scan in 2D, and the Fermat-vs-Miller-Rabin inverse in 3D.",
      "seal": "112b874eb3d7b3b35b1c142dac49c12410161f571029d46891952866baeafff0",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-pell",
      "title": "THE PELL",
      "kicker": "one seed solution breeds infinitely many — x²−2y²=1",
      "accent": "#60e0c0",
      "blurb": "Pell's equation in the 5-window house format — x^2 - D y^2 = 1 for non-square D. For D=2 the smallest solution is (3,2) since 9-8=1, and from that fundamental solution all others cascade by (x,y)->(3x+4y, 2x+3y): (17,12),(99,70),(577,408),... infinitely many. Equivalently they are the powers (3+2√2)^k, and each ratio x/y is a razor-sharp approximation to √2 (the convergents of [1;2,2,2,...]). It runs from Brahmagupta and Bhaskara's chakravala through Fermat to Lagrange, and is misnamed after John Pell. See the ratios in 1D, the cascade in 2D, and the group-inverse on the hyperbola in 3D.",
      "seal": "93cd51cba57eb9c203c2c2f8d914387331c6abe6e2e835f14729361003ab816e",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-farey",
      "title": "THE FAREY",
      "kicker": "fractions in order, mediants, and kissing Ford circles",
      "accent": "#ffd0ff",
      "blurb": "the Farey sequence in the 5-window house format — all fractions in [0,1] in lowest terms with denominator <= n, in order of size. Two miracles: neighbours a/b < c/d satisfy bc - ad = 1 (unimodular, as tight as coprime fractions get), and the first fraction to appear between two neighbours is their mediant (a+c)/(b+d). Ford circles visualize it: a circle of radius 1/(2q^2) on each p/q tangent to the line, and two circles kiss exactly when their fractions are Farey neighbours. See the ordered sequence in 1D, the Ford-circle packing in 2D, and the mediant/un-mediant inverse in 3D.",
      "seal": "c41e72e696d20727c9957faf354da32ba5996acfb8d56292a331eb702cc8eaa5",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-pythagorean-tree",
      "title": "THE PYTHAGOREAN TREE",
      "kicker": "every primitive right triangle grown from (3,4,5)",
      "accent": "#70ff90",
      "blurb": "the Barning-Hall Pythagorean tree in the 5-window house format — every primitive Pythagorean triple (right triangle with coprime integer sides) grown from the single seed (3,4,5). Each triple has exactly three children, obtained by multiplying its column vector by three fixed 3x3 integer matrices A, B, C: (3,4,5) -> (5,12,13),(21,20,29),(15,8,17), and so on. The theorem: this ternary tree contains every primitive triple exactly once — none missing, none repeated. It is the additive cousin of Euclid's m,n formula (Barning 1963, Hall 1970). See the seed and children in 1D, the tree in 2D, and the infinite-descent inverse in 3D.",
      "seal": "0b7c1d23dbabce5631818daa08e213036bf3a28a9860333f9353d2949563d4bf",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-reciprocity",
      "title": "THE RECIPROCITY",
      "kicker": "is p a square mod q? — Gauss's golden theorem links it to q mod p",
      "accent": "#ffe070",
      "blurb": "quadratic reciprocity in the 5-window house format — Gauss's golden theorem. For two odd primes, 'is p a square mod q?' and 'is q a square mod p?' have a hidden link: the answers are the same, unless both p and q are 3 mod 4, in which case they are opposite. With the Legendre symbol (p/q) = +1 if p is a square mod q else -1, the law is (p/q)(q/p) = (-1)^(((p-1)/2)((q-1)/2)). Gauss proved it eight different ways; it underlies algebraic number theory and cryptography. Symbols compute fast by Euler's criterion a^((p-1)/2) mod p. See the residues in 1D, the reciprocity check in 2D, and the p<->q transpose inverse in 3D.",
      "seal": "9d5cc71fb582f4c4d869967ef653d23e66191822e25ff25ed0edf067edbd7944",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "slug": "the-euler",
      "title": "THE EULER",
      "kicker": "cross every bridge once — the birth of graph theory",
      "accent": "#70c0ff",
      "blurb": "the Seven Bridges of Konigsberg in the 5-window house format — in 1736 Euler asked whether you could walk across all seven bridges of Konigsberg exactly once, proved it impossible, and invented graph theory. Turning land masses into vertices and bridges into edges, an Euler path (every edge once) exists iff the graph is connected with 0 or exactly 2 odd-degree vertices; with 0 odd you can return home (Euler circuit). Konigsberg's four land masses had degrees 5,3,3,3 (all odd), so no walk exists. See the land-mass degrees in 1D, the live bridge graph in 2D, and the Euler-vs-Hamilton inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "3373a91dad45c6c6120cc915d907c37f6ae3798a4a0c895453174fd928644f50"
    },
    {
      "slug": "the-ramsey",
      "title": "THE RAMSEY",
      "kicker": "among any 6 people, 3 friends or 3 strangers — unavoidable",
      "accent": "#ff9060",
      "blurb": "Ramsey's theorem in the 5-window house format — complete disorder is impossible. Among any 6 people there must be 3 who all know each other or 3 who are all mutual strangers; with 5 people you can avoid both (friendships as a pentagon, strangerhoods as the pentagram). The threshold is the Ramsey number R(3,3)=6: every 2-colouring of the complete graph K6 contains a monochromatic triangle, while some 2-colouring of K5 contains none. Ramsey numbers explode and are mostly unknown (even R(5,5) is only pinned between 43 and 48). See the counts in 1D, the forced triangle in 2D, and the order-from-disorder inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "3ce1b90ae1846b4d4438dee32fce9056682037705ddab144faee446f74187e4e"
    },
    {
      "slug": "the-prufer",
      "title": "THE PRUFER",
      "kicker": "a labeled tree ⟷ a short number sequence — Cayley's n^(n-2)",
      "accent": "#b0e070",
      "blurb": "Cayley's formula and the Prufer sequence in the 5-house format — the number of labeled trees on n numbered vertices is exactly n^(n-2) (5 vertices give 125 trees, 6 give 1296). Prufer's bijection proves it: encode a tree by repeatedly pruning the smallest-labeled leaf and recording its neighbour (n-2 steps, values 1..n); decode by reversing. Since there are exactly n^(n-2) sequences and each names one tree, Cayley's formula follows. Each vertex's degree equals its count in the sequence plus one. See the pruning in 1D, the encode/decode in 2D, and the codec bijection in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "6c4ae2af8a2ace6a43f5f04113bcae13d65b0c85b5dc6edaa8b85aa9528f3c80"
    },
    {
      "slug": "the-hall",
      "title": "THE HALL",
      "kicker": "everyone can be matched iff no k suitors share only k-1 options",
      "accent": "#ff90b0",
      "blurb": "Hall's marriage theorem in the 5-window house format — when can everyone be paired to an acceptable partner, none shared? A perfect matching exists if and only if Hall's condition holds: every group of k people together has at least k acceptable partners between them. One direction is pigeonhole (k people with k-1 options are stuck); the deep half is that this condition is also sufficient. Equivalently the system-of-distinct-representatives theorem. See the pooled-options test in 1D, the bipartite matcher in 2D, and the obstruction-certificate inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "2147f5037c60578080ab7bc3d4f08d66a3b8d0b20eb8aa19d99e871e983ce4dd"
    },
    {
      "slug": "the-matrix-tree",
      "title": "THE MATRIX-TREE",
      "kicker": "count every spanning tree with one determinant",
      "accent": "#90d0ff",
      "blurb": "Kirchhoff's Matrix-Tree theorem in the 5-window house format — a graph can have astronomically many spanning trees, but counting them collapses to one determinant. Build the Laplacian (degree on the diagonal, -1 per edge), delete any one row and its matching column, take the determinant — that number is exactly the count of spanning trees. For the complete graph K_n it returns Cayley's n^(n-2) (the count Prufer proves another way); for a cycle, n; for a tree, 1. Kirchhoff found it in 1847 analysing electrical circuits. See the Laplacian in 1D, the compute-vs-count in 2D, and the enumerate-vs-compute inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "2f215af4e1c649859dcacb5d158ac73b7baf80176ee108bb6c2873ef15c56865"
    },
    {
      "slug": "the-diffie-hellman",
      "title": "THE DIFFIE-HELLMAN",
      "kicker": "agree a secret over an open channel — never sent",
      "accent": "#70e0a0",
      "blurb": "Diffie-Hellman key exchange in the 5-window house format — two strangers agree on a shared secret while eavesdroppers listen, without ever sending it. Public: a prime p and base g. Alice publishes g^a mod p, Bob publishes g^b mod p; Alice computes (g^b)^a and Bob computes (g^a)^b, both equal g^(ab) mod p. An eavesdropper with g^a and g^b must solve the discrete logarithm, believed intractable for large p. It launched public-key cryptography and secures much of the internet. See the exchange in 1D, the channel with Eve in 2D, and the one-way-function inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "775842baa7b11dbef035f396e4642f06251932f862b7e35db445799f0adb906f"
    },
    {
      "slug": "the-rsa",
      "title": "THE RSA",
      "kicker": "a lock anyone can close, only the key-holder opens",
      "accent": "#ffb060",
      "blurb": "RSA public-key encryption in the 5-window house format — pick primes p,q; publish n=pq and exponent e. Anyone encrypts a message m as c = m^e mod n; the owner decrypts with a private exponent d where c^d mod n = m. d is the inverse of e modulo phi(n)=(p-1)(q-1), and by Euler's theorem m^(ed) = m mod n, so encrypt and decrypt undo each other. Finding d requires phi, which requires factoring n into p*q — believed intractable. Anyone can lock (public e); only the owner unlocks (private d). See the keys in 1D, the round-trip in 2D, and the multiply-vs-factor trapdoor in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "9a0b723e16bc6530d976c2a1672b0cf02bf9bf5d5f652048f2cf398463f67726"
    },
    {
      "slug": "the-elliptic-curve",
      "title": "THE ELLIPTIC-CURVE",
      "kicker": "the group hidden in a cubic — modern crypto's engine",
      "accent": "#a070ff",
      "blurb": "elliptic curves in the 5-window house format — the points on y^2 = x^3 + ax + b, where you can add two points by a chord-and-tangent rule (line through them meets the curve a third time, reflect over the x-axis). This turns the points into a group: associative, with an identity (point at infinity) and inverses. Over a finite field the group is finite; multiplying a point by a scalar k is fast, but recovering k (the elliptic-curve discrete log) is believed even harder than ordinary discrete log, so ECC matches RSA security with far smaller keys. Secures TLS, signatures, Bitcoin. See the multiples in 1D, the point grid in 2D, and the multiply-vs-log inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "221fa06458eb5f966e7b82dd4e1c8a0e625387d9addc844b488b1b5008553b62"
    },
    {
      "slug": "the-one-time-pad",
      "title": "THE ONE-TIME-PAD",
      "kicker": "the only provably unbreakable cipher — used once",
      "accent": "#ffe0a0",
      "blurb": "the one-time pad in the 5-window house format — the only cipher proven unbreakable, not merely hard. With a key that is truly random, at least as long as the message, and used once, encrypt by XOR (c = m XOR k) and decrypt by XOR again. Shannon proved (1949) this gives perfect secrecy: given the ciphertext, every plaintext is exactly equally likely, so it leaks zero information — no assumption required. The costs: the key must be as long as the message and never reused; reuse is catastrophic (c1 XOR c2 = m1 XOR m2, the key cancels). See the XOR in 1D, the secrecy and the reuse break in 2D, and the information-theoretic inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "95e3ba39aa9cb6e2e26da655f3d2b4e8520210448113cdf7e42e5e8459cd754b"
    },
    {
      "slug": "the-blum-blum-shub",
      "title": "THE BLUM-BLUM-SHUB",
      "kicker": "random bits provably as hard to predict as factoring",
      "accent": "#70b0ff",
      "blurb": "Blum-Blum-Shub in the 5-window house format — a cryptographically secure random-bit generator whose next bit is provably as hard to predict as factoring. Pick primes p,q both = 3 mod 4, set M=pq, and repeat x -> x^2 mod M, emitting the least significant bit each step. An attacker who sees any run of output cannot predict the next bit better than chance unless they can factor M. It also allows direct random access: the i-th state is x0^(2^i mod lambda(M)) mod M without iterating. It trades speed for a real security reduction to factoring. See the state stream in 1D, the generator and direct jump in 2D, and the square-vs-square-root inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "0ce659213e3bc1e61e8aab5ae3309a79483ccad97cd9734d1f350d27911d45c6"
    },
    {
      "slug": "the-feigenbaum",
      "title": "THE FEIGENBAUM",
      "kicker": "the universal constant of the road to chaos — δ ≈ 4.669",
      "accent": "#ff7040",
      "blurb": "the Feigenbaum constant in the 5-window house format — turn up r in the logistic map x -> r x(1-x) and the stable value splits into a 2-cycle at r=3, a 4-cycle at 3.449, then 8, 16, 32, the period doubling forever with windows shrinking geometrically toward chaos at r~3.5699. Feigenbaum found the shrink ratio approaches a universal constant delta=4.6692016..., the SAME number for any system that reaches chaos by period-doubling (faucets, hearts, circuits). See the shrinking gaps in 1D, the bifurcation fig-tree in 2D, and the universality across maps in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "9fe9e471080ea1ffc9e74c88a50f0968d103cf1e203a29f944247f40a1efc64f"
    },
    {
      "slug": "the-arnold-cat",
      "title": "THE ARNOLD-CAT",
      "kicker": "scramble an image to noise — it returns exactly",
      "accent": "#ffa0e0",
      "blurb": "Arnold's cat map in the 5-window house format — send each pixel (x,y) of an N x N image to ((2x+y) mod N, (x+y) mod N). It stretches and folds any picture into total noise within a few steps (a chaotic mixing map), but because it is a bijection on a finite grid it loses nothing and must return exactly to the original after a finite number of steps (Poincare recurrence). The return period depends on N with no simple formula: N=101 returns after 25 steps, N=50 after 150. Chaos and perfect predictability in one map. See the jagged period in 1D, the live scramble-and-return in 2D, and the reversible-permutation inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "8b6e243369f4478453230759b1958f198d4a0662d42036d3076d97819ac99815"
    },
    {
      "slug": "the-koch",
      "title": "THE KOCH",
      "kicker": "infinite perimeter, finite area — dimension log4/log3",
      "accent": "#90e0ff",
      "blurb": "the Koch snowflake in the 5-window house format — start with an equilateral triangle and on the middle third of every edge erect a smaller triangle, turning 1 segment into 4 of a third the length, forever. Each step multiplies the perimeter by 4/3, so the boundary grows without bound, yet the area converges to exactly 8/5 of the starting triangle (2*sqrt(3)/5). A curve of infinite length bounding finite area, continuous but nowhere differentiable, with fractal dimension log4/log3 ~ 1.2619 between a line and a plane. One of the first fractals (1904). See the diverging/converging limits in 1D, the drawn snowflake in 2D, and the box-counting inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "3fe11d9e35140795da715f67efa9703f2e068660c9555e74f9e1a19b945cdd7f"
    },
    {
      "slug": "the-cantor",
      "title": "THE CANTOR",
      "kicker": "measure zero, yet uncountable — the dust that remains",
      "accent": "#d0a0ff",
      "blurb": "the Cantor set in the 5-window house format — remove the open middle third of [0,1], then of each remaining piece, forever. The removed length sums to 1/3+2/9+4/27+... = 1, so the surviving dust has measure zero. Yet it is uncountable: a point survives exactly when its base-3 expansion uses only digits 0 and 2 (no 1), and halving those digits maps onto every binary number in [0,1] — a bijection with the whole interval. So it has as many points as [0,1] while occupying no length. Fractal dimension log2/log3 ~ 0.6309. See the two sizes in 1D, the construction + membership in 2D, and the measure-vs-cardinality inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "30997d42eca59549d308aad547f779f460bd57a1bc4e198b796d8968d939b909"
    },
    {
      "slug": "the-henon",
      "title": "THE HENON",
      "kicker": "a strange attractor — bounded forever, chaotic always",
      "accent": "#70ffb0",
      "blurb": "the Henon map in the 5-window house format — iterate x' = 1 - 1.4 x^2 + y, y' = 0.3 x. The points never settle or escape; they wander forever inside a bounded region tracing a banana-shaped curve that, zoomed in, reveals layer upon layer of fine strands — a strange attractor (fractal, dimension ~1.26). It is deterministic yet unpredictable: two starts a billionth apart diverge within ~40 steps (positive Lyapunov exponent ~0.42, the fingerprint of chaos), and each step shrinks area by 0.3 so orbits collapse onto a zero-area fractal. Michel Henon built it (1976) to stand in for the Lorenz attractor. See a coordinate in 1D, the attractor in 2D, and the invertible contract/expand inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "2db98c24224897c296f31de064d2767516e4d66309dd260a657057b3ead98820"
    },
    {
      "slug": "the-fenwick",
      "title": "THE FENWICK",
      "kicker": "running totals in O(log n) by the low-bit trick",
      "accent": "#80d0a0",
      "blurb": "the Fenwick tree (Binary Indexed Tree) in the 5-window house format — maintain running prefix sums and point updates both in O(log n) using one array and the low-bit i&(-i), which isolates the lowest set bit. Each slot holds the sum of a range whose length is that low-bit (slot 12 covers 9-12, slot 8 covers 1-8). A prefix query hops down by subtracting the low-bit across disjoint ranges; an update hops up by adding it, each touching about log n slots. The most elegant structure for dynamic running totals. See the coverage ranges in 1D, the hop animation in 2D, and the update/query inverse walks in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "c5d2a7a6329f029355381da4f31323796ed2905b2a76273585df32e78d2cf95e"
    },
    {
      "slug": "the-union-find",
      "title": "THE UNION-FIND",
      "kicker": "merge sets & test connectivity in near-constant time",
      "accent": "#ffb090",
      "blurb": "Union-Find (Disjoint Set Union) in the 5-window house format — track items grouped into non-overlapping sets with UNION (merge two sets) and FIND (which set?). Each set is a tree; FIND follows parents to the root, UNION links one root under another. Union by rank (attach shorter under taller) and path compression (after a FIND, point every node straight at the root) give an amortized cost of alpha(n), the inverse Ackermann function, which is <= 4 for any real n. Backbone of Kruskal's MST, connected components, percolation. See the forest in 1D, the path-compressing find in 2D, and the one-way-merge inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "4ad4cbe467c22738b86f4e7035bbf443c8ae14c3ba4d20507f19551f9e546ae3"
    },
    {
      "slug": "the-hyperloglog",
      "title": "THE HYPERLOGLOG",
      "kicker": "count billions of distinct items in ~1.5 KB",
      "accent": "#a0e0ff",
      "blurb": "HyperLogLog in the 5-window house format — count the number of DISTINCT items in a stream using fixed tiny memory (~1.5 KB to count into the billions with ~2% error), storing no items. Hash each item; a hash starting with k zeros suggests ~2^k distinct items seen (a k-zero run happens once in 2^k). Keep the maximum run length; split into m buckets by the first bits, track each max, and combine with a harmonic mean and bias correction. Sketches over the same set are identical and merge for free across machines. See the registers in 1D, the live estimate in 2D, and the count-vs-members inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "d09da037d813a6b0314054cb34e0dddb626ff808d038009bcb8c709c0ec910ca"
    },
    {
      "slug": "the-morris",
      "title": "THE MORRIS",
      "kicker": "count to N in ~log log N bits — unbiased",
      "accent": "#ffc0a0",
      "blurb": "the Morris counter in the 5-window house format — count up to N using only about log log N bits (roughly 5 bits for a billion instead of 30) by storing the logarithm of the count. Keep a small exponent c; to increment, bump c only with probability 2^-c, so increments thin out as c grows. The estimate is 2^c - 1, and its expected value is exactly the true count — the counter is unbiased. A single counter is high-variance, but averaging many independent counters homes in on the truth. Robert Morris built it in 1978 at Bell Labs. See the thinning increments in 1D, the estimate and spread in 2D, and the store-the-log inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "6d0df74e6ba49e94aa0a770ad2f86874e9f714b5a5700138ccc1e6ef33954b4c"
    },
    {
      "slug": "the-treap",
      "title": "THE TREAP",
      "kicker": "a search tree balanced by random priorities — tree + heap",
      "accent": "#c0ffa0",
      "blurb": "the treap in the 5-window house format — a binary search tree that stays balanced by chance. Each node holds a key (obeying BST order: smaller left, larger right) and a random priority (obeying heap order: parent beats children). For any keys, once priorities are fixed exactly one tree shape satisfies both, and because priorities are random it is balanced with high probability (expected height O(log n)) with far simpler code than a red-black tree — just rotations on insert. The shape is a function of the pairs alone, so the same keys and priorities build the identical tree regardless of insertion order. See the two orders in 1D, the rotating tree in 2D, and the order-independence inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "67578a84080b2cc4940dcf9ec5612b8379606a8997bd760d5308c72d73d18215"
    },
    {
      "slug": "the-condorcet",
      "title": "THE CONDORCET",
      "kicker": "rational voters, an irrational majority — A>B>C>A",
      "accent": "#ff80a0",
      "blurb": "the Condorcet paradox in the 5-window house format — even when every voter has consistent (transitive) preferences, the group's majority preference can cycle. Three voters ranking (A>B>C),(B>C>A),(C>A>B) give a majority for A over B, B over C, and C over A. There is no Condorcet winner (no candidate beating all others head-to-head), and the outcome depends entirely on the agenda: a chairman setting the order of pairwise votes picks the winner. Discovered by Condorcet in 1785, it is the seed of Arrow's impossibility theorem. See the ballots in 1D, the beats-cycle and agenda control in 2D, and the transitive-vs-cyclic inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "96aab34c06594baea35e68df5b32342106a54bf042730c2e121260513d3c54f7"
    },
    {
      "slug": "the-shapley",
      "title": "THE SHAPLEY",
      "kicker": "the unique fair split — average marginal contribution",
      "accent": "#f0c060",
      "blurb": "the Shapley value in the 5-window house format — the unique fair way to divide a cooperating group's value: average each player's marginal contribution over every order of joining. It is the only allocation obeying efficiency (shares sum to the whole), symmetry (interchangeable players equal), dummy (a null player gets nothing), and additivity. In voting it becomes power: with weights 50/30/20 and a 51% quota the biggest party holds 2/3 of the power on half the seats, while the 30 and 20 parties are exactly equal. Underlies cost-sharing, credit attribution, and SHAP for explaining ML. See the marginal contributions in 1D, the pivot count in 2D, and the power-vs-weight inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "e05f25dcaf642093f3697818b4607a50679217af34eb55ee7e286153753bcac7"
    },
    {
      "slug": "the-alabama",
      "title": "THE ALABAMA",
      "kicker": "add a seat to the house — a state loses one",
      "accent": "#ffa070",
      "blurb": "the Alabama apportionment paradox in the 5-window house format — under Hamilton's largest-remainder method, adding a seat to the legislature can make a state lose one. Each state gets quota = pop/total x house-size, rounded down, and leftover seats go to the largest fractional remainders; growing the house rescales every quota at once, so a state can have its leftover seat snatched away. Named for the 1880 census, where Alabama would get 8 seats in a 299-member House but only 7 in a 300-member House. Balinski & Young proved no method escapes all such paradoxes. See the seat drop in 1D, the live apportionment in 2D, and the monotonicity-impossibility inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "a020206c4335e6e0e051d21f1a5ad638782da81d7993032a8742dd7022f00433"
    },
    {
      "slug": "the-borda",
      "title": "THE BORDA",
      "kicker": "same ballots, different rule — plurality crowns the loser",
      "accent": "#90c0ff",
      "blurb": "the rule-dependence of elections in the 5-window house format — who wins depends entirely on the counting rule. For 10 voters (4: A>B>C, 3: B>C>A, 3: C>B>A), plurality elects A with 4 first-place votes, but a majority prefers B to A (6-4) and C to A (6-4), so A is the Condorcet loser who loses to everyone, yet plurality crowns them. The Borda count (points by rank) instead elects B, which is exactly the Condorcet winner. First-past-the-post hands victory to the universally-rejected candidate while a full-ballot rule reverses it (Borda 1770). See the tallies in 1D, the three rules and head-to-head grid in 2D, and the discarded-preference inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "11d1c53f86b8a8b880008917e8708b7d3f579504fe860f9417ba8a7eead1d4b5"
    },
    {
      "slug": "the-banzhaf",
      "title": "THE BANZHAF",
      "kicker": "voting power by swing votes — weight 49 can equal weight 1",
      "accent": "#ffb0d0",
      "blurb": "the Banzhaf power index in the 5-window house format — measure a voter's real power in a weighted body by counting swing votes: winning coalitions where the voter is critical (leaving flips pass to fail). Power is each voter's share of total swings. It is almost never proportional to weight: with weights 50,49,1 and a majority quota of 50, the weight-49 and weight-1 parties have exactly equal power, and a large weight can be a dummy with zero power. Banzhaf devised it in 1965 for a lawsuit over a malapportioned county board. See the swing counts in 1D, the critical-coalition scan in 2D, and the Banzhaf-vs-Shapley inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "06d1714fc903b9598f411798dd72294acf299d866dfae220b63a92a360d5878b"
    },
    {
      "slug": "the-bell",
      "title": "THE BELL",
      "kicker": "entanglement beats every classical bound — CHSH 2√2 > 2",
      "accent": "#b090ff",
      "blurb": "Bell's theorem via the CHSH game in the 5-window house format — two entangled particles go to distant labs; each picks one of two measurement settings and records +-1, with no signal between them. If the particles carried predetermined answers (local hidden variables), the CHSH combination S of their correlations can never exceed 2. Quantum mechanics reaches S = 2*sqrt(2) ~ 2.828, breaking the bound — correlations stronger than any classical mechanism, without hidden coordination or signalling. Bell proved it in 1964; the 2022 Nobel honored the experiments. See the CHSH wall in 1D, the angle dials in 2D, and the no-local-explanation inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "01ca230b55226f383560fb19daeab65b857dba7c0fc6f1e442c2b6f8ce8625a2"
    },
    {
      "slug": "the-deutsch",
      "title": "THE DEUTSCH",
      "kicker": "one quantum query where classical needs two",
      "accent": "#70d0ff",
      "blurb": "Deutsch's algorithm in the 5-window house format — the first proof a quantum computer beats a classical one. Given a black box computing an unknown function f from one bit to one bit, decide if it is constant (same output both inputs) or balanced (different). Classically you must query it twice; quantumly, once. A Hadamard puts the input in superposition, the box runs f once, and a second Hadamard makes the two paths interfere — constructively if constant, destructively if balanced — so measuring the input qubit reads 0 for constant, 1 for balanced. Quantum speedup from interference, the seed of Shor and Grover. See the four verdicts in 1D, the running circuit in 2D, and the relation-vs-values inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "73564e21541100d5eae30744b43d8f9b9b635048008b0e19c11043c6274711d0"
    },
    {
      "slug": "the-grover",
      "title": "THE GROVER",
      "kicker": "search N items in √N — the quantum shortcut",
      "accent": "#ffa0e0",
      "blurb": "Grover's algorithm in the 5-window house format — search an unsorted database of N items for a marked one in about sqrt(N) steps, a quadratic speedup over the N/2 a classical scan averages. Start with all items in equal superposition (amplitude 1/sqrt(N)) and repeat two moves: an oracle that flips the marked amplitude's sign, and a diffusion that reflects every amplitude about the average. Each round rotates the state toward the marked item, its probability growing as sin^2((2k+1)theta) with sin(theta)=1/sqrt(N), peaking near (pi/4)sqrt(N) rounds. Overshoot and it comes back down. See the amplitudes in 1D, the amplification and overshoot in 2D, and the rotation inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "1b601654d235f20548e8217da1ed8633b3a2960b0018b586c503e48fdf08c3fd"
    },
    {
      "slug": "the-ghz",
      "title": "THE GHZ",
      "kicker": "three qubits refute local realism with certainty",
      "accent": "#90ffd0",
      "blurb": "the GHZ paradox in the 5-window house format — three particles entangled in the GHZ state (|000>+|111>)/sqrt2, sent to distant labs, refute local realism with certainty in one measurement round (no statistics, unlike Bell). Quantum predicts XXX = +1 while XYY = YXY = YYX = -1. If the particles had predetermined values, XXX = x1x2x3 would equal the product (XYY)(YXY)(YYX) = x1x2x3(y1y2y3)^2 = x1x2x3 — the same. But quantum gives +1 vs -1, so no assignment of definite values fits all four predictions. The sharpest form of Bell's theorem: all or nothing. See the observables in 1D, the impossible assignment in 2D, and the no-hidden-reality inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "41aa43e217bec6fb8a1e198f4d0788dca08c3a269942c5125b7783c36e7a9a4d"
    },
    {
      "slug": "the-teleportation",
      "title": "THE TELEPORTATION",
      "kicker": "move a qubit's state with entanglement + 2 classical bits",
      "accent": "#b0d0ff",
      "blurb": "quantum teleportation in the 5-window house format — move an unknown quantum state from Alice to Bob without sending the qubit, and without either learning the state. Alice and Bob pre-share an entangled Bell pair; Alice does a joint Bell measurement on her mystery qubit and her half of the pair, getting 2 random classical bits and destroying her copy (no-cloning). Bob applies one of four corrections (I, X, Z, XZ) chosen by those bits, and his half becomes exactly the original state. No faster-than-light: without the bits, Bob's qubit is noise. It transfers information, not matter, and underlies quantum networks. See the protocol in 1D, the four-outcome recovery in 2D, and the classical-plus-quantum inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "c2e994cf700ad38a66cac0153e3c90dfe71e261885dbc4b896cda9eb919056df"
    },
    {
      "slug": "the-busy-beaver",
      "title": "THE BUSY BEAVER",
      "kicker": "the longest-running halter — and the edge of the computable",
      "accent": "#ff9a3c",
      "blurb": "the busy beaver in the 5-window house format — among all n-state 2-symbol Turing machines that halt on a blank tape, which runs longest (S(n)) and prints the most 1s (Sigma(n))? These record functions grow faster than any computable function: Sigma is definable but not computable. For n=3 the champion prints Sigma(3)=6 ones. Verified live: the canonical 3-state champion halts in 14 steps with exactly 6 ones, and the 2-state champion halts in 6 steps with 4 ones. See the tape step in 1D, run the champions to a halt in 2D, and the uncomputability horizon in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "e31fc15e0b0f316d237c6495b445eda86bf24887f2be4a9d41bd049e6aa1a35f"
    },
    {
      "slug": "the-margolus",
      "title": "THE MARGOLUS MIRROR",
      "kicker": "a reversible CA — run it back to the exact seed",
      "accent": "#7ad0b0",
      "blurb": "the Margolus mirror in the 5-window house format — a reversible block cellular automaton on the Margolus neighbourhood: the grid is cut into 2x2 blocks whose partition shifts by one cell every other step, and each block is transformed by a bijection on block states, so the whole update is invertible. Run forward any number of steps, then backward, and you recover the exact seed bit-for-bit. The rule rotates each block 180 degrees, conserving the live-cell count exactly. Verified live: forward 12 then backward 12 reproduces the seed, and the count is constant every step. See the alternating partitions in 1D, forward/rewind in 2D, and the no-arrow-of-time braid in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "d24ca90b4f1c3f7b3943e2e424c01a16e675dce2f5568189337184b85fa1bfd8"
    },
    {
      "slug": "the-reed-solomon",
      "title": "THE REED-SOLOMON",
      "kicker": "lose any n-k symbols, recover the data exactly",
      "accent": "#e05a7a",
      "blurb": "Reed-Solomon erasure coding in the 5-window house format — treat k data symbols as coefficients of a degree-(k-1) polynomial over a finite field and evaluate at n>k points to get n codeword symbols. Lose any n-k of them and from any k survivors a unique polynomial still fits (Lagrange), so its coefficients hand back the original data exactly. Two points make a line; k points make a degree-(k-1) curve; the curve remembers the lost points. Runs CDs, QR codes, RAID, deep-space telemetry. Verified live over GF(257): 200 trials, encode 4 into 8, erase 4, recover exactly. See the sampled curve in 1D, click-to-erase recovery in 2D, and the holographic spread in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "50d623402834d0801ce461c80cd956dd634b6c44996fbc2eb29a13c806c134a8"
    },
    {
      "slug": "the-minsky",
      "title": "THE MINSKY MACHINE",
      "kicker": "multiply with only INC and decrement-or-branch",
      "accent": "#6ab0e8",
      "blurb": "the Minsky counter machine in the 5-window house format — only unbounded counters and two instructions: INC(r) adds one, JZDEC(r) jumps if zero else decrements and continues. No arithmetic at all, yet Turing-complete. A short program over counters X,Y,Z,T computes X*Y using nothing but +1 and -1-or-branch: it adds X to Z, Y times, restoring X through a temp each round, and halts with Z the exact product. Verified live: the program halts with Z=m*n for every pair m,n in 0..12. See the counters and program in 1D, run a multiply in 2D, and the power-vs-efficiency inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "42a02e8c9e837eeb592a26eff3845b1ee43c017581ae7cb647dfbe8dd8f5f154"
    },
    {
      "slug": "the-crc",
      "title": "THE CRC",
      "kicker": "append check bits so corruption can't hide",
      "accent": "#d4b03c",
      "blurb": "the cyclic redundancy check in the 5-window house format — treat a bit-string as a polynomial over GF(2) (XOR, no carries), pick a generator g(x), and append the remainder of (message*x^r)/g so the codeword is exactly divisible by g. The receiver re-divides: remainder 0 = intact, nonzero = corrupted. A good g catches every single-bit error (one flip is x^i, never divisible by a g with two+ terms) and, if g has factor (x+1), every odd number of errors. Verified live with CRC-8 (g=0x107): 300 messages divisible, and every single-bit flip detected. See the shift-register division in 1D, flip-and-detect in 2D, and the syndrome-points-at-the-error inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "bb2d5075c9c0e759ec028d19b6029d39a9a6c04fcec51f346151f9595704dca3"
    },
    {
      "slug": "the-fractran",
      "title": "THE FRACTRAN",
      "kicker": "a whole language made of fractions — universal, unreadable",
      "accent": "#c060ff",
      "blurb": "Conway's FRACTRAN in the 5-window house format — a program is a list of fractions and the data is one integer: multiply by the first fraction that keeps it whole, repeat, halt when none does. That is the entire (Turing-complete) language, with prime exponents as registers. [2/3] adds: 2^a*3^b halts at 2^(a+b). The 14-fraction PRIMEGAME from 2 emits powers of 2 whose exponents are exactly the primes. Verified live with BigInt: the adder is exact for all a,b in 0..6 and PRIMEGAME emits 2,3,5,7. See the prime-register integer in 1D, run adder/PRIMEGAME in 2D, and the universality-vs-opacity inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "84d9bb0464dc6304719bdd329b8a6fe61328ee0075f71e2c296c1e4c56351f06"
    },
    {
      "slug": "the-verhoeff",
      "title": "THE VERHOEFF",
      "kicker": "a check digit that catches every transposition — via a non-abelian group",
      "accent": "#e0705a",
      "blurb": "the Verhoeff check digit in the 5-window house format — it catches every single wrong digit AND every adjacent transposition, which mod-10 checksums (Luhn, ISBN-10) provably cannot, by doing arithmetic in the non-commutative dihedral group D5 so that 09 and 90 differ. A permutation table scrambles each digit by position, a fixed table combines them, and the check digit forces the running product to the identity. Verified live: over a range, every valid number checks to 0, all single-digit errors caught, all adjacent transpositions caught (Luhn shown missing one). See the D5 product march in 1D, flip/swap detection in 2D, and the non-commutativity inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "016b9a2f5c67bdb316f973ae8b9edf04a984c19ccbc5e34e4e44335740da8e05"
    },
    {
      "slug": "the-fisher-yates",
      "title": "THE FISHER-YATES",
      "kicker": "a provably-uniform shuffle — n! paths onto n! orderings",
      "accent": "#58b0e0",
      "blurb": "the Fisher-Yates shuffle in the 5-window house format — a uniform random permutation in one pass: from the last item to the first, swap each with a uniformly random item at or before it. Unbiasedness is exact, not statistical: the map from random choices to permutations is a bijection — n! choice-sequences, n! permutations, each hit exactly once. The tempting swap-with-any-index variant makes n^n paths, not divisible by n!, and is provably biased. Verified live: for n=5, enumerating all 120 choice-sequences yields all 120 permutations, each exactly once. See the shrinking swap range in 1D, enumerate+bias in 2D, and the paths-equal-outcomes inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "e14311f4c9b2241d0ba4eb342d8beb0ce8e8d417f1203b3d937f9fabc2b3a182"
    },
    {
      "slug": "the-boustrophedon",
      "title": "THE BOUSTROPHEDON",
      "kicker": "an ox-plough triangle that grows the zigzag numbers",
      "accent": "#f0b048",
      "blurb": "the boustrophedon transform in the 5-window house format — 'ox-turning,' a triangle read back and forth like a plough, each entry the running sum of the one before plus the one across from the row above. Fed the seed (1,0,0,...) it grows the zigzag/Euler numbers 1,1,1,2,5,16,61,272,1385 — which count alternating (up-down) permutations and are the Taylor coefficients of tan+sec. Verified live: the transform reproduces A000111 exactly, and the alternating-permutation counts match for small n. See the plough fill in 1D, build+verify in 2D, and the one-triangle-three-worlds inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "d1ca25cff65b1af073702928439014d16f4bacc61a627ccce4a82b4dbef02bb5"
    },
    {
      "slug": "the-montgomery",
      "title": "THE MONTGOMERY",
      "kicker": "multiply mod N with shifts, never dividing by N",
      "accent": "#6ad0a0",
      "blurb": "Montgomery multiplication in the 5-window house format — compute a*b mod N without dividing by N, replacing modular reduction with shifts and a multiply (why every RSA/ECC chip uses it). Work in Montgomery form scaled by R=2^k>N and reduce with REDC, which divides by R (a shift) not N; a precomputed N' = -N^-1 mod R makes the leftover vanish exactly. Verified live (N=1000003, R=2^20): 300 random pairs, convert to Montgomery form, REDC-multiply, convert back = a*b mod N exactly, no division by N. See the REDC steps in 1D, the round trip in 2D, and the change-of-coordinates inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "3b24eea5cb16ed4622d2e79858db57d2d6c78e21901d2049dd092f4c226c7ab8"
    },
    {
      "slug": "the-fast-inverse-sqrt",
      "title": "THE FAST INVERSE SQRT",
      "kicker": "1/sqrt(x) with a bit-hack and one Newton step — no divide",
      "accent": "#c8a020",
      "blurb": "the Quake III fast inverse square root in the 5-window house format — compute 1/sqrt(x) with no division and no sqrt, using a bit-level trick: reinterpret the float's bits as an integer, do i = 0x5f3759df - (i>>1), reinterpret back, and you have 1/sqrt(x) to ~3.4%; one Newton step y=y(1.5-0.5xy^2) sharpens it to ~0.17%. The shift halves the exponent (a square root) and the constant corrects the mantissa and bias. Verified live: over a sweep the raw hack is within ~3.4% and post-Newton within ~0.18% of true 1/sqrt(x). See the float bit layout in 1D, the estimate vs truth in 2D, and the log-in-the-bits inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "1a02c6a3f271ef6b42c0e36c4a800b4a25e30cd9c07f1ae64d0e3d1d9be1a2a3"
    },
    {
      "slug": "the-bresenham",
      "title": "THE BRESENHAM",
      "kicker": "draw a line with integers only — every pixel within half a pixel",
      "accent": "#5aa0e0",
      "blurb": "Bresenham's line algorithm in the 5-window house format — draw a straight line on a pixel grid using only integer add/subtract/compare, no float, no division, no multiply in the loop. An integer error term decides at each step whether to move straight or diagonally, always picking the pixel nearest the true line, so every pixel lands within half a pixel of the ideal. It rasterized every early display and GPU. Verified live: over 500 random lines, every plotted pixel is within perpendicular distance 0.5 of the exact line, integer-only. See the error accumulator in 1D, drag-to-draw in 2D, and the bounded-error staircase inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "911295717e672acf3fdf46591bc328add1ea7ddfa9def9569048d7306cca6338"
    },
    {
      "slug": "the-peterson",
      "title": "THE PETERSON",
      "kicker": "mutual exclusion with plain reads and writes — model-checked",
      "accent": "#e06060",
      "blurb": "Peterson's algorithm in the 5-window house format — two threads share a critical section with no special hardware, only reads/writes to two 'want' flags and one 'turn' variable: each raises its flag, yields the turn, and enters only when the other isn't interested or it's this thread's turn. Verified live by exhaustive model-checking: over every reachable interleaving, both threads are never in the critical section at once (mutual exclusion) and no non-trivial state is stuck (deadlock-free). See the flags and turn in 1D, step/model-check in 2D, and the memory-ordering inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "efb2b74717981e72e07ed13263fc75b0adf9290a472254a1a1e39d8596c8246d"
    },
    {
      "slug": "the-metropolis",
      "title": "THE METROPOLIS",
      "kicker": "sample any distribution knowing only ratios — detailed balance",
      "accent": "#d060a0",
      "blurb": "Metropolis-Hastings MCMC in the 5-window house format — sample from any distribution you can only evaluate up to a constant, by a random walk that accepts each proposed move with probability min(1, pi(new)/pi(old)). The chain's stationary distribution is exactly the target, guaranteed by detailed balance (flow i->j equals flow j->i). Verified live: the constructed transition matrix satisfies detailed balance to ~1e-17, and its power-iterated stationary distribution equals the normalized target to ~1e-15. See the accept/reject walk in 1D, build+verify+run in 2D, and the normalizer-cancels inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "a0329b056d0188977a1b030446cf65e572421fba746904eafe74376f98039b18"
    },
    {
      "slug": "the-move-to-front",
      "title": "THE MOVE-TO-FRONT",
      "kicker": "recency becomes rank — a cache and a compressor in one",
      "accent": "#60c090",
      "blurb": "the move-to-front transform in the 5-window house format — keep a list of all symbols; for each input symbol output its current position, then move it to the front. Recently-seen symbols cluster near the front, turning locally-repetitive data into small numbers that compress well (the middle stage of bzip2). It is literally an LRU cache. Verified live: encode then decode round-trips exactly, and clustered input yields a far smaller mean code than uniform-random input. See the list reordering in 1D, encode/decode + locality in 2D, and the recency-equals-rank inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "90ea29d945027550f5bdc2d7eecc5284feabc3a34ce12697cce9e28070c79a41"
    },
    {
      "slug": "the-goodstein",
      "title": "THE GOODSTEIN",
      "kicker": "unbounded growth that always crashes to 0 — unprovable in PA",
      "accent": "#d07050",
      "blurb": "Goodstein sequences in the 5-window house format — write a number in hereditary base 2 (exponents in base 2 too, all the way down), bump every 2 to a 3 and subtract 1, then bump 3 to 4 and subtract 1, and so on. The values rocket upward, yet Goodstein's theorem says every sequence eventually crashes to 0 — and this true fact about integers is unprovable in Peano arithmetic (Kirby-Paris 1982), needing ordinals below epsilon-0. Verified live with BigInt: G(1),G(2),G(3) reach 0 in 1,3,5 steps; G(4) terminates but astronomically far off. See hereditary base in 1D, run to 0 in 2D, and the ordinal-countdown inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "bd591e62dd501b3f62080479071cb40c4b754eb40225d6f6dcca3e49c6d660be"
    },
    {
      "slug": "the-morton",
      "title": "THE MORTON",
      "kicker": "interleave the bits of x and y — 2D into one address",
      "accent": "#50b0c0",
      "blurb": "the Morton (Z-order) code in the 5-window house format — map a 2D coordinate to one number by interleaving the bits of x and y, and de-interleave to decode: a perfect bijection between the grid and 0..N^2-1, where points close in 2D tend to stay close in the code. It linearizes space for databases, GPU textures, and caches; drawing cells in code order traces a recursive Z. It is the cheap cousin of the Hilbert curve. Verified live: over a 16x16 grid, encode then decode round-trips for all 256 cells and every code is distinct. See the bit-zipper in 1D, click-a-cell + Z-path in 2D, and the partial-locality seams inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "558f7956a9eb75c5fadae79e1aaf5e3d98cbba8433c37c2641575b8ba9a2ed8b"
    },
    {
      "slug": "the-burnside",
      "title": "THE BURNSIDE",
      "kicker": "count necklaces by averaging fixed points, not by dedup",
      "accent": "#b088e0",
      "blurb": "Burnside's lemma in the 5-window house format — how many distinct necklaces from n beads in k colors, if rotation doesn't count as new? Naively k^n colorings, but rotations collapse many. Burnside counts the distinct ones exactly by averaging the colorings fixed by each rotation: (1/n) sum over d|n of phi(d)*k^(n/d). For 6 beads, 2 colors: 14 necklaces, not 64. Verified live: the closed form equals a brute-force orbit count for all n=1..8, k=1..4. See a necklace rotating in 1D, formula vs brute in 2D, and the count-by-fixed-points inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "0341c38887229929b1d8cdd71ee212ab5462a539982aee4b703093e9cef7995a"
    },
    {
      "slug": "the-lights-out",
      "title": "THE LIGHTS OUT",
      "kicker": "a light puzzle is a linear system over GF(2)",
      "accent": "#e0c040",
      "blurb": "Lights Out in the 5-window house format — press a light and it toggles itself and its four neighbors; goal all-off. It looks like trial and error but is linear algebra over GF(2): each board is a vector, each press a column of a fixed matrix A, and solving is A*p=b (mod 2) by Gaussian elimination. Pressing twice cancels (order never matters), and solvability depends only on A: the classic 5x5 matrix has rank 23, so exactly 2^23 of 2^25 boards are solvable. Verified live: 200 random solvable boards are cleared exactly by the GF(2) solution; rank is 23. See a plus-toggle in 1D, click+solve in 2D, and the vector-space inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "c20c1d9861d6d03509f7e8255195c70983ecf860f9bc98c1678368e72c90be65"
    },
    {
      "slug": "the-fifteen-puzzle",
      "title": "THE 15-PUZZLE",
      "kicker": "a conserved parity walls off half the arrangements",
      "accent": "#60c0a0",
      "blurb": "the 15-puzzle parity invariant in the 5-window house format — 15 tiles and a gap in a 4x4 frame, slid to sort them. Sam Loyd's $1000 to swap only tiles 14 and 15 was safe: impossible. Every slide is a transposition of a tile with the gap and moves the gap one row, and a hidden quantity — parity of tile inversions plus the gap's row — is unchanged by every move. So arrangements split into two classes; you can only reach the half sharing the solved board's parity. Verified live: the invariant is unchanged across 5000 random legal moves. See a slide's two parity flips in 1D, scramble/swap in 2D, and the two-components inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "27992b4869b00919902d63c19cd453b548a7a8a136e027362a1298f17eacc4ba"
    },
    {
      "slug": "the-rsk",
      "title": "THE RSK",
      "kicker": "permutations become two tableaux — order becomes shape",
      "accent": "#70b0d0",
      "blurb": "the Robinson-Schensted correspondence in the 5-window house format — a bijection between permutations of 1..n and pairs of standard Young tableaux (P,Q) of the same shape, built by row insertion: each number bumps the smallest larger element down a row, cascading. Schensted's theorem: P's first-row length equals the longest increasing subsequence. Verified live: over all 720 permutations of 6, RSK maps each to a distinct (P,Q) pair (a bijection) and the first-row length equals the LIS every time. See row-insertion bumping in 1D, tableaux building in 2D, and the reversible order-becomes-shape inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "5d7ceacb5ebc54228cc37ef3b010dd9c2c711157f765878604262b92bc83166f"
    },
    {
      "slug": "the-faulhaber",
      "title": "THE FAULHABER",
      "kicker": "sum of p-th powers is one polynomial — via Bernoulli numbers",
      "accent": "#d0a840",
      "blurb": "Faulhaber's formula in the 5-window house format — the sum 1^p+2^p+...+n^p is always a polynomial in n of degree p+1, with coefficients from the Bernoulli numbers. p=1 gives n(n+1)/2; p=2 gives n(n+1)(2n+1)/6; p=3 gives [n(n+1)/2]^2, so sum of cubes = square of the sum (Nicomachus). Bernoulli numbers B0=1,B1=-1/2,B2=1/6,B4=-1/30 recur in tan, zeta, and Euler-Maclaurin. Verified live with exact BigInt rationals: closed forms match, Nicomachus holds, Bernoulli numbers compute correctly, and the general Faulhaber polynomial equals the direct sum for p=1..6. See Nicomachus tiling in 1D, direct-vs-polynomial in 2D, and the sum-becomes-integral inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "4328d83c823a5627edc84f1cde1b8612fd8d92bd0d2e4430f42e9d8af7695c29"
    },
    {
      "slug": "the-floyd-steinberg",
      "title": "THE FLOYD-STEINBERG",
      "kicker": "dither by broadcasting rounding error to neighbors",
      "accent": "#a0a0c0",
      "blurb": "Floyd-Steinberg dithering in the 5-window house format — make few colors look like many: quantize each pixel to the nearest level, then spread the rounding error to not-yet-processed neighbors (7/16,3/16,5/16,1/16). The eye averages the dots back into the original shade, so a 1-bit image shows smooth gradients. Error is never discarded, only passed on, so average brightness is preserved (up to a boundary residual). Verified live: dithering a smooth signal to 2 levels, |sum_in - sum_out| equals the leftover residual and the running error never exceeds one quantization step. See the carried error in 1D, dither vs threshold in 2D, and the conserved-brightness inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "14e7a82e0cb593bd832548fc4c577548c3c74210209972763234410ad4abf2d5"
    },
    {
      "slug": "the-three-distance",
      "title": "THE THREE-DISTANCE",
      "kicker": "step by an irrational forever — gaps take only 3 sizes",
      "accent": "#60c0b0",
      "blurb": "the three-distance (Steinhaus / three-gap) theorem in the 5-window house format — mark {alpha},{2alpha},...,{n alpha} around a circle for irrational alpha; however many points, the gaps between neighbors take at most THREE distinct lengths, and when three appear the largest equals the sum of the other two. It underlies golden-ratio spacing, phyllotaxis, and low-discrepancy sampling. Verified live: for several irrationals and every n from 2 to 60, the distinct gap count is at most 3 and the largest is the sum of the other two. See points and colored gaps in 1D, the circle in 2D, and the split-a-largest-gap inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "6fce16c8df2092bdeb1960a446a11bd66e666a01e3f3e10396daae14847160f1"
    },
    {
      "slug": "the-rotating-calipers",
      "title": "THE ROTATING CALIPERS",
      "kicker": "polygon diameter in O(n) — only antipodal pairs matter",
      "accent": "#e08040",
      "blurb": "rotating calipers in the 5-window house format — find a convex polygon's diameter (farthest vertex pair) in O(n) instead of O(n^2): two parallel lines grip the polygon on opposite sides and rotate together, and the farthest pair is always a pair of antipodal vertices they touch, swept in one loop. The same trick gives width, min-area bounding box, and closest distance between two convex polygons. Verified live: over 200 random convex hulls, the calipers diameter equals the brute-force max over all vertex pairs every time. See the rotating calipers in 1D, hull+diameter in 2D, and the only-antipodal-pairs-matter inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "8d0deabd616af3804d81f67701a3f4bb9c248f00de1ffee66d4a4350e734c137"
    },
    {
      "slug": "the-catalan",
      "title": "THE CATALAN",
      "kicker": "one number counts a hundred structures — and /(n+1) is a mirror",
      "accent": "#6cc0d0",
      "blurb": "the Catalan numbers in the 5-window house format — 1,1,2,5,14,42,... count balanced parentheses, Dyck paths, polygon triangulations, binary trees, and dozens more, all equal to C(2n,n)/(n+1). The division by n+1 is the reflection principle: the bad lattice paths that cross the diagonal biject with paths to a reflected endpoint C(2n,n-1), so C_n = C(2n,n) - C(2n,n-1). Verified live: the closed form equals a brute count of balanced strings for n=0..10, and the reflection identity holds. See a Dyck path in 1D, three counts agreeing in 2D, and the reflection inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "a5d54b01e6b090c718f4b6bc54512a792750da1c72170f7e52988f42527336a1"
    },
    {
      "slug": "the-strassen",
      "title": "THE STRASSEN",
      "kicker": "multiply 2x2 with 7 products, not 8 — bending O(n^3)",
      "accent": "#e0704a",
      "blurb": "Strassen's algorithm in the 5-window house format — multiply two 2x2 matrices with 7 scalar multiplications instead of 8 (via combined products like M1=(a+d)(e+h)), trading a multiply for additions. Recursively on n x n quadrants, 7 vs 8 turns O(n^3) into O(n^log2 7) ~ O(n^2.807), the first sub-cubic matrix multiply. Verified live: the recursive Strassen product equals the naive product exactly for random integer matrices, using 7 scalar mults per 2x2 (naive uses 8). See the 7 products in 1D, matrices multiplied in 2D, and the exponent-bending recursion inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "6504889552300c15b5a70b11de8fee3338d802b716a26e14ec4ff763d5428935"
    },
    {
      "slug": "the-permanent",
      "title": "THE PERMANENT",
      "kicker": "the determinant's all-plus twin — and it's #P-hard",
      "accent": "#c05090",
      "blurb": "the matrix permanent in the 5-window house format — same sum over permutations as the determinant but with all plus signs, no alternating minus. That change makes it #P-hard (Valiant 1979): the determinant is O(n^3) by Gaussian elimination, the permanent has no known polynomial algorithm. Ryser's formula beats naive n! via inclusion-exclusion in O(2^n n), and the permanent counts perfect matchings of a bipartite graph. Verified live: Ryser's formula equals the brute permutation-sum for 100 random matrices up to size 6. See permanent vs determinant in 1D, a 0/1 matrix computed two ways in 2D, and the sign-is-hardness inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "b810468e5b974e7338541a84f166fe521e63e35918c8f1c23abf32478b0f1618"
    },
    {
      "slug": "the-lzw",
      "title": "THE LZW",
      "kicker": "build a dictionary on the fly — and never send it",
      "accent": "#d0b040",
      "blurb": "LZW compression in the 5-window house format — build a dictionary of substrings on the fly: extend the current match while it's in the dictionary, and when it breaks, emit the code for the longest match, add (match + next char) to the dictionary, restart. The decoder rebuilds the identical dictionary from the codes alone, so no table is transmitted. It ran GIF, UNIX compress, and PDF/TIFF. Verified live: encode then decode reproduces the input exactly across fixed strings (including the self-referential KwKwK case) and 200 random strings. See the dictionary growing in 1D, encode/decode in 2D, and the lockstep-derivation inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "8d72c6958c10d4f7ba8d4cdace95f3b4312a8f11ceb440dfd511313396ba656e"
    },
    {
      "slug": "the-haar",
      "title": "THE HAAR",
      "kicker": "average and difference, recurse — perfectly invertible",
      "accent": "#50c0a0",
      "blurb": "the Haar wavelet in the 5-window house format — the simplest multiresolution transform: replace each pair of samples with their average and difference (scaled by 1/sqrt2), then recurse on the averages, yielding a coarse approximation plus detail coefficients at every scale. Being orthonormal, it is perfectly invertible (inverse-averaging resurrects the exact signal) and preserves energy: sum(coeffs^2) = sum(signal^2) (Parseval). It is the ancestor of wavelet image compression. Verified live: for 200 random length-8 signals the inverse reproduces the original to ~1e-12 and Parseval holds. See average/difference in 1D, decompose+reconstruct in 2D, and the self-inverse pyramid in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "69866da886a78d9fa011dc3f9184146b957d13c665cd7a4787bff0b50db5bdfd"
    },
    {
      "slug": "the-eulerian",
      "title": "THE EULERIAN",
      "kicker": "count permutations by descents — a bell inside n!",
      "accent": "#d09040",
      "blurb": "the Eulerian numbers in the 5-window house format — <n,k> counts permutations of 1..n with exactly k descents (a value followed by a smaller one). The triangle 1;1,1;1,4,1;1,11,11,1;1,26,66,26,1 is symmetric and each row sums to n!, obeying <n,k>=(k+1)<n-1,k>+(n-k)<n-1,k-1>. Verified live: the recurrence matches a brute descent-tally over all n! permutations for n=1..7, and rows sum to n!. See a permutation's descents in 1D, recurrence vs brute in 2D, and the uniformity-into-distribution inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "d7e2c6c6d390ea0afe97d690d781e3ac84d49246e7db0f59737aeff727675dfa"
    },
    {
      "slug": "the-rudin-shapiro",
      "title": "THE RUDIN-SHAPIRO",
      "kicker": "a deterministic +-1 sequence with random-walk-flat sums",
      "accent": "#7090d0",
      "blurb": "the Rudin-Shapiro sequence in the 5-window house format — r_n = (-1)^(number of '11' pairs in binary of n) is a +-1 sequence engineered so its partial sums grow like sqrt(N) not N (a nearly flat power spectrum, very low autocorrelation), which is exactly what radar pulse-compression and spread-spectrum need. Verified live: over N up to 200000 the ratio |S_N|/sqrt(N) stays bounded (measured max ~2.45, within the proven bound 2+sqrt2~3.41). See the +-1 walk in 1D, the sqrt(N) envelope in 2D, and the deterministic-looks-random inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "9b42d358556b433b5c24e7f12e07f7e798534441249867071e32a7ac9ee1ba98"
    },
    {
      "slug": "the-lyndon",
      "title": "THE LYNDON",
      "kicker": "unique factorization of a string into Lyndon words",
      "accent": "#c0a050",
      "blurb": "Lyndon words in the 5-window house format — a Lyndon word is strictly smaller than all its rotations (aab yes, aba/baa no), and the Chen-Fox-Lyndon theorem says every string factors uniquely into non-increasing Lyndon words (a prime factorization for strings), found by Duval's O(n) algorithm. banana -> b.an.an.a. Verified live: for 500 random strings, Duval's factorization concatenates back to the original, every factor is Lyndon, and the factors are non-increasing. See the factored blocks in 1D, live Duval in 2D, and the unique-factorization/necklace inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "63aa8b12b709a16a09b83c3b0ac04121150f9dcaa7e66c988e30a2f77e151014"
    },
    {
      "slug": "the-z-algorithm",
      "title": "THE Z-ALGORITHM",
      "kicker": "all prefix matches in O(n) — a pointer that never retreats",
      "accent": "#e06050",
      "blurb": "the Z-algorithm in the 5-window house format — the Z-array gives, at each position i, the longest substring starting at i that matches a prefix of the string; computed in O(n) (not naive O(n^2)) by keeping the rightmost match interval [l,r] and reusing earlier Z-values inside it. Concatenate pattern + separator + text and the Z-array finds every pattern occurrence in linear time. Verified live: for 500 random strings, every Z[i] equals a brute prefix-match. See the [l,r] window in 1D, Z-bars in 2D, and the mirror-reuse inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "dbfe9bf99c76848be9635b7f221eb15e405a50da6dfd83ecb60e2acccf438a16"
    },
    {
      "slug": "the-euler-tour",
      "title": "THE EULER TOUR",
      "kicker": "flatten a tree so every subtree is a contiguous range",
      "accent": "#50b0a0",
      "blurb": "the Euler tour technique in the 5-window house format — flatten a tree by a DFS that records each node's entry (tin) and exit (tout) times, so the subtree of any node v is exactly the contiguous range [tin[v], tout[v]] in the array. Subtree-sum, descendant-of, and subtree-size all become O(1)-O(log n) range queries, letting segment/Fenwick trees answer tree problems. Verified live: for 200 random trees, each subtree equals the set of nodes with entry time in [tin[v], tout[v]] and size tout-tin+1. See DFS timestamps in 1D, click-a-node in 2D, and the hierarchy-is-interval-nesting inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "c303b1faf930b0998a8efd15d6065c37f6498cfa5c6256ec2c7a6c4756032fe0"
    },
    {
      "slug": "the-stirling",
      "title": "THE STIRLING",
      "kicker": "count set partitions — and translate powers to falling factorials",
      "accent": "#cf9838",
      "blurb": "the Stirling numbers of the second kind in the 5-window house format — S(n,k) counts partitions of n labeled items into exactly k non-empty blocks, via S(n,k)=k*S(n-1,k)+S(n-1,k-1) (a new item joins a block or starts one). Row sums give the Bell number. And the same numbers change basis: x^n = sum_k S(n,k)*(x)_k, powers into falling factorials. Verified live: the recurrence matches brute set-partition counts n=1..7, rows sum to Bell numbers, and the basis identity holds exactly. See the recurrence in 1D, recurrence vs brute in 2D, and the counting-is-coordinates inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "80b98b4bf9dd3fc7d882fcee17dfed73d383fc251c6831a959ff556dc8c704d2"
    },
    {
      "slug": "the-hierholzer",
      "title": "THE HIERHOLZER",
      "kicker": "cross every edge once — decided by counting odd corners",
      "accent": "#b05868",
      "blurb": "Hierholzer's algorithm in the 5-window house format — an Eulerian trail crosses every edge of a graph exactly once (the Seven Bridges of Konigsberg). Euler proved one exists iff the graph is connected with 0 odd-degree vertices (a circuit) or exactly 2 (an open trail); Hierholzer finds it in O(E) by walking till stuck then splicing in detours. Verified live: on 0/2-odd graphs the trail uses every edge exactly once; on K4 (four odd vertices) it correctly reports no trail. See the degree-parity rule in 1D, a traced trail in 2D, and the local-parity-decides-global inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "4f56684774776686d958cdffe9673c44bbaa3c45a40658386375b35b2ca34c82"
    },
    {
      "slug": "the-two-sat",
      "title": "THE 2-SAT",
      "kicker": "satisfiability in linear time — a contradiction is a cycle",
      "accent": "#7048c0",
      "blurb": "2-SAT in the 5-window house format — decide whether clauses, each an OR of two literals like (x OR not y), can all be satisfied. Full SAT is NP-complete, but 2-SAT is linear: each clause (a OR b) means not-a implies b and not-b implies a; build the implication graph, find strongly-connected components, and it is satisfiable iff no variable and its negation share a component (the assignment reads off in reverse topological order). Verified live: the SCC verdict matches brute force over all 2^n assignments for 300 formulas, and satisfiable ones return a valid assignment. See a clause become implications in 1D, the SCC solver in 2D, and the contradiction-is-a-cycle inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "3f2aed3cdee19af2439949be7b0251ffe899eb56a44aa684d2e5000099a858d9"
    },
    {
      "slug": "the-cartesian-tree",
      "title": "THE CARTESIAN TREE",
      "kicker": "one tree, two orders — range-min and LCA are the same",
      "accent": "#40a8c8",
      "blurb": "the Cartesian tree in the 5-window house format — built from a sequence to satisfy two orders at once: in-order traversal reproduces array positions (a BST on indices) and every parent's value <= its children (a min-heap on values), in O(n) with a stack. The payoff: the lowest common ancestor of positions i,j is exactly the position of the minimum in a[i..j], so range-minimum and LCA become the same problem. Verified live: in-order equals 0..n-1, the heap property holds, and LCA(i,j) value equals the range-minimum. See the min rising to the root in 1D, click-two-positions in 2D, and the RMQ-equals-LCA inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "33684a406c173afe12df109d43e881392fc12804210138d468ef924d0c9ea4b1"
    },
    {
      "slug": "the-gauss-legendre",
      "title": "THE GAUSS-LEGENDRE",
      "kicker": "n samples integrate polynomials of degree 2n-1 exactly",
      "accent": "#d08840",
      "blurb": "Gaussian quadrature in the 5-window house format — instead of many evenly-spaced samples, place n points at the roots of the n-th Legendre polynomial with matching weights and integrate every polynomial up to degree 2n-1 exactly. Two points nail cubics, three nail quintics: the accuracy of ~2n even samples from only n, because you choose where to sample, not just how heavily. Verified live: 2-point Gauss is exact for x^0..x^3 (fails at degree 4) and 3-point through degree 5 (fails at 6), matching the analytic integrals on [-1,1]. See the Legendre-root nodes in 1D, integrate rising powers in 2D, and the choose-where-doubles-reach inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "064310fef72ab6904c6a0ee23236828eb091d3f96e9eff02663a832bd3a40f3c"
    },
    {
      "slug": "the-motzkin",
      "title": "THE MOTZKIN",
      "kicker": "paths that may rest — Catalan hiding under the flats",
      "accent": "#56b8c0",
      "blurb": "the Motzkin numbers in the 5-window house format — 1,1,2,4,9,21,51,127 count lattice paths from (0,0) to (n,0) using up, down, and LEVEL steps that never dip below the axis (Dyck paths that may rest). Recurrence M_n = M_{n-1} + sum M_k M_{n-2-k}, and M_n = sum_k C(n,2k) Cat_k ties them to Catalan. Verified live: the recurrence matches a brute path count for n=0..10 and the Catalan relation holds. See a Motzkin path in 1D, three counts agreeing in 2D, and the Catalan-under-flats inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "4412886952611256f416236eb109524b1b8e6821cbb283139d60992cdad1e0f4"
    },
    {
      "slug": "the-dutch-flag",
      "title": "THE DUTCH FLAG",
      "kicker": "sort three colors in one pass — correct by invariant",
      "accent": "#d06868",
      "blurb": "Dijkstra's Dutch national flag problem in the 5-window house format — sort an array of three values (0/1/2) in one pass, O(n) time O(1) space, with three pointers low/mid/high: a 0 swaps down, a 2 swaps up (mid not advancing), a 1 stays. The invariant keeps 0s before low, 1s to mid, unknown [mid,high], 2s after high. It is the heart of 3-way quicksort. Verified live: for 500 random 0/1/2 arrays the single pass yields a sorted array that is a permutation of the input. See the four regions in 1D, step the partition in 2D, and the correct-by-invariant inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "5a235d325ba3982e885076bdae342593662832a1fb63f2d02a13fa855412d647"
    },
    {
      "slug": "the-sturm",
      "title": "THE STURM",
      "kicker": "count real roots in an interval without finding them",
      "accent": "#b06840",
      "blurb": "Sturm's theorem in the 5-window house format — count the real roots of a polynomial in [a,b] without finding them: build the Sturm chain (p, p', then successive negated polynomial-division remainders), count sign changes V(a) and V(b), and the number of distinct real roots in (a,b] is exactly V(a)-V(b). Verified live: for 200 polynomials built from distinct integer roots, the sign-variation count equals the actual roots in the interval, over full and sub-intervals. See the chain's signs in 1D, a slidable interval in 2D, and the count-not-locate inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "bb27b54de06397fe98a8e50fc79a5fd32944d78bdd0afd8a1245c88476c5c210"
    },
    {
      "slug": "the-closest-pair",
      "title": "THE CLOSEST PAIR",
      "kicker": "nearest two points in O(n log n) — geometry bounds the strip",
      "accent": "#e08850",
      "blurb": "the closest-pair problem in the 5-window house format — find the two nearest of n points in O(n log n) instead of O(n^2): sort by x, split at the median, recurse in each half (distance delta), then only points in a vertical strip of width 2delta around the split can beat delta, and each such point compares to at most a constant number of y-neighbors (a packing bound). Verified live: for 200 random point sets, the divide-and-conquer closest distance equals the brute-force minimum over all pairs. See the strip in 1D, closest pair drawn in 2D, and the packing-bounds-candidates inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "db7f8d10538bf26fcf4b32180c283d94450416e7372ea4cb12c27781444583a9"
    },
    {
      "slug": "the-newton-identities",
      "title": "THE NEWTON IDENTITIES",
      "kicker": "power sums <-> polynomial coefficients, no roots needed",
      "accent": "#a898d8",
      "blurb": "Newton's identities in the 5-window house format — connect the power sums p_k = sum x_i^k of a polynomial's roots to the elementary symmetric polynomials e_k (its coefficients by Vieta) via p_k = e1 p_{k-1} - e2 p_{k-2} + ... +- k e_k. So the sums of powers of the unknown roots reconstruct the polynomial's coefficients without ever finding the roots. Verified live: for 300 random root-sets, Newton's identities recover e_k from the power sums, matching Vieta exactly. See the two summaries in 1D, recover-from-power-sums in 2D, and the moments-are-coefficients inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "d02000ae4b0ccb10d88f002c15eef89e561da510bd5edf0c8357f3d8318d18b4"
    },
    {
      "slug": "the-patience-sorting",
      "title": "THE PATIENCE SORTING",
      "kicker": "deal cards to piles — the pile count is the longest increasing run",
      "accent": "#c8a848",
      "blurb": "patience sorting in the 5-window house format — deal cards onto piles, each on the leftmost pile whose top is >= it (else a new pile); the number of piles equals the longest increasing subsequence of the deck, computed in O(n log n) by binary search. Back-pointers recover the actual subsequence, and the structure ties to RSK and the Ulam-Hammersley problem. Verified live: for 500 random sequences the pile count equals the LIS length from an independent O(n^2) method. See cards dealt to piles in 1D, a deck sorted in 2D, and the greedy-is-optimal inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "31f94ab433e516e75b8c14c46730bb296ed5b4fa73e0c59d83aa7f2bbb98c0e9"
    },
    {
      "slug": "the-shunting-yard",
      "title": "THE SHUNTING YARD",
      "kicker": "infix to RPN in one pass — precedence resolved once",
      "accent": "#58b0a0",
      "blurb": "Dijkstra's shunting-yard algorithm in the 5-window house format — convert infix (3+4*2) to postfix (3 4 2 * +) in one left-to-right pass using an operator stack that respects precedence and parentheses, then evaluate postfix trivially on a value stack. It is how calculators and compilers turn human math into machine order, in O(n) with no recursion. Verified live: for 500 random expressions the shunting-yard postfix evaluates to the same value as an independent recursive-descent evaluator. See the operator stack in 1D, tokens shunting in 2D, and the order-encodes-grammar inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "95097b682a053df0008fbab87102d0e515f6ac376959ed611ba28ce0bb11d9f5"
    },
    {
      "slug": "the-thompson-nfa",
      "title": "THE THOMPSON NFA",
      "kicker": "regex to NFA — match by advancing a whole state set, no backtracking",
      "accent": "#d05858",
      "blurb": "Thompson's construction in the 5-window house format — compile a regular expression into an NFA from four gadgets (literal, concatenation, alternation, star) glued by epsilon-transitions, then match a string by tracking the SET of reachable states, stepping the whole set per character. This runs in O(nm) with no catastrophic backtracking (the guarantee grep and RE2 give). Verified live: for several patterns over {a,b}, the NFA set-simulation's accept/reject matches a reference regex across all strings up to length 6. See the gadgets in 1D, accept/reject in 2D, and the advance-all-paths inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "003ad900f0e2edde95babcac58720378d6ff6573f6a492850a6e7f1fcf73f051"
    },
    {
      "slug": "the-kadane",
      "title": "THE KADANE",
      "kicker": "max subarray in one pass — forget a prefix when it turns negative",
      "accent": "#b878d0",
      "blurb": "Kadane's algorithm in the 5-window house format — find the maximum-sum contiguous subarray in one pass, O(n) time O(1) space: the best subarray ending here is either this element or this element plus the best ending previously, whichever is larger; reset when the running sum goes negative. It is dynamic programming distilled to two scalars. Verified live: for 500 random arrays with negatives, Kadane's result equals a brute maximum over all O(n^2) subarrays. See the running best in 1D, the winning subarray in 2D, and the optimal-substructure inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "6de9667f6e7af589bfee7d99dc64317c0924289ca1a7b5e29905e89e5f461761"
    },
    {
      "slug": "the-ford-fulkerson",
      "title": "THE FORD-FULKERSON",
      "kicker": "max flow equals min cut — the bottleneck found by filling it",
      "accent": "#5090d0",
      "blurb": "Ford-Fulkerson max-flow in the 5-window house format — push flow from source to sink along augmenting paths of unsaturated pipes (with residual back-edges to reroute) until none remain; the maximum flow equals the minimum cut, the smallest total capacity severing source from sink (LP duality made concrete). Verified live: for 300 random networks the max flow equals the min-cut capacity (the residual-reachable set) and flow is conserved at every node. See an augmenting path in 1D, a network solved with its cut in 2D, and the max-equals-min inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "5068358c68a6c99ba3b3c75f3b24058a2810ddf1e9cbd7e359ef989bb734d8b6"
    },
    {
      "slug": "the-feistel",
      "title": "THE FEISTEL",
      "kicker": "a reversible cipher from a one-way function",
      "accent": "#c05868",
      "blurb": "the Feistel network in the 5-window house format — build a reversible block cipher from ANY function, even a non-invertible one: split the block into halves L,R; each round the new left is old R and the new right is old L XOR F(R, round-key); decrypt by running the same structure with round keys reversed. It is the skeleton of DES and Blowfish. Verified live: with a deliberately non-invertible round function F, decrypt(encrypt(x)) reproduces x exactly for 1000 random blocks and key schedules. See the round in 1D, encrypt/decrypt in 2D, and the reversibility-from-architecture inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "5ad7f97a58a89b47df4371fde8121c2aab23534fee4c2953c95e36d2cc17b989"
    },
    {
      "slug": "the-beatty",
      "title": "THE BEATTY",
      "kicker": "two irrational sequences tile the integers exactly once",
      "accent": "#58b8a8",
      "blurb": "Beatty sequences in the 5-window house format — for irrational alpha>1 and its conjugate beta with 1/alpha+1/beta=1, the floor-sequences floor(n*alpha) and floor(n*beta) together contain every positive integer exactly once (Rayleigh-Beatty theorem). For alpha=golden ratio these are the Wythoff sequences behind Wythoff Nim. Verified live: for five irrationals, the two sequences partition 1..2000 exactly (no gaps, no overlaps). See the colored integer line in 1D, the partition in 2D, and the densities-sum-to-one inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "1bce59cbcf2d3223a53e549044bb6552fc514605b939df24a7a81908b25b3828"
    },
    {
      "slug": "the-enigma",
      "title": "THE ENIGMA",
      "kicker": "a cipher that is its own inverse — and could never encrypt a letter to itself",
      "accent": "#b09050",
      "blurb": "the Enigma machine in the 5-window house format — a rotor cipher whose reflector makes it reciprocal (if A encrypts to K, K encrypts to A), so one machine and setting both encrypt and decrypt. But the reflector also guaranteed no letter ever encrypts to itself, and that constraint was Enigma's fatal weakness (the foothold the Bombe exploited). Verified live: on a simplified 3-rotor + reflector machine, encrypting the ciphertext with the same start settings returns the plaintext, and no character ever equals its plaintext letter, across 200 random settings. See the signal path in 1D, encrypt-twice in 2D, and the symmetry-was-the-crack inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "73faa0f2ae012cd2a0a09732ffffd2beffdf9333f7ab7dd0d44321f445fdce2f"
    },
    {
      "slug": "the-boyer-moore",
      "title": "THE BOYER-MOORE",
      "kicker": "search by skipping — learn most from a mismatch",
      "accent": "#d07850",
      "blurb": "Boyer-Moore string search in the 5-window house format — match the pattern right-to-left and, on a mismatch, skip ahead (often by the whole pattern length) using the bad-character rule: if the mismatched text character is absent from the pattern, jump entirely past it; else align its last occurrence. This can be sublinear, examining fewer characters than the text length. Verified live: over 500 random texts/patterns, the bad-character (Horspool) search returns exactly the same match positions as a naive scan. See a skip in 1D, the leaping search in 2D, and the mismatch-informs-most inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "87a19b167ec92b9e85e76694cb4a8750e1116b202e57b1b29c4f10b5fc8a9373"
    },
    {
      "slug": "the-aitken",
      "title": "THE AITKEN",
      "kicker": "accelerate convergence by cancelling the error's shape",
      "accent": "#7098d8",
      "blurb": "Aitken's delta-squared process in the 5-window house format — accelerate a slowly-converging sequence: from x_n -> L form x'_n = x_n - (dx_n)^2 / d^2 x_n, which homes in on L far faster. If the sequence converges geometrically (x_n = L + c*r^n), Aitken returns L exactly in one step by cancelling the error term. Verified live: Aitken returns L to ~1e-13 for a geometric sequence, and on the x=cos(x) fixed-point iteration reaches 1e-8 accuracy in 17 accelerated terms versus 43 raw. See three-terms-to-one in 1D, raw vs accelerated in 2D, and the model-the-error inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "2a69b1cf3596d7ac12c7a4a68bda9fec4eed3c68fe121af5f48576242eadfe9b"
    },
    {
      "slug": "the-van-der-corput",
      "title": "THE VAN DER CORPUT",
      "kicker": "reverse the bits of n — points that fill the interval evenly",
      "accent": "#60b0c8",
      "blurb": "the van der Corput sequence in the 5-window house format — fill [0,1) far more evenly than random by reversing the binary digits of n around the radix point: 1->0.5, 2->0.25, 3->0.75, 4->0.125. The first 2^m points are exactly the dyadic rationals {j/2^m} scrambled, so its discrepancy shrinks like log(N)/N versus random's 1/sqrt(N). It underlies quasi-Monte Carlo integration. Verified live: the first 2^6 points equal the dyadic grid and the star discrepancy is far below matched random points. See bit-reversal in 1D, vdc vs random in 2D, and the bisection inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "5c38f5c51a6e2c1268ac325651fa31b5d5d18074ddaf5681b8d87f1916001d7c"
    },
    {
      "slug": "the-floyd-warshall",
      "title": "THE FLOYD-WARSHALL",
      "kicker": "all-pairs shortest paths by admitting one waypoint at a time",
      "accent": "#6890d0",
      "blurb": "Floyd-Warshall in the 5-window house format — shortest path between every pair of vertices in one triple loop: dist[i][j] = min(dist[i][j], dist[i][k]+dist[k][j]), sweeping k over all vertices as intermediate stops. O(V^3), handles negative edges, flags negative cycles. Verified live: over 200 random weighted graphs, the Floyd-Warshall distance matrix matches shortest paths from an independent per-source Bellman-Ford. See the relaxation in 1D, the evolving matrix in 2D, and the DP-over-waypoint-sets inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "bad5631435b110ff3669a297e9a241e1f88c6fe5555dfd313653219a66fbe4f1"
    },
    {
      "slug": "the-durand-kerner",
      "title": "THE DURAND-KERNER",
      "kicker": "all polynomial roots at once — estimates that repel into place",
      "accent": "#d08858",
      "blurb": "the Durand-Kerner (Weierstrass) method in the 5-window house format — find ALL n roots of a degree-n polynomial simultaneously by iterating each estimate z_i <- z_i - p(z_i)/prod_{j!=i}(z_i - z_j); the denominator divides out the other roots so estimates repel toward distinct roots. From evenly-spread complex guesses it converges to all roots at once, no deflation. Verified live: over 100 polynomials from known integer roots, it converges so |p(z)| < 1e-4 at every returned root. See the repulsion in 1D, complex convergence in 2D, and the coupled-fixed-point inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "b5546188d9f4a94eb3e40b8eadc01c9ee2fa4639d7e0d602d507ccf64c075fa0"
    },
    {
      "slug": "the-cayley-hamilton",
      "title": "THE CAYLEY-HAMILTON",
      "kicker": "every matrix satisfies its own characteristic polynomial",
      "accent": "#b07858",
      "blurb": "the Cayley-Hamilton theorem in the 5-window house format — every square matrix satisfies its own characteristic polynomial: compute p(lambda)=det(lambda*I - A), substitute the matrix A for lambda, and get the zero matrix p(A)=0. A consequence: any power of A, and A^-1, is a polynomial in A of degree < n, so a matrix's whole behavior is n coefficients. Verified live: for 200 random integer matrices (2x2, 3x3), substituting A into its characteristic polynomial (via Faddeev-LeVerrier) yields the zero matrix. See char-poly-to-zero in 1D, p(A) computed in 2D, and the finite-basis inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "578a02388bb5e85e39adce4d53a1984cb049fce5f4b75091562b839f7b1d7eb2"
    },
    {
      "slug": "the-hopcroft-karp",
      "title": "THE HOPCROFT-KARP",
      "kicker": "maximum bipartite matching = minimum vertex cover",
      "accent": "#58b878",
      "blurb": "Hopcroft-Karp in the 5-window house format — find a maximum matching in a bipartite graph (largest set of edges sharing no endpoint) in O(E sqrt V) by repeatedly finding augmenting paths (alternating unmatched/matched, free at both ends) and flipping them to grow the matching by one. By Konig's theorem the max matching size equals the min vertex cover. Verified live: over 300 random bipartite graphs, the augmenting-path matching size equals a brute-force maximum matching. See an augmenting path in 1D, the matching in 2D, and the max-matching-equals-min-cover inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "cc4d3cbfcf9479f940a750bc30ba7ef86c7444ff7b81d45616d06ba3403ef763"
    },
    {
      "slug": "the-stirling-cycles",
      "title": "THE STIRLING CYCLES",
      "kicker": "count permutations by cycles — the inverse of set partitions",
      "accent": "#c88848",
      "blurb": "the unsigned Stirling numbers of the first kind in the 5-window house format — c(n,k) counts permutations of n items with exactly k cycles (mirror of the second kind, which counts set partitions), via c(n,k)=(n-1)c(n-1,k)+c(n-1,k-1). Rows sum to n!, and they are the rising-factorial coefficients: x(x+1)...(x+n-1)=sum_k c(n,k) x^k. Verified live: the recurrence matches a brute cycle count for n=1..7, rows sum to n!, and the rising-factorial identity holds. See a permutation's cycles in 1D, recurrence vs brute in 2D, and the two-kinds-are-inverse-matrices inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "7ae4c5dee88ebfb2c13552ac68f7209a055b3aeaef1ae5ac9291d987c1dd7eaf"
    },
    {
      "slug": "the-inversions",
      "title": "THE INVERSIONS",
      "kicker": "count disorder in O(n log n) — additive across a divide",
      "accent": "#d0687a",
      "blurb": "inversion counting in the 5-window house format — an inversion is a pair out of order (a[i]>a[j], i<j); the count measures distance from sorted and equals the minimum adjacent swaps to sort. Naively O(n^2), but a modified merge sort counts them in O(n log n): each time a right-half element is taken before the left is exhausted, it inverts with every remaining left element. Verified live: over 500 random arrays, the merge-sort count equals a brute O(n^2) count. See a merge counting crossings in 1D, an array's inversions in 2D, and the additive-across-a-divide inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "c24181dff551eb7637bc0a261eb9737b55e7b126ec636d6d317cfd65b44f3ab3"
    },
    {
      "slug": "the-shoelace",
      "title": "THE SHOELACE",
      "kicker": "polygon area from vertex coordinates — signs cancel the outside",
      "accent": "#58b0c0",
      "blurb": "the shoelace formula in the 5-window house format — the area of any simple polygon from its vertices: A = (1/2)|sum (x_i y_{i+1} - x_{i+1} y_i)|, the criss-cross pattern like lacing a shoe. The signed sum also gives orientation (CCW positive, CW negative). It works for convex or non-convex simple polygons and drops out of Green's theorem. Verified live: for 200 polygons the shoelace area equals a triangle-fan sum, and for lattice polygons it matches Pick's theorem (A = I + B/2 - 1). See the criss-cross in 1D, a lattice polygon in 2D, and the signed-triangles-cancel inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "dfeb485a51fae4dd8124f92f5d375da770bb4558908c1bfc6a5f609527b7bfec"
    },
    {
      "slug": "the-horner",
      "title": "THE HORNER",
      "kicker": "evaluate in n multiplications — and it's synthetic division",
      "accent": "#c0a048",
      "blurb": "Horner's method in the 5-window house format — evaluate a degree-n polynomial in n multiplications (vs naive ~2n) by nesting: a_n x^n+...+a_0 = (...((a_n)x+a_{n-1})x+...)x+a_0. The same nesting is synthetic division: intermediate values are the quotient coefficients dividing by (x-r), and the final value is the remainder = p(r) (remainder theorem). Verified live: over 500 cases Horner equals naive evaluation, and synthetic division gives q,rem with q(x)(x-r)+rem=p(x) and rem=p(r). See the nested eval in 1D, eval+division in 2D, and the evaluating-is-dividing inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "59749f33118936f1315224f1c2f7746327ff062958834d2c96beaedb1ece3910"
    },
    {
      "slug": "the-jacobi-symbol",
      "title": "THE JACOBI SYMBOL",
      "kicker": "a residue test computed by reciprocity — without factoring",
      "accent": "#a06890",
      "blurb": "the Jacobi symbol in the 5-window house format — the Legendre symbol (a/p) says if a is a quadratic residue mod prime p (+1) or not (-1); the Jacobi symbol extends it to odd n as the product of Legendre symbols over n's prime factors, (a/n)=prod (a/p_i)^{e_i}. It is computed fast by quadratic reciprocity WITHOUT factoring n (the engine of Solovay-Strassen primality). Caveat: for composite n, (a/n)=+1 does not guarantee a is a residue. Verified live: the reciprocity value equals the Legendre product over the factorization for odd n, and equals Legendre (predicting residues) for primes. See the reciprocity ladder in 1D, symbol vs product in 2D, and the answer-without-factoring inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "9a083feb4494132b6969942ac048b16cf692ed33d002c754f576db831649bf59"
    },
    {
      "slug": "the-lucas-theorem",
      "title": "THE LUCAS THEOREM",
      "kicker": "a giant binomial mod p from base-p digits alone",
      "accent": "#b08850",
      "blurb": "Lucas' theorem in the 5-window house format — compute C(m,n) mod a prime p using only the base-p digits: C(m,n) mod p equals the product of C(m_i,n_i) mod p over corresponding digits. So C(1000,500) mod 7 comes from a handful of tiny binomials. Corollary: C(m,n) is odd exactly when n's binary digits are a subset of m's, which is why Pascal mod 2 is the Sierpinski triangle. Verified live: the digit-product equals a direct C(m,n) mod p across 300 cases for primes 2,3,5,7,11. See base-p digits align in 1D, digit-product vs direct in 2D, and the local-digits-decide-the-global inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "56f54e64ae8d830042302ae931deb78d521fb022654ac8da900d83a8de0f40c9"
    },
    {
      "slug": "the-quickselect",
      "title": "THE QUICKSELECT",
      "kicker": "the k-th smallest in O(n) — median of medians",
      "accent": "#d06858",
      "blurb": "quickselect in the 5-window house format — find the k-th smallest element without fully sorting by partitioning around a pivot and recursing into only the side containing the k-th (expected O(n)). Median-of-medians (Blum-Floyd-Pratt-Rivest-Tarjan) guarantees O(n) worst case: split into groups of 5, take each median, recursively find the median of medians as pivot. It computes a median in guaranteed linear time. Verified live: over 300 random arrays, median-of-medians quickselect returns exactly sorted[k] for every k. See partition-and-recurse in 1D, select-the-k-th in 2D, and the answer-without-order inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "fddbe20070971d4ba2eedab3614966287a9f720701bce5335f2457f4ec7e6c42"
    },
    {
      "slug": "the-suffix-array",
      "title": "THE SUFFIX ARRAY",
      "kicker": "sort every suffix — index every substring in O(n) space",
      "accent": "#c05868",
      "blurb": "the suffix array in the 5-window house format — the starting positions of all suffixes of a string, sorted lexicographically (banana -> [5,3,1,0,4,2]). With the LCP array (longest common prefix of adjacent suffixes) it answers 'does pattern P occur?' by binary search in O(m log n), finds longest repeats, and does much of a suffix tree's job in far less memory. Verified live: over 300 strings the doubling-built suffix array matches a brute lexicographic sort and the LCP array is correct. See suffixes sorted in 1D, SA+LCP+search in 2D, and the every-substring-is-a-suffix-prefix inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "96a7809006013fbd74f3898664f4f6e22ab6fdcdc0df96a20c639d977b3cd1dd"
    },
    {
      "slug": "the-point-in-polygon",
      "title": "THE POINT IN POLYGON",
      "kicker": "inside or outside decided by ray-crossing parity",
      "accent": "#58b0a0",
      "blurb": "point-in-polygon (ray casting) in the 5-window house format — shoot a ray from the point and count polygon-edge crossings: odd = inside, even = outside. It works for any simple polygon (convex or concave) as a consequence of the Jordan curve theorem, and the winding-number method agrees. Verified live: over 500 points and polygons, ray-casting parity, the winding number, and a convex ground-truth test all agree. See crossings flip inside/outside in 1D, a movable point in 2D, and the direction-independent-parity inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "3f9f42972075b3897fe130b7f925c9dc81cc874090e4b2d8e1050272bd90fe90"
    },
    {
      "slug": "the-thomas",
      "title": "THE THOMAS",
      "kicker": "solve a tridiagonal system in O(n) — sparsity conserved",
      "accent": "#c0a058",
      "blurb": "the Thomas algorithm in the 5-window house format — solve a tridiagonal linear system (each equation touches a variable and its two neighbours) in O(n) instead of O(n^3): a forward sweep eliminates the sub-diagonal, then back-substitution reads off the answers. Tridiagonal systems drive cubic splines, the 1D heat equation, and Crank-Nicolson. Verified live: over 300 random tridiagonal systems, the Thomas solution recovers the true x with max error ~1e-16. See the forward sweep in 1D, a solved system with zero residual in 2D, and the sparsity-conserved inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "8752eb25b275a2fa648150b84d571e59a0a306f307bc73e9ed345a1dbf0b4583"
    },
    {
      "slug": "the-sprague-grundy",
      "title": "THE SPRAGUE-GRUNDY",
      "kicker": "every impartial game is secretly a Nim heap",
      "accent": "#c05868",
      "blurb": "the Sprague-Grundy theorem in the 5-window house format — every impartial game position equals a single Nim heap, its Grundy number (nimber), computed as the mex (minimum excludant) of the successors' Grundy numbers. A position is losing for the mover iff Grundy=0, and the Grundy of a sum of games is the XOR of the parts. Verified live: a subtraction game's Grundy values equal n mod 4, and a Nim position is losing iff the XOR of heap sizes is 0 (matching brute minimax over 300 positions). See mex in 1D, Nim XOR win/loss in 2D, and the composition-becomes-XOR inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "c45082e544b249f092a7be58140dbcb81619240ffc7a63e5c934793b20f19f5e"
    },
    {
      "slug": "the-game-of-life",
      "title": "THE GAME OF LIFE",
      "kicker": "two rules, a glider crawling (1,1) every 4 generations",
      "accent": "#58c080",
      "blurb": "Conway's Game of Life in the 5-window house format — a grid cellular automaton with two rules (B3/S23): a dead cell is born with exactly 3 live neighbours, a live cell survives with 2 or 3. From these emerge the blinker (period 2), the block (still), and the glider that crawls diagonally, returning to its shape shifted by (1,1) every 4 generations. Life is Turing-complete. Verified live: blinker period 2, block stationary, glider translates (1,1) per 4 gens under B3/S23. See the rule in 1D, the glider stepping in 2D, and the undecidable-future inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "0b4517afb75fec93f23b730896cf7572c5a277af612187c7ecea3790b5573c8d"
    },
    {
      "slug": "the-hook-length",
      "title": "THE HOOK LENGTH",
      "kicker": "count Young tableaux as n! over a product of hooks",
      "accent": "#c8a050",
      "blurb": "the hook length formula in the 5-window house format — the number of standard Young tableaux of a shape (fillings of a Young diagram with 1..n increasing along rows and down columns) equals n! divided by the product of hook lengths, where a cell's hook is 1 + its arm (cells right) + its leg (cells below). A global count of intricate fillings collapses to one product; these counts are the dimensions of the symmetric group's irreducible representations. Verified live: n!/prod(hooks) equals a brute count of standard Young tableaux for every partition of n=1..7. See a cell's hook in 1D, a diagram's hooks + count in 2D, and the global-count-from-local-geometry inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "edb8e769294144c0b0fe399c8d78ac0e967572142981fbbab7f4aa05c383fa6f"
    },
    {
      "slug": "the-midpoint-circle",
      "title": "THE MIDPOINT CIRCLE",
      "kicker": "draw a circle with integers and 8-fold symmetry",
      "accent": "#5a90d0",
      "blurb": "the midpoint circle algorithm in the 5-window house format — draw a circle on a pixel grid with only integer arithmetic (no float, trig, or sqrt) via a decision variable that picks the pixel nearest the true circle, exploiting 8-fold symmetry to plot 8 pixels per octant step. It is the integer companion to the Bresenham line. Verified live: over radii 3..40, every plotted pixel is within 0.5 of the true radius (measured max ~0.49), integer-only. See the decision variable in 1D, a drawn circle in 2D, and the octant-mirrored-8-ways inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "a21ee9b37fcfd5218c0c067c7c3de73feb4b1432c76b2e92d4f37a6ffe5d1728"
    },
    {
      "slug": "the-dirichlet-convolution",
      "title": "THE DIRICHLET CONVOLUTION",
      "kicker": "arithmetic functions form a ring — Mobius is the inverse of 1",
      "accent": "#b078a0",
      "blurb": "the Dirichlet convolution in the 5-window house format — combine arithmetic functions by (f*g)(n) = sum over divisors d of n of f(d)g(n/d), making them a ring with identity epsilon (1 at n=1). The Mobius function is the inverse of the constant-1 (mu*1=epsilon), which is Mobius inversion; Euler's totient gives phi*1=Id; divisor count tau=1*1; divisor sum sigma=1*Id. Number theory's identities become algebra. Verified live: mu*1=epsilon, phi*1=Id, 1*1=tau, 1*Id=sigma for all n<=100. See divisor pairs in 1D, the four identities in 2D, and the Mobius-inversion-is-a-group-inverse inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "145041343305acddb32fda13503c631f2971531d7fd30a0b9e6bcce16db761e5"
    },
    {
      "slug": "the-hungarian",
      "title": "THE HUNGARIAN",
      "kicker": "minimum-cost assignment by reducing to zeros",
      "accent": "#6088c0",
      "blurb": "the Hungarian algorithm in the 5-window house format — solve the assignment problem (match n workers to n jobs at minimum total cost) in O(n^3) instead of checking n! matchings, by subtracting row and column constants (which never change the optimal assignment) until a zero-cost complete matching appears. It is the workhorse of scheduling, tracking, and allocation. Verified live: over 200 random cost matrices (n=2..6), the Hungarian assignment's total cost equals the brute-force minimum over all permutations. See row reduction in 1D, an optimal assignment in 2D, and the reduction-reveals-the-answer inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "633c4090a48de598586fceaa2d30b0823a1db4ffafc4d9d2453d865b3e5b4c4a"
    },
    {
      "slug": "the-cholesky",
      "title": "THE CHOLESKY",
      "kicker": "a matrix square root — A = L·Lᵀ, half the work of LU",
      "accent": "#58a8b0",
      "blurb": "the Cholesky decomposition in the 5-window house format — factor a symmetric positive-definite matrix A into A = L*L^T with L lower-triangular (a matrix square root), about twice as fast as general LU. It solves SPD systems, least squares, and draws correlated Gaussian samples (Sigma = LL^T, transform normals by L). If A is not positive-definite the algorithm fails under a negative square root, so it doubles as a definiteness test. Verified live: for 300 random SPD matrices L*L^T reconstructs A to ~1e-15, L is lower-triangular, and a non-SPD matrix is rejected. See the triangular build in 1D, factor+reconstruct in 2D, and the symmetry-halves-the-work inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "58856c7b716106dbc257b37b1a67fc919cafee164a94b47a124401621c45baef"
    },
    {
      "slug": "the-narayana",
      "title": "THE NARAYANA",
      "kicker": "Catalan sliced by peaks — a refinement that sums back",
      "accent": "#c090a0",
      "blurb": "the Narayana numbers in the 5-window house format — N(n,k) counts Dyck paths (balanced-paren strings) of semilength n with exactly k peaks (an up-step then a down-step, '()'), refining the Catalan numbers: sum_k N(n,k) = C_n. The closed form is N(n,k)=(1/n)C(n,k)C(n,k-1), and the triangle 1;1,1;1,3,1;1,6,6,1 is symmetric. Verified live: N(n,k) equals a brute count of Dyck paths with k peaks and the rows sum to the Catalan number for n=1..8. See peaks on a path in 1D, formula vs brute in 2D, and the Catalan-decomposed-by-a-statistic inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "f5fe9761de5a63b92b94950689702ea5868fe9c508925f4d6a84d1e36dc2b11f"
    },
    {
      "slug": "the-pancake-sorting",
      "title": "THE PANCAKE SORTING",
      "kicker": "sort by prefix flips — Bill Gates' only paper",
      "accent": "#d0a048",
      "blurb": "pancake sorting in the 5-window house format — sort a stack when the only move is a prefix flip (insert a spatula, flip the top portion). Greedy (bring the largest unsorted pancake up, then flip it down) always sorts in at most 2n-3 flips; finding the true minimum (the pancake number) is NP-hard. The famous fact: Bill Gates' only research paper (with Papadimitriou, 1979) improved the bound. Verified live: over 300 random stacks, greedy pancake sorting produces a sorted stack using at most 2n-3 flips. See a prefix flip in 1D, a stack sorted in 2D, and the restricted-moves-make-the-optimum-hard inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "b52754da238dec5f4832a2f817d61db0203f905ef585f2a31cf780a892b8da5e"
    },
    {
      "slug": "the-fermat-factorization",
      "title": "THE FERMAT FACTORIZATION",
      "kicker": "factor n as a difference of squares — the seed of the sieves",
      "accent": "#b06858",
      "blurb": "Fermat's factorization method in the 5-window house format — split an odd n as a difference of squares n = a^2 - b^2 = (a-b)(a+b): start a at ceil(sqrt(n)) and increase until a^2 - n is a perfect square b^2, then (a-b),(a+b) are factors. It is fast when the factors are close to sqrt(n), slow when far apart (why secure RSA uses primes of very different sizes). Verified live: for a range of odd composites, the search returns nontrivial factors whose product is n (e.g. 5959 = 59 x 101). See a^2-n climbing in 1D, factors found in 2D, and the factoring-is-square-hunting inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "a8a53c36e0459a361dabecacf5f5c0aa7f43e21d435ec3453a14f41d94e6ddb4"
    },
    {
      "slug": "the-manacher",
      "title": "THE MANACHER",
      "kicker": "longest palindrome in linear time — reflection is the memory",
      "accent": "#7088c8",
      "blurb": "Manacher's algorithm in the 5-window house format — find the longest palindromic substring in O(n) instead of the naive O(n^2), by keeping the rightmost palindrome and reusing each new center's mirror radius so symmetry is never rechecked. A '#'-separator transform unifies even and odd palindromes. Verified live: over 300 random strings, Manacher's answer matches a brute-force longest palindrome in length, is itself a palindrome, and occurs in the string. See the radius array in 1D, the highlighted palindrome in 2D, and the mirror-reuse inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "a03efac6dd0b21f05d682fd46f739f6acbabeea79b90e4e05c98d61be3c0e6ef"
    },
    {
      "slug": "the-aho-corasick",
      "title": "THE AHO-CORASICK",
      "kicker": "match a whole set of patterns in one linear pass",
      "accent": "#c07850",
      "blurb": "the Aho-Corasick automaton in the 5-window house format — find all occurrences of a SET of patterns in a text in one linear pass, by building a trie of the patterns and adding failure links (jump to the longest proper suffix that is still a live prefix) so the scan never restarts. It is grep's multi-string engine and every signature scanner. Verified live: over 300 random (pattern-set, text) cases the automaton's complete match list equals a brute-force search, and {he,she,his,hers} in 'ushers' is recovered exactly. See the single scan in 1D, the match list in 2D, and the failure-link-is-the-inverse-of-a-trie-edge inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "5fb687e22ececb2e7e59b9ce1086e46dec4e764645edf2b6b183b9922e6de020"
    },
    {
      "slug": "the-reservoir",
      "title": "THE RESERVOIR",
      "kicker": "a uniform sample from a stream of unknown length, O(1) memory",
      "accent": "#58a878",
      "blurb": "reservoir sampling (Algorithm R) in the 5-window house format — draw a uniform random sample from a stream of unknown length in O(1) memory: keep the first item, and replace the kept item with probability 1/(i+1) when the i-th arrives; every item ends with probability exactly 1/n. Verified live: over 100000 trials on a length-8 stream, each position is selected with empirical frequency within ~2% of 1/8. See the 1/(i+1) acceptance in 1D, the histogram flattening to the uniform line in 2D, and the no-n-needed inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "16d3a6e368b550f6405b57fd54ae8804a32c4bc3b93f0aded0fd52184cc46650"
    },
    {
      "slug": "the-ducci",
      "title": "THE DUCCI",
      "kicker": "absolute differences around a ring — power-of-2 always burns to zero",
      "accent": "#b878c0",
      "blurb": "the Ducci sequence (diffy game) in the 5-window house format — replace each number in a ring by the absolute difference with its neighbor, and iterate: (a,b,c,d)->(|a-b|,|b-c|,|c-d|,|d-a|). When the ring length is a power of two, it ALWAYS collapses to all-zeros from any start; for other lengths it can cycle forever. Verified live: over 300 random 4-tuples every Ducci sequence reaches (0,0,0,0), while n=3 (1,2,3) does not reach zero within 200 steps. See a step in 1D, the ring collapsing in 2D, and the length-decides-the-fate inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "9469e2d39ce6d737e05fd25e1894ee1e45c5f4026fd22c8e53240b4f3228744b"
    },
    {
      "slug": "the-computus",
      "title": "THE COMPUTUS",
      "kicker": "the date of Easter by pure integer arithmetic",
      "accent": "#c0a048",
      "blurb": "the computus in the 5-window house format — compute Easter Sunday (first Sunday after the first ecclesiastical full moon on/after 21 March) by the Anonymous Gregorian algorithm (Gauss/Butcher/Meeus): a handful of integer divisions and remainders encoding the 19-year Metonic moon cycle, the epact, and Gregorian century corrections. Verified live: the algorithm reproduces a table of 20 known Easter dates (2000-2049) exactly, e.g. 2024 -> 31 March, 2025 -> 20 April. See the Metonic cycle in 1D, Easter placed on a calendar in 2D, and the arithmetic-stands-in-for-astronomy inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "854d4b796478126d183d8fbe69a483b79e7a33c532be6686bf97725c82a14d81"
    },
    {
      "slug": "the-bbp",
      "title": "THE BBP",
      "kicker": "the n-th hex digit of pi, without the digits before it",
      "accent": "#c0a048",
      "blurb": "the Bailey-Borwein-Plouffe formula in the 5-window house format — compute the n-th hexadecimal digit of pi WITHOUT computing any earlier digit, by isolating one digit through modular exponentiation (16^(n-k) mod (8k+j)) so no giant number is built. It made pi random-access. Verified live: BBP's hex digits for n=0..23 match pi's reference hex expansion (243F6A8885A308D313198A2E) exactly. See pi's hex digits in 1D, a single addressed digit in 2D, and the random-access-into-an-irrational inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "a3e7ab012d8fdd030593c01048e80f4d632866fe076062a6ea35b68b9e86acbd"
    },
    {
      "slug": "the-hamming",
      "title": "THE HAMMING",
      "kicker": "parity that locates the error, not just detects it",
      "accent": "#6098c0",
      "blurb": "the Hamming(7,4) code in the 5-window house format — protect 4 data bits with 3 parity bits (positions 1,2,4) so any single-bit error is corrected: recompute the three parities and the 3-bit syndrome, read as a binary number, is the position of the flipped bit. The first error-correcting code (1950). Verified live: for all 16 messages, every single-bit flip (7 positions) is corrected and the clean codeword decodes exactly — 128 cases pass. See the overlapping parity sets in 1D, a flip located and repaired in 2D, and the syndrome-is-the-error's-address inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "f56e3a9215243ec7121910e9f33d98505b15e6243acde59aa2a22db3cc5520a0"
    },
    {
      "slug": "the-chinese-remainder",
      "title": "THE CHINESE REMAINDER",
      "kicker": "rebuild a number uniquely from its residues",
      "accent": "#78b070",
      "blurb": "the Chinese Remainder Theorem in the 5-window house format — from a number's remainders modulo pairwise-coprime moduli, reconstruct the number uniquely modulo their product, built from modular inverses. The map x -> (x mod m1, x mod m2, ...) is a ring isomorphism, so nothing is lost. It powers RSA-CRT decryption, secret sharing, and residue-number-system arithmetic. Verified live: over 300 random coprime-modulus systems the reconstructed x satisfies every congruence and lies in [0,M); x=2mod3,3mod5,2mod7 -> 23. See three modular rings in 1D, reconstruction in 2D, and the bijection-loses-nothing inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "86e059540b6645a90d3faf7540cee09fb54bb08f7c12f5f274f3cce4e8adf14e"
    },
    {
      "slug": "the-berlekamp-massey",
      "title": "THE BERLEKAMP-MASSEY",
      "kicker": "recover the shortest LFSR from its output alone",
      "accent": "#b06890",
      "blurb": "the Berlekamp-Massey algorithm in the 5-window house format — from a bit sequence, recover the SHORTEST linear-feedback shift register that produces it, inferring the hidden taps from the output in O(n^2). This is why linear stream ciphers fall: an L-stage LFSR is fully exposed by just 2L output bits. It is also the decoding core of BCH and Reed-Solomon codes. Verified live: over 300 sequences from random LFSRs, the recovered register has length <= the generator's and exactly regenerates the whole sequence. See an LFSR shifting in 1D, taps recovered from a stream in 2D, and the recover-the-generator-from-the-output inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "682ea49a0d8cc95cd4e9783b6e5bea509df941531aa62ee172d71cd9e3766fda"
    },
    {
      "slug": "the-karger",
      "title": "THE KARGER",
      "kicker": "find the global min cut by random contraction",
      "accent": "#c07850",
      "blurb": "Karger's algorithm in the 5-window house format — find a graph's global minimum cut by RANDOM DESTRUCTION: repeatedly contract a random edge (merge endpoints, keep parallels) until two super-nodes remain; the edges between them are a cut. A single run finds the true min cut with probability >= 2/n^2, so best-of-O(n^2 log n) runs succeeds with high probability. A hard optimum, found by random merging. Verified live: over 40 random graphs the best of 200 contraction runs equals the brute-force global min cut. See one contraction in 1D, best-vs-brute on a graph in 2D, and the answer-is-what-resists-destruction inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "48de8598016b8fd4579f7c2fb8c9520180eb4e91635b5263974b2344911986b1"
    },
    {
      "slug": "the-lucas-lehmer",
      "title": "THE LUCAS-LEHMER",
      "kicker": "a deterministic primality verdict for Mersenne numbers",
      "accent": "#c05858",
      "blurb": "the Lucas-Lehmer test in the 5-window house format — decide with certainty whether M_p = 2^p - 1 is prime: set s0=4, iterate s -> (s^2 - 2) mod M_p exactly p-2 times; M_p is prime iff the final s is 0. No randomness, no witnesses — one deterministic recurrence, which is why the largest known primes are all Mersenne. Verified live: the test passes for prime exponents {3,5,7,13,17,19,31,61} and fails for {11,23,29,37,41,43} (composite M_p). See the s^2-2 sequence in 1D, the verdict in 2D, and the special-form-earns-certainty inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "03e634eb4968b4bf385e39c89c2786ffce2c2253d51eeaa4aee55ff42a267bcf"
    },
    {
      "slug": "the-arithmetic-coding",
      "title": "THE ARITHMETIC CODING",
      "kicker": "the whole message as one number, at the entropy limit",
      "accent": "#c0a048",
      "blurb": "arithmetic coding in the 5-window house format — compress a whole message into a single number in [0,1) by narrowing the interval to each symbol's probability sub-interval; the final width equals the product of symbol probabilities, so the code length equals the Shannon entropy exactly — beating Huffman's whole-bit-per-symbol floor. Verified live: over 300 random strings the exact BigInt coder round-trips (decode(encode(s))=s) and the final interval width equals the exact product of symbol frequencies. See the interval narrowing in 1D, encode+decode with entropy in 2D, and the one-number-at-the-entropy-limit inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "b6fdeb6c59cc9b12c2789d75a8ce78c6f1a9f1cf1697a514535ae3567c76212c"
    },
    {
      "slug": "the-needleman-wunsch",
      "title": "THE NEEDLEMAN-WUNSCH",
      "kicker": "optimal global alignment by dynamic programming",
      "accent": "#58a0b0",
      "blurb": "the Needleman-Wunsch algorithm in the 5-window house format — find the optimal global alignment of two sequences (matches, mismatches, gaps maximizing a score) by filling a DP grid where each cell is the best score aligning two prefixes; the corner is the optimum and a traceback reconstructs the alignment. It founded biological sequence comparison. Verified live: over 300 random pairs the DP score equals a brute-force optimum over all alignments, and the traceback re-scores to the DP value with gaps removed giving back the originals. See the alignment in 1D, the score grid + traceback in 2D, and the table-is-the-answer-and-the-map inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "e86b9786bd0e2d24ff58870a3ab8e8d3bd1e9f51d4733125f91fe7b789ce8653"
    },
    {
      "slug": "the-goertzel",
      "title": "THE GOERTZEL",
      "kicker": "one DFT bin from a tiny resonant filter",
      "accent": "#a078c0",
      "blurb": "the Goertzel algorithm in the 5-window house format — compute a single DFT frequency bin without the whole transform, using a second-order IIR filter (s = x + 2cos(w)s1 - s2, w = 2*pi*k/N) and reading the magnitude from the last two states; O(N) for one bin versus O(N log N) for all. It is how phones decode DTMF touch-tones. Verified live: over 200 random signals Goertzel's magnitude for every bin equals the direct DFT magnitude to ~1e-13. See the filter accumulating in 1D, one bin against the full spectrum in 2D, and the filter-for-one-tone inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "fe7223d6a61e75b2b71324e6c1fbc477f2374e78c721733e36615abb3fad44f9"
    },
    {
      "slug": "the-sutherland-hodgman",
      "title": "THE SUTHERLAND-HODGMAN",
      "kicker": "clip a polygon to a window, one edge at a time",
      "accent": "#60a870",
      "blurb": "the Sutherland-Hodgman algorithm in the 5-window house format — clip a polygon to a convex window by clipping against one edge at a time: for each clip edge keep the inside vertices and insert intersection points where the boundary crosses, then pipe the result to the next edge. It is the graphics pipeline's viewport clip. Verified live: over 200 random polygons every clipped-output vertex lies inside the convex window and clipping is idempotent (area unchanged on re-clip); a convex subject's clipped area matches a Monte-Carlo estimate of the true intersection. See a half-plane clip in 1D, a polygon clipped to a window in 2D, and the factor-2D-into-1D-cuts inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "4c1109e1c7d8bca9fb60e3cd564dc26cbc0c12133a090d99966ab6a87e58abaa"
    },
    {
      "slug": "the-stoer-wagner",
      "title": "THE STOER-WAGNER",
      "kicker": "the global min cut, deterministically, no source/sink",
      "accent": "#c05868",
      "blurb": "the Stoer-Wagner algorithm in the 5-window house format — find a graph's global minimum cut deterministically, without max-flow and without choosing a source and sink: each phase does a maximum-adjacency ordering (add the most tightly connected vertex), the last vertex's weight is a valid cut, and merging the last two and repeating covers all pairs in n-1 phases. Verified live: over 80 random weighted graphs the Stoer-Wagner cut equals the brute-force minimum over all bipartitions. See the adjacency ordering in 1D, cut-vs-brute on a graph in 2D, and the no-source-sink-needed inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "833d301d07f8429d267bc2ec59f7f2dbcec781bf13a69e53d25c71f8924bfd90"
    },
    {
      "slug": "the-misra-gries",
      "title": "THE MISRA-GRIES",
      "kicker": "frequent items from a stream in k-1 counters",
      "accent": "#c0a048",
      "blurb": "the Misra-Gries algorithm in the 5-window house format — find the frequent items in a stream using only k-1 counters: increment on a match, open a counter on a free slot, and decrement ALL counters on overflow (dropping zeros); every item with true frequency > n/k is guaranteed to survive. It is the streaming heavy-hitters primitive and generalizes Boyer-Moore majority. Verified live: over 500 random streams every item with freq > n/k survives, every reported count <= the true count, and the summary never exceeds k-1 entries. See the counters in 1D, summary-vs-truth in 2D, and the keep-what-matters inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "0469a36bdc067e90b23b57fac5c1ddd8da22c64e37a9414f90a6eb530ca35d4f"
    },
    {
      "slug": "the-dancing-links",
      "title": "THE DANCING LINKS",
      "kicker": "exact cover by O(1) reversible unlink/relink",
      "accent": "#9068c0",
      "blurb": "Dancing Links (DLX) in the 5-window house format — Knuth's technique for exact cover (choose rows of a 0/1 matrix covering each column exactly once), which Algorithm X searches by backtracking. The matrix is a mesh of circular doubly-linked nodes, so removing a row/column is O(1) and, crucially, putting it back on backtrack is O(1) too. It solves Sudoku, pentomino tilings, n-queens. Verified live: the real DLX (with cover/uncover pointers) returns a valid exact cover on 200 constructed instances and agrees with brute force on solvability of random instances. See reversible unlink in 1D, a solved matrix in 2D, and the deletion-is-its-own-inverse inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "49cf8b48cc0223243197acdc3136e6db06f7e85e3bfafce06413ee6fb8ebd0e5"
    },
    {
      "slug": "the-plain-changes",
      "title": "THE PLAIN CHANGES",
      "kicker": "all n! permutations, each one adjacent swap apart",
      "accent": "#58a0a8",
      "blurb": "plain changes (Steinhaus-Johnson-Trotter) in the 5-window house format — list all n! permutations so each differs from the last by a single adjacent swap, the minimal change: track a direction per element and repeatedly move the largest mobile element, flipping directions. English change-ringers have rung bells in this order for centuries; it is a Gray code for permutations. Verified live: for n=1..7 the algorithm produces all n! permutations, every one distinct, each consecutive pair differing by exactly one adjacent transposition. See the changes in 1D, stepping the sequence in 2D, and the Hamiltonian-path inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "afe03ba910906a180565f59a26eab26a13a5e75dc07c245812e6426cf466f5d1"
    },
    {
      "slug": "the-bitonic",
      "title": "THE BITONIC",
      "kicker": "a fixed, data-oblivious sorting network",
      "accent": "#60a870",
      "blurb": "bitonic sort in the 5-window house format — a sorting network: a fixed sequence of compare-and-swap operations that sorts any input of n=2^m elements, building a bitonic (up-then-down) sequence then merging halves. Which positions are compared never depends on the data, so it is data-oblivious — ideal for GPUs and hardware, and the antidote to race conditions (no data-dependent decisions). Verified live: the network sorts all 2^8 binary inputs (the 0-1 principle, which guarantees it sorts every input) and matches a reference sort on random arrays. See a comparator in 1D, an 8-element sort in 2D, and the data-oblivious-no-races inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "3d06342e72c656be147ae5b7db1b18cae61a94059b66b83e8219ee5dc41add91"
    },
    {
      "slug": "the-baby-step-giant-step",
      "title": "THE BABY-STEP GIANT-STEP",
      "kicker": "discrete log by meeting in the middle, O(sqrt n)",
      "accent": "#c06868",
      "blurb": "baby-step giant-step in the 5-window house format — solve the discrete logarithm g^x = h (mod p) by meeting in the middle: write x = im + j with m = ceil(sqrt(n)), tabulate the baby steps g^0..g^(m-1), then take giant steps h*(g^-m)^i until one lands in the table; O(sqrt n) time and space instead of O(n). It is the classic generic attack on discrete-log crypto. Verified live: over 300 random (p,g,x) the recovered exponent satisfies g^x = h mod p; 2^x=9 mod 23 gives x=5. See the split exponent in 1D, a solved instance in 2D, and the meet-in-the-middle inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "c14bda900aa42cdec83c2cedca0379ccd951f4801aae1282f103adc299ac2ee5"
    },
    {
      "slug": "the-toom-cook",
      "title": "THE TOOM-COOK",
      "kicker": "multiplication as evaluate-multiply-interpolate (~n^1.46)",
      "accent": "#c0a048",
      "blurb": "Toom-Cook (Toom-3) multiplication in the 5-window house format — multiply big numbers faster than n^2 by treating each as a polynomial: split into 3 parts, evaluate both at 5 points, multiply those 5 values (not 9), then interpolate the product polynomial and recombine, giving ~n^1.46. It generalizes Karatsuba (Toom-2) toward FFT multiplication. Verified live: over 300 random polynomial pairs the Toom-3 product equals the direct convolution exactly. See the 5 sample points in 1D, evaluate/interpolate in 2D, and the multiplication-as-interpolation inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "4ff876143f73902718668263b2f60d88e31a1724ec7b224fb1f145d170553522"
    },
    {
      "slug": "the-welzl",
      "title": "THE WELZL",
      "kicker": "the smallest enclosing circle, pinned by <=3 points",
      "accent": "#58a0b8",
      "blurb": "Welzl's algorithm in the 5-window house format — find the smallest enclosing circle of a point set in expected linear time by adding points one at a time: while a new point is inside, nothing changes; when it falls outside it must lie on the boundary, so the circle is rebuilt from the <=3 known boundary points. The circle-through-3 formula divides by a determinant that vanishes for collinear points (the guard). Verified live: over 200 random point sets Welzl's circle contains every point and its radius equals the brute-force minimum. See 2- and 3-point circles in 1D, a bounded cloud in 2D, and the <=3-points-decide inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "022c1d05cd2537c9ede68fcec6275b84e5fed8ebd4248397e8e8c2a5b3ff55fb"
    },
    {
      "slug": "the-kosaraju",
      "title": "THE KOSARAJU",
      "kicker": "strongly connected components in two DFS passes",
      "accent": "#70a860",
      "blurb": "Kosaraju's algorithm in the 5-window house format — find a directed graph's strongly connected components (maximal mutually-reachable groups) with two DFS passes: DFS the graph for finish times, then DFS the REVERSED graph in decreasing finish order; each tree is one SCC. SCCs reveal cycles and deadlocks, and a reference-counting garbage collector needs them because it cannot free a reference cycle. Verified live: over 200 random digraphs two nodes share a Kosaraju component iff they are mutually reachable (checked against the transitive closure). See the two passes in 1D, coloured components in 2D, and the reverse-the-arrows inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "b7ab5707bfbc896373b8f40ef0460f71467516aa0f67cd20774a4520af707219"
    },
    {
      "slug": "the-e-spigot",
      "title": "THE E-SPIGOT",
      "kicker": "digits of e that drip, one per pass, no big number",
      "accent": "#a878c0",
      "blurb": "the e-spigot in the 5-window house format — pour out the decimal digits of Euler's number e one at a time using only small integers, never forming a big high-precision value. In the factorial (mixed-radix) number system e-2 = 1/2!+1/3!+1/4!+..., so a 1 in each factorial place, multiplied by 10 with carries in bases 2,3,4,..., emits one decimal digit per pass. Verified live: the algorithm reproduces the first 30 digits of e (2.718281828459045235360287471352) matching a reference exactly. See the factorial places in 1D, dripping digits in 2D, and the digits-drip-not-accumulate inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "cf87fea035754b2bf6a4ada4e80fc20cd058fe974af6e2967d2a3f458d8546a2"
    },
    {
      "slug": "the-splay-tree",
      "title": "THE SPLAY TREE",
      "kicker": "a search tree that reshapes itself around what you use",
      "accent": "#c0a048",
      "blurb": "the splay tree in the 5-window house format — a self-adjusting binary search tree: every access splays the touched node to the root by rotations, with no balance rules or stored heights, so recently and frequently used keys drift to the top and operations are amortized O(log n). It is a self-optimizing cache in tree form. Verified live: over 300 random operation sequences the in-order traversal stays sorted, the key set matches a reference, every key is found, and after each access that key is at the root. See a splay in 1D, a self-adjusting tree in 2D, and the access-reshapes-the-tree inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "16e358ee69e8b6dbe71c826cd835bf437790ad2bbbd8d14f899dbefbcf16b34e"
    },
    {
      "slug": "the-interpolation-search",
      "title": "THE INTERPOLATION SEARCH",
      "kicker": "guess the position from the value — O(log log n) on uniform data",
      "accent": "#c06868",
      "blurb": "interpolation search in the 5-window house format — find a value in a sorted array by guessing its position from its value (position proportional to how far x sits between the endpoints), leaping most of the way in one step; O(log log n) expected on uniform data, beating binary search's O(log n). It is how you find a name near the front of a phone book. Verified live: over 400 random sorted arrays it finds every present key, rejects every absent one, and always agrees with binary search on membership. See the value-interpolated probe in 1D, probes leaping to the target in 2D, and the split-by-value inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "baa68aff15b30c6dbe19e081413362e6ce10011d616b00a757d17b46fcb2a332"
    },
    {
      "slug": "the-suffix-automaton",
      "title": "THE SUFFIX AUTOMATON",
      "kicker": "the smallest machine recognizing every substring, O(n) states",
      "accent": "#58a0b0",
      "blurb": "the suffix automaton in the 5-window house format — the smallest deterministic machine recognizing exactly the substrings of a string, with only O(n) states despite up to n(n+1)/2 substrings; every substring is a path, and equivalent end-positions are merged via suffix links. It counts distinct substrings, answers membership, and finds longest common substrings in linear time. Verified live: over 300 random strings the distinct-substring count Sigma(len - len[link]) equals a brute-force count, and it accepts substrings while rejecting non-substrings. See the length ranges in 1D, states in 2D, and the linear-machine-holds-quadratic-set inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "ef3e6f84f5af2e2376a142b59a85b70952f98101d7a148f1db016660ed369531"
    },
    {
      "slug": "the-perfect-hash",
      "title": "THE PERFECT HASH",
      "kicker": "zero collisions, O(n) space, one probe per lookup",
      "accent": "#70a860",
      "blurb": "perfect hashing (the FKS scheme) in the 5-window house format — store a fixed set of n keys with zero collisions and O(n) space using two levels: a top hash spreads keys into n buckets, and each bucket of b keys gets a secondary table of size b^2 with a collision-free hash; Sigma b^2 is O(n) in expectation. Every lookup is a single probe. It builds static dictionaries (keywords, Unicode tables) with guaranteed constant-time lookup. Verified live: over 200 random key sets every key resolves to a unique slot (exact for members, rejecting non-members) and total space stays O(n). See the two levels in 1D, buckets and b^2 secondaries in 2D, and the choose-hashes-with-no-collisions inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "61109d6fe83995dc0e042b4aeba7bf1875f0b9129f2a617b9c632080c7c4f752"
    },
    {
      "slug": "the-lz77",
      "title": "THE LZ77",
      "kicker": "compress by pointing backward into your own past",
      "accent": "#a878c0",
      "blurb": "LZ77 in the 5-window house format — compress by pointing backward: when upcoming text has already appeared within a sliding window of the recent past, emit a (distance, length) reference to that earlier copy plus the next new character, so the file describes itself in terms of its own history. It is the core of gzip, PNG, and ZIP (LZ77 + Huffman = DEFLATE). Verified live: over 300 random strings decompress(compress(s)) reproduces s exactly, and repetitive text collapses to few tokens (abracadabraabracadabra -> 9 tokens). See a back-reference in 1D, tokenized text in 2D, and the point-back-not-restore inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "69af2513b075de12a114495cd553ff15dd336d3a61c9dc6c252080dc6db30754"
    },
    {
      "slug": "the-held-karp",
      "title": "THE HELD-KARP",
      "kicker": "exact TSP by bitmask DP — n! tours in 2^n states",
      "accent": "#c05868",
      "blurb": "the Held-Karp algorithm in the 5-window house format — solve the travelling salesman problem exactly by dynamic programming: dp[set][city] is the cheapest way to start at the origin, visit that set, and end at that city; since the future depends only on which cities remain and where you are, 2^n*n states replace n! tours. It is the founding example of dynamic programming. Verified live: over 80 random graphs (n=3..7) the Held-Karp optimal tour equals the brute-force minimum over all permutations. See the state-merge in 1D, an optimal tour in 2D, and the exponential-collapses-to-DP inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "bf880fa43767cefb4c31d30d6c6706174f999bde8eb558fe550495e9f326ef20"
    },
    {
      "slug": "the-conjugate-gradient",
      "title": "THE CONJUGATE GRADIENT",
      "kicker": "solve SPD systems in n steps via A-orthogonal directions",
      "accent": "#c0a048",
      "blurb": "the conjugate gradient method in the 5-window house format — solve a symmetric positive-definite system Ax=b (minimize the quadratic bowl) by choosing A-orthogonal (conjugate) search directions, so each step's progress is never undone; in exact arithmetic it reaches the exact solution in at most n steps using only matrix-vector products. It is the workhorse for huge sparse systems. Verified live: over 200 random SPD systems it reaches the solution within n steps (residual ~1e-15) and matches a direct Gaussian solve. See conjugate vs zig-zag in 1D, the bowl path in 2D, and the A-orthogonality inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "1f23a40d829d1cd552a3b2a42ca4b9fbee448c447a850f59a2e524dd1eaa1f51"
    },
    {
      "slug": "the-pagerank",
      "title": "THE PAGERANK",
      "kicker": "importance as the stationary distribution of a random surfer",
      "accent": "#58a0b0",
      "blurb": "PageRank in the 5-window house format — rank nodes by importance defined recursively (a page is important if important pages link to it): the ranking is the stationary distribution of a random surfer who follows links with probability d and teleports otherwise, i.e. the dominant eigenvector of the Google matrix, found by power iteration. It was the original engine of Google search. Verified live: over 200 random graphs the PageRank vector sums to 1, is a fixed point (M*pi=pi), and converges to the same vector regardless of the starting distribution. See rank flow in 1D, node sizes in 2D, and the reputation-as-fixed-point inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "24125d127158e19055864e43582bf035ca8d10874d7394313243fb30b51ca89c"
    },
    {
      "slug": "the-clenshaw",
      "title": "THE CLENSHAW",
      "kicker": "evaluate a Chebyshev series by a stable backward recurrence",
      "accent": "#70a860",
      "blurb": "Clenshaw's algorithm in the 5-window house format — evaluate a sum of orthogonal polynomials (Sigma c_k T_k(x), the Chebyshev series) without building the polynomials, by running their three-term recurrence backward from the highest degree, carrying two running values. It is Horner's method for Chebyshev series, and the backward direction is numerically stable. Verified live: over 400 random coefficient sets and points in [-1,1] Clenshaw's result equals the direct term-by-term Chebyshev sum to ~1e-15. See the backward sweep in 1D, the series plotted in 2D, and the stable-backward-recurrence inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "42dd5d4390d616ae6fae40ed7249f7bc9b4663ba6756d559592a37e93e3a9a03"
    },
    {
      "slug": "the-bellman-ford",
      "title": "THE BELLMAN-FORD",
      "kicker": "shortest paths with negative edges — and the impossible loop",
      "accent": "#a878c0",
      "blurb": "the Bellman-Ford algorithm in the 5-window house format — find shortest paths from a source even with negative edge weights (which Dijkstra cannot) by relaxing every edge n-1 times; that many passes always suffice. Then one extra pass is the tell: if any edge can still be relaxed, a negative cycle is reachable and shortest paths are undefined. It underlies distance-vector routing and arbitrage detection. Verified live: on 200 non-negative graphs its distances match Floyd-Warshall, and on graphs with a reachable negative cycle it detects the cycle every time. See a relaxation in 1D, distances + neg-cycle flag in 2D, and the extra-pass-detects-the-impossible-loop inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "a0869000ac88b90558512d24baee91c0c03386b0107177a36de16e1cdba44c96"
    },
    {
      "slug": "the-fractional-cascading",
      "title": "THE FRACTIONAL CASCADING",
      "kicker": "search many sorted lists with one search plus bridges",
      "accent": "#58a0b8",
      "blurb": "fractional cascading in the 5-window house format — answer the same query against many sorted lists with a single binary search: weave a fraction of each list into the previous one and add bridge pointers, so once you locate the query in the first list, every other list's answer is a constant-time hop away, turning k searches of O(log n) into O(log n + k). It is the classic iterated-search speedup in computational geometry. Verified live: over 300 random setups of k sorted lists, the successor it reports in each list equals an independent binary search. See promoted elements + bridges in 1D, a multi-list query in 2D, and the one-search-k-handoffs inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "bacbac1a611356f6249eab5befddf9d7fa08bb92cf063fe7e43f2fb42fc6031e"
    },
    {
      "slug": "the-dinic",
      "title": "THE DINIC",
      "kicker": "max flow by leveled blocking flows — max flow = min cut",
      "accent": "#c05868",
      "blurb": "Dinic's algorithm in the 5-window house format — compute maximum flow by organizing the graph into levels via BFS and pushing a blocking flow that saturates many shortest paths at once; only O(V) phases are needed, giving O(V^2 E). By max-flow-min-cut, the value equals the minimum cut, the network's true bottleneck. Verified live: over 60 random capacitated graphs Dinic's max flow equals the brute-force minimum s-t cut. See the level graph in 1D, flow vs min cut in 2D, and the leveled-blocking-flow inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "3626cd558228a9585a763eb34684dec74f995bd815150b5b2099803d243be65c"
    },
    {
      "slug": "the-cyk",
      "title": "THE CYK",
      "kicker": "context-free recognition, bottom-up in O(n^3)",
      "accent": "#70a860",
      "blurb": "the CYK algorithm in the 5-window house format — decide whether a string is in a context-free language, bottom-up: with the grammar in Chomsky normal form (rules A->BC or A->terminal), fill a table of which nonterminals generate each substring, combining small spans into larger ones, so an exponential derivation search becomes an O(n^3) dynamic program. It is a foundation of parsing. Verified live: with a balanced-parentheses grammar in CNF, CYK accepts a non-empty string iff it is balanced, matching an independent balance check over 400 random bracket strings. See a binary join in 1D, the span table in 2D, and the bottom-up-from-spans inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "cf7662587d9ca52a4719f008ee32b3856181bb4f6c4cc488c9ed5565ab8bcd37"
    },
    {
      "slug": "the-hirschberg",
      "title": "THE HIRSCHBERG",
      "kicker": "optimal alignment in linear space via midpoints",
      "accent": "#c0a048",
      "blurb": "Hirschberg's algorithm in the 5-window house format — compute an optimal global alignment (same result as Needleman-Wunsch) in linear space instead of O(nm): the optimal path must cross the middle column somewhere, found from two linear-space score sweeps (forward to the middle, backward from the end), then recurse on the halves. It makes genome-length alignment feasible in memory. Verified live: over 300 random pairs Hirschberg's alignment scores identically to Needleman-Wunsch and de-gaps back to the originals. See the midpoint crossing in 1D, an alignment in 2D, and the midpoint-divide-and-conquer inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "fd1bae09e852c945a158b543b9fc6a2ba78c3f0e0cd13ade2d10b8da984eac6a"
    },
    {
      "slug": "the-kasai",
      "title": "THE KASAI",
      "kicker": "the LCP array in linear time by reusing the last overlap",
      "accent": "#a878c0",
      "blurb": "Kasai's algorithm in the 5-window house format — compute the LCP array (longest common prefix between adjacent suffixes in a suffix array) in linear time: process suffixes in TEXT order and reuse the previous answer, because dropping the first character shortens a suffix's LCP with its neighbor by at most one, so a running length falls by <=1 per step and rises at most n times total. The LCP array powers substring search and longest-repeated-substring. Verified live: over 300 random strings Kasai's O(n) LCP array equals a brute-force pairwise-prefix computation. See the running length in 1D, sorted suffixes + LCP in 2D, and the reuse-the-last-overlap inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "9440f8f6afd9090a66f690d4e65a3f476ad403b7f1ffbba61fa76f59dfa09109"
    },
    {
      "slug": "the-gaussian-quadrature",
      "title": "THE GAUSSIAN QUADRATURE",
      "kicker": "n sample points integrate degree 2n-1 exactly",
      "accent": "#c0a048",
      "blurb": "Gaussian quadrature in the 5-window house format — approximate an integral by a weighted sum at cleverly-chosen points: n nodes at the roots of the Legendre polynomial, with matching weights, integrate every polynomial up to degree 2n-1 EXACTLY, twice what a fixed grid of n points could. The placement, not the count, buys the accuracy. It is the backbone of numerical integration. Verified live: for n=2..5 the n-point Gauss-Legendre rule reproduces the exact integral of random polynomials of degree <=2n-1 and is not exact at degree 2n. See the Legendre nodes in 1D, exact integration in 2D, and the optimal-placement inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "45fdcfeb50088e90c0823cfb80bb9a29da4c951a2ad5574761e805eff26f88f8"
    },
    {
      "slug": "the-boruvka",
      "title": "THE BORUVKA",
      "kicker": "the minimum spanning tree, built by parallel merges",
      "accent": "#58a0b0",
      "blurb": "Boruvka's algorithm in the 5-window house format — build a minimum spanning tree in parallel: every component simultaneously finds its cheapest outgoing edge, all are added at once, and components merge; each round at least halves the component count, finishing in O(log V) rounds. It is the oldest MST algorithm (1926) and the most naturally parallel. Verified live: over 200 random connected weighted graphs Boruvka's MST weight equals Kruskal's. See a parallel round in 1D, the MST on a graph in 2D, and the parallel-merge inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "c18a4b208a36ebcd322486f0a1d3f8227eec8151aa7d163e8ca00a33225f5cec"
    },
    {
      "slug": "the-bloom-filter",
      "title": "THE BLOOM FILTER",
      "kicker": "probabilistic membership with one-sided error",
      "accent": "#70a860",
      "blurb": "the Bloom filter in the 5-window house format — test set membership with a bit array and k hash functions in tiny memory, without storing the elements: add sets k bits, a query passes iff all k are 1; any 0 means definitely absent, all 1 means probably present. The only error is a false POSITIVE, never a false negative. It is everywhere: databases, caches, spell-checkers, crypto clients. Verified live: over 200 filters every inserted element queries positive (zero false negatives, one-sided error); the false-positive rate is measured live against the ideal (1-e^-kn/m)^k. See the k-bit set in 1D, insert+query in 2D, and the one-sided-error inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "3d1a055b05055ecc76076eee9de28e66660e3e39a2a2d31782df31fa3ef31cd6"
    },
    {
      "slug": "the-halley",
      "title": "THE HALLEY",
      "kicker": "cubic-convergence root finding via the second derivative",
      "accent": "#c05868",
      "blurb": "Halley's method in the 5-window house format — find a root even faster than Newton by using the second derivative (curvature): the step x - 2ff'/(2f'^2 - ff'') fits a better local model, so the error cubes each iteration instead of squaring, roughly 3x the correct digits per step versus Newton's 2x. Verified live: over 200 cases Halley converges to the true cube root and reaches tolerance in no more iterations than Newton (usually fewer) - e.g. cbrt(50) in 3 Halley steps vs 4 Newton. See tangent vs curve in 1D, iterates in 2D, and the curvature-triples-the-digits inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "10fc60dc8427d573babe02da9d6062f187f9842625b486e68021a4c1610db9ad"
    },
    {
      "slug": "the-bluestein",
      "title": "THE BLUESTEIN",
      "kicker": "the DFT of any length via a chirp convolution",
      "accent": "#a878c0",
      "blurb": "Bluestein's algorithm in the 5-window house format — compute the DFT of ANY length N (not just a power of two) by turning it into a convolution with a chirp: the identity kn = (k^2+n^2-(k-n)^2)/2 makes the transform a convolution of the chirp-premultiplied signal with a chirp kernel, which can be padded to a power of two and done by FFT, so a prime-length DFT runs at FFT speed. Verified live: for arbitrary lengths (5,7,11,13 and non-powers 6,9,15) Bluestein's chirp transform equals the direct DFT to ~1e-14. See the chirp in 1D, an arbitrary-N spectrum in 2D, and the chirp-frees-the-length inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "239dc4982e29a7fb142490445295c5c3d3ac94b07a1ccf7a72d9bd4aa11c59c1"
    },
    {
      "slug": "the-ntt",
      "title": "THE NTT",
      "kicker": "the FFT over a finite field — exact, no rounding",
      "accent": "#c0a048",
      "blurb": "the number-theoretic transform in the 5-window house format — the FFT done over a finite field (integers mod a prime with a root of unity of the right order, here 998244353, generator 3): the same butterfly structure computes exact convolutions of integer sequences with NO floating-point error. Transform, multiply pointwise, inverse-transform, and the product is exact. It is how huge polynomials and integers are multiplied exactly. Verified live: over 200 random integer polynomial pairs the NTT convolution equals the exact naive convolution. See field butterflies in 1D, exact product in 2D, and the transform-in-a-finite-field inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "9ecdf78b6b61a8ffce56103ccc30a7a5b8e3c92e79e39db2decb97629517fa41"
    },
    {
      "slug": "the-runge-kutta",
      "title": "THE RUNGE-KUTTA",
      "kicker": "fourth-order ODE steps by four slope samples",
      "accent": "#58a0b0",
      "blurb": "the Runge-Kutta method (RK4) in the 5-window house format — advance a differential equation one step by sampling the slope four times (start, two midpoints, end) and taking a weighted average (1,2,2,1)/6; the sampling errors cancel to fourth order, so halving the step cuts the error ~16x. One clever RK4 step is as accurate as thousands of Euler steps. It is the default workhorse for simulating physical systems. Verified live: RK4 solves y'=y to reproduce e to ~1e-5, its error shrinks ~16x when the step halves (fourth order), and y'=cos t reproduces sin t. See the four slopes in 1D, RK4 vs Euler in 2D, and the four-slope-errors-cancel inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "41f5fd5d3127aa5212b2fe2eb1c77bb4a9f658fef2c2e6327ac5780806ee1d8f"
    },
    {
      "slug": "the-ear-clipping",
      "title": "THE EAR CLIPPING",
      "kicker": "triangulate a polygon by snipping one ear at a time",
      "accent": "#70a860",
      "blurb": "ear clipping in the 5-window house format — triangulate a simple polygon by repeatedly snipping an ear (a convex corner whose triangle contains no other vertex); each snip removes one triangle and one vertex until a triangle remains. The Two Ears Theorem guarantees an ear always exists, so it never gets stuck and yields exactly n-2 triangles. It is the standard way to turn a polygon into renderable triangles. Verified live: for 300 convex polygons and a set of non-convex reflex test shapes, ear clipping produces n-2 triangles whose areas sum to the polygon. See an ear in 1D, a triangulation in 2D, and the snip-one-safe-corner inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "227254d5cd348597fabbfec49b9586f1a8d5ae0232cde0d03d96afdb9dd22d20"
    },
    {
      "slug": "the-binary-lifting",
      "title": "THE BINARY LIFTING",
      "kicker": "ancestor and LCA queries in log time by doubling",
      "accent": "#c05868",
      "blurb": "binary lifting in the 5-window house format — jump to any ancestor in a tree in logarithmic time: precompute each node's 2^k-th ancestor for every k, then a jump of d steps follows the binary digits of d (O(log depth) hops instead of d). The same table answers lowest-common-ancestor queries: level the two nodes, then jump both up in decreasing powers of two until they meet. It is the standard ancestor/LCA tool. Verified live: over 200 random trees the binary-lifting LCA equals a brute parent-walk for every query. See a binary jump in 1D, an LCA on a tree in 2D, and the leap-in-powers-of-two inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "bfe565cf6b8fe95806aaa735ca2f9196d6ea8adf8ff4bfa3d6f19798052097ae"
    },
    {
      "slug": "the-eertree",
      "title": "THE EERTREE",
      "kicker": "every distinct palindrome in a linear-size tree",
      "accent": "#a878c0",
      "blurb": "the eertree (palindromic tree) in the 5-window house format — an automaton holding every distinct palindromic substring of a string; remarkably a length-n string has at most n distinct palindromic substrings, so the structure has <=n+2 nodes. Each palindrome grows from a shorter one by adding a matching character at both ends, built online one character at a time. It counts palindromic substrings and finds the longest in linear space. Verified live: over 500 random strings the eertree's node count equals a brute count of distinct palindromic substrings. See the nesting in 1D, the count vs brute in 2D, and the palindromes-are-a-tree inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "b83a032cf6d7025b61b2e3baec00fe3e3c7ffe363562806c48689beb17bf61fc"
    },
    {
      "slug": "the-linear-sieve",
      "title": "THE LINEAR SIEVE",
      "kicker": "primes in O(n) — each composite struck once, by its least prime",
      "accent": "#c0a048",
      "blurb": "the linear sieve (Euler's sieve) in the 5-window house format — list primes up to n in true O(n), strictly better than Eratosthenes which crosses out numbers many times, by marking each composite EXACTLY ONCE by its smallest prime factor; as a bonus it computes the smallest-prime-factor of every number for instant factorization. Verified live: up to 2000 the linear sieve's prime list equals Eratosthenes' and its stored smallest-prime-factor matches the true one for every number. See the mark-once rule in 1D, numbers by smallest prime factor in 2D, and the each-composite-struck-once inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "28e18369bbbde43936b35263719d09979f82121f5c3a78f778cab3479d558443"
    },
    {
      "slug": "the-zeta-transform",
      "title": "THE ZETA TRANSFORM",
      "kicker": "all subset-sums at once, invertible by Mobius",
      "accent": "#58a0b0",
      "blurb": "the zeta transform (sum-over-subsets DP) in the 5-window house format — compute for every set S at once the sum of f over all subsets of S (F[S]=Sigma_{T subset of S} f[T]), in n*2^n instead of 3^n, by sweeping one bit at a time; its exact inverse is the Mobius transform (subtract instead of add), inclusion-exclusion made fast and invertible. Verified live: over 300 random functions the SOS transform equals the brute subset-sum for every S, and the Mobius transform inverts it exactly. See the bitwise sweep in 1D, transform vs brute in 2D, and the invertible-inclusion-exclusion inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "1c698e73b29b8b8a5d3a40091c8cfcdb67d38347562c2cd92113dbc2afae38e1"
    },
    {
      "slug": "the-verlet",
      "title": "THE VERLET",
      "kicker": "a symplectic integrator — energy bounded for millions of steps",
      "accent": "#70a860",
      "blurb": "Verlet integration in the 5-window house format — advance a physical system time-symmetrically so it is symplectic (preserves phase-space area), keeping total energy BOUNDED for millions of steps where forward Euler pumps energy in and the orbit explodes. It updates position from the average of old and new force. It is the integrator behind molecular dynamics and game physics. Verified live: for the oscillator x''=-x over 20000 steps Verlet's relative energy drift stays under 0.001 while forward Euler's energy grows astronomically. See the symmetric step in 1D, phase-space orbits in 2D, and the symmetry-conserves-energy inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "bb7577f41dec31b9a420ebbd787fcdbbeb449892227fc949a203581e6bb3c9d2"
    },
    {
      "slug": "the-separating-axis",
      "title": "THE SEPARATING AXIS",
      "kicker": "convex collision by looking for one separating line",
      "accent": "#c05868",
      "blurb": "the separating axis theorem in the 5-window house format — decide whether two convex shapes overlap: they are disjoint iff there exists a line (axis) onto which their projections do not overlap, and you only need to test each shape's own edge normals as candidate axes; if projections overlap on all of them, they collide. It is the standard fast 2D collision test in game physics. Verified live: over 500 random convex-polygon pairs SAT's verdict matches an independent overlap oracle (a vertex inside the other, or crossing edges) every time. See a projection gap in 1D, a collide/separate verdict in 2D, and the find-the-gap inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "950500b39cb2bcaf6e10720a526ff81305226de22c63bc2a134865ca9eaa1aa2"
    },
    {
      "slug": "the-dilworth",
      "title": "THE DILWORTH",
      "kicker": "min chains to cover a poset = max antichain",
      "accent": "#a878c0",
      "blurb": "Dilworth's theorem in the 5-window house format — a min-max duality on a partial order: the minimum number of chains (comparable sequences) needed to cover all elements equals the maximum antichain (largest set of pairwise-incomparable elements), so a covering optimum is read off a packing optimum; the min chain cover is found by bipartite matching (min chains = n - max matching). It underlies scheduling bounds and Erdos-Szekeres. Verified live: over 300 random partial orders the min chain cover (via matching) equals a brute-force maximum antichain. See chains vs antichains in 1D, a Hasse diagram in 2D, and the covering-equals-packing inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "1e76a81fe1f32130dc8069daead7ddc736cfa4e8393454e1f7b57739616d7bc4"
    },
    {
      "slug": "the-alpha-beta",
      "title": "THE ALPHA-BETA",
      "kicker": "minimax value, pruning the provably-irrelevant branches",
      "accent": "#c05868",
      "blurb": "alpha-beta pruning in the 5-window house format — compute the exact minimax value of a game tree while skipping branches that cannot change the result: carry bounds alpha (best assured to the maximizer) and beta (best assured to the minimizer), and cut off a branch the moment it is proven worse than one already found. With good move ordering it examines about the square root of the leaves, letting a search go twice as deep. It is the engine inside classical chess and checkers programs. Verified live: over 300 random game trees alpha-beta returns the same value as full minimax while visiting no more nodes. See a cutoff in 1D, a searched tree in 2D, and the prune-the-irrelevant inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "49e6b24edf94a200aec67b9051e2a34345868aee1bb5596627bb5c7756e5769a"
    },
    {
      "slug": "the-sparse-table",
      "title": "THE SPARSE TABLE",
      "kicker": "O(1) range-min from two overlapping precomputed blocks",
      "accent": "#58a0b0",
      "blurb": "the sparse table in the 5-window house format — answer range-minimum queries in O(1) after O(n log n) preprocessing by storing the minimum of every power-of-two block; any range is covered by just two overlapping blocks, and the overlap is harmless because min is idempotent (min(x,x)=x). It is the classic static range-min/max structure. Verified live: over 200 random arrays the sparse-table range-minimum equals a brute scan for every query. See two-block coverage in 1D, a range query in 2D, and the idempotence-lets-blocks-overlap inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "9ee8be8016d3cb1af238f5500411f23a8a5444ff4f934fe08661f549f5a5ce6d"
    },
    {
      "slug": "the-johnson-apsp",
      "title": "THE JOHNSON APSP",
      "kicker": "all-pairs shortest paths with negative edges, via reweighting",
      "accent": "#70a860",
      "blurb": "Johnson's algorithm in the 5-window house format — find all-pairs shortest paths even with negative edge weights (which Dijkstra alone cannot): run one Bellman-Ford to get a node potential h(v), reweight every edge to w'(u,v)=w(u,v)+h(u)-h(v) (now non-negative, same shortest paths), then run fast Dijkstra from every source and undo the shift. Verified live: over 100 random graphs with negative edges (no negative cycle) Johnson's distances match Floyd-Warshall exactly. See the reweighting telescoping in 1D, all-pairs distances in 2D, and the reweight-away-the-negatives inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "3b97c69f0dcb8355168c7c76f7fa7457e62e8138fbd2ec9703b679d9aee9a83f"
    },
    {
      "slug": "the-kd-tree",
      "title": "THE KD-TREE",
      "kicker": "nearest-neighbor search by descend-and-prune",
      "accent": "#c0a048",
      "blurb": "the k-d tree in the 5-window house format — organize points in space by splitting alternately along each axis (x, then y, then x...), so a nearest-neighbor query descends to the query's cell and only backtracks into sibling regions that could still hold something closer; most of the space is pruned by the current best distance, so a query is typically O(log n). It is the standard structure for nearest-neighbor and range search. Verified live: over 200 random point sets the k-d tree's nearest neighbor equals a brute scan for every query. See alternating splits in 1D, the partition + query in 2D, and the descend-and-prune inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "f17efcffd07908bebec6269732e759047b967cab768fe8f7c34f014cf590d249"
    },
    {
      "slug": "the-barnes-hut",
      "title": "THE BARNES-HUT",
      "kicker": "N-body forces in O(n log n) — a faraway crowd is one point",
      "accent": "#a878c0",
      "blurb": "the Barnes-Hut algorithm in the 5-window house format — simulate gravity among n bodies in O(n log n) instead of O(n^2) by approximating a distant cluster with its single center of mass: build a quadtree, and if a cell's size/distance is below a threshold theta, treat the whole cell as one body, else recurse. Accuracy is a dial: theta->0 recovers the exact sum. It is the foundation of large-scale astrophysical N-body simulation. Verified live: at theta=0.3 the Barnes-Hut force is within a few percent of the direct O(n^2) sum, and the error shrinks as theta decreases. See the theta criterion in 1D, a quadtree of bodies in 2D, and the faraway-crowd-is-one-point inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "1533c968a636f82463a0f67c36500c0744382ccb86a4d7310c7a831db4a1f2f2"
    },
    {
      "slug": "the-romberg",
      "title": "THE ROMBERG",
      "kicker": "integration accelerated by cancelling the error terms",
      "accent": "#c0a048",
      "blurb": "Romberg integration in the 5-window house format — take the trapezoid rule and converge ferociously fast: the trapezoid error is a known series in h^2, so combine estimates at h and h/2 to cancel the leading error term (Richardson extrapolation), then the next, in a triangular table that leaps from O(h^2) to O(h^2k) after k refinements. Verified live: with 7 levels Romberg matches pi (via 4/(1+x^2)), e-1, integral of sin, and integral of x^4 to under 1e-8, where a 64-panel trapezoid is still off by ~1e-5. See the extrapolation table in 1D, a converging corner in 2D, and the cancel-the-error-terms inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "98f475ad82269260a8b28a969f09b9d0478fc93298ff3245fdb55d030b6013a7"
    },
    {
      "slug": "the-pohlig-hellman",
      "title": "THE POHLIG-HELLMAN",
      "kicker": "discrete log broken by smooth order + CRT",
      "accent": "#c05868",
      "blurb": "the Pohlig-Hellman algorithm in the 5-window house format — solve the discrete logarithm g^x = h (mod p) quickly whenever the group order p-1 is smooth (factors into small primes): solve the log separately in each prime-power subgroup (where it is tiny) and stitch the pieces with the Chinese Remainder Theorem. So a hard log in a huge group becomes many easy logs in small ones. It is why cryptographic groups need a large prime factor in their order. Verified live: over 200 cases with smooth-order primes the recovered exponent satisfies g^x = h (mod p). See the order factoring in 1D, subgroup logs recombined in 2D, and the logs-in-small-subgroups inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "6432d877a69f33086c901ce483d99abceb53383297337935755a8282c55bc7b9"
    },
    {
      "slug": "the-householder-qr",
      "title": "THE HOUSEHOLDER QR",
      "kicker": "QR by reflections — a mirror per column, stably",
      "accent": "#58a0b0",
      "blurb": "Householder QR in the 5-window house format — factor a matrix A into an orthonormal Q and an upper-triangular R using a sequence of reflections: each Householder reflection is a mirror that flips one column onto a coordinate axis, zeroing everything below the diagonal in a single stroke, and is far more numerically stable than Gram-Schmidt's repeated subtractions. It is the workhorse behind least-squares and the QR eigenvalue algorithm. Verified live: over 300 random matrices Q*R reconstructs A to ~1e-15, R is upper-triangular, and Q^T*Q equals the identity. See a reflection in 1D, A=QR in 2D, and the mirror-per-column inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "7e0b20a77274cf2d9671ee509a3e5ac567f5862c8b41012c7b50054e60d6cf1d"
    },
    {
      "slug": "the-power-iteration",
      "title": "THE POWER ITERATION",
      "kicker": "the dominant eigenvector by repeated multiplication",
      "accent": "#70a860",
      "blurb": "power iteration in the 5-window house format — find the dominant eigenvector of a matrix by repeated multiplication: start with any vector, multiply by the matrix, normalize, repeat; any vector is a mix of eigenvectors, and each multiply amplifies each component by its eigenvalue, so the largest-magnitude eigenvalue's direction takes over. The Rayleigh quotient then reads off the eigenvalue. It is the seed of PageRank and the QR eigenvalue method. Verified live: over 200 random symmetric matrices it converges to a genuine eigenpair (residual < 1e-6) and the dominant |lambda| is the same from any start. See eigen-components scaling in 1D, the iterate rotating in 2D, and the multiplication-is-a-filter inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "491a0882dbcea8bba2093025efead2b3ad9371c7e00feb1de9b26a8cf2d7a1d5"
    },
    {
      "slug": "the-brent-cycle",
      "title": "THE BRENT CYCLE",
      "kicker": "cycle detection in O(1) memory, fewer evals than Floyd",
      "accent": "#a878c0",
      "blurb": "Brent's cycle detection in the 5-window house format — find the loop in a sequence x, f(x), f(f(x)), ... using constant memory: keep one saved value and compare the moving value to it at exponentially-spaced checkpoints (powers of two); when the value repeats, the gap reveals the cycle length lambda, and a short second scan finds where the cycle starts (mu). It uses fewer function evaluations than tortoise-and-hare, and is the cycle-finder inside Pollard's rho. Verified live: over 300 random functional graphs Brent's (lambda, mu) equals a brute record-every-value computation. See doubling checkpoints in 1D, a rho-shaped graph in 2D, and the one-teleporting-checkpoint inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "3a0425b086c153283e34cfbf1425159775e55a540e26f44e8452a789f1d9dc4e"
    },
    {
      "slug": "the-pollard-p1",
      "title": "THE POLLARD P-1",
      "kicker": "factoring surfaced by a gcd when p-1 is smooth",
      "accent": "#c05868",
      "blurb": "Pollard's p-1 algorithm in the 5-window house format — factor a composite n when a prime factor p has a smooth p-1: compute a^(k!) mod n for growing k; by Fermat's little theorem, once k! is a multiple of p-1, a^(k!)=1 (mod p), so a^(k!)-1 is a multiple of p and gcd(a^(k!)-1, n) reveals p, without ever knowing p. It is why RSA primes avoid a smooth p-1. Verified live: for 90 constructed n=p*q where p-1 is 15-smooth (q-1 not), the algorithm returns a nontrivial factor dividing n. See the running gcd in 1D, a factored n in 2D, and the gcd-surfaces-the-smooth-prime inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "167e67738dd26c6cf1ce79bc10fdbdc91af7e9efa238c2855314410f00fa01d9"
    },
    {
      "slug": "the-pairing-heap",
      "title": "THE PAIRING HEAP",
      "kicker": "a lazy, self-adjusting priority queue",
      "accent": "#58a0b0",
      "blurb": "the pairing heap in the 5-window house format — a priority queue that stays fast by being lazy: merge links the larger root under the smaller (one comparison, O(1)), and the real work is deferred to delete-min, which does a two-pass pairing of the orphaned children; decrease-key cuts a node and re-merges it. In practice it rivals the Fibonacci heap while being far simpler. Verified live: over 300 sequences repeated delete-min yields the keys in sorted order, and 200 decrease-key operations produce the correct extraction order. See a merge in 1D, an interactive heap in 2D, and the link-lazily-pair-on-extract inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "642eb53e0787a7caabff0c299c1f97c6d305928723a5f38e9998590293c15484"
    },
    {
      "slug": "the-gauss-seidel",
      "title": "THE GAUSS-SEIDEL",
      "kicker": "iterative linear solve with immediate feedback",
      "accent": "#70a860",
      "blurb": "Gauss-Seidel in the 5-window house format — solve Ax=b iteratively by sweeping the variables, setting each from the current best estimate of the others, and using each fresh value IMMEDIATELY within the same sweep (unlike Jacobi); for a diagonally-dominant system this relaxation converges to the exact solution, faster than Jacobi. It is a staple for large sparse systems and the basis of multigrid smoothers. Verified live: over 200 random diagonally-dominant systems Gauss-Seidel converges to a direct Gaussian solve to ~1e-15. See a sweep in 1D, the residual shrinking in 2D, and the relax-with-immediate-feedback inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "d0c1f47a08dd50c9bbfcedb75276c5d4e4c5d65520e75677364ed10056e703be"
    },
    {
      "slug": "the-delannoy",
      "title": "THE DELANNOY",
      "kicker": "king-path counting — Pascal with a third, diagonal term",
      "accent": "#c0a048",
      "blurb": "the Delannoy numbers in the 5-window house format — D(m,n) counts lattice paths from (0,0) to (m,n) using east (1,0), north (0,1), and the diagonal (1,1) (a king's moves); that extra diagonal makes the recurrence D(m,n)=D(m-1,n)+D(m,n-1)+D(m-1,n-1), Pascal-like with a third term. The central values D(n,n) are 1,3,13,63,321,1683,... Verified live: the recurrence equals a brute enumeration of all king-paths for m,n<=5, and the central Delannoy numbers match the known sequence. See the three predecessors in 1D, the Delannoy grid in 2D, and the three-predecessors inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "1d0c227235e5fed9cdad550566bf70dc4d1a0b9b65ef9f025eebb7f2095a87cc"
    },
    {
      "slug": "the-gjk",
      "title": "THE GJK",
      "kicker": "convex collision by asking if the origin is in A minus B",
      "accent": "#a878c0",
      "blurb": "the GJK algorithm in the 5-window house format — decide whether two convex shapes overlap by a reframing: they intersect iff the ORIGIN lies inside their Minkowski difference A-B; it never builds that difference, probing it with a support function and evolving a tiny simplex (point->edge->triangle) toward the origin, deciding in a few steps. It is the collision engine of physics and robotics libraries. Verified live: over 500 random convex-polygon pairs GJK's collide/separate verdict matches an independent overlap oracle (vertex inside the other, or crossing edges) every time. See a support probe in 1D, a collide/separate verdict in 2D, and the origin-in-the-Minkowski-difference inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "44957a5922ae9dc8ac54db7434850b9ca03a49670eccdd290da6a7bd35a56ef4"
    },
    {
      "slug": "the-push-relabel",
      "title": "THE PUSH-RELABEL",
      "kicker": "max flow by pushing excess downhill by height",
      "accent": "#c05868",
      "blurb": "the push-relabel algorithm in the 5-window house format — compute maximum flow by a local rule with no augmenting paths: maintain a preflow (nodes may hold excess) and a height label per node; push flow only downhill across an admissible edge (h[u]=h[v]+1), and relabel (lift) a stuck node with excess so it can drain; the excess settles at the sink and equals the minimum cut. It is often the fastest max-flow method in practice. Verified live: over 60 random capacitated graphs push-relabel's max flow equals the brute-force minimum s-t cut. See downhill pushes in 1D, flow vs min cut in 2D, and the flow-downhill-by-height inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "631eff0634fe708dc5e1c1a3e2b0ecbff3a952904c2618aa1e28e2fef8abee3c"
    },
    {
      "slug": "the-cycle-sort",
      "title": "THE CYCLE SORT",
      "kicker": "sorting with the minimum possible number of writes",
      "accent": "#58a0b0",
      "blurb": "cycle sort in the 5-window house format — sort an array with the minimum possible number of writes: a permutation decomposes into disjoint cycles, and cycle sort follows each cycle, placing every element directly into its final slot, so each out-of-place element is written exactly once; the total writes are provably minimal. It matters when writing is expensive, as on flash memory or EEPROM. Verified live: over 300 random permutations cycle sort produces the sorted array, and its write count equals the theoretical minimum computed from the permutation's cycle structure. See cycles in 1D, colored cycles in 2D, and the one-write-per-cycle inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "84c674f40714b8b3223aa16ca18365572e03d4e829819d2d6b98d5fc3a13e3df"
    },
    {
      "slug": "the-steffensen",
      "title": "THE STEFFENSEN",
      "kicker": "quadratic fixed-point convergence with no derivative",
      "accent": "#70a860",
      "blurb": "Steffensen's method in the 5-window house format — find a fixed point of g (a root of g(x)-x) with quadratic convergence (Newton's speed) but no derivative: from x compute x1=g(x), x2=g(x1), then apply Aitken's delta-squared extrapolation x - (x1-x)^2/(x2-2x1+x); three plain iterations folded into one accelerated step. Verified live: for cos x (the Dottie number 0.739085), a sqrt(2) map, and e^-x, Steffensen reaches the fixed point in a handful of steps, far fewer than plain fixed-point iteration. See the delta-squared jump in 1D, iterates in 2D, and the extrapolate-three-iterates inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "34950653352c84ddc399e0c62f373b9e3a1714421eebd8e930a103ceb2308d9b"
    },
    {
      "slug": "the-chebyshev",
      "title": "THE CHEBYSHEV",
      "kicker": "interpolate at clustered nodes to defeat Runge",
      "accent": "#c0a048",
      "blurb": "Chebyshev interpolation in the 5-window house format — sample a function not on an even grid but at the Chebyshev nodes (clustered toward the ends, the projected roots of the Chebyshev polynomials); this defeats Runge's phenomenon, where interpolating on an even grid diverges wildly at the edges as points are added while Chebyshev interpolation converges. It underlies spectral methods and high-accuracy approximation. Verified live: on Runge's function 1/(1+25x^2), the Chebyshev max error shrinks as nodes are added (~0.02 at n=21) while the equispaced error explodes (~60). See the semicircle projection in 1D, both interpolants in 2D, and the cluster-the-samples inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "46506fd06efd8236c3bcb9927f400b3b8dc9f37570822ad108fac4998be2793f"
    },
    {
      "slug": "the-schroder",
      "title": "THE SCHRODER",
      "kicker": "Catalan with a flat step — super-Catalan path counts",
      "accent": "#a878c0",
      "blurb": "the (large) Schröder numbers in the 5-window house format — count lattice paths from (0,0) to (2n,0) using up (1,1), down (1,-1), and FLAT (2,0) steps, never dipping below the axis; they are the Catalan numbers with a flat step allowed (a super-Catalan count) and satisfy a convolution recurrence. The sequence is 1,2,6,22,90,394,1806,... Verified live: the recurrence equals a brute enumeration of all such Schröder paths for n<=5, and the values match the known large-Schröder sequence 1,2,6,22,90,394. See a Schröder path in 1D, recurrence vs brute in 2D, and the split-at-first-return inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "29c48ec568068d6b43c136e07ea8eb8a51279f34f2240497e969fa59f10f9a11"
    },
    {
      "slug": "the-hopcroft",
      "title": "THE HOPCROFT",
      "kicker": "the minimal DFA by merging indistinguishable states",
      "accent": "#c05868",
      "blurb": "Hopcroft's algorithm in the 5-window house format — minimize a deterministic finite automaton to the smallest DFA recognizing the same language by partition refinement: split accepting from non-accepting states, then repeatedly split any group whose members transition into different groups, until stable; the final classes are the Myhill-Nerode equivalence classes. Splitting by the smaller half gives O(n log n). Verified live: over 200 random DFAs the minimized automaton accepts the same language (all strings up to length 6) and is truly minimal (every pair of states distinguishable). See the refinement in 1D, a minimized DFA in 2D, and the merge-the-indistinguishable inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "1d376f37f1ffd2b5bee76d0ccc989a338adc21620709ed8017a4c4f7af446c39"
    },
    {
      "slug": "the-polya",
      "title": "THE POLYA",
      "kicker": "counting up to symmetry by averaging fixed points",
      "accent": "#58a0b0",
      "blurb": "Polya enumeration in the 5-window house format — count distinct objects up to symmetry without listing them: for a necklace of n beads in k colors, Burnside's lemma says the number of distinct necklaces equals the AVERAGE number of colorings FIXED by each rotation, which works out to (1/n) Sum_{d|n} phi(d) k^(n/d). It is the counting engine behind chemical isomers, graph enumeration, and combinatorial design. Verified live: for all n<=8 and k<=3 the necklace formula equals a brute count of rotation orbits. See fixed-point averaging in 1D, a necklace count in 2D, and the average-the-fixed-points inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "efd3c43253b48a9aefe137376ef65324ceb805f1774aaefe9ebefdccf25a658c"
    },
    {
      "slug": "the-bk-tree",
      "title": "THE BK-TREE",
      "kicker": "fuzzy string search pruned by the triangle inequality",
      "accent": "#70a860",
      "blurb": "the BK-tree in the 5-window house format — index strings for fuzzy search (all words within edit distance k of a query) without comparing against every word: store each string as a child labelled by its edit distance to the parent, and query by the triangle inequality (a child at distance d can only hold matches within [d-k, d+k] of the query), pruning most branches. It is the classic structure behind spell-checkers. Verified live: over 200 random string sets the BK-tree's within-distance-k results exactly match a brute scan. See a pruned child in 1D, a fuzzy query in 2D, and the triangle-inequality-prune inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "6a12eef6a1569c51b8487134d04ca2909711456cc96039e60e298e09bd64f948"
    },
    {
      "slug": "the-descartes",
      "title": "THE DESCARTES",
      "kicker": "bound the positive roots by counting sign changes",
      "accent": "#c0a048",
      "blurb": "Descartes' rule of signs in the 5-window house format — read a bound on a polynomial's positive real roots straight off its coefficients: the number of positive roots is at most the number of sign changes in the coefficient sequence, and differs from it by an even number (x -> -x gives the negative-root bound). You learn about the roots before computing any. Verified live: over 300 polynomials built from known real roots, the true count of positive roots is always <= the sign-change count and has the same parity. See the sign-change walk in 1D, a polynomial in 2D, and the read-roots-off-the-signs inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "bc4812bd0047969a6e35a42e57e89d530d0fb832153f88ff85a53319ce536f31"
    },
    {
      "slug": "the-givens",
      "title": "THE GIVENS",
      "kicker": "QR by plane rotations — a rotation per entry",
      "accent": "#a878c0",
      "blurb": "Givens rotations in the 5-window house format — factor a matrix into orthonormal Q and upper-triangular R by zeroing below-diagonal entries one at a time: each Givens rotation is a 2x2 plane rotation that annihilates a single element while preserving lengths (orthogonal), touching only two rows, ideal for sparse matrices and incremental updates. It is the QR method of choice for sparse and streaming problems. Verified live: over 300 random matrices Q*R reconstructs A to ~1e-15, R is upper-triangular, and Q^T*Q is the identity. See a plane rotation in 1D, A=QR in 2D, and the rotation-per-entry inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "0cb2a2a9847061b38215a9f1be161d4197bfc995f1f670894aa96fd5855561b0"
    },
    {
      "slug": "the-tonelli-shanks",
      "title": "THE TONELLI-SHANKS",
      "kicker": "the square root modulo a prime",
      "accent": "#c0a048",
      "blurb": "the Tonelli-Shanks algorithm in the 5-window house format — compute a square root modulo a prime: given n and prime p, find r with r^2 = n (mod p) whenever one exists. It tests whether n is a quadratic residue via the Legendre symbol; if p = 3 (mod 4) the root is n^((p+1)/4), and otherwise a loop descends the 2-adic tower of p-1 using a known non-residue. It underpins elliptic-curve point decompression and Rabin cryptography. Verified live: over every prime below 2000 and every residue, the returned r satisfies r^2 = n, and null is returned exactly for non-residues. See residues vs squares in 1D, a modular root in 2D, and the invert-squaring inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "566e7bb153538af9617ace91a444dc9c013ba1177ed9cd66437a38da9c71ef39"
    },
    {
      "slug": "the-tarjan-scc",
      "title": "THE TARJAN SCC",
      "kicker": "every strongly connected component in one DFS",
      "accent": "#c05868",
      "blurb": "Tarjan's SCC algorithm in the 5-window house format — find the strongly connected components of a directed graph (maximal groups where every vertex reaches every other) in a single depth-first search. It tracks each vertex's discovery index and the lowest index reachable from its subtree (the low-link); when a vertex's low-link equals its own index it roots an SCC, and the component is popped off a stack. One pass, linear time. Verified live: over 400 random digraphs Tarjan's component partition equals a brute partition by mutual reachability. See the low-link idea in 1D, a colored SCC graph in 2D, and the one-DFS inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "f07c66fb4a51aa8e84c4af88d286a807e8f189d708ab0a7985d62e65890c36b0"
    },
    {
      "slug": "the-gray-code",
      "title": "THE GRAY CODE",
      "kicker": "count so only one bit flips per step",
      "accent": "#70a860",
      "blurb": "the reflected binary (Gray) code in the 5-window house format — order all 2^n binary strings so consecutive ones differ in exactly one bit, cyclically (last and first differ by one bit too). The i-th code is i XOR (i>>1). Because only one bit flips per step it eliminates the transient glitches of ordinary counters, which is why rotary encoders, Karnaugh maps, and error-tolerant ADCs use it. Verified live: for up to 12 bits every consecutive pair (including wrap-around) has Hamming distance exactly 1, and all 2^n codes are distinct. See the single-bit steps in 1D, the full code in 2D, and the walk-the-cube inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "acf739e65fb074ecc1cdc4e674899a27f08ac15f33924b6f867d7366f0b30e3e"
    },
    {
      "slug": "the-alias-method",
      "title": "THE ALIAS METHOD",
      "kicker": "O(1) weighted sampling by flattening the odds",
      "accent": "#d4a017",
      "blurb": "Vose's alias method in the 5-window house format — turn any weighted distribution over k outcomes into a table that samples in O(1) per draw (one uniform bucket pick plus one coin flip). It flattens the uneven probabilities into k equal-area buckets, each holding at most two outcomes: a primary and an alias. Building the table is O(k); after that every draw is constant-time no matter how skewed the weights. It is the standard engine behind fast weighted random selection — like a loot drop table. Verified live: over 500 random weight sets the assembled table reconstructs the exact input probabilities (error < 1e-9). See the pour-to-level idea in 1D, a flattened drop table in 2D, and the flatten-for-O(1) inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "fec86d961b77b10d24033690e620238df1278258079e7ea4ea98090998ac3634"
    },
    {
      "slug": "the-stern-brocot",
      "title": "THE STERN-BROCOT",
      "kicker": "every positive rational, once, in lowest terms",
      "accent": "#58a0b0",
      "blurb": "the Stern-Brocot tree in the 5-window house format — an infinite binary tree containing every positive rational exactly once, each already in lowest terms. Each node is the mediant (a+c)/(b+d) of the two fractions bracketing it; descending left or right narrows the interval, and the L/R path spells the fraction's continued-fraction expansion. It is at once a perfect enumeration of the rationals and an optimal search for the simplest fraction in an interval. Verified live: every node down to depth 11 is in lowest terms (gcd=1) and all are distinct, and every reduced p/q with p,q<=20 is found by binary search in the tree. See the mediant insertion in 1D, the tree + search in 2D, and the grow-mediants inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "89ee3d9b89a42eb6235192af00be6ab9a340f7d91c6ba82f3d085c8f04c5dfca"
    },
    {
      "slug": "the-walsh-hadamard",
      "title": "THE WALSH-HADAMARD",
      "kicker": "a Fourier-like transform from only +1 and -1",
      "accent": "#58a0b0",
      "blurb": "the Walsh-Hadamard transform in the 5-window house format — a Fourier-like transform built entirely from +1 and -1, no sines, no complex numbers, no rounding. Its matrix is recursively [[H,H],[H,-H]], every row orthogonal to every other; the fast version (FWHT) uses only additions and subtractions, and applying it twice returns N times the original, exactly, in integer arithmetic. It is the backbone of Hadamard codes, CDMA spread-spectrum, and Boolean-function analysis. Verified live: for sizes up to 256, FWHT applied twice equals N times the input bit-for-bit, and all Hadamard rows are mutually orthogonal (H.H^T = N.I). See the butterfly in 1D, a spectrum in 2D, and the sign-only-basis inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "6aa32e075bb2e75e8cf413e32094c149dc35877a5e1db0619cc1fd32429bc4fb"
    },
    {
      "slug": "the-hilbert-curve",
      "title": "THE HILBERT CURVE",
      "kicker": "one line that fills the plane, keeping neighbors near",
      "accent": "#d4a017",
      "blurb": "the Hilbert curve in the 5-window house format — a space-filling curve: a single continuous line that visits every cell of a 2^k x 2^k grid exactly once, and consecutive cells on the line are always grid-neighbors (one step apart). That locality means points close along the 1-D curve are usually close in 2-D, which is why databases and image formats use the Hilbert index. The map index<->(x,y) is a pure bit-twiddle with quadrant rotations. Verified live: for grids up to 64x64 the index<->(x,y) map is a bijection, and every pair of consecutive indices lands on cells at Manhattan distance 1. See the recursive U-shapes in 1D, the curve in 2D, and the locality-preserving inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "9a89fb1e879002edf059d8b16c7ebdf9893128edd12f38690f38086ee423172c"
    },
    {
      "slug": "the-aes-sbox",
      "title": "THE AES S-BOX",
      "kicker": "the cipher's non-linearity from one field inversion",
      "accent": "#a878c0",
      "blurb": "the AES S-box in the 5-window house format — the single non-linear step of the Advanced Encryption Standard. It maps each byte to another by two operations in the finite field GF(2^8): take the multiplicative inverse of the byte (0->0), then apply a fixed affine bit-mix. The inverse step is what gives AES its resistance to linear and differential cryptanalysis; every nonzero byte has a unique inverse b^-1 with b (x) b^-1 = 1. Verified live: b (x) b^-1 = 1 for all 255 nonzero bytes, the S-box is a bijection with S^-1 . S = identity over 256 bytes, and it matches the published AES values (00->63, 01->7c, 53->ed). See the byte->inverse->affine chain in 1D, the 16x16 table in 2D, and the field-inversion inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "5c189bc93a9e3d1e75b089417eb53f47ba6f96f33918d232bc503aba774f376c"
    },
    {
      "slug": "the-lambda-calculus",
      "title": "THE LAMBDA CALCULUS",
      "kicker": "numbers, and arithmetic, from pure functions",
      "accent": "#70a860",
      "blurb": "the lambda calculus in the 5-window house format — all of computation from a single idea: functions. No numbers are built in; a number is encoded as a function. The Church numeral n is 'apply f, n times': 0 = Lf.Lx.x, and SUCC wraps one more f. Addition, multiplication, and exponentiation are then just ways of composing these functions (MULT m n = Lf. m (n f), EXP m n = n m) — arithmetic falls out of function application alone. Verified live: encoding then decoding gives back 0..10; PLUS, MULT, EXP of Church numerals equal ordinary a+b, a*b, a^b; and SUCC(SUCC 0) = 2. See a numeral as repetition in 1D, composition in 2D, and the numbers-as-functions inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "894d86f33927a41ba97f890c9c9b180537b2177fbc537daf99386124cf25e9cf"
    },
    {
      "slug": "the-balanced-ternary",
      "title": "THE BALANCED TERNARY",
      "kicker": "base 3 with digits -1,0,+1 — no sign bit",
      "accent": "#c0a048",
      "blurb": "balanced ternary in the 5-window house format — base 3 with the digit set {-1,0,+1} (T,0,1) instead of {0,1,2}. Every integer, positive or negative, has a unique representation with no sign bit at all, because the negative digit carries the sign internally. Negating a number is just flipping every digit's sign; rounding to nearest is truncation; and it is the most efficient integer base by radix economy. Knuth called it 'perhaps the prettiest number system.' Verified live: every integer from -40 to 40 has a unique balanced-ternary string over {-1,0,1} that evaluates back exactly, and negation equals flipping every digit. See the pan-balance places in 1D, an encoding in 2D, and the signless inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "d2656a8a112db0bfe91b657c9010c071916d3405e7555be0f7929d98223dd423"
    },
    {
      "slug": "the-viterbi",
      "title": "THE VITERBI",
      "kicker": "the most likely message through the noise",
      "accent": "#58a0b0",
      "blurb": "the Viterbi algorithm in the 5-window house format — decode a convolutional code by finding the single most likely transmitted sequence given a noisy received one, not by trying all 2^L messages but by a dynamic program over a trellis of encoder states. At each step it keeps only the best surviving path into each state; a traceback reads off the maximum-likelihood message. It is the decoder in Wi-Fi, GSM, satellite links, and Voyager. Verified live: over 400 noisy trials the Viterbi path metric equals the brute-force minimum Hamming distance to any codeword (it truly finds the nearest), and it corrects a single bit error exactly. See the trellis in 1D, an encode/noise/decode in 2D, and the survivor-path inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "8e5df5231a174a4bdfd83e9ef638e3f0667d66992bd52a43edb966f7e4d491df"
    },
    {
      "slug": "the-combinadics",
      "title": "THE COMBINADICS",
      "kicker": "index any subset by a single number",
      "accent": "#d4a017",
      "blurb": "the combinatorial number system (combinadics) in the 5-window house format — give every k-element subset a unique integer index and back, a bijection between 0..C(n,k)-1 and the k-subsets of an n-set. The index of {c1>...>ck} is C(c1,k)+C(c2,k-1)+...+C(ck,1); unranking runs it backwards with a greedy binomial peel. It lets you store, shuffle, or address combinations by a single number — a subset odometer. Verified live: over all C(10,5)=252 subsets, rank of unrank is the identity, and every unrank yields a valid distinct 5-subset. See the binomial rank in 1D, an index-to-subset in 2D, and the single-index inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "b7ded2761f619df6cac660228835c18ca9419c82bfff978a446c8f9a15050b5d"
    },
    {
      "slug": "the-kahan",
      "title": "THE KAHAN SUM",
      "kicker": "sum a million floats without losing the crumbs",
      "accent": "#a878c0",
      "blurb": "Kahan compensated summation in the 5-window house format — add a long list of floating-point numbers while recovering the rounding error a naive running total silently discards. It carries a tiny compensation variable: each step computes what was lost to rounding and feeds it back into the next addition, so the sum stays accurate to nearly the last bit even when the naive total has drifted. Verified live: summing 0.1 one million times, naive float64 drifts by ~1e-6, while Kahan matches the accurately-rounded sum to ~0. See the compensation step in 1D, naive-vs-Kahan drift in 2D, and the feed-back-the-error inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "e0cb7ba258c38bce51f74fb906f7b2fc251eddf0139e5b05773a2b1f5b9b50f6"
    },
    {
      "slug": "the-residue-number-system",
      "title": "THE RESIDUE NUMBER SYSTEM",
      "kicker": "carry-free arithmetic in parallel modular lanes",
      "accent": "#c0a048",
      "blurb": "a residue number system (RNS) in the 5-window house format — represent an integer not by its digits but by its remainders modulo a set of coprime bases (n <-> (n mod 3, n mod 5, n mod 7)). By the Chinese Remainder Theorem every value from 0 to the product minus one has a unique such triple, and addition and multiplication work independently, in parallel, with no carries between channels. It is used for fast carry-free arithmetic in DSP and cryptographic hardware. Verified live: over 0..104 the triples are unique and CRT reconstructs n exactly; componentwise + and x match ordinary arithmetic mod 105. See the three lanes in 1D, a channel-wise add/multiply in 2D, and the many-moduli-at-once inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "a72dc9f76eadb8eb2dd05a76d289489df2b79d3286c58af8416fd19d9c3d18c1"
    },
    {
      "slug": "the-non-adjacent-form",
      "title": "THE NON-ADJACENT FORM",
      "kicker": "the sparsest signed-binary representation",
      "accent": "#70a860",
      "blurb": "the non-adjacent form (NAF) in the 5-window house format — a signed-binary representation with digits {-1,0,+1} in which no two adjacent digits are both nonzero. Every integer has a unique NAF, and it has the fewest nonzero digits of any signed-binary representation (on average only a third nonzero, versus half for ordinary binary). That sparsity speeds up the double-and-add used in elliptic-curve and modular exponentiation: fewer nonzero digits means fewer additions. Verified live: every integer from -128 to 127 has a unique NAF over {-1,0,1} with no two adjacent nonzeros, it evaluates back exactly, and its weight never exceeds the ordinary binary weight. See a run collapse in 1D, NAF vs binary in 2D, and the sparse-signed inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "e9b1891a5f3a46254a9851675d037d07c5359fed4b155f81f1dab673f68221d2"
    },
    {
      "slug": "the-lehmer",
      "title": "THE LEHMER CODE",
      "kicker": "index any permutation by a single integer",
      "accent": "#c0a048",
      "blurb": "the Lehmer code and factorial number system in the 5-window house format — give every permutation a unique integer and back, a bijection between the n! orderings of n items and the numbers 0..n!-1. The Lehmer code records at each position how many later elements are smaller; reading it in the factorial base (place values (n-1)!, (n-2)!, ..., 1) yields the permutation's rank. It is how you index, shuffle, or store a permutation as one number. Verified live: over all 720 permutations of 6 items, rank of unrank is the identity and every rank yields a distinct permutation. See the inversion counts in 1D, a rank-to-permutation in 2D, and the single-integer inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "4253124347994ee12ca877784f3339da3a3f45f5e69107a295f619b03126ba3d"
    },
    {
      "slug": "the-tunstall",
      "title": "THE TUNSTALL CODE",
      "kicker": "variable strings to fixed-length codes — Huffman's dual",
      "accent": "#58a0b0",
      "blurb": "Tunstall coding in the 5-window house format — Huffman's mirror image: where Huffman maps variable-length symbols to variable-length codes, Tunstall maps variable-length source strings to fixed-length codes. It builds a dictionary by starting with the alphabet and repeatedly splitting the most probable leaf into its children until it has 2^R entries, then gives every entry the same R-bit codeword. Long likely strings get a whole codeword each, so common runs compress into one fixed block. Verified live: over 200 random streams, greedy parse + fixed-code encode round-trips exactly, and every codeword is the same length. See the leaf-splitting in 1D, a parse in 2D, and the fixed-code inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "c4f672ef44af258794c9e6777816949a7d5cc96b7f7434806566347b54c24eba"
    },
    {
      "slug": "the-poisson-disk",
      "title": "THE POISSON DISK",
      "kicker": "random points that never crowd — blue noise",
      "accent": "#70a860",
      "blurb": "Poisson-disk sampling in the 5-window house format — scatter points that are random but never closer than a radius r to one another (blue noise). Bridson's algorithm does it in O(n): a background grid (cell r/sqrt2, so each cell holds at most one point) and an active list; for each active point it throws k candidates into the annulus [r, 2r] and accepts the first with no neighbor closer than r. The even, gap-respecting spread is used for stippling, texture, and sensor placement. Verified live: across 30 runs every pair of accepted samples is at least r apart (minimum gap equals r, never less). See the candidate ring in 1D, an exclusion-disk scatter in 2D, and the blue-noise inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "f80a13e74ff52d6e399432cc694f684fdc209dcddeebe5dfe73b8921cdd6151e"
    },
    {
      "slug": "the-langtons-ant",
      "title": "THE LANGTON ANT",
      "kicker": "order emerges from two rules after 10,000 steps of chaos",
      "accent": "#a878c0",
      "blurb": "Langton's ant in the 5-window house format — a two-rule cellular automaton: an ant turns right on a white cell (then flips it black) and left on a black cell (then flips it white), and steps forward. From an all-white grid its path looks chaotic for about ten thousand steps, then locks into a periodic 'highway' that repeats every 104 steps, marching off to infinity. Order emerges from two trivial rules with no hint of it in between. Verified live: the ant's displacement is constant every 104 steps once the highway forms (net move -2,+2 per period), while the early chaotic phase has no such regularity. See the two rules in 1D, the trail in 2D, and the emergent-order inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "4ead46e40359c94fa2d5c85f6abafba2cfdee6dac72d690a2d894b6855e4133f"
    },
    {
      "slug": "the-count-min-sketch",
      "title": "THE COUNT-MIN SKETCH",
      "kicker": "count a huge stream in a tiny fixed table",
      "accent": "#d4a017",
      "blurb": "the Count-Min sketch in the 5-window house format — estimate how often each item appears in a huge stream using a tiny fixed table (d rows x w columns of counters), far smaller than the number of distinct items. Each item is hashed into one counter per row and increments them; its estimate is the minimum of those d counters. Because collisions can only add to a counter, the estimate is never an underestimate, and taking the min squeezes out most collision noise. Verified live: over an 8000-item stream drawn from a 400-symbol alphabet the sketch's estimate is >= the true count for every distinct item — it never underestimates. See the per-row hashing in 1D, a query in 2D, and the fixed-table inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "3c6297e9b19d7ff3b6e5c280248fc7954f624e7c8678d29b6d1ae4c70abc195c"
    },
    {
      "slug": "the-minhash",
      "title": "THE MINHASH",
      "kicker": "set similarity from a fistful of minimums",
      "accent": "#58a0b0",
      "blurb": "MinHash in the 5-window house format — estimate the Jaccard similarity of two sets (intersection over union) from tiny fixed-size signatures instead of the sets themselves. For each of k hash functions keep only the minimum hash value over a set; the fraction of signature positions that agree between two sets is an unbiased estimate of their Jaccard similarity. It is the engine behind near-duplicate detection in web-scale document sets. Verified live: identical sets agree in all k positions (estimate exactly 1); across 200 random pairs with k=256 the estimate stays within ~0.07 of the true Jaccard and averages ~0.02 error. See the single-hash minimum in 1D, two signatures in 2D, and the thumbprint inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "31fb7c23ff453aec172267180e74a7f17dcb103015e5081f9feb371ceec56983"
    },
    {
      "slug": "the-skew-binary",
      "title": "THE SKEW BINARY",
      "kicker": "a number base where +1 costs O(1)",
      "accent": "#c0a048",
      "blurb": "skew binary in the 5-window house format — a positional number system with place values 2^(k+1)-1 (1, 3, 7, 15, 31, ...) and digits {0,1,2}, where at most one digit is a 2 and it must be the lowest nonzero digit. Its magic is that +1 changes at most two digits — a genuine O(1) increment with no carry ripple — while ordinary binary can cascade carries across every bit. It is the number system behind skew-binary random-access lists and purely functional numeric structures. Verified live: over 0..5000 each canonical rep equals the counter, stays canonical (<= one 2, lowest), and every increment touches at most 2 digits. See the increment rule in 1D, a stepping counter in 2D, and the no-ripple inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "e46bb78164f5cf335cd8e2a78a013b25cd1bae1444eb6e698df60dccac1b53cf"
    },
    {
      "slug": "the-delta-sigma",
      "title": "THE DELTA-SIGMA",
      "kicker": "a whole waveform in a river of single bits",
      "accent": "#70a860",
      "blurb": "the delta-sigma modulator in the 5-window house format — turn a smooth analog signal into a stream of single bits (+-1) whose local average tracks the input. A first-order loop integrates the difference between input and the last output bit, then emits the sign; feedback keeps the running error near zero. Crucially it shapes the quantization noise — pushing it up to high frequencies where a lowpass filter removes it — so one bit at a high sample rate reconstructs the signal accurately. It is how most audio and sensor ADCs actually work. Verified live: modulating a 0.5*sine to a +-1 stream, a zero-phase lowpass reconstructs it to RMS < 0.02, and the quantization-noise energy is far larger at high frequencies than low. See bit density in 1D, sine-vs-reconstruction in 2D, and the noise-shaping inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "035ce5061a964c4978fa1e6ab8c31009c4446444602880d8b80cbe299d6cc5dc"
    },
    {
      "slug": "the-cuckoo-filter",
      "title": "THE CUCKOO FILTER",
      "kicker": "deletable membership with never a false negative",
      "accent": "#d4a017",
      "blurb": "the cuckoo filter in the 5-window house format — answer 'have I seen this item?' using tiny fingerprints in a compact table, like a Bloom filter but also supporting deletion. Each item has two candidate buckets (the second reachable from the first by XOR-ing a hash of its fingerprint), so an item can be relocated cuckoo-style to make room. It can return a false positive (a fingerprint collision) but never a false negative: anything inserted and not deleted is always found. Verified live: across 20 filled filters, every inserted item is found (no false negatives), and the false-positive rate on non-members is ~2%. See the two-bucket addressing in 1D, a filled filter in 2D, and the fingerprint-nest inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "4d12a98cf108290df17aefecc6999ca2d1adf0fbcb266f06d2da085e3751ff10"
    },
    {
      "slug": "the-collatz",
      "title": "THE COLLATZ",
      "kicker": "the hailstone that (so far) always lands on 1",
      "accent": "#a878c0",
      "blurb": "the Collatz map in the 5-window house format — the simplest unsolved problem in mathematics: take any positive integer; if even, halve it; if odd, triple and add one; repeat. The Collatz conjecture says this hailstone sequence always reaches 1, no matter the start — a claim tested to astronomical bounds but still unproven. The sequence bounces wildly up and down before it falls, and its stopping time is famously unpredictable. Verified live: for every n from 1 to 100000 the iteration reaches 1 (the conjecture holds throughout the tested range); the longest is n=77031 at 350 steps. See the hailstone of 27 in 1D, a trajectory in 2D, and the unproven-order inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "dde3b5c9972cec4bcbe16162597aa59889891c435d27a92e56e49080e058c2db"
    },
    {
      "slug": "the-rans",
      "title": "THE rANS",
      "kicker": "compress a whole message into one big integer",
      "accent": "#58a0b0",
      "blurb": "rANS (range Asymmetric Numeral System) in the 5-window house format — a modern entropy coder that encodes a whole message into a single very large integer, reaching near-optimal compression like arithmetic coding but with table-lookup speed. Each symbol folds into the state x by x <- floor(x/f_s)*M + (x mod f_s) + c_s using its frequency and cumulative; decoding peels symbols back off in reverse. It powers Zstandard, LZFSE, and modern image codecs. Verified live: over 300 random messages and frequency tables, encode of decode reproduces the exact message (BigInt state, no loss). See the slot carving in 1D, an encode/decode in 2D, and the one-integer inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "0dcf2091537fe5af2f99808a443668d4e35b5e46aebc2f8f3623b685b51d3f47"
    },
    {
      "slug": "the-xiaolin-wu",
      "title": "THE XIAOLIN WU LINE",
      "kicker": "an antialiased line — one unit of ink per column",
      "accent": "#70a860",
      "blurb": "Xiaolin Wu's line algorithm in the 5-window house format — draw an antialiased line (smooth, no jaggies) almost as fast as Bresenham's aliased one. At each step along the major axis it lights the two pixels straddling the true line, with brightnesses proportional to how close the line passes to each. The two brightnesses always sum to 1: exactly one pixel's worth of ink per column, split by coverage — so total intensity (energy) is conserved and the edge looks feathered instead of stepped. Verified live: over 500 random lines, at every step the two pixel intensities sum to exactly 1 (energy conserved to floating precision). See the coverage split in 1D, a magnified line in 2D, and the conservation inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "10374852faf182b8a5330c9ad06cf0b8d49308beca833614c02a3e6915d6b576"
    },
    {
      "slug": "the-negafibonacci",
      "title": "THE NEGAFIBONACCI",
      "kicker": "one signless code across the whole number line",
      "accent": "#c0a048",
      "blurb": "negaFibonacci coding in the 5-window house format — represent every integer, positive and negative, as a unique sum of non-consecutive negaFibonacci numbers F(-1), F(-2), ... = 1, -1, 2, -3, 5, -8, 13, ... with digits {0,1}, no two adjacent 1s, and no sign bit. Because the base sequence already alternates sign, negatives are reached for free. It is Zeckendorf's theorem extended across zero. Verified live: every integer from -50 to 50 has exactly one such representation, with no two adjacent 1s, that evaluates back to it. See the alternating base in 1D, an encoding in 2D, and the signless inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "4b484c83ede4893b1ae75dc05b617b1da3c86d251c170c587c0c5391b505ec32"
    },
    {
      "slug": "the-wang-tiles",
      "title": "THE WANG TILES",
      "kicker": "an edge-matching rule that makes tiling undecidable",
      "accent": "#a878c0",
      "blurb": "Wang tiles in the 5-window house format — unit squares with a color on each edge; you may place them (no rotation) only if touching edges share a color. Simple as they look, deciding whether a given set can tile the plane is undecidable, and some sets tile only aperiodically, never repeating. A backtracking solver fills a finite grid respecting the edge rule, or reports that no legal tiling exists. Verified live: a tileset extracted from a real tiling fills the grid with every shared edge matching, while an over-constrained instance (a corner color no tile provides) yields zero solutions. See the edge rule in 1D, a tiled grid in 2D, and the undecidability inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "f6709a3403551587ae7412763e0acecad27fb5de2ae8fe85210e21c27af1287f"
    },
    {
      "slug": "the-van-emde-boas",
      "title": "THE VAN EMDE BOAS",
      "kicker": "integer successor in O(log log u)",
      "accent": "#d4a017",
      "blurb": "the van Emde Boas tree in the 5-window house format — store integers from a universe {0..u-1} and answer successor, predecessor, and membership in O(log log u) time, faster than any comparison tree's log u. It recursively splits the universe into sqrt(u) clusters plus a summary structure over which clusters are non-empty, and stores each node's min/max lazily so most queries short-circuit. It is the classic structure for very fast integer successor search. Verified live: over 40 random trees on a 256-element universe, member(x) matches a reference set and successor(x) matches the sorted-set successor for every x. See the cluster+summary split in 1D, successor queries in 2D, and the log-log inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "9b13fb2feb9139853e5ee7854ea07e6db48730b482235cd3afe79d91852b1082"
    },
    {
      "slug": "the-paillier",
      "title": "THE PAILLIER",
      "kicker": "add two numbers without ever decrypting them",
      "accent": "#a878c0",
      "blurb": "the Paillier cryptosystem in the 5-window house format — additively homomorphic encryption: add two encrypted numbers without ever decrypting them. Multiplying two ciphertexts yields an encryption of the sum of the plaintexts, and raising a ciphertext to a power k yields an encryption of the plaintext times k. Encryption is Enc(m,r)=g^m*r^n mod n^2 with fresh random r, so every encryption of the same number looks different yet the algebra lines up. It underpins private voting and encrypted aggregation. Verified live: over 200 trials, Dec(Enc(a)*Enc(b))=a+b and Dec(Enc(a)^k)=k*a, all mod n. See the ciphertext product in 1D, an encrypted add in 2D, and the compute-under-lock inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "290e484970789bfb8531ba627afee62b3475758131bfc9385542ed2ea688275d"
    },
    {
      "slug": "the-post-correspondence",
      "title": "THE POST CORRESPONDENCE",
      "kicker": "a domino puzzle that is undecidable",
      "accent": "#58a0b0",
      "blurb": "the Post Correspondence Problem in the 5-window house format — a deceptively simple puzzle that is undecidable. Given dominoes, each with a top string and a bottom string, find a sequence of them (repeats allowed) so the concatenated tops exactly equal the concatenated bottoms. No algorithm can decide in general whether a given set has a solution, yet for specific sets a bounded search either finds one or exhausts all short sequences. Verified live: a known solvable set yields a sequence whose tops equal its bottoms, while a top-heavy set (every top longer than its bottom) provably has no solution up to the search depth. See stacked dominoes in 1D, a search in 2D, and the undecidability inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "8a79f7725fbd868de4772f56310d81849c8ec3d7ac87d718880f4918b9993ddf"
    },
    {
      "slug": "the-wavelet-tree",
      "title": "THE WAVELET TREE",
      "kicker": "rank a symbol in O(log sigma) by halving the alphabet",
      "accent": "#d4a017",
      "blurb": "the wavelet tree in the 5-window house format — store a sequence over an alphabet so it answers rank (how many times symbol c appears in the first i positions) and access (what symbol is at position i) in O(log sigma) time, using near the sequence's entropy in space. It recursively splits the alphabet in half: a bitvector marks whether each symbol went to the lower or upper half, and the halves recurse; rank becomes a walk down the tree counting bits. It is a cornerstone of compressed text indexing (FM-indexes). Verified live: over 200 random sequences, access(i) returns the true symbol and rank_c(i) equals a brute prefix count for every position and symbol. See the root bit split in 1D, a rank query in 2D, and the halving inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "37fbefb954052c1e4f6a2c8b7eb4f70aebab17483ce5e9f36928694f9bf5566a"
    },
    {
      "slug": "the-delaunay",
      "title": "THE DELAUNAY",
      "kicker": "triangles with empty circumcircles — the fattest mesh",
      "accent": "#70a860",
      "blurb": "Delaunay triangulation in the 5-window house format — connect points into triangles so that no point lies inside any triangle's circumcircle (the empty-circle property). Equivalently it maximizes the smallest angle, avoiding slivers, which is why it is the mesh of choice for interpolation, terrain, and finite elements. The Bowyer-Watson algorithm builds it incrementally: insert each point, delete every triangle whose circumcircle now contains it, and retriangulate the hole. Verified live: over 40 random point sets, every triangle's circumcircle is empty — no other input point falls inside it (thousands of checks). See the empty-circle test in 1D, a triangulation in 2D, and the empty-circumcircle inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "015770bb8abb4cb489601a6fd5e33a902600912bec402f802bcc146fa21dcc30"
    },
    {
      "slug": "the-factorial-base",
      "title": "THE FACTORIAL BASE",
      "kicker": "a mixed-radix odometer with factorial place values",
      "accent": "#c0a048",
      "blurb": "the factorial number system in the 5-window house format — a mixed-radix positional system where the place values are factorials (1!, 2!, 3!, ...) and the digit in place i may range only from 0 up to i. Every non-negative integer below m! has a unique such representation — an odometer whose wheels have different sizes (2, 3, 4, ... positions). It is the natural index for permutations and the backbone of the Lehmer code. Verified live: over all 5040 integers 0..7!-1, encode of decode is the identity, each digit stays within its rising radix, and all representations are distinct. See the factorial place values in 1D, a stepping counter in 2D, and the growing-wheels inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "80585c2ea52865c41568abdd47d5c755defa167bcc191167f90a7cc5104d2f75"
    },
    {
      "slug": "the-cyclic-tag",
      "title": "THE CYCLIC TAG",
      "kicker": "a universal computer from three strings and one rule",
      "accent": "#a878c0",
      "blurb": "a cyclic tag system in the 5-window house format — one of the tiniest known universal computers. It has a fixed cyclic list of production strings and a growing data string. Each step: remove the first data symbol; if it was a 1, append the current production; if a 0, append nothing; then advance to the next production, cycling. From this almost-nothing, Rule 110's universality was proven — cyclic tag systems can emulate any computation. Verified live: running productions (010, 000, 1111) on the seed '11' reproduces the exact documented state sequence 11 -> 1010 -> 010000 -> 10000 -> ... matching an independent hand derivation. See the step rule in 1D, the evolving data in 2D, and the minimal-engine inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "c6049ec0b00d09a086165294dba8acd6db6ffba9c64d9b877222c59e7bf9f1b7"
    },
    {
      "slug": "the-de-casteljau",
      "title": "THE DE CASTELJAU",
      "kicker": "a Bezier curve from nested interpolation",
      "accent": "#70a860",
      "blurb": "de Casteljau's algorithm in the 5-window house format — evaluate a Bezier curve by repeated linear interpolation: take the control points, interpolate each adjacent pair by parameter t to get one fewer point, and repeat until a single point remains — that point is on the curve. It is numerically stable and needs only midpoint-style blends, no polynomial powers. Remarkably, it computes exactly the same result as the Bernstein polynomial sum C(n,i) t^i (1-t)^(n-i) P_i. Verified live: over 300 random curves, de Casteljau's nested interpolation matches the Bernstein polynomial to floating precision (worst deviation ~1e-14). See the interpolation pyramid in 1D, a curve in 2D, and the blend-not-power inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "d204593eff3f549b73124f97034072ff281f863a1b226c8793f95ba0ea826334"
    },
    {
      "slug": "the-continued-fraction",
      "title": "THE CONTINUED FRACTION",
      "kicker": "a number as a ladder of nested reciprocals",
      "accent": "#c0a048",
      "blurb": "continued fractions in the 5-window house format — write a number as a0 + 1/(a1 + 1/(a2 + ...)), a ladder of nested reciprocals. For a rational p/q the ladder is finite and the integer parts [a0; a1, a2, ...] come straight from the Euclidean algorithm. Folding the ladder back up (the convergents) reconstructs p/q exactly, in lowest terms. Continued fractions give the best rational approximations of any number and underlie lattice reduction and Pell's equation. Verified live: over 500 random rationals, the continued fraction then reconstructed via convergents returns the exact reduced fraction. See the Euclidean quotients in 1D, the convergents in 2D, and the ladder inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "0751bac45b4d4065ee9c82991503da4ec98db9b04545fc00b540e7459a9f61d0"
    },
    {
      "slug": "the-rope",
      "title": "THE ROPE",
      "kicker": "a long string as a balanced tree of pieces",
      "accent": "#d4a017",
      "blurb": "a rope in the 5-window house format — store a long string as a balanced binary tree whose leaves hold small pieces and whose internal nodes cache the length of their left subtree. This makes concatenation and splitting O(log n) — just re-link a few nodes, no copying — and indexing a walk down the tree using the cached lengths. It is the structure behind fast text editors and immutable string libraries, where inserting into a huge document must not copy it whole. Verified live: over 200 random strings, ropeIndex(i) returns the same character as the flat string at every position, and concatenating two ropes flattens to the exact concatenation (length and content). See the length-cache walk in 1D, a rope tree in 2D, and the no-copy inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "129dda0b2d4e5d8bc2b9dfd1055ec972dc8724374eebfae7fe2cd709f5af6ff3"
    },
    {
      "slug": "the-ldpc",
      "title": "THE LDPC CODE",
      "kicker": "a codeword that heals its own errors",
      "accent": "#58a0b0",
      "blurb": "an LDPC code in the 5-window house format — low-density parity-check codes protect data with a sparse set of parity checks: a valid codeword c satisfies H*c = 0 (mod 2) for the parity-check matrix H. The classic bit-flipping decoder is beautifully simple: compute which checks fail, count for each bit how many failing checks it touches, and flip the bit in the most failing checks; repeat until all checks pass. LDPC codes approach the Shannon limit and protect Wi-Fi, 5G, and deep-space links. Verified live: over 400 valid codewords each hit by a single bit error, bit-flipping restores H*c = 0 and recovers the original codeword — 400/400. See the accused bit in 1D, a decode in 2D, and the self-repair inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "813a18e3bdd3ac565a01fd121c39b3548fb07b5bd0e366beebbdcdff9d5e6cad"
    },
    {
      "slug": "the-burrows-wheeler",
      "title": "THE BURROWS-WHEELER",
      "kicker": "a reversible scramble that makes text compress",
      "accent": "#58a0b0",
      "blurb": "the Burrows-Wheeler transform in the 5-window house format — reversibly reorder a string so similar characters cluster together (making it far more compressible), yet the original can be perfectly reconstructed from the transform plus one index. It takes the last column of the sorted table of all rotations of the string; astonishingly that scrambled last column holds enough to invert the whole thing. It is the heart of bzip2 and the FM-index. Verified live: over 200 random strings, inverting the BWT (last column + index) returns the exact original. See the sorted rotations in 1D, a transform+inverse in 2D, and the lossless-scramble inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "a4a11437fb45a7395eb18c654e4c495359fa6359128a5c94db6d9a7ef62f37f3"
    },
    {
      "slug": "the-shamir",
      "title": "THE SHAMIR SHARING",
      "kicker": "split a secret so any k of n rebuild it, fewer learn nothing",
      "accent": "#a878c0",
      "blurb": "Shamir's secret sharing in the 5-window house format — split a secret into n shares so that any k of them reconstruct it exactly, but any k-1 reveal nothing at all. The trick: hide the secret as the constant term of a random degree-(k-1) polynomial over a finite field, and hand out points on it. k points pin down the polynomial (and its constant) by Lagrange interpolation; fewer leave the constant completely undetermined — every possible secret equally consistent. Verified live: over 200 schemes, every k-subset reconstructs the secret while k-1 shares leave it undetermined. See the polynomial through points in 1D, a k-of-n split in 2D, and the trust-split inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "e857b20738ffbd327a3a59e32a704eefb2dff9a30ffca9d6c2347ec9addedd8a"
    },
    {
      "slug": "the-neville",
      "title": "THE NEVILLE",
      "kicker": "polynomial interpolation by a triangle of blends",
      "accent": "#c0a048",
      "blurb": "Neville's algorithm in the 5-window house format — evaluate the unique polynomial through a set of data points at a query x by a triangle of linear blends, never forming the polynomial explicitly. Start with the y-values; each step combines two neighboring lower-degree interpolants, weighted by distance to x, into one of higher degree, until a single value remains. It is numerically friendly and, like de Casteljau, replaces coefficients with repeated interpolation. Verified live: over 300 random datasets, Neville's value passes exactly through every data point and matches Lagrange interpolation at random x (to ~1e-13). See the blend triangle in 1D, an interpolating curve in 2D, and the no-coefficients inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "2727df9409ae741cf51a9997bfdc74c89ed1e03795dc517ec2ba0d301f57d3c7"
    },
    {
      "slug": "the-booth",
      "title": "THE BOOTH",
      "kicker": "signed multiplication by recoding the bits",
      "accent": "#d4a017",
      "blurb": "Booth's algorithm in the 5-window house format — multiply two signed binary numbers directly in two's complement, with no special-casing of the sign. It recodes the multiplier by looking at adjacent bit pairs: a 0->1 boundary means subtract the multiplicand, a 1->0 boundary means add it, inside a run do nothing. A run of ones like 0111 becomes 'add once, subtract once' instead of three adds — fewer operations, and negatives handled for free. Verified live: over 2000 random 8-bit signed multipliers against arbitrary multiplicands, Booth recoding's result equals a*b exactly. See the bit-pair recoding in 1D, a recoded multiply in 2D, and the boundary-only inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "2083f7d5586071c9d6453e0cf6d4f8d14ce73e53b00c03a591fec184f52d1eab"
    },
    {
      "slug": "the-fibonacci-heap",
      "title": "THE FIBONACCI HEAP",
      "kicker": "a priority queue that pays for order only when it must",
      "accent": "#70a860",
      "blurb": "the Fibonacci heap in the 5-window house format — a priority queue that makes insert and decrease-key cost only O(1) amortized, deferring all the real work to extract-min, which then lazily consolidates trees of equal degree. That fast decrease-key is what lets Dijkstra and Prim hit their best textbook bounds. It keeps a forest of heap-ordered trees and a pointer to the minimum root; the 'pay later' laziness is the whole idea. Verified live: over 100 random heaps, repeated extract-min returns the keys in exact sorted order. See the lazy insert in 1D, insert+extract in 2D, and the pay-later inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "68178ca1909c6a8336aa69e54bd44607ff46c2be1d89c43142973f7c79cd4406"
    },
    {
      "slug": "the-graham-scan",
      "title": "THE GRAHAM SCAN",
      "kicker": "the convex hull by keeping only left turns",
      "accent": "#70a860",
      "blurb": "the Graham scan in the 5-window house format — compute the convex hull of a set of points (the smallest convex polygon enclosing them all, like a rubber band snapped around nails). It sorts the points, then walks them keeping only left turns: whenever three consecutive points make a right turn, the middle one is popped. What remains is the hull, in O(n log n). It is a workhorse of computational geometry. Verified live: over 300 random point sets the hull is convex (every turn a left turn) and every input point lies inside or on it. See the turn test in 1D, a hull in 2D, and the rubber-band inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "70dab64725feb302eced7d631e8349bb2286e97b045261673d22053b0f8d3742"
    },
    {
      "slug": "the-rabin-karp",
      "title": "THE RABIN-KARP",
      "kicker": "string search by a rolling hash",
      "accent": "#58a0b0",
      "blurb": "the Rabin-Karp algorithm in the 5-window house format — find a pattern in text using a rolling hash: hash the pattern once, then slide a window over the text, updating the window's hash in O(1) per step by removing the leaving character and adding the entering one (polynomial hashing modulo a large prime). Only when hashes match does it verify character-by-character. It shines at multi-pattern search and plagiarism detection. Verified live: over 300 random text/pattern pairs, the rolling-hash match positions (with verification) exactly equal a brute-force scan. See the rolling window in 1D, a search in 2D, and the fingerprint inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "5e57ef73597579f44e1512ae0fb9bb8b0a9b7d4729fa4e5cc154d10cc23260e7"
    },
    {
      "slug": "the-tea",
      "title": "THE TEA CIPHER",
      "kicker": "a whole block cipher from add, shift, xor",
      "accent": "#a878c0",
      "blurb": "TEA (the Tiny Encryption Algorithm) in the 5-window house format — a block cipher famous for being tiny (a few lines) yet a real Feistel-style cipher. It encrypts a 64-bit block with a 128-bit key over 32 rounds, each mixing the two halves with shifts, additions, and XORs and a magic constant (the golden-ratio delta 0x9E3779B9). Decryption runs the same operations in reverse. Its simplicity made it a teaching classic (and spurred XTEA after weaknesses were found). Verified live: over 500 random blocks and keys, decrypt(encrypt(x)) returns x exactly, and the ciphertext differs from the plaintext. See a round in 1D, an encrypt/decrypt in 2D, and the add-shift-xor inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "882aa414d94c272c46bd1083c48ce2c80c77ad4651019e3c34c2b8c0289b92de"
    },
    {
      "slug": "the-lloyd",
      "title": "THE LLOYD",
      "kicker": "k-means: cluster by moving to the mean",
      "accent": "#c0a048",
      "blurb": "Lloyd's algorithm (k-means) in the 5-window house format — pick k centers, then alternate two steps: assign each point to its nearest center, and update each center to the mean of its assigned points. Repeat until nothing moves. Each step can only lower the total squared distance (the distortion), so it converges monotonically to a local optimum. It is the workhorse of clustering, quantization, and color reduction. Verified live: over 80 runs, every updated center is exactly the mean of its assigned points, and the distortion is non-increasing at every step. See the two moves in 1D, settling clusters in 2D, and the move-to-mean inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "785e6e4da9f2bbbc787816836c42718f8bfb89937742f8e0e9bdc5eba9f62ca0"
    },
    {
      "slug": "the-freivalds",
      "title": "THE FREIVALDS",
      "kicker": "verify a matrix product in O(n^2) with a random probe",
      "accent": "#d4a017",
      "blurb": "Freivalds' algorithm in the 5-window house format — check whether a claimed matrix product A*B = C is correct in O(n^2), far faster than the O(n^3) to recompute A*B. The trick: pick a random 0/1 vector r and test whether A(Br) = Cr. If A*B = C the test always passes; if not, it fails with probability >= 1/2 each round, so a few rounds catch any error with overwhelming confidence. It is the seminal randomized verifier. Verified live: over 200 trials the correct product passes all rounds, and a product with a single wrong entry is caught within 10 rounds. See the random probe in 1D, an accept/reject in 2D, and the check-without-redoing inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "c6fd0eb5e083e9ea628ae2d209f27d5c6f72c4670fa2c6ff61a814abbf0bfb16"
    },
    {
      "slug": "the-stein",
      "title": "THE STEIN",
      "kicker": "GCD with only shifts and subtractions",
      "accent": "#c0a048",
      "blurb": "Stein's binary GCD in the 5-window house format — compute the greatest common divisor using only subtraction, comparison, and bit shifts, no division or modulo (slow in hardware). It rests on three facts: gcd(2a,2b)=2*gcd(a,b), gcd(2a,b)=gcd(a,b) when b is odd, and gcd(a,b)=gcd(|a-b|,min(a,b)) for two odds. Strip common factors of two, halve evens, subtract odds, restore the twos at the end. It is the GCD of choice on hardware without a divide unit. Verified live: over 3000 random pairs, Stein's shift-and-subtract GCD equals the Euclidean GCD exactly. See a reduction trace in 1D, a computed GCD in 2D, and the division-free inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "fd6a0efea876ebaa8ffdb497651c180468d3d482074b1dc0fa76d529ea122906"
    },
    {
      "slug": "the-consistent-hashing",
      "title": "THE CONSISTENT HASHING",
      "kicker": "node churn that moves only ~1/n of the keys",
      "accent": "#58a0b0",
      "blurb": "consistent hashing in the 5-window house format — map keys to servers so that adding or removing a server moves only a small fraction of keys (roughly 1/n) instead of remapping everything as plain modulo hashing would. Nodes and keys are placed on a hash ring; a key belongs to the first node clockwise from it. Remove a node and only its keys spill to the next; everyone else stays put. Virtual nodes smooth the load. It is the backbone of distributed caches and databases. Verified live: over 50 rings, every key maps to a node, and removing a node moves only that node's keys. See the ring in 1D, a node drop in 2D, and the churn-without-chaos inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "02b69f0d23d0bfad34d79e7b71469920bc157b36b57bd7bab3304048a73ad606"
    },
    {
      "slug": "the-marching-squares",
      "title": "THE MARCHING SQUARES",
      "kicker": "trace an isoline through a grid of values",
      "accent": "#70a860",
      "blurb": "marching squares in the 5-window house format — extract a contour (isoline) from a grid of scalar values. For each cell it looks at which of the four corners are above the threshold: a 4-bit case index (0-15) selects, from a small lookup table, which cell edges the contour crosses. The exact crossing point on each edge is found by linear interpolation between the two corner values. Stitched together, the segments trace the level set. It is how weather maps draw isobars and metaballs get outlines. Verified live: over 60 random fields, every contour vertex sits on a grid edge whose two endpoints straddle the threshold. See a cell case in 1D, a contour in 2D, and the interpolated-crossing inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "10c498acbae5da6eb0a22f95415e3b9d2e33a0fcc226d227ae961c0c0093d440"
    },
    {
      "slug": "the-catmull-rom",
      "title": "THE CATMULL-ROM",
      "kicker": "a smooth spline through every control point",
      "accent": "#a878c0",
      "blurb": "the Catmull-Rom spline in the 5-window house format — an interpolating cubic curve: unlike a Bezier, it passes exactly through every control point, using each point's neighbors to set the tangent there. Each segment between Pi and Pi+1 is a cubic in t with C(0)=Pi and C(1)=Pi+1, and the tangent at Pi is (Pi+1 - Pi-1)/2. It gives smooth, natural-looking paths, which is why it is everywhere in animation and camera motion. Verified live: over 300 random point sets, each segment's endpoints land exactly on the two control points it spans. See a segment in 1D, a curve in 2D, and the interpolating inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "30dfc5c50154b56678cc98887f356f107e84701b9abfc07b967be4d107611a1b"
    },
    {
      "slug": "the-barrett",
      "title": "THE BARRETT",
      "kicker": "reduce mod n without dividing",
      "accent": "#d4a017",
      "blurb": "Barrett reduction in the 5-window house format — compute x mod n without dividing, replacing the expensive division with one multiplication and a shift using a precomputed constant mu = floor(4^k/n) (k = bit length of n). The quotient is estimated as floor(x*mu / 4^k), then x - q*n is corrected by at most two subtractions. For inputs x < n^2 (a modular product), this is exact. It is a cornerstone of fast modular exponentiation in RSA and elliptic-curve crypto, where the modulus is fixed and reused millions of times. Verified live: over thousands of pairs with x < n^2, Barrett reduction equals x mod n exactly, needing at most 2 corrections. See the multiply-shift steps in 1D, a reduction in 2D, and the no-divide inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "7adb6162a686b039d4687929bc36c2ebb85d3db6608d66fb88cbe488b9bf2146"
    },
    {
      "slug": "the-dragon-curve",
      "title": "THE DRAGON CURVE",
      "kicker": "a fold that fills space and never crosses itself",
      "accent": "#70a860",
      "blurb": "the dragon curve in the 5-window house format — the shape from folding a strip of paper in half repeatedly, then unfolding every crease to a right angle. Its turn sequence is the regular paperfolding sequence: at step n, turn left if the odd part of n is congruent to 1 (mod 4), else right. Though it packs into a fractal that tiles the plane, the curve never crosses itself — every unit segment is traversed at most once. It is a classic of graphics and number theory. Verified live: up to order 14 (16384 segments) the dragon curve is self-avoiding, every unit edge distinct. See the fold sequence in 1D, the curve in 2D, and the fold-not-route inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "1e721c13f1f1b9bd02ab335cf523d1ea059a40e949e5fbaf3e4551001c8a3e8d"
    },
    {
      "slug": "the-logistic-map",
      "title": "THE LOGISTIC MAP",
      "kicker": "chaos from a one-line rule, by period-doubling",
      "accent": "#a878c0",
      "blurb": "the logistic map in the 5-window house format — x -> r*x*(1-x), the simplest equation that becomes chaotic. As the growth rate r rises, the long-run behavior doubles: one steady value, then an oscillation between two, then four, then eight (the period-doubling cascade), and past r ~ 3.5699 it dissolves into deterministic chaos. The intervals between doublings shrink by the universal Feigenbaum constant delta ~ 4.669, the same for a huge class of systems. Verified live: the attractor has period 1 at r=2.8, 2 at 3.2, 4 at 3.5, 8 at 3.55, and no short period at 3.9 (chaos). See the doublings in 1D, the bifurcation diagram in 2D, and the deterministic-chaos inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "7a355c1f6da19db87091c5689399c029eb461ed3ca78ca5ced2aed3b26a5917e"
    },
    {
      "slug": "the-rendezvous-hashing",
      "title": "THE RENDEZVOUS HASHING",
      "kicker": "assign by highest random weight, no ring",
      "accent": "#58a0b0",
      "blurb": "rendezvous hashing (Highest Random Weight) in the 5-window house format — assign each key to a server with no ring and no coordination: hash the key with every candidate server and pick the server with the highest combined hash. Every party computes the same winner independently. When a server leaves, only the keys that had it as their top choice move (to their second choice); no other key is disturbed. It cleanly handles weighted servers and small clusters. Verified live: over 80 clusters, every key maps to a definite node, and removing a node moves only that node's keys. See the weight pick in 1D, a node drop in 2D, and the shared-computation inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "8511318664688cd4bf6e872425aa157a500ccfe311b06233031b814405f20ea6"
    },
    {
      "slug": "the-lu-decomposition",
      "title": "THE LU DECOMPOSITION",
      "kicker": "factor a matrix into two triangles once, solve forever",
      "accent": "#c0a048",
      "blurb": "LU decomposition in the 5-window house format — factor a square matrix A into a lower-triangular L and an upper-triangular U (with a row-permutation P for stability), so that P*A = L*U. It is Gaussian elimination, remembered: once you have L and U you can solve A*x=b for many right-hand sides cheaply by two triangular sweeps, and read the determinant off U's diagonal. Partial pivoting swaps in the largest pivot each step to keep the arithmetic stable. Verified live: over 300 random matrices, P*A equals L*U to ~1e-15, and L is lower-triangular with unit diagonal. See elimination in 1D, the factors in 2D, and the factor-once inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "91d08f656f16a8476b05c5040adafbf0236229a0e7a7dfd9a9ddb663e432376a"
    },
    {
      "slug": "the-cubic-spline",
      "title": "THE CUBIC SPLINE",
      "kicker": "the smoothest curve through the points (C2)",
      "accent": "#d4a017",
      "blurb": "the natural cubic spline in the 5-window house format — the smoothest curve through a set of points: a separate cubic on each interval, joined so that value, slope, and curvature all match at every knot (C2 continuity), with zero curvature at the two ends (the natural condition). Those matching conditions reduce to a tridiagonal linear system for the second derivatives, solved in O(n). It is the smooth interpolant of choice for data fitting and font/animation curves. Verified live: over 200 datasets, the spline passes through every point, its first derivative is continuous at every interior knot, and the second derivative is zero at both ends. See matched knots in 1D, a spline in 2D, and the match-the-curvature inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "7fbca702bd5f934067024b5874b27d8eead33be05a2e5864178d5f0cfd085723"
    },
    {
      "slug": "the-2-sat",
      "title": "THE 2-SAT",
      "kicker": "satisfy two-literal clauses in linear time",
      "accent": "#58a0b0",
      "blurb": "2-SAT in the 5-window house format — decide whether clauses, each an OR of two literals, can all be satisfied, in linear time (unlike NP-complete general SAT). Each clause (a OR b) becomes two implications, not-a -> b and not-b -> a, forming a graph; the formula is satisfiable iff no variable and its negation land in the same strongly connected component, and a valid assignment is read from the component order. Verified live: over 300 random instances, the SCC-based verdict matches a brute-force check of all 2^n assignments, and the extracted assignment satisfies every clause. See the implication in 1D, a formula in 2D, and the reachability inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "003ac753c7cc79e586376e15b7992a3656e227526bc309acbec060b84a858ce7"
    },
    {
      "slug": "the-gershgorin",
      "title": "THE GERSHGORIN",
      "kicker": "trap every eigenvalue in a disc, without solving",
      "accent": "#c0a048",
      "blurb": "Gershgorin's circle theorem in the 5-window house format — pin down where a matrix's eigenvalues can be without computing them: every eigenvalue lies within at least one Gershgorin disc, centered at a diagonal entry a_ii with radius equal to the sum of the absolute off-diagonal entries in that row. A few cheap sums bound the whole spectrum, invaluable for stability analysis and preconditioning. Verified live: over 300 random symmetric matrices, every (Jacobi-computed) eigenvalue falls inside a Gershgorin disc, and the eigenvalues sum to the trace. See a row-disc in 1D, discs on the line in 2D, and the localize-by-rows inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "6318edc914c3527dc569dc0f32c05cf75a2b7cae53827c0d273a823b3dc70298"
    },
    {
      "slug": "the-least-rotation",
      "title": "THE LEAST ROTATION",
      "kicker": "the canonical rotation of a necklace, in O(n)",
      "accent": "#a878c0",
      "blurb": "Booth's least-rotation algorithm in the 5-window house format — find the lexicographically smallest rotation of a string in linear time, the canonical form of a necklace where all rotations are equivalent. Instead of trying every rotation (O(n^2)), it runs a KMP-style failure-function scan over the doubled string, sliding a candidate start and jumping past mismatches. It is how you canonicalize cyclic sequences (circular DNA, polygon encodings, necklace enumeration). Verified live: over 500 random strings, Booth's rotation index gives the exact same rotation as a brute-force minimum over all rotations. See the necklace rotations in 1D, a canonical form in 2D, and the one-name inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "c664c8c1daaa4005c628d3df252bbfba5dfbe0a4b1d27976830c61771781db07"
    },
    {
      "slug": "the-sieve-of-atkin",
      "title": "THE SIEVE OF ATKIN",
      "kicker": "primes from quadratic forms, not multiples",
      "accent": "#70a860",
      "blurb": "the sieve of Atkin in the 5-window house format — find primes using quadratic forms instead of marking multiples. A number (with small primes handled separately) is prime if it solves one of three modular equations an odd number of times: 4x^2+y^2 (n mod 12 in {1,5}), 3x^2+y^2 (n mod 12 = 7), or 3x^2-y^2 with x>y (n mod 12 = 11), then multiples of prime squares are removed. It is asymptotically faster than Eratosthenes. Verified live: the sieve of Atkin's prime list up to 5000 is identical to trial division (669 primes). See the quadratic forms in 1D, a prime grid in 2D, and the parabola-sieve inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "6e59793807cfe4b6bde09e6d1bc4c8fd82d0017ad54cba6af7e3e56115b1ec9b"
    },
    {
      "slug": "the-li-chao",
      "title": "THE LI CHAO TREE",
      "kicker": "the lowest line at any x, in log time",
      "accent": "#d4a017",
      "blurb": "the Li Chao tree in the 5-window house format — maintain a set of lines and answer 'which line is lowest at this x?' in O(log) time, storing the lower envelope of a pencil of lines. Each node owns the line that dominates the middle of its x-range; a new line either replaces it or is pushed to the half where it might win. It powers the convex-hull trick for speeding up dynamic programming, turning O(n^2) DP transitions into O(n log n). Verified live: over 200 trees, the Li Chao query returns exactly the minimum of all inserted lines at each x, matching a brute-force scan. See the lower envelope in 1D, a query in 2D, and the stored-envelope inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "d478c2d6c7fb46ac8f419a0ddacd0cacd021d9d9ce256e19dce7b2bed99df777"
    },
    {
      "slug": "the-extended-euclid",
      "title": "THE EXTENDED EUCLID",
      "kicker": "the GCD carries a Bézout certificate",
      "accent": "#c0a048",
      "blurb": "The extended Euclidean algorithm in the 5-window house format — run the ordinary Euclidean division loop, but carry the coefficients along so the GCD arrives with a proof of how to build it: integers x, y with a·x + b·y = gcd(a,b), Bézout's identity. Those coefficients are exactly what give modular inverses and power RSA. Verified live: over 5000 random pairs, the returned (x,y) satisfy a·x + b·y = gcd(a,b) exactly and the gcd matches the ordinary Euclidean one. See the division loop in 1D, a pair certified in 2D, and the remainder-that-proves-itself inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "c2acec9a46c912e67c788a9ac2b0dd3bc94201270c85277dea1fe57529f68a9c"
    },
    {
      "slug": "the-hough",
      "title": "THE HOUGH",
      "kicker": "a point becomes a curve to vote for lines",
      "accent": "#a878c0",
      "blurb": "The Hough transform in the 5-window house format — detect lines in a scatter of points by a duality: each point (x,y) becomes a sinusoid ρ = x·cosθ + y·sinθ in parameter space, and the sinusoids of collinear points all cross at one (ρ,θ) — the line's own parameters. Accumulate votes and lines appear as peaks, robust to gaps and noise. Verified live: collinear points exactly satisfy ρ₀ = x·cosθ₀ + y·sinθ₀ (worst error ~1e-13), and the accumulator peak recovers the line across 200 trials at 720×720 resolution. See the sinusoids crossing in 1D, an accumulator in 2D, and detection-as-voting in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "3453be90d7bd389e236ac9fc95c950c496362311f3539eea5b67dfc6308fca6e"
    },
    {
      "slug": "the-savitzky-golay",
      "title": "THE SAVITZKY-GOLAY",
      "kicker": "smooth the noise without blurring the shape",
      "accent": "#58a0b0",
      "blurb": "The Savitzky–Golay filter in the 5-window house format — smooth noisy data without flattening its features by fitting a low-degree polynomial to each sliding window by least squares and taking the fitted center value, instead of averaging (which crushes peaks). The whole operation collapses to one fixed convolution kernel. Verified live: a filter of order d reproduces any polynomial of degree ≤ d exactly (worst ~1e-12) over 200 random cases — it does not distort what it should preserve. See a window fit in 1D, Savitzky–Golay vs a moving average in 2D, and shape-preserving smoothing in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "ed0a5079d66df7a4ffe29d2147bf3699fa8ed5fe663390d6354b2543f3bc7b96"
    },
    {
      "slug": "the-kalman",
      "title": "THE KALMAN",
      "kicker": "fuse guess and measurement optimally, recursively",
      "accent": "#d4a017",
      "blurb": "The Kalman filter in the 5-window house format — optimally fuse a prediction with a noisy measurement: keep an estimate and its uncertainty, and blend each reading in by the Kalman gain K = P/(P+R), trusting the measurement more when the estimate is uncertain. For a static value under Gaussian noise, the running estimate equals the precision-weighted mean of all readings, with posterior variance 1/Σ(precisions). Verified live: over 300 runs, the recursion's estimate exactly equals the batch precision-weighted mean and its variance equals 1/Σprecision. See the gain blend in 1D, a converging estimate in 2D, and memoryless-yet-optimal fusion in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "9ecd6f2cc07e670677369a9cdc8ca2a4edb89aaf04d4c5b30867d7e440c2cead"
    },
    {
      "slug": "the-seam-carving",
      "title": "THE SEAM CARVING",
      "kicker": "carve out the least-noticed seam",
      "accent": "#70a860",
      "blurb": "Seam carving in the 5-window house format — resize an image by removing the least noticeable connected paths of pixels rather than scaling or cropping. Assign each pixel an energy, then find the top-to-bottom seam (one pixel per row, each within one column of the row above) of minimum total energy; dynamic programming finds that optimal seam in one pass. Verified live: over 200 random energy grids, the DP minimum vertical seam has exactly the same total energy as an exhaustive search over all seams. See a seam's connectivity in 1D, an energy grid's minimum seam in 2D, and the path-of-least-attention inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "fd9fccaa3cdc7fcd0fdd542de932bf08d293c6027c3f6b9eab18600c71b60929"
    },
    {
      "slug": "the-miller-rabin",
      "title": "THE MILLER-RABIN",
      "kicker": "a witness names the composite, no factor needed",
      "accent": "#b088d0",
      "blurb": "The Miller–Rabin primality test in the 5-window house format — decide primality by interrogating witnesses. Write n−1 = 2^r·d; a base a is a witness to compositeness if aᵈ ≠ 1 and none of aᵈ, a²ᵈ, … equals n−1 (mod n). Rabin proved at least 3/4 of bases witness any odd composite n > 9, so a few random bases catch composites with overwhelming probability and small fixed base sets are deterministic below known bounds. Verified live: a fixed 12-base test agrees with trial division for every n below 100000, and over random odd composites the witness fraction never drops below 3/4. See witnesses vs liars in 1D, a verdict in 2D, and the prove-composite-without-factoring inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "9391dcc66eafaff371227ff0f3f7bd0b2b80e23d88a5b0d5abbf2adbe0a02853"
    },
    {
      "slug": "the-walker-alias",
      "title": "THE WALKER ALIAS",
      "kicker": "loaded dice drawn in one step, no search",
      "accent": "#e0b020",
      "blurb": "Walker's alias method in the 5-window house format — draw from any discrete distribution in O(1) per sample. Preprocess the probabilities into n equal-height columns, each holding at most two outcomes (a primary and an alias) split at a threshold; to sample, pick a column uniformly then flip a biased coin for primary-or-alias. The construction repeatedly pairs an under-full outcome with an over-full one until every column is exactly full, so the reconstructed probabilities are exact. Verified live: over 500 random distributions, each outcome's probability reconstructed from the alias table equals the target exactly (worst ~1e-16). See columns filling in 1D, a table checked in 2D, and the O(1) two-choice draw in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "afb68668b23e3ffbef795e4dca46e6354c880a1933ae84793109c112e58bb01d"
    },
    {
      "slug": "the-l-system",
      "title": "THE L-SYSTEM",
      "kicker": "a plant grown at the golden rate from one seed",
      "accent": "#58b878",
      "blurb": "The Lindenmayer system in the 5-window house format — grow a string by rewriting every symbol at once, in parallel, by fixed rules (a model of plant and shell development). The classic Fibonacci L-system uses A → AB and B → A: from A you get A, AB, ABA, ABAAB, ABAABABA… and the length of each generation is a Fibonacci number, because each A becomes an A and a B while each B becomes an A — exactly the Fibonacci recurrence. Verified live: for generations 0–25 the string length equals the matching Fibonacci number exactly, and the A/B counts obey the recurrence. See the parallel rewrite in 1D, growth tracking Fibonacci in 2D, and numbers-as-morphology in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "260b505b102546cf2f0c3d10710f4051fb7729eb91e7fd6524a117125d63796b"
    },
    {
      "slug": "the-lazy-lord",
      "title": "THE LAZY LORD",
      "kicker": "defer the work, still answer exactly",
      "accent": "#58a0b0",
      "blurb": "The segment tree with lazy propagation in the 5-window house format — answer range questions and apply range updates on an array, both in O(log n). The trick is laziness: adding a value to a whole range doesn't touch every element — it marks the covering nodes with a pending update and only pushes it down to children when a later query actually needs to descend. Work is deferred until it matters, yet every answer is exactly what a naive per-element array would give. Verified live: over 60 trees and thousands of interleaved range-add / range-sum operations, every query matches a naive array element-for-element. See the pending mark in 1D, live checks in 2D, and the owed-work inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "7ee2782ac5e6e870d05665dc0bae734895749950f7fa266c0c7467fbd452407f"
    },
    {
      "slug": "the-minsky-counters",
      "title": "THE MINSKY COUNTERS",
      "kicker": "the smallest machine that can multiply",
      "accent": "#7088c0",
      "blurb": "The Minsky counter machine in the 5-window house format — one of the smallest things that can compute anything: a few unbounded counters and just two instruction kinds, increment-and-jump or decrement-if-nonzero-and-branch. With only two counters it is already Turing-complete. Here a fixed program of INC / DEC-branch instructions multiplies: fed m and n in two counters, it halts with their product in a third, having only ever added and subtracted one. Verified live: the multiply program halts with counter C = m·n for all m,n in 0–14 (over 400 pairs), matching direct multiplication. See the whole instruction set in 1D, the running counters in 2D, and multiplication-from-two-moves in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "548fdac88eca2fc8ae6914a554a304133396d792745b80e3af72416d991a6a58"
    },
    {
      "slug": "the-picks-theorem",
      "title": "THE PICK'S THEOREM",
      "kicker": "area from counting fenceposts and interior dots",
      "accent": "#c07850",
      "blurb": "Pick's theorem in the 5-window house format — the exact area of any simple lattice polygon by counting dots: A = I + B/2 − 1, where I is the interior lattice points and B the boundary ones. No calculus — count interior dots, count the boundary fenceposts, and the area falls out exactly, tying a continuous quantity to two discrete counts. Verified live: over hundreds of random lattice polygons the shoelace area equals I + B/2 − 1 exactly. See the fenceposts and interior dots in 1D, a polygon checked in 2D, and the area-from-dots inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "28bf072132a633e3128db024a5875fff21b2e4f9c024dcd471c8f9bd662ec50d"
    },
    {
      "slug": "the-cantor-pairing",
      "title": "THE CANTOR PAIRING",
      "kicker": "weave two numbers into one, and back",
      "accent": "#58a0b0",
      "blurb": "The Cantor pairing function in the 5-window house format — weave two naturals into one, reversibly: π(x,y) = (x+y)(x+y+1)/2 + y walks the grid along diagonals, assigning 0,1,2,… to each cell so every pair gets a unique number and every number decodes to exactly one pair. It is a genuine bijection ℕ²→ℕ — a proof in one formula that the plane of pairs is no bigger than the line of counting numbers. Verified live: over all pairs in 0–200, unpair(pair(x,y)) returns exactly (x,y) and no two pairs collide. See the diagonal enumeration in 1D, a round-trip in 2D, and the lossless-merge inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "c34695338629e8445ecf5d7eca2b3fe58deacd407c9713260eabd201d2d24979"
    },
    {
      "slug": "the-egyptian-fraction",
      "title": "THE EGYPTIAN FRACTION",
      "kicker": "a fraction split into distinct unit shares, greedily",
      "accent": "#70a860",
      "blurb": "The Egyptian fraction in the 5-window house format — write a proper fraction as a sum of distinct unit fractions (4/13 = 1/4 + 1/18 + 1/468). The Fibonacci–Sylvester greedy algorithm grabs the largest unit fraction that fits, 1/⌈q/p⌉, and subtracts, until nothing remains; it always terminates and, because the remaining numerator strictly shrinks, the denominators come out strictly increasing, hence distinct. Verified live (exact BigInt): for every reduced p/q with q ≤ 60, the greedy unit fractions are strictly increasing and sum exactly back to p/q. See the greedy split in 1D, pieces summed in 2D, and the distinct-unit-parts inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "7de6351478bb792ece574d7eb7c405389e30e65f3ece2277122867597421c576"
    },
    {
      "slug": "the-aliquot",
      "title": "THE ALIQUOT",
      "kicker": "the number that equals the sum of its parts",
      "accent": "#e0b020",
      "blurb": "The aliquot sum in the 5-window house format — s(n) adds up all of a number's proper divisors, sorting the integers into deficient (s<n), abundant (s>n), and the rare perfect (s=n): 6 = 1+2+3, 28 = 1+2+4+7+14. Two numbers form an amicable pair when each is the aliquot sum of the other — 220 and 284, known since antiquity. Verified live: exhaustively to 10000, the only perfect numbers are 6, 28, 496, 8128, and the amicable pairs include 220&284, each confirmed by summing divisors. See perfect and amicable numbers in 1D, a number weighed against its parts in 2D, and the identity-from-parts inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "5b0fcdaaf6d351f41dcc65add07d4dddc92dda15d5516e5f83401bfff82a31d4"
    },
    {
      "slug": "the-bulgarian-solitaire",
      "title": "THE BULGARIAN SOLITAIRE",
      "kicker": "any pile grinds down to the staircase",
      "accent": "#d06858",
      "blurb": "Bulgarian solitaire in the 5-window house format — deal n cards into piles of any sizes; each move, take one card from every pile and gather them into a single new pile. When n is triangular, n = 1+2+…+k, this process from any starting configuration always settles into the same staircase {k, k−1, …, 1} — a stable attractor it can never leave. Verified live: for triangular n up to 36, hundreds of random starts all converge to the staircase, which is itself a fixed point of the move. See the take-one-from-each move in 1D, a start funneling in 2D, and the self-organizing-attractor inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "b45be9faa921c7e0feead410dfa06eb57af67c2a46cbae0255817908db2e90c7"
    },
    {
      "slug": "the-gale-shapley",
      "title": "THE GALE-SHAPLEY",
      "kicker": "proposals settle into a matching no pair wants to break",
      "accent": "#58a0b0",
      "blurb": "The Gale–Shapley algorithm in the 5-window house format — pair two sides, each with ranked preferences, into a stable matching where no unmatched pair both prefer each other to their partners. Proposers propose in order; each reviewer holds the best offer and rejects the rest; rejects try their next choice. It always terminates with everyone matched, and is proposer-optimal: every proposer gets the best partner they could have in any stable matching. Verified live: over hundreds of random instances the output has no blocking pair and matches everyone, and for small n it is exactly the proposer-optimal stable matching found by brute force. See the propose-hold-reject loop in 1D, a matching in 2D, and the deferred-acceptance inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "f7f0f6f634bfadff74bcd547202314c294fc8e2b3b802c9a195d2422912eeae4"
    },
    {
      "slug": "the-stern-diatomic",
      "title": "THE STERN DIATOMIC",
      "kicker": "a sequence that lists every fraction exactly once",
      "accent": "#6ab0d0",
      "blurb": "Stern's diatomic sequence (fusc) in the 5-window house format — built by a(0)=0, a(1)=1, a(2n)=a(n), a(2n+1)=a(n)+a(n+1): 1,1,2,1,3,2,3,1,4,3,5,… It hides a miracle: the consecutive ratios a(n)/a(n+1) list every positive rational exactly once, in lowest terms, never repeating — the Stern–Brocot enumeration read straight off a sequence. Verified live: consecutive terms are always coprime, the ratios up to n=8000 are distinct and reduced, and every reduced p/q with p,q≤8 appears among them. See the recurrence in 1D, the ratios in 2D, and the all-fractions-on-one-thread inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "5ec5bd9a2a0497fb888db88838a90fda11e0b56be8f29a666a2c70cec48bd09d"
    },
    {
      "slug": "the-zeller",
      "title": "THE ZELLER",
      "kicker": "a formula that names any day of the week",
      "accent": "#c0a048",
      "blurb": "Zeller's congruence in the 5-window house format — a closed-form formula returning the day of the week for any date, no calendar lookup or day counting. It packs the irregular Gregorian rules (month lengths, leap years, century correction) into one modular expression: h = (d + ⌊13(m+1)/5⌋ + K + ⌊K/4⌋ + ⌊J/4⌋ + 5J) mod 7, treating January and February as months 13 and 14 of the prior year. Verified live: over 20000 random Gregorian dates (1901–2099), Zeller's congruence matches the reference calendar's weekday every time. See the formula in 1D, a date checked in 2D, and the calendar-as-formula inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "3d0b1a9ea9e848caf1431f7c21742244327b0a9a6b5c1f8db7702ac75c62657f"
    },
    {
      "slug": "the-karplus-strong",
      "title": "THE KARPLUS-STRONG",
      "kicker": "noise fed through a loop becomes a plucked note",
      "accent": "#a878c0",
      "blurb": "Karplus–Strong synthesis in the 5-window house format — a realistic plucked string from almost nothing: fill a length-N buffer with noise, then repeatedly output the front sample and feed back the average of two neighbours. The delay line sets the pitch; the averaging is a low-pass that lets high harmonics die faster than low ones, just as a real string decays. The fundamental sits near fs/(N+0.5), the delay length plus the filter's half-sample lag. Verified live: for several N the measured pitch (by autocorrelation) matches fs/(N+0.5) within ~2%, and the signal energy decays monotonically. See the ring buffer in 1D, a waveform in 2D, and the string-from-a-delay-line inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "5506df8bc53b84a33c3ec91575ac4f3f14b98d1c4cc5ea7a9f7aec1b13d5c6c0"
    },
    {
      "slug": "the-hofstadter-q",
      "title": "THE HOFSTADTER Q",
      "kicker": "a recurrence that feeds on itself, maybe off the edge",
      "accent": "#c86868",
      "blurb": "Hofstadter's Q-sequence in the 5-window house format — a recurrence that feeds on its own recent values as indices: Q(1)=Q(2)=1, Q(n) = Q(n − Q(n−1)) + Q(n − Q(n−2)). Unlike Fibonacci's fixed look-back, Q's look-back distance depends on itself, making it chaotic. Astonishingly, whether it stays well-defined forever (never reading an index ≤ 0) is an open problem; it merely appears to, as far as anyone has computed. Verified live: computed to n = 100000, every lookback index stays in range, and the first ten values match the hand-derived reference 1,1,2,3,3,4,5,5,6,6. See the self-reference in 1D, the chaotic plot in 2D, and the order's-edge inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "a24b79b8d9126cbe8e6bd43925d51b8709112a7c0ddd503ee838ad36eba755d7"
    },
    {
      "slug": "the-lehmer-code",
      "title": "THE LEHMER CODE",
      "kicker": "number a permutation with a mixed-radix odometer",
      "accent": "#e0b020",
      "blurb": "The Lehmer code (factorial number system) in the 5-window house format — give every permutation a unique index 0..k!−1, and every index its permutation back, a perfect bijection. It records for each position how many smaller elements sit to its right, then reads those counts as digits in a mixed radix whose place values are factorials (…,3!,2!,1!,0!) instead of powers of ten — an odometer whose wheels have different sizes. Verified live: for every permutation of up to 7 elements, encode-then-decode returns the original, and the indices cover 0..k!−1 exactly once. See the factoradic odometer in 1D, a permutation ranked in 2D, and the ordering-as-index inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "327e325d034d26342c07914949d74620a9e5ebb8fa3d6f126a4a9c3b20428d85"
    },
    {
      "slug": "the-havel-hakimi",
      "title": "THE HAVEL-HAKIMI",
      "kicker": "decide if a wiring diagram can exist, and build it",
      "accent": "#9d78c0",
      "blurb": "The Havel–Hakimi algorithm in the 5-window house format — given a wish-list of vertex degrees, can a simple graph (no loops, no double edges) deliver it? Its move is greedy and exact: take the hungriest vertex, connect it to the next-hungriest, cross those off, repeat. Run out of partners or go negative → impossible; reach all zeros → graphical, and the same steps build a realizing graph. Verified live: over hundreds of random degree sequences the verdict matches the independent Erdős–Gallai criterion, and when graphical the construction yields a simple graph with exactly those degrees. See the reduction in 1D, a built graph in 2D, and the existence-by-construction inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "b0820e1eab4c87cbbbd1779cfadd505b31d9fca69794029beb9ecfe79a0f1647"
    },
    {
      "slug": "the-pollard-rho",
      "title": "THE POLLARD RHO",
      "kicker": "crack a number open by walking a cycle",
      "accent": "#c86868",
      "blurb": "Pollard's rho algorithm in the 5-window house format — find a factor of a composite without trial-dividing to its square root. It iterates x ← x²+c (mod n) and watches for a collision modulo a hidden factor p: two iterates agreeing mod p (not mod n) reveal p as gcd(|x−y|, n). By the birthday paradox a mod-p collision appears after only ~√p steps, and Floyd's tortoise-and-hare finds it with no extra memory. Verified live: over hundreds of semiprimes n = p·q, the returned d satisfies 1 < d < n and divides n exactly. See the ρ-shaped cycle in 1D, a factorization in 2D, and the collide-mod-the-factor inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "6ed441a7e56b6bec0caf966296941b6f68dca2c36c5ea6ba28830b6b8c6d4bfb"
    },
    {
      "slug": "the-lagrange-four-square",
      "title": "THE LAGRANGE FOUR-SQUARE",
      "kicker": "every whole number is four squares",
      "accent": "#6ab0a0",
      "blurb": "Lagrange's four-square theorem in the 5-window house format — every non-negative integer is a sum of four squares: n = a²+b²+c²+d². Three squares is not enough (7 and 15 cannot be written with three), but four always suffice, no exceptions. It is a startling completeness: the squares 0,1,4,9,16,… are sparse, yet four of them (with repeats) tune to hit every whole number exactly. Verified live: for every n from 0 to 3000 a representation is found and its sum checked to equal n exactly. See the decompositions in 1D, four-square tiles in 2D, and the universal-quaternary-basis inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "82dc95c434fa13d634a50c38b7a9b335577eda104fc75385a53c1286bb2a9b2e"
    },
    {
      "slug": "the-cuckoo-hashing",
      "title": "THE CUCKOO HASHING",
      "kicker": "a key kicks out its neighbour and lands safe",
      "accent": "#58a0b0",
      "blurb": "Cuckoo hashing in the 5-window house format — worst-case O(1) lookup: any key lives in one of just two possible slots, given by two hash functions in two tables, so membership always reads at most two cells, never a long probe chain. Insertion borrows the cuckoo's trick: if your slot is taken, kick the occupant out and re-home it in its other slot, which may kick the next; rarely the chain loops and the tables rebuild with fresh hashes. Verified live: after building many tables at moderate load, every inserted key is found in ≤2 probes, none lost. See the two-nest eviction in 1D, filling tables in 2D, and the constant-time-by-construction inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "93f364ea975464f1584971e04cc94314c44f4db84a921387ab81186493d93ca7"
    },
    {
      "slug": "the-kummer",
      "title": "THE KUMMER",
      "kicker": "count the carries to know how many times p divides",
      "accent": "#c07850",
      "blurb": "Kummer's theorem in the 5-window house format — a hidden bridge between addition and divisibility: the number of times a prime p divides the binomial C(m+n, n) equals exactly the number of carries when adding m and n in base p. A divisibility question is answered by the schoolyard mechanics of carrying digits — no factorials needed, just add in base p and count carries. Verified live: over 20000 random (m,n,p), the p-adic valuation of C(m+n,n) by Legendre's formula equals the carry count of m+n in base p. See a base-p addition's carries in 1D, the count checked in 2D, and the divisibility-inside-addition inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "90e146afa06086ff78c9feb6ba2b0874b0773a11d051a5f9a924a16ea3228a30"
    },
    {
      "slug": "the-combinatorial-number-system",
      "title": "THE COMBINATORIAL NUMBER SYSTEM",
      "kicker": "number a subset with a descending choice",
      "accent": "#e0b020",
      "blurb": "The combinatorial number system in the 5-window house format — give every k-element subset of {0,…,n−1} a unique rank in 0..C(n,k)−1, and every rank its subset back, a bijection between combinations and integers. For a subset written descending c_k>…>c_1 the rank is C(c_k,k)+…+C(c_1,1); a 'factorial base for choosing.' Verified live: for every k-subset with n≤12, rank-then-unrank returns the original, and the ranks cover 0..C(n,k)−1 exactly once. See the descending-binomial rank in 1D, a subset ranked in 2D, and the selection-as-one-number inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "ce3a257ed5cfa382a597afbafe1a44e4d8e81335394a938a6ff99072c64b4521"
    },
    {
      "slug": "the-rodrigues",
      "title": "THE RODRIGUES",
      "kicker": "spin a vector about an axis by one formula",
      "accent": "#6ab0d0",
      "blurb": "Rodrigues' rotation formula in the 5-window house format — rotate a vector v about a unit axis k by angle θ in one expression: v' = v cosθ + (k×v) sinθ + k(k·v)(1−cosθ). It splits v into the part along the axis (untouched) and the perpendicular part (spun in its plane), reassembling the rotated vector without forming a full matrix — the axis–angle rotation in closed form. Verified live: over 5000 random axes, angles, and vectors, Rodrigues equals the equivalent rotation matrix, preserves length, inverts under −θ, and fixes the axis. See the split-and-spin in 1D, a rotation checked in 2D, and the rotation-without-a-matrix inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "729cbbe65ac6ab3122bff137b23c4345a088595f3febc4ad68d5505eee4a7bde"
    },
    {
      "slug": "the-leftist-heap",
      "title": "THE LEFTIST HEAP",
      "kicker": "two heaps fuse in log time",
      "accent": "#58a0b0",
      "blurb": "The leftist heap in the 5-window house format — a priority queue whose defining trick is a fast merge: two heaps combine in O(log n). Every node stores an s-value (distance to the nearest empty slot) and the heap keeps every node's left child at least as tall as its right (s(left) ≥ s(right)), so the right spine stays ≤ log n long. Merging walks two right spines and swaps children to restore the invariant; insert and extract-min are merges in disguise. Verified live: over hundreds of random heaps, extract-min yields fully sorted order, the leftist invariant holds at every node, and the heap property holds. See the right-spine merge in 1D, a heap tree in 2D, and the built-to-fuse inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "bdd3c1141d97d950f5556edd5c7899fac18c2e091b35cf3f1e0387ffd2934e62"
    },
    {
      "slug": "the-quadratic-reciprocity",
      "title": "THE QUADRATIC RECIPROCITY",
      "kicker": "a golden law linking two primes' squares",
      "accent": "#b06868",
      "blurb": "Quadratic reciprocity in the 5-window house format — Gauss's golden theorem links two questions that look independent: is p a square mod q, and is q a square mod p? With the Legendre symbol (a/p) = +1 if a is a nonzero square mod p else −1, the law says for distinct odd primes (p/q)·(q/p) = (−1)^((p−1)/2·(q−1)/2): the two answers agree unless both primes are ≡ 3 (mod 4), when they flip. Verified live: for every pair of distinct odd primes below 200 the reciprocity identity holds (Legendre symbols by Euler's criterion), and the −1 and 2 supplements hold too. See the flip rule in 1D, a Legendre grid in 2D, and the flip-the-question inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "c8266748db4925d1052e508709219d67a35ff365d4924f6d56346d1506fd3243"
    },
    {
      "slug": "the-descartes-circle",
      "title": "THE DESCARTES CIRCLE",
      "kicker": "four kissing circles bound by one curvature law",
      "accent": "#58a0b0",
      "blurb": "Descartes' circle theorem in the 5-window house format — four mutually tangent (kissing) circles are bound by one law on their curvatures k=1/r: (k₁+k₂+k₃+k₄)² = 2(k₁²+k₂²+k₃²+k₄²). Given three tangent circles the fourth's curvature is k₁+k₂+k₃ ± 2√(k₁k₂+k₂k₃+k₃k₁) — two solutions, an inner and outer kiss — and a complex version gives the fourth center too. Verified live: over thousands of tangent triples the fourth curvature satisfies the identity and the computed fourth circle is genuinely tangent to all three. See the kissing circles in 1D, a fourth circle solved in 2D, and the tangency-as-arithmetic inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "5d43600a17ec38a2c7dccce5d2951bee95f4b7f043f524b27d27235ad8e70d39"
    },
    {
      "slug": "the-golomb-sequence",
      "title": "THE GOLOMB SEQUENCE",
      "kicker": "a sequence that counts its own values",
      "accent": "#c07850",
      "blurb": "Golomb's self-describing sequence in the 5-window house format — the unique non-decreasing sequence of positive integers where a(n) is the number of times n appears in the sequence itself: 1,2,2,3,3,4,4,4,5,5,5,… The single 1 says '1 appears once'; the two 2s say '2 appears twice'; the three 4s say '4 appears three times' — and a(4)=3. It bootstraps itself via a(n) = 1 + a(n − a(a(n−1))). Verified live: the sequence is non-decreasing and for every value v (whose full run lies in range) the count of v equals a(v) exactly. See the self-narration in 1D, run-length counts in 2D, and the self-authoring inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "58f2aec586d31915294cfd99e01e46f4b9b730a2eb89f02f3d8ca8084c0e91ec"
    },
    {
      "slug": "the-padovan",
      "title": "THE PADOVAN",
      "kicker": "numbers grown at the plastic ratio",
      "accent": "#6ab0d0",
      "blurb": "The Padovan sequence in the 5-window house format — Fibonacci's quieter cousin: P(n) = P(n−2) + P(n−3), starting 1,1,1,2,2,3,4,5,7,9,12,16,… Instead of summing the two previous terms it skips one, and its growth ratio converges not to the golden ratio but to the plastic number ρ ≈ 1.324718 — the unique real root of x³=x+1, the smallest Pisot number. It also satisfies the identity P(n)=P(n−1)+P(n−5). Verified live: the recurrence holds, the identity holds, and P(n)/P(n−1) converges to the plastic number, the exact root of x³−x−1. See the sequence in 1D, the triangle spiral in 2D, and the different-reach inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "bebd00841aa737c153d7bb111b76f4d033e5e9d853d213514623eabb5879ee5f"
    },
    {
      "slug": "the-primitive-root",
      "title": "THE PRIMITIVE ROOT",
      "kicker": "one root that generates every residue",
      "accent": "#e0b020",
      "blurb": "The primitive root in the 5-window house format — a single number g mod a prime p whose powers g¹,…,g^(p−1) run through every nonzero residue 1,…,p−1 exactly once before returning to 1: a generator of the multiplicative group, one seed reaching every residue by repeated multiplication. Primitive roots underlie discrete logarithms, Diffie–Hellman, and RNGs. Verified live: for primes below 300, a primitive root's powers form a permutation of {1,…,p−1}, and the count of primitive roots equals φ(p−1) exactly. See a generator's cycle in 1D, the orbit around the residues in 2D, and the all-from-one-seed inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "c9f8d11aeeb4b29b3e1a33172a60646c200c65cdeb4a9e992f666bbe61062201"
    },
    {
      "slug": "the-eulerian-numbers",
      "title": "THE EULERIAN NUMBERS",
      "kicker": "count permutations by their climbs",
      "accent": "#b06868",
      "blurb": "Eulerian numbers in the 5-window house format — A(n,k) counts the permutations of {1,…,n} with exactly k ascents (positions where the next element is larger). They form a triangle like Pascal's: A(n,k) = (k+1)·A(n−1,k) + (n−k)·A(n−1,k−1). Each row sums to n! (every permutation has some ascents), and the triangle is symmetric A(n,k)=A(n,n−1−k), since reversing a permutation swaps ascents and descents. Verified live: the recurrence matches a brute-force count of permutations by ascents for n≤8, each row sums to n!, and the symmetry holds. See the triangle in 1D, a row bar chart in 2D, and the count-don't-list inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "89a0b616a760cc962979d627d44d8a0ae9534068bba5cca03ed9dabfcbd88ec6"
    },
    {
      "slug": "the-negabinary",
      "title": "THE NEGABINARY",
      "kicker": "count in base minus-two, no sign needed",
      "accent": "#6ab0d0",
      "blurb": "Negabinary in the 5-window house format — base −2: the same digits {0,1} as binary, but place values are powers of −2 (1, −2, 4, −8, 16, …). The alternating signs let a single unsigned digit string represent every integer, positive and negative, with no sign bit and no two's-complement: −6 is 1110 (= −8 + 4 − 2). Encoding just repeatedly takes n mod 2 and divides by −2, and each integer's representation is unique. Verified live: over every integer from −2000 to 2000, decode(encode(n)) returns n and all representations are distinct. See the signed place values in 1D, a number's digits in 2D, and the sign-folded-into-the-base inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "10d98451663c4f68851b02eb4473e06cacf03a2e8358939abe144cacb04615d7"
    },
    {
      "slug": "the-boyer-moore-majority",
      "title": "THE BOYER-MOORE MAJORITY",
      "kicker": "one survivor of pairwise cancellation",
      "accent": "#b06868",
      "blurb": "The Boyer–Moore majority vote in the 5-window house format — find an element appearing in more than half a stream using O(1) memory: one candidate and one counter. Sweep once: if the counter is zero, adopt the current element; if the next matches, increment, else decrement. Matching and non-matching elements cancel in pairs, so a true majority cannot be fully cancelled — it is the last one standing, confirmed by a single verification pass. Verified live: over 3000 random arrays (with and without a majority), the O(1)-space vote plus verify pass returns exactly what a brute frequency count does. See the cancellation in 1D, a live stream in 2D, and the majority-by-cancellation inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "a70e410b31963e68e31986c37ec2639cf55bc2f898a89ec6fe3b4f90bca5d4b4"
    },
    {
      "slug": "the-lyndon-factorization",
      "title": "THE LYNDON FACTORIZATION",
      "kicker": "factor a string into non-increasing necklaces",
      "accent": "#58a0b0",
      "blurb": "The Lyndon factorization (Chen–Fox–Lyndon) in the 5-window house format — split any string uniquely into a non-increasing sequence of Lyndon words. A Lyndon word is strictly smaller than all its rotations — an aperiodic necklace with a canonical start. Duval's algorithm computes it in linear time and constant extra space, in one left-to-right scan; every string has exactly one such factorization, and 'banana' becomes b·an·an·a. Verified live: over 3000 random strings, Duval's factors concatenate back to the input, each factor is a Lyndon word, and the factors are non-increasing. See a Lyndon word vs its rotations in 1D, a factored string in 2D, and the canonical-cut inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "d585bf13b51335b8aca8011a1c10fed06d47f9ba2f51ea56c9327b6805a28ba1"
    },
    {
      "slug": "the-lifting-the-exponent",
      "title": "THE LIFTING THE EXPONENT",
      "kicker": "count how many times p divides a power difference",
      "accent": "#e0b020",
      "blurb": "The Lifting the Exponent lemma in the 5-window house format — an olympiad power tool: for an odd prime p, if p divides a−b but neither a nor b, then vₚ(aⁿ−bⁿ) = vₚ(a−b) + vₚ(n). The whole exponent of p in a huge power difference is the exponent in the base difference plus the exponent in n — the extra factors of p come only from n itself. Verified live (exact BigInt): over 3000 random valid (a,b,n,p), vₚ of aⁿ−bⁿ computed directly equals vₚ(a−b)+vₚ(n). See the lemma in 1D, a case checked in 2D, and the exponent-lifted inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "4085a8feadf9565093e073f0ae0d21b7ac5803bf65f25d6dd7bbad917922114d"
    },
    {
      "slug": "the-liang-barsky",
      "title": "THE LIANG-BARSKY",
      "kicker": "clip a line to a window by four parameters",
      "accent": "#70a860",
      "blurb": "The Liang–Barsky algorithm in the 5-window house format — clip a line segment to a rectangular window using its parametric form P(t) = P₀ + t·(P₁−P₀), t∈[0,1]. Each of the four window edges gives an inequality p·t ≤ q; the algorithm tightens the entry parameter u₁ and exit u₂ against all four, rejecting if u₁ > u₂. The surviving [u₁,u₂] gives the clipped endpoints — no repeated edge intersections, just parameter bookkeeping. Verified live: over thousands of random segments, the clipped part lies entirely inside the rectangle and its endpoints sit exactly on the original line. See the parameter interval in 1D, clipped segments in 2D, and the one-shrink inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "aa97704b201ccc1b0368fb630d3eda1c966cbe79e51315959a4c77c216d1448c"
    },
    {
      "slug": "the-continued-fraction-sqrt",
      "title": "THE CONTINUED FRACTION OF ROOT N",
      "kicker": "the square root's fraction repeats in a palindrome",
      "accent": "#c0a048",
      "blurb": "The continued fraction of √n in the 5-window house format — for non-square n it is eventually periodic with a striking shape: √n = [a₀; a₁,…,a_L] where the repeating block ends in 2a₀ and the part before it, (a₁,…,a_{L−1}), is a palindrome. So √7 = [2; 1,1,1,4] and √19 = [4; 2,1,3,1,2,8]. The convergent just before the period closes gives the fundamental solution of Pell's equation x²−n·y² = ±1. Verified live: for every non-square n up to 1000 the period ends in 2a₀, its front is a palindrome, and the pre-period convergent solves Pell (exact BigInt). See the periods in 1D, one expansion in 2D, and the Pell-from-rhythm inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "418e599d2a927d12a305366201efef85de74191aa6b919f1e7b44c5e30658172"
    },
    {
      "slug": "the-haar-wavelet",
      "title": "THE HAAR WAVELET",
      "kicker": "average and difference a signal reversibly",
      "accent": "#58a0b0",
      "blurb": "The Haar wavelet transform in the 5-window house format — the simplest multiresolution analysis: repeatedly replace pairs of samples by their average and their difference. The averages form a coarser signal; the differences capture the detail lost at each scale. With the √2 normalization the transform is orthonormal — a rotation into a wavelet basis — so it preserves energy and is perfectly invertible. Verified live: over 2000 random signals, the inverse Haar reconstructs the input exactly and the sum of squared coefficients equals the sum of squared samples. See the average/difference step in 1D, coefficients in 2D, and the multiscale inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "e1d4ebc885b4e0a3ad76fa1713548c878fc32f2e72a48dcaf597dcd0cd0663fa"
    },
    {
      "slug": "the-bell-numbers",
      "title": "THE BELL NUMBERS",
      "kicker": "count the ways to partition a set",
      "accent": "#b06868",
      "blurb": "The Bell numbers in the 5-window house format — B(n) counts the ways to partition a set of n elements into non-empty unlabeled blocks: 1,1,2,5,15,52,203,877,… Three elements split 5 ways, four split 15. They are built with almost no arithmetic by Bell's triangle (Aitken's array): start each row with the last entry of the previous, then each next is the one to its left plus the one above-left; B(n) is the first number in row n — and equals Σ_k S(n,k), the Stirling numbers of the second kind. Verified live: the triangle matches a brute-force count of set partitions for n≤8, and B(n) equals the Stirling sum. See Bell's triangle in 1D, a row checked in 2D, and the count-without-listing inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "a394bf8bb98f0fbf3cd1162bf1fd4cf86eca11f613324107c94e8357c9b2bbdc"
    },
    {
      "slug": "the-median-of-medians",
      "title": "THE MEDIAN OF MEDIANS",
      "kicker": "pick the k-th smallest in guaranteed linear time",
      "accent": "#70a860",
      "blurb": "Median of medians in the 5-window house format — find the k-th smallest element in guaranteed linear time (O(n) worst case). The trick is a provably good pivot: split into groups of five, take each group's median, then recursively take the median of those medians. That pivot beats at least 30% of the elements on each side, so the recursion shrinks fast enough to stay linear — no adversarial input can force it slow. Verified live: over 3000 random arrays, the element selected for rank k equals the true k-th smallest from a full sort. See the groups-of-five pivot in 1D, a selection in 2D, and the certified-pivot inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "ef23d57fd703213f412c8a875e7214f98b2b09a1d8435860dac1b426ce988074"
    },
    {
      "slug": "the-jarvis-march",
      "title": "THE JARVIS MARCH",
      "kicker": "wrap a hull around points like a gift",
      "accent": "#6ab0d0",
      "blurb": "The Jarvis march (gift wrapping) in the 5-window house format — find the convex hull the way you'd wrap a present: start at the guaranteed-extreme leftmost point, then repeatedly pick the next hull vertex as the one making every other point lie to its left (the most clockwise turn). Each step wraps one more edge around the outside until you return to the start. It runs in O(n·h) time (h = hull vertices), fast when the hull is small. Verified live: over 1500 random point sets, the gift-wrapped hull matches an independent monotone-chain hull and every point lies inside or on it. See the wrapping in 1D, a hull in 2D, and the pull-it-taut inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "bf8b7e65f75e788763beeef84c947fc81e96c55a926ad775fd43ae450f2ae6a3"
    },
    {
      "slug": "the-edwards-curve",
      "title": "THE EDWARDS CURVE",
      "kicker": "a curve whose addition never fails",
      "accent": "#b06868",
      "blurb": "The twisted Edwards curve in the 5-window house format — a·x² + y² = 1 + d·x²·y² carries an addition law with a rare virtue: it is complete, the same formula works for every pair of points with no special cases (no separate doubling rule, no point-at-infinity). The neutral element is the ordinary point (0,1) and the inverse of (x,y) is (−x,y); when a is a square and d is a non-square mod p, the points form an abelian group with the addition never breaking. This is why Ed25519 uses Edwards curves. Verified live: over a small curve, every sum is on the curve, (0,1) is the identity, (−x,y) inverts, and the addition is associative over thousands of triples. See the complete formula in 1D, points added in 2D, and the exception-free group law in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "71413a02faf092c0d97ebb18306aaae4dff810cfd89e1ca47140ad9788510314"
    },
    {
      "slug": "the-cornacchia",
      "title": "THE CORNACCHIA",
      "kicker": "represent a number as x squared plus d y squared",
      "accent": "#e0b020",
      "blurb": "Cornacchia's algorithm in the 5-window house format — solve x² + d·y² = m in integers (when a solution exists) astonishingly fast. It first finds a square root r of −d modulo m (r² ≡ −d), then runs a Euclidean-style descent on (m, r), stopping the moment the remainder drops below √m; that remainder is x, and y follows from (m − x²)/d being a perfect square. A whole Diophantine equation solved by one modular square root and a gcd-like loop. Verified live: over hundreds of primes m with a representation, the returned (x,y) satisfies x² + d·y² = m exactly. See representations in 1D, squares summing to m in 2D, and the reduce-don't-search inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "826107b8a27b16c97a5cc3ebc610d04ae082d4edbf9574e3369e2a2573aca610"
    },
    {
      "slug": "the-quadtree",
      "title": "THE QUADTREE",
      "kicker": "quarter the plane recursively to query it fast",
      "accent": "#c07850",
      "blurb": "The point-region quadtree in the 5-window house format — index 2D points by recursively quartering the plane. Each node holds a small bucket; when it overflows it splits into four children (NW, NE, SW, SE), redistributing its points. To answer a range query — which points fall in a rectangle? — you descend only into children whose regions intersect the query, skipping vast empty or far-away quadrants. Verified live: over 1000 random point sets and query rectangles, the quadtree returns exactly the same points as a brute-force scan. See the subdivision in 1D, a query in 2D, and the pruned-descent inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "395dfcea88c711d6a4f4dfa3126a0ba405ce21083972e54c9e5a7f50e1df7952"
    },
    {
      "slug": "the-hofstadter-female-male",
      "title": "THE HOFSTADTER FEMALE-MALE",
      "kicker": "two sequences that define each other",
      "accent": "#a878c0",
      "blurb": "Hofstadter's Female and Male sequences in the 5-window house format — defined by mutual recursion, each needing the other to take a step: F(0)=1, M(0)=0, F(n) = n − M(F(n−1)), M(n) = n − F(M(n−1)). Neither can be computed alone; they must be unrolled together, each new term reaching into the other. From this tangle emerge two interleaving sequences: F = 1,1,2,2,3,3,4,5,5,6,6,… and M = 0,0,1,2,2,3,4,4,5,6,6,… Verified live: computed to 100000, both are well-defined and the opening values match the reference sequences. See the mutual recurrence in 1D, both plotted in 2D, and the entanglement inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "b977116f06cbca6e0b910f79cc278ec5478709a10053a3543156babfb2dc9fe2"
    },
    {
      "slug": "the-dutch-national-flag",
      "title": "THE DUTCH NATIONAL FLAG",
      "kicker": "sort three colours in one pass",
      "accent": "#70a860",
      "blurb": "The Dutch national flag problem in the 5-window house format — Dijkstra's one-pass, in-place, three-pointer partition of an array of three values (red/white/blue, or <,=,> a pivot) into three contiguous bands. A low and mid pointer advance from the front, a high from the back; each element mid meets is swapped into the correct band and the pointers close in — no counting, no second pass. It is the heart of three-way quicksort. Verified live: over 3000 random arrays and pivots, the one-pass partition leaves everything < pivot, then =, then >, and the output is a permutation of the input. See the three pointers in 1D, a partition in 2D, and the one-sweep inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "24aba7520c1f50809e48f8fc0e64af2c1f68f52b0a1693875b55c64c53f02915"
    },
    {
      "slug": "the-sociable-numbers",
      "title": "THE SOCIABLE NUMBERS",
      "kicker": "numbers whose divisor-sums loop back in a chain",
      "accent": "#d06858",
      "blurb": "Sociable numbers in the 5-window house format — the aliquot dynamics: repeatedly replace n by s(n), the sum of its proper divisors, and watch the orbit. A perfect number is a fixed point (1-cycle: s(6)=6); an amicable pair is a 2-cycle (220→284→220); sociable numbers close a longer loop: 12496 → 14288 → 15472 → 14536 → 14264 → back to 12496, a 5-cycle where each is the divisor-sum of the last. Verified live: iterating s(n) from 6, 220, and 12496 closes cycles of length 1, 2, and 5 exactly. See the chains in 1D, an orbit closing in 2D, and the identity-as-cycle inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "49044cbed442a480c87def2065ad97ddf22874cdf1a362bbd1b0872c20ea5ab6"
    },
    {
      "slug": "the-pepin",
      "title": "THE PEPIN",
      "kicker": "one test decides a Fermat prime",
      "accent": "#b06868",
      "blurb": "Pépin's test in the 5-window house format — decide whether a Fermat number F_k = 2^(2^k)+1 is prime with a single modular exponentiation: for k ≥ 1, F_k is prime if and only if 3^((F_k−1)/2) ≡ −1 (mod F_k). One test, no factoring, and it is an iff, not a probabilistic guess. Fermat conjectured every F_k prime; Pépin's test (with Euler's factor) shows F_5 = 2³²+1 is composite = 641 × 6700417. Verified live (exact BigInt): F_1…F_4 pass (prime), F_5 fails (composite), and F_5 = 641 × 6700417. See the Fermat numbers in 1D, verdicts in 2D, and the one-exponentiation inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "fdc65a8386d75542bbf37d3bb7fe894029128f83cbb62c2136cbd89963c288fb"
    },
    {
      "slug": "the-continued-fraction-e",
      "title": "THE CONTINUED FRACTION OF E",
      "kicker": "the number e written as a patterned fraction",
      "accent": "#c0a048",
      "blurb": "The continued fraction of e in the 5-window house format — where π's CF looks random, e = [2; 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, …] follows a clean pattern: a 2, then repeating triples (1, 2m, 1) for m = 1, 2, 3, … The even numbers 2, 4, 6, 8 march through, each flanked by ones. Truncating gives rational convergents that rush toward e. Verified live: the generated terms obey the 2;(1,2m,1) pattern, and the convergent p/q (exact BigInt) matches e to floating precision. See the pattern in 1D, convergents approaching e in 2D, and the order-in-a-transcendental inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "909c12897988f12de4eb17eb550bf51db57c0f0f801c8dad4ad25819528470fa"
    },
    {
      "slug": "the-interval-tree",
      "title": "THE INTERVAL TREE",
      "kicker": "query which intervals overlap fast",
      "accent": "#58a0b0",
      "blurb": "The interval tree in the 5-window house format — answer 'which stored intervals overlap this query?' quickly by augmenting a binary search tree. Nodes are keyed by each interval's left endpoint; every node also caches the maximum right endpoint in its subtree. That cache lets a query prune whole branches: if a subtree's max-high is below the query's low, nothing there can overlap, so skip it. A scan of every interval becomes a guided descent. Verified live: over 2000 random interval sets and queries, the tree returns exactly the intervals a brute-force scan finds. See the max-high pruning in 1D, an overlap query in 2D, and the cached-reach inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "cf519b1669a1948a62b19ac8dd27856e435ee0335d6bfb841b83397a172dbfd5"
    },
    {
      "slug": "the-cohen-sutherland",
      "title": "THE COHEN-SUTHERLAND",
      "kicker": "clip a line by four boundary bits",
      "accent": "#c07850",
      "blurb": "The Cohen–Sutherland algorithm in the 5-window house format — clip a line to a rectangular window using 4-bit region codes (outcodes). The plane is divided into 9 regions around the window; each endpoint gets a 4-bit code (left, right, below, above). If both codes are 0 the segment is fully inside (accept); if their bitwise AND is non-zero both endpoints share an outside half-plane, so the segment misses (reject); otherwise clip against one crossed boundary and repeat. A few bit tests replace geometric case analysis. Verified live: over thousands of segments, the clipped part lies inside the window, its endpoints sit on the original line, and the result matches the Liang–Barsky clip. See the outcode regions in 1D, clipped segments in 2D, and the four-bits inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "039b09d0b68e1daafbd2de46e4a1ffba9c90c43d18f949a860ea5e6257361c9a"
    },
    {
      "slug": "the-gaussian-primes",
      "title": "THE GAUSSIAN PRIMES",
      "kicker": "primes of the complex plane, split or inert",
      "accent": "#e0b020",
      "blurb": "Gaussian primes in the 5-window house format — the primes of the complex integers ℤ[i] = {a+bi}. A rational prime doesn't always stay prime: p=2 and every p ≡ 1 (mod 4) splits into two conjugate Gaussian primes (5=(2+i)(2−i), 13=(3+2i)(3−2i)), because such p is a sum of two squares (Fermat); every p ≡ 3 (mod 4) stays inert. The norm N(a+bi)=a²+b² is multiplicative, tying factorization together. Verified live: the norm is multiplicative over thousands of pairs, and a rational prime splits iff p=2 or p ≡ 1 (mod 4). See split vs inert in 1D, the lattice in 2D, and the primality-in-the-plane inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "e2b89afb775b6d8fc6b25cf94ca8eadab2f9e3e86919bb583820f9ff49defc6f"
    },
    {
      "slug": "the-tribonacci",
      "title": "THE TRIBONACCI",
      "kicker": "a sequence at the tribonacci ratio",
      "accent": "#6ab0d0",
      "blurb": "The tribonacci sequence in the 5-window house format — Fibonacci with a three-term memory: each term is the sum of the previous three. 0,0,1,1,2,4,7,13,24,44,81,149,… The ratio of consecutive terms converges not to the golden ratio but to the tribonacci constant η ≈ 1.839286755 — the unique real root of x³ = x² + x + 1. Sum the last two → φ; sum the last three → η. Verified live: the three-term recurrence holds, and T(n)/T(n−1) converges to the real root of x³−x²−x−1. See the sequence in 1D, growth converging in 2D, and the deeper-memory inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "6c62a0c1039eb7e0eec9b00a311b4e2cb3d47f38e5f524515a7d5ee14e2c4566"
    },
    {
      "slug": "the-b-tree",
      "title": "THE B-TREE",
      "kicker": "a balanced tree that keeps all leaves level",
      "accent": "#c0a048",
      "blurb": "The B-tree in the 5-window house format — the balanced search tree that runs databases and filesystems. Unlike a binary tree, each node holds many keys and children, so the tree stays short and bushy (ideal when each node is a disk block). It self-balances by splitting a full node and pushing its median key up, keeping every leaf at exactly the same depth regardless of insertion order; keys stay sorted, and every non-root node stays between half-full and full. Verified live: after random insertions, an in-order walk yields the sorted keys, all leaves share one depth, and every node's key count stays within the B-tree bounds. See a node split in 1D, the tree balancing in 2D, and the split-not-rotate inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "70907047b996b737a4d13e095903863ea399075a083b9bb7a549969475ca9399"
    },
    {
      "slug": "the-happy-number",
      "title": "THE HAPPY NUMBER",
      "kicker": "digit-squares that reach 1 or loop",
      "accent": "#d06858",
      "blurb": "Happy numbers in the 5-window house format — a simple game: replace n by the sum of the squares of its digits, and repeat. If you reach 1, n is happy (7→49→97→130→10→1). If not, you fall into a single unavoidable 8-cycle: 4→16→37→58→89→145→42→20→4. Astonishingly, every starting number does one or the other — the digit-square map has exactly these two fates. Verified live: for every n up to 100000, the orbit reaches 1 or enters the 4-cycle, and the unhappy 8-cycle is confirmed step by step. See the two fates in 1D, an orbit in 2D, and the two-destinations inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "38e4b1c58bac7918512aad596abec61391d060fd861fa2fc68674602bc89a20d"
    },
    {
      "slug": "the-bridges",
      "title": "THE BRIDGES",
      "kicker": "the edges whose loss disconnects",
      "accent": "#c07850",
      "blurb": "Bridge-finding in the 5-window house format — a bridge is an edge whose removal disconnects the graph, a single link with no backup path. Tarjan's algorithm finds all bridges in one depth-first traversal using low-link values: as DFS explores, each vertex records the earliest node reachable from its subtree via a back edge, and a tree edge (u,v) is a bridge iff low(v) > discovery(u) — v's subtree can climb no higher than v. No re-checking, no edge-by-edge removal. Verified live: over hundreds of random graphs, the low-link bridges are exactly the edges whose removal increases the connected-component count. See the low-link test in 1D, bridges highlighted in 2D, and the one-descent inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "08ef7758ae5c483e15108cd505b8d56a2ba82f41653569a9200e0d52d94d21e8"
    },
    {
      "slug": "the-hensel-lifting",
      "title": "THE HENSEL LIFTING",
      "kicker": "lift a root to higher and higher prime power",
      "accent": "#6ab0d0",
      "blurb": "Hensel's lemma in the 5-window house format — Newton's method for p-adic numbers: a simple root of a polynomial mod a prime p can be lifted to a root mod p², then p³, then any p^k, each step uniquely refining the solution. If f(r) ≡ 0 (mod p) and f'(r) ≢ 0 (mod p), one correction r ← r − f(r)·f'(r)⁻¹ sharpens the root by a full power of p. It builds modular square roots and p-adic solutions digit by p-adic digit. Verified live: lifting a root of x²−A from mod p to mod p^k yields r with r² ≡ A (mod p^k) exactly, over thousands of cases. See the lift chain in 1D, levels checked in 2D, and the p-adic-ascent inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "12c3c017ed96efd7739f9eda31f22e5d813272b4c4b9019d0cf0e686bc5ead0f"
    },
    {
      "slug": "the-quickhull",
      "title": "THE QUICKHULL",
      "kicker": "wrap a hull by divide and conquer",
      "accent": "#70a860",
      "blurb": "Quickhull in the 5-window house format — the convex hull by divide and conquer, the quicksort of geometry. Take the two extreme points (leftmost, rightmost); the line between them splits the rest into two sides. On each side, find the point farthest from the line — it must be a hull vertex; its triangle's interior is discarded and the two outer sub-regions recurse. Points far from the current hull are found first, so interior clusters are culled quickly. Verified live: over 1500 random point sets, quickhull's hull (collinear vertices canonicalized) equals an independent monotone-chain hull, and every point lies inside or on it. See the split in 1D, a hull in 2D, and the discard-the-interior inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "fbac4c6c6e6120cf89a25dc4e8023986d972f2abda788df675871a417fdf8d1e"
    },
    {
      "slug": "the-lucas-number",
      "title": "THE LUCAS NUMBER",
      "kicker": "Fibonacci's companion sequence",
      "accent": "#58a0b0",
      "blurb": "The Lucas numbers in the 5-window house format — Fibonacci's companion: same recurrence L(n)=L(n−1)+L(n−2), but starting 2, 1 instead of 0, 1, giving 2,1,3,4,7,11,18,29,47,76,… They shadow the Fibonacci numbers with elegant identities: L(n) = F(n−1) + F(n+1), and L(n)² − 5·F(n)² = 4·(−1)ⁿ. Their ratio also tends to the golden ratio φ, and L(n) = φⁿ + ψⁿ exactly. Verified live (exact BigInt): the recurrence holds, L(n)=F(n−1)+F(n+1), and L(n)²−5F(n)²=4(−1)ⁿ for n up to 80. See both sequences in 1D, the identities in 2D, and the companion inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "73c4d9e7d1d8e5f080c326d823ace28ec3d8b65a58949236f14099e29243f849"
    },
    {
      "slug": "the-mo-algorithm",
      "title": "THE MO ALGORITHM",
      "kicker": "reorder queries to answer them fast",
      "accent": "#c0a048",
      "blurb": "Mo's algorithm in the 5-window house format — answer many range queries offline (e.g. 'how many distinct values in a[l..r]?') fast by reordering the questions. Sort the queries so consecutive ones have nearly the same window, then slide two pointers (l and r), adding and removing one element at a time while maintaining a running answer. Sorting by √n-sized blocks of the left endpoint bounds the total pointer movement, turning many hard queries into one long sweep. Verified live: over hundreds of instances, Mo's distinct-count answers exactly match a naive per-query recount. See the block-sort in 1D, sliding queries in 2D, and the reorder-and-sweep inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "5eeedb84c0761013ef77c7de84134a71d4a65d3aa2cd0b0f507934694f4f18b1"
    },
    {
      "slug": "the-armstrong",
      "title": "THE ARMSTRONG",
      "kicker": "numbers that rebuild themselves from digit-powers",
      "accent": "#d06858",
      "blurb": "Armstrong (narcissistic) numbers in the 5-window house format — numbers that rebuild themselves from their own digits: raise each digit to the power of the number of digits, sum, and get the number back. 153 = 1³+5³+3³. 9474 = 9⁴+4⁴+7⁴+4⁴. Every single-digit number is trivially one, and beyond that they are rare and finite in each base — only 88 exist in base ten, the largest a 39-digit number. Verified live: the Armstrong numbers up to 100000 are exactly 1–9, 153, 370, 371, 407, 1634, 8208, 9474, 54748, 92727, 93084. See the decompositions in 1D, a number checked in 2D, and the self-reflection inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "9aef00eafa3573c31311049b50dea9084c228df6d9df4baedf388cab28a90974"
    },
    {
      "slug": "the-jacobsthal",
      "title": "THE JACOBSTHAL",
      "kicker": "a sequence doubling its two-back term",
      "accent": "#6ab0d0",
      "blurb": "The Jacobsthal numbers in the 5-window house format — Fibonacci's shape with a twist: J(n) = J(n−1) + 2·J(n−2), the two-back term doubled. From J(0)=0, J(1)=1 they run 0,1,1,3,5,11,21,43,85,171,… alternately just below and above the powers of two. They have a clean closed form J(n) = (2ⁿ − (−1)ⁿ)/3 and a striking identity: J(n)+J(n+1) = 2ⁿ — consecutive Jacobsthal numbers sum exactly to a power of two. Verified live (exact BigInt): the recurrence holds, the closed form holds, and J(n)+J(n+1)=2ⁿ, for n up to 90. See the sequence in 1D, the power-of-two sums in 2D, and the doubled-weight inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "42b7b137144925e2222c4f5c1a52e2c37d9d56edb960485ffe4a6872eea7e37f"
    },
    {
      "slug": "the-proth",
      "title": "THE PROTH",
      "kicker": "one witness decides a Proth prime",
      "accent": "#b06868",
      "blurb": "Proth's theorem in the 5-window house format — a fast, exact primality test for Proth numbers, those of the form N = k·2ⁿ+1 with k odd and k < 2ⁿ. N is prime if and only if there exists an integer a with a^((N−1)/2) ≡ −1 (mod N); such an a is a witness to primality, and for a prime Proth number half of all bases work, so a small search finds one fast. It powers the search for many of the largest known primes. Verified live: over thousands of Proth numbers, 'a witness exists' matches primality (by trial division) exactly, and known Proth primes are witnessed. See the witness test in 1D, a verdict in 2D, and the one-witness inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "73b57f8a4125b2d0f03a35f950b8b5d58ba633ef997a8e0dbe7d225d37ae9383"
    },
    {
      "slug": "the-engel-expansion",
      "title": "THE ENGEL EXPANSION",
      "kicker": "a number as a sum of ascending unit fractions",
      "accent": "#c0a048",
      "blurb": "The Engel expansion in the 5-window house format — write any real in (0,1] as a sum of ascending unit fractions with nested denominators: x = 1/a₁ + 1/(a₁a₂) + 1/(a₁a₂a₃) + …, where the a's are non-decreasing integers ≥ 2. It is built greedily: take aₖ = ⌈1/u⌉, subtract, continue with u·aₖ − 1. Every real has one; rationals terminate. It is an 'ascending continued fraction' — and e − 1 has the simple expansion [1, 1, 2, 3, 4, 5, …]. Verified live (exact BigInt): for random rationals the terms are non-decreasing and reconstruct the number exactly. See the greedy peel in 1D, nested fractions in 2D, and the ascending-unit-steps inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "c3698bdd7347f93cac0da67c4c8eb48bbac98560a546ee5b6fa5370780517d24"
    },
    {
      "slug": "the-heron",
      "title": "THE HERON",
      "kicker": "triangle area from its three sides",
      "accent": "#e0b020",
      "blurb": "Heron's formula in the 5-window house format — a triangle's area from its three side lengths alone, no height or angle: with s = (a+b+c)/2, Area = √(s(s−a)(s−b)(s−c)). It leads to a rare species: Heronian triangles, with integer sides and integer area — 3-4-5 (area 6), 13-14-15 (area 84), 5-5-6 (area 12). Most integer-sided triangles have irrational area (2-3-4 does not qualify); Heronian ones are the exception where both are whole. Verified live: Heron's formula matches the coordinate area over thousands of triangles, and the Heronian triangles have exactly the integer areas claimed. See the formula in 1D, a triangle checked in 2D, and the area-from-sides inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "36e99fb9e0c2f916f027c775330386f7b114f86057d21c2bf774c9ce18cfd6c2"
    },
    {
      "slug": "the-keith-number",
      "title": "THE KEITH NUMBER",
      "kicker": "numbers that appear in their own digit-sequence",
      "accent": "#58b878",
      "blurb": "Keith numbers (repfigits — 'replicating Fibonacci digits') in the 5-window house format — numbers that appear in a sequence seeded by their own digits. Take an n-digit number, start a Fibonacci-like sequence with its n digits, and let each new term be the sum of the previous n terms; if the original number turns up, it is a Keith number. 197 has digits 1,9,7; the sequence 1,9,7,17,33,57,107,197 — and there it is. They are startlingly rare: only a handful below each power of ten. Verified live: the Keith numbers up to 100000 are exactly 14,19,28,47,61,75,197,742,1104,1537,…,93993. See a self-seeded sequence in 1D, a number reappearing in 2D, and the self-genesis inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "2e82e4d79087f4d3c9c13d20b3f744da38d44f57947bce43cef35a24fb2b7418"
    },
    {
      "slug": "the-solovay-strassen",
      "title": "THE SOLOVAY-STRASSEN",
      "kicker": "test primality by the Jacobi symbol",
      "accent": "#b06868",
      "blurb": "The Solovay–Strassen test in the 5-window house format — decide primality using the Jacobi symbol (a/n), a generalization of the Legendre symbol computable by a fast quadratic-reciprocity recursion without factoring. Euler's criterion: for a prime n, a^((n−1)/2) ≡ (a/n) (mod n) for every a coprime to n; for an odd composite this fails for at least half of all bases, so a few random bases catch composites. Verified live: the Jacobi symbol equals the Legendre symbol for primes, the test matches trial division for odd n below 50000, and the composite witness fraction is always ≥ 1/2. See witnesses vs liars in 1D, a verdict in 2D, and the symbol-not-factor inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "0c047fa29317b509ddd7826d6269548e17b39efdc1047fa3fd3b0d1e5fe31863"
    },
    {
      "slug": "the-narayana-cow",
      "title": "THE NARAYANA COW",
      "kicker": "a sequence at the supergolden ratio",
      "accent": "#6ab0d0",
      "blurb": "Narayana's cows sequence in the 5-window house format — from a 14th-century puzzle: a cow produces one calf a year, and each calf, from its fourth year, does the same. The herd grows by a(n) = a(n−1) + a(n−3): 1,1,1,2,3,4,6,9,13,19,28,41,… The ratio of consecutive terms converges to the supergolden ratio ψ ≈ 1.4655712 — the unique real root of x³ = x² + 1, a cousin of the golden and plastic ratios. Verified live: the recurrence holds, and a(n)/a(n−1) converges to the real root of x³−x²−1. See the sequence in 1D, growth converging in 2D, and the its-own-constant inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "cb4f33d051eafd0877407ae619d38e0c033c5d2ecadf493db0dbcf015a65969b"
    },
    {
      "slug": "the-lca",
      "title": "THE LCA",
      "kicker": "the meeting point of two nodes in one leap",
      "accent": "#58a0b0",
      "blurb": "The lowest common ancestor by binary lifting in the 5-window house format — the LCA of two tree nodes is their deepest shared ancestor, where their paths to the root first meet. Binary lifting answers LCA queries in O(log n) after O(n log n) preprocessing: for each node it stores its 2^k-th ancestors. To find the LCA, lift the deeper node to the other's depth, then jump both upward in powers of two as far as possible without meeting — one step above lands on the LCA. Verified live: over hundreds of random trees and node pairs, the binary-lifting LCA equals the naive ancestor-walk LCA. See the jumps in 1D, a tree LCA in 2D, and the convergence-by-doubling inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "e080cb24cad1360b06f139761b91b66e62520e6ce0da417455c737c89b88bbd5"
    },
    {
      "slug": "the-baum-sweet",
      "title": "THE BAUM-SWEET",
      "kicker": "a bit-pattern sequence read by 0-blocks",
      "accent": "#c07850",
      "blurb": "The Baum–Sweet sequence in the 5-window house format — a string of 0s and 1s read from the binary digits of each index: b(n)=1 if the binary of n contains no block of consecutive 0s of odd length, else 0. So b(2)=0 (binary 10 has a single 0, odd) and b(9)=1 (1001, the 00 block is even). It is an automatic sequence — generated by a finite automaton reading binary — and obeys a clean recurrence: b(2n+1)=b(n), b(4n)=b(n), b(4n+2)=0. Verified live: the recurrence generates exactly the direct definition for every n up to 100000. See the sequence in 1D, a number's 0-blocks in 2D, and the spoken-in-binary inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "0817598a0a152d6510984edb9c059770258644f6598202b186ddf8c3dbfadb2e"
    },
    {
      "slug": "the-wieferich",
      "title": "THE WIEFERICH",
      "kicker": "the vanishingly rare Wieferich primes",
      "accent": "#e0b020",
      "blurb": "Wieferich primes in the 5-window house format — primes so rare that only two are known. Fermat's little theorem gives 2^(p−1) ≡ 1 (mod p) for every odd prime; a Wieferich prime satisfies the far stronger congruence modulo p²: 2^(p−1) ≡ 1 (mod p²). Only 1093 and 3511 qualify below 6.7×10¹⁵ — despite vast searches, no third is known. They are tied to Fermat's Last Theorem: any first-case prime-exponent counterexample would have to be Wieferich. Verified live (exact BigInt): among all primes below 20000, exactly 1093 and 3511 satisfy 2^(p−1) ≡ 1 (mod p²). See the sharpened congruence in 1D, a prime tested in 2D, and the one-more-power inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "8d9a311017b793c54ddc28f3226ba7f8fcaa5d7bb42a24a5d44c0a0551262019"
    },
    {
      "slug": "the-leonardo",
      "title": "THE LEONARDO",
      "kicker": "a Fibonacci-plus-one sequence",
      "accent": "#6ab0d0",
      "blurb": "The Leonardo numbers in the 5-window house format — Fibonacci's numbers with a +1: L(0)=L(1)=1, L(n) = L(n−1) + L(n−2) + 1, running 1,1,3,5,9,15,25,41,67,109,… They connect to Fibonacci by the exact identity L(n) = 2·F(n+1) − 1. They matter in computing: Dijkstra used them for smoothsort, an in-place sort whose heap sizes are Leonardo numbers, giving adaptive O(n) behavior on nearly-sorted input. Verified live (exact BigInt): the +1 recurrence holds, and L(n) = 2·F(n+1) − 1 for n up to 90. See the sequence in 1D, the identity in 2D, and the constant-nudge inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "4d58370d3b44b355bc4d97f7085627275e590e1be1210b779f43f8beb078124b"
    },
    {
      "slug": "the-vampire-number",
      "title": "THE VAMPIRE NUMBER",
      "kicker": "numbers that factor into fangs from their own digits",
      "accent": "#b06868",
      "blurb": "Vampire numbers in the 5-window house format — numbers that hide a factorization inside their own digits. A number with 2k digits is a vampire if it equals the product of two k-digit 'fangs' that together use exactly the original digits, in some order, and the fangs aren't both multiples of ten. The smallest: 1260 = 21 × 60 (digits 1,2,6,0 rearranged); also 1395 = 15 × 93, 1435 = 35 × 41. The number wears the very digits of its factors. Verified live: the four-digit vampire numbers are exactly 1260, 1395, 1435, 1530, 1827, 2187, 6880. See the fang split in 1D, a candidate checked in 2D, and the digit-preserving-factor inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "a6fe8a96a7b6b2181b98011ec97b64046c86275340849a1cf995cfe239d6ec2e"
    },
    {
      "slug": "the-de-boor",
      "title": "THE DE BOOR",
      "kicker": "evaluate a B-spline by nested interpolation",
      "accent": "#58a0b0",
      "blurb": "De Boor's algorithm in the 5-window house format — evaluate a B-spline curve at a parameter t by repeated linear interpolation, the B-spline analogue of de Casteljau for Béziers. Given control points and a knot vector, it finds the active knot span, takes the handful of control points influencing t, and blends them in successive rounds of interpolation (weights from the knots) until one point remains: the curve at t. It is numerically stable and needs no explicit basis functions. Verified live: over 1000 random B-splines and parameters, de Boor's result equals the direct Cox–de Boor basis-function sum Σ N_i,p(t)·P_i. See the blend in 1D, a curve in 2D, and the corner-cutting inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "9fdcb45183cd97eac476b2ca393ee0201651a88068204022372f5ea9ec56dc9f"
    },
    {
      "slug": "the-mertens",
      "title": "THE MERTENS",
      "kicker": "a conjecture that holds then fails",
      "accent": "#c07850",
      "blurb": "The Mertens function in the 5-window house format — the running sum of the Möbius function: M(n) = μ(1)+μ(2)+…+μ(n), where μ(k) is +1, −1, or 0 by the parity and squarefreeness of k's prime factorization. It jitters around zero. The famous Mertens conjecture claimed |M(n)| < √n for all n — it holds for every n anyone can compute, yet Odlyzko and te Riele proved it FALSE for some enormous n. A conjecture true as far as the eye can see, but ultimately wrong. Verified live: the μ sieve matches direct factorization, and |M(n)| < √n holds for every n from 2 to 10000. See μ and M in 1D, M vs ±√n in 2D, and the evidence-is-not-proof inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "26dbd6de46f487f89ba4882efd117d8310e156bf324796d3cdab0e0c3c72a781"
    },
    {
      "slug": "the-sparse-set",
      "title": "THE SPARSE SET",
      "kicker": "a set with no array to initialize",
      "accent": "#e0b020",
      "blurb": "The sparse set in the 5-window house format — store a set of small integers with O(1) insert, remove, and membership, needing no array initialization. It keeps two arrays: dense, a packed list of members, and sparse, indexed by value, pointing back into dense. Membership is a double-lookup: x is present iff sparse[x] points to a slot in dense that holds x. Because both directions must agree, uninitialized garbage in sparse can never falsely report membership, and iteration is just walking dense. Verified live: over hundreds of random insert/remove/query sequences, the sparse set's membership and dense contents exactly match a reference set. See the double-index in 1D, the arrays in 2D, and the self-validating inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "f8c8164d2c9f3171c099bb3a6be41e314dc6fb7c57492378d7d6234cf18bc693"
    },
    {
      "slug": "the-wolstenholme",
      "title": "THE WOLSTENHOLME",
      "kicker": "a binomial congruence mod p-cubed for primes five and up",
      "accent": "#9a6ad0",
      "blurb": "Wolstenholme's theorem in the 5-window house format — for every prime p ≥ 5, the central binomial coefficient satisfies C(2p, p) ≡ 2 (mod p³). Ordinary primality only forces this modulo p; Wolstenholme lifts it two full powers higher, to p-cubed, and only for primes five and up (it fails for 2 and 3). Equivalently, the numerator of 1 + 1/2 + … + 1/(p−1) is divisible by p². Verified live with exact BigInt: C(2p,p) ≡ 2 (mod p³) for every prime 5..101, while p=3 gives residue 18. See the congruence in 1D, a prime checked in 2D, and the depth-of-a-congruence inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "5cdba86360364e0b7beeb46c94529093b197e7dacc97c4ac6017d04b3a53c76d"
    },
    {
      "slug": "the-weird-number",
      "title": "THE WEIRD NUMBER",
      "kicker": "abundant numbers no subset of divisors can total",
      "accent": "#c85a5a",
      "blurb": "Weird numbers in the 5-window house format — a number is abundant when its proper divisors sum to more than itself, and semiperfect when some subset of those divisors sums to exactly itself. A weird number is abundant but NOT semiperfect: it overflows, yet no combination of its parts reconstructs it. The smallest is 70 (divisors 1,2,5,7,10,14,35 sum to 74 > 70, but no subset totals 70). They are scarce — only sixteen below 13000. Verified live: an exhaustive scan (abundance test, then a subset-sum DP) finds exactly 70, 836, 4030, 5830, …, 12670. See the definition in 1D, a number tested in 2D, and the excess-without-expressibility inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "53f5d324952400348ceec19cf7ebf59b5f1713b91d4d25c801a13016267b5271"
    },
    {
      "slug": "the-smith-number",
      "title": "THE SMITH NUMBER",
      "kicker": "numbers whose digit sum equals their factors'",
      "accent": "#d4a020",
      "blurb": "Smith numbers in the 5-window house format — a composite whose digit sum equals the sum of the digits of all its prime factors (with multiplicity). The smallest is 4 = 2×2 (digitsum 4 = 2+2). Also 22 = 2×11 (4 = 2+1+1), 27 = 3³ (9 = 3+3+3), 58 = 2×29 (13 = 2+2+9). Primes are excluded (they would match trivially). Named after Harold Smith, whose phone number 4937775 is one. Verified live: an exhaustive scan (factor each composite, compare digit sums) reproduces the known fifty Smith numbers below 1100. See the coincidence in 1D, a number factored in 2D, and the digits-of-the-factors inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "c633138eb43672e4a88e6b13520d787958c7d34166ab84126baaab05592f9c55"
    },
    {
      "slug": "the-hull-dobell",
      "title": "THE HULL–DOBELL",
      "kicker": "when a linear congruential generator hits full period",
      "accent": "#40b0a0",
      "blurb": "The Hull–Dobell theorem in the 5-window house format — exactly when a linear congruential generator x → (a·x + c) mod m visits every residue before repeating (full period m, from any seed). The three conditions: (1) gcd(c, m) = 1; (2) a − 1 is divisible by every prime factor of m; (3) if 4 divides m, then 4 divides a − 1. Meet all three and the generator permutes all m residues; miss one and it stalls into a short cycle. Verified live: for every LCG with modulus m ≤ 60 and all (a, c), the theorem's full-period prediction matches the actually measured cycle length. See the conditions in 1D, an orbit drawn in 2D, and the predict-vs-measure inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "bbb2f38f1d608fe399ee0379c0786a2af223a9aac3363330bbbcf18a9314271b"
    },
    {
      "slug": "the-ekg-sequence",
      "title": "THE EKG SEQUENCE",
      "kicker": "a sequence walking by shared factors",
      "accent": "#e0609a",
      "blurb": "The EKG sequence in the 5-window house format — start 1, 2, and each next term is the smallest positive integer not yet used that shares a common factor with the previous term. It runs 1, 2, 4, 6, 3, 9, 12, 8, 10, 5, 15, … Its plot looks like a heartbeat trace. It is conjectured (and largely proven) to be a permutation of all positive integers, and a proven structural fact holds: every prime p first appears immediately after 2p and is immediately followed by 3p. Verified live: over 3000 terms, consecutive terms always share a factor > 1, all are distinct, every integer 1..1000 appears, and each interior prime p sits between 2p and 3p. See the walk in 1D, the heartbeat in 2D, and the reordered-by-factors inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "757545abe69dd6b0e7376cb78ce8a6da39dfcd9ace8bd7566545b056b659fcc6"
    },
    {
      "slug": "the-legendre-formula",
      "title": "THE LEGENDRE FORMULA",
      "kicker": "count how many times a prime divides a factorial",
      "accent": "#a0b040",
      "blurb": "Legendre's formula in the 5-window house format — the exact exponent of a prime p in n! without ever building the factorial: v_p(n!) = ⌊n/p⌋ + ⌊n/p²⌋ + ⌊n/p³⌋ + …, terminating once pᵏ exceeds n. There is a striking closed form too: v_p(n!) = (n − s_p(n))/(p − 1), where s_p(n) is n's digit sum in base p. A consequence: the trailing zeros of n! equal v₅(n!). Verified live: the floor-sum equals the true exponent and equals the digit-sum form for all n≤2000 and primes 2,3,5,7,11,13, and trailing zeros of n! (exact BigInt) equal v₅(n!). See the sum in 1D, both forms in 2D, and the count-without-the-product inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "a634fc9f6282264c6e45f3a29c175f25bd5f7f952a6d2aeabf0bfcc1add6dcd1"
    },
    {
      "slug": "the-tetration",
      "title": "THE TETRATION",
      "kicker": "a power tower reduced modulo m settles down",
      "accent": "#b060c0",
      "blurb": "Tetration in the 5-window house format — iterated exponentiation, the power tower. ᵏa = a^a^…^a with k copies, evaluated top-down: ³2 = 2^(2^2) = 16, ⁴2 = 2^16 = 65536, ⁵2 dwarfs the universe. Yet modulo m the tower stops growing: ¹a, ²a, ³a, … (mod m) becomes constant after a small height, because exponents reduce mod φ(m) (generalized Euler) and φ iterated on m reaches 1 in a few steps. Verified live: the tower-mod recursion (exact BigInt for short towers, generalized-Euler lift for tall ones) matches the direct tower where feasible, and ᵏa mod m is constant for all tall k across many (a,m). See the freeze in 1D, the climb-then-freeze in 2D, and the finite-fingerprint inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "0a968ffc914f65e1a0d3db3061395ec00a68194fb92946065ec5fe6a45f6a1eb"
    },
    {
      "slug": "the-continued-fraction-of-e",
      "title": "THE CONTINUED FRACTION OF e",
      "kicker": "the continued fraction of e and its convergents",
      "accent": "#48b0a8",
      "blurb": "The continued fraction of e in the 5-window house format — while e = 2.71828… looks random in decimal, its continued fraction is perfectly regular: e = [2; 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, …] — a 2, then repeating triples (1, 2m, 1) for m = 1, 2, 3, … Truncations give the convergents, the best rational approximations to e: 2, 3, 8/3, 11/4, 19/7, 87/32, 106/39, 193/71, … Verified live: the pattern a[3m−1]=2m holds, and the convergents (exact BigInt) approach e to within 1e−12, checked against e = Σ 1/j! computed as an exact fraction. See the terms in 1D, the shrinking error in 2D, and the order-beneath-the-constant inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "9fa0fd1f6013e8b8b360248219323ad82d41c2622c1168708190d77f47a3da08"
    },
    {
      "slug": "the-vector-clock",
      "title": "THE VECTOR CLOCK",
      "kicker": "vector clocks and the shape of causality",
      "accent": "#5a90d0",
      "blurb": "Vector clocks in the 5-window house format — capturing causality in a distributed system with no shared clock. Each of N processes keeps a vector of N counters; it bumps its own entry on every event, and on receiving a message takes the componentwise maximum of its vector and the sender's, then bumps its own. The payoff is exact: event a happened-before b if and only if VC(a) < VC(b) componentwise; if neither dominates, the events are concurrent. Verified live: over hundreds of random event graphs (process timelines plus messages), VC(a) < VC(b) matches graph reachability for every pair of events. See the merge in 1D, an event graph in 2D, and the order-without-a-clock inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "17fbdd34a16e9d3caad58f1b34972927178bd12650a52021a7d0c2e76ab402e2"
    },
    {
      "slug": "the-dudeney",
      "title": "THE DUDENEY",
      "kicker": "numbers equal to the cube of their own digit sum",
      "accent": "#e0a828",
      "blurb": "Dudeney numbers in the 5-window house format — a positive integer that is a cube whose cube root equals the sum of its own digits: n = (digit sum of n)³. The example that started it: 512 = 8³, and 5+1+2 = 8. There are exactly six: 1, 512, 4913 (= 17³), 5832 (= 18³), 17576 (= 26³), and 19683 (= 27³). After that, cubes grow faster than any digit sum can reach, so the list is complete and finite. Verified live: scanning cubes k³ and keeping those whose digit sum equals k yields exactly {1, 512, 4913, 5832, 17576, 19683}. See the coincidence in 1D, a cube tested in 2D, and the self-closing-cube inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "22da1c3dca6f3824ac68cc6d8386f401871c7252942114e4948c59b7ce5c340d"
    },
    {
      "slug": "the-liouville",
      "title": "THE LIOUVILLE",
      "kicker": "a sign that flips by the parity of prime factors",
      "accent": "#c060a0",
      "blurb": "The Liouville function in the 5-window house format — λ(n) = (−1)^Ω(n), where Ω(n) counts prime factors with multiplicity: +1 for an even count, −1 for odd. Its beautiful divisor identity: Σ_{d|n} λ(d) = 1 if n is a perfect square, 0 otherwise — a flawless square-detector. Its running total L(n) drives Pólya's conjecture (L(n) ≤ 0 for n ≥ 2), true for hundreds of millions of terms yet ultimately false (first counterexample n = 906150257). Verified live: the divisor-sum equals [n is square] for n≤2000, λ is multiplicative, and L(n)≤0 for 2≤n≤600. See the square-detector in 1D, the L-walk in 2D, and the evidence-is-not-proof inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "dd447e0ce7991fc670ce024e675cc691c2969a35c2f403453959824ead382f8b"
    },
    {
      "slug": "the-erdos-gallai",
      "title": "THE ERDŐS–GALLAI",
      "kicker": "when a list of degrees can be a real graph",
      "accent": "#5ab0c0",
      "blurb": "The Erdős–Gallai theorem in the 5-window house format — deciding whether a list of numbers can be the degrees of a real simple graph. A non-increasing sequence is graphical iff its sum is even and, for every k, Σ_{i≤k} d_i ≤ k(k−1) + Σ_{i>k} min(d_i, k). The left side is the demand of the top k vertices; the right is the most those edges can be absorbed. It is the exact companion to the Havel–Hakimi reduction by a different route. Verified live: over 3000 random sequences the Erdős–Gallai verdict matches the independent Havel–Hakimi reduction, and every graphical sequence is realized by a constructed simple graph with exactly those degrees. See the criterion in 1D, a realizing graph in 2D, and the degrees-back-into-a-graph inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "3dd062c29bd8bc4882c010cc726eed3e2b77fa856f812c2d2e354654a20e33bf"
    },
    {
      "slug": "the-three-squares",
      "title": "THE THREE SQUARES",
      "kicker": "which numbers are sums of three squares",
      "accent": "#b0a040",
      "blurb": "Legendre's three-square theorem in the 5-window house format — a non-negative integer n is a sum of three squares a²+b²+c² if and only if n is NOT of the form 4^a(8b+7). So 7, 15, and 28=4·7 fail, but everything else works, from 6=1+1+4 to 30=1+4+25. It is the sharp companion to Lagrange's four-square theorem (four squares always suffice): three suffice for all but a thin, precisely-described family. Verified live: an exhaustive search for a²+b²+c²=n agrees with the arithmetic test 'n is not 4^a(8b+7)' for every n up to 3000. See the forbidden forms in 1D, a representation in 2D, and the name-the-exception inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "af4c27b378a546943e41ac9455fd8cee65033d7cf859c90603a2d277040b2afb"
    },
    {
      "slug": "the-lah",
      "title": "THE LAH",
      "kicker": "counting partitions into ordered lists",
      "accent": "#d08840",
      "blurb": "Lah numbers in the 5-window house format — L(n,k) counts the ways to sort n labelled items into k non-empty ordered lists, where the order within each list matters. They have a closed form L(n,k) = C(n−1,k−1)·n!/k! and a recurrence L(n,k) = L(n−1,k−1) + (n+k−1)L(n−1,k). They are the exact coefficients converting between the two factorial bases: the rising factorial equals a Lah-weighted sum of falling factorials, x^(n rising) = Σ_k L(n,k) x^(k falling). Verified live (exact BigInt): closed form equals recurrence for n≤12, and the rising=Σ L·falling identity holds for integer x. See the triangle in 1D, the identity in 2D, and the ordered-blocks inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "c55b6d02cda0584a89b13588c98699a7ab63642a28b438ce8fb9a0187e0d2bcb"
    },
    {
      "slug": "the-fubini",
      "title": "THE FUBINI",
      "kicker": "counting the ways to rank things with ties",
      "accent": "#7aa0e0",
      "blurb": "The Fubini (ordered Bell) numbers in the 5-window house format — counting the ways to rank n items allowing ties, i.e. the weak orderings, or ordered set partitions. They run 1, 1, 3, 13, 75, 541, 4683, … For 3 items there are 13 outcomes. Two formulas produce them: a(n) = Σ_k k!·S(n,k) (Stirling numbers of the second kind times the k! orderings of the blocks), and the recurrence a(n) = Σ_{i=1}^n C(n,i) a(n−i). Verified live (exact BigInt): Σ k!·S(n,k) equals the binomial recurrence for n≤9, and both equal a direct brute-force count of weak orderings for n≤6. See the sequence in 1D, the three computations in 2D, and the ranked-blocks inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "bc4c370235fc107904039eff88099cffc7608add7a70ec533f27f85c6ee086ce"
    },
    {
      "slug": "the-euler-partition",
      "title": "THE EULER PARTITION",
      "kicker": "two ways to break a number into parts that always agree",
      "accent": "#5ab0e0",
      "blurb": "Euler's partition theorem in the 5-window house format — the number of ways to write n as a sum of distinct parts equals the number using only odd parts. For n=6: distinct partitions 6, 5+1, 4+2, 3+2+1 (four); odd partitions 5+1, 3+3, 3+1+1+1, 1×6 (also four). Euler proved it with generating functions: ∏(1+x^k) = ∏1/(1−x^(2k−1)), and Glaisher gave a direct bijection. Verified live: an exact partition-counting DP shows distinct-part count equals odd-part count for every n up to 60. See the two families in 1D, the matching counts in 2D, and the two-restrictions-one-number inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "f48a964a74f366360078b264f4d8f0ef6e1eb67e644e3607a81bf863dde54300"
    },
    {
      "slug": "the-zolotarev",
      "title": "THE ZOLOTAREV",
      "kicker": "a coin-flip sign hidden in modular multiplication",
      "accent": "#b070c0",
      "blurb": "Zolotarev's lemma in the 5-window house format — for an odd prime p and a coprime to p, the Legendre symbol (a/p) (whether a is a quadratic residue mod p) equals the sign of the permutation x → a·x mod p on {1,…,p−1}. Whether a has a square root mod p is exactly whether multiplication by a shuffles the residues evenly or oddly. Verified live: for every odd prime p<80 and every a, the permutation sign (from cycle structure) equals the Legendre symbol (from Euler's criterion a^((p−1)/2)). See the shuffle in 1D, cycles and signs in 2D, and the residue-as-parity inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "c6c417d4bba3b6f4f2991945b7b71dd404ebbaa31609791418aadb1880051268"
    },
    {
      "slug": "the-q-binomial",
      "title": "THE Q-BINOMIAL",
      "kicker": "counting subspaces with a q-analog of the binomial",
      "accent": "#d0a030",
      "blurb": "The Gaussian binomial coefficient in the 5-window house format — [n,k]_q is the q-analog of C(n,k): replace each integer m by 1+q+…+q^(m−1). It has a product form, equals Σ q^(inversions) over binary words with k ones, and when q is a prime power it counts the k-dimensional subspaces of F_q^n. Setting q=1 recovers C(n,k). Verified live (exact BigInt): the product equals the inversion-sum for q∈{2,3,5} and n≤7, and equals the brute-force count of k-subspaces of F_2^n for n≤4. See the three faces in 1D, all cross-checked in 2D, and the choosing-into-subspaces inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "ea4b0c2c74042612dead0472017d921a9443106ad52917819acb1ca14e1d1725"
    },
    {
      "slug": "the-lindstrom-gessel-viennot",
      "title": "THE LINDSTRÖM–GESSEL–VIENNOT",
      "kicker": "non-crossing paths counted by a determinant",
      "accent": "#50b070",
      "blurb": "The Lindström–Gessel–Viennot lemma in the 5-window house format — counting families of non-crossing lattice paths with a single determinant. With sources A_i and sinks B_j and M_ij = #paths A_i→B_j, det(M) equals the signed count of vertex-disjoint path families A_i→B_σ(i). In the planar arrangement where only the identity matching avoids crossings, the determinant counts exactly the non-intersecting families — the crossing pairs cancel in the signed sum. Verified live: for 2-source and 3-source planar placements, det(M) (from binomial path counts) equals a brute-force enumeration of vertex-disjoint families. See the 2×2 case in 1D, the grid and matrix in 2D, and the crossings-cancel inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "8597b3d6f147c7ead7f6a21e5f3cb9a59e86599f399706c637f45ec20acdc2bf"
    },
    {
      "slug": "the-sperner",
      "title": "THE SPERNER",
      "kicker": "the widest layer of the subset lattice",
      "accent": "#c06890",
      "blurb": "Sperner's theorem in the 5-window house format — the largest family of subsets of an n-set with none containing another (an antichain) has size C(n,⌊n/2⌋), the middle layer of the subset lattice. The clean proof partitions all 2^n subsets into exactly C(n,⌊n/2⌋) symmetric chains (nested runs); since an antichain meets each chain at most once, it has at most that many members, and the middle layer achieves it. Verified live: a recursive symmetric chain decomposition is built for n≤8 — the chains partition all subsets, each is a genuine chain, and their count equals C(n,⌊n/2⌋). See the middle layer in 1D, the chain-decomposed lattice in 2D, and the antichains-bounded-by-chains inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "df47461d833ad859227c4996881bd74af91e66f4423908fa9602a0feff696a9f"
    },
    {
      "slug": "the-markov-triple",
      "title": "THE MARKOV TRIPLE",
      "kicker": "a Diophantine equation whose solutions grow on a tree",
      "accent": "#5ad0c0",
      "blurb": "Markov triples in the 5-window house format — the positive-integer solutions of x²+y²+z²=3xyz. The smallest is (1,1,1), then (1,1,2), (1,2,5), (1,5,13), (2,5,29), … and every one is reachable from (1,1,1) by Vieta jumping: the equation is quadratic in each variable, so (x,y,z)→(x,y,3xy−z) hops to another solution, unfolding an infinite binary tree. The numbers that appear — 1,2,5,13,29,34,89,… — are the Markov numbers. Verified live (exact BigInt): every Vieta-jump triple from (1,1,1) satisfies the equation, the known Markov numbers all appear, and jumping twice returns the original. See the seed jumps in 1D, the tree in 2D, and the solutions-breed-solutions inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "33faddbd3393da9fd6c7b4e01004fb52d53db5710b78492af10cb4d2d6c46f2f"
    },
    {
      "slug": "the-mantel",
      "title": "THE MANTEL",
      "kicker": "how many edges before a triangle is forced",
      "accent": "#d07850",
      "blurb": "Mantel's theorem in the 5-window house format — a triangle-free graph on n vertices has at most ⌊n²/4⌋ edges, attained only by the complete balanced bipartite graph K_{⌊n/2⌋,⌈n/2⌉}. Split the vertices in two halves and join every cross-pair: no triangle forms (a triangle needs two vertices on one side, never adjacent), giving exactly ⌊n²/4⌋ edges; one more edge forces a triangle. It is the n=3 case of Turán's theorem. Verified live: an exhaustive search over all graphs on up to 6 vertices finds the max triangle-free edge count equals ⌊n²/4⌋, and the balanced bipartite graph attains it. See K₃,₃ in 1D, a graph tested in 2D, and the pack-to-the-edge inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "09d1d55d99f9fb9ebb7dc061423524a6f72ddc74e9a760bce6cb817fc0e07d6d"
    },
    {
      "slug": "the-radon",
      "title": "THE RADON",
      "kicker": "four points that always split into two overlapping halves",
      "accent": "#6890d8",
      "blurb": "Radon's theorem in the 5-window house format — any d+2 points in d-dimensional space split into two groups whose convex hulls overlap. In the plane (d=2), any four points partition into two sets sharing a common point, the Radon point: either one point lies inside the triangle of the other three, or the four form a quadrilateral whose diagonals cross. The proof is linear algebra: four planar points always have an affine dependence ΣλᵢPᵢ=0 with Σλᵢ=0; grouping by the sign of λ gives the two overlapping sets. Verified live: for thousands of random 4-point sets, the sign-split's two weighted barycentres coincide at a shared Radon point. See the two cases in 1D, the partition in 2D, and the enough-points-force-overlap inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "c333662e7ed5c15934c2c8d0aef1fb0b126f570c81871071d94e2017a3c656a5"
    },
    {
      "slug": "the-von-mangoldt",
      "title": "THE VON MANGOLDT",
      "kicker": "weighting the primes so divisor-sums give a logarithm",
      "accent": "#b878d0",
      "blurb": "The von Mangoldt function in the 5-window house format — Λ(n) equals ln p when n is a prime power p^k (like 8=2³ or 25=5²), and 0 otherwise. This weighting produces a clean identity: Σ_{d|n} Λ(d) = ln n — the divisor-sum rebuilds a logarithm exactly. Its running total, the Chebyshev function ψ(x)=Σ_{n≤x}Λ(n), grows like x, a statement equivalent to the Prime Number Theorem. Verified live: Σ_{d|n}Λ(d) equals ln n to floating precision for every n≤2000, Λ fires only on prime powers, and ψ(x)/x hovers near 1. See Λ on prime powers in 1D, the divisor-sum in 2D, and the logarithm-redistributed inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "432fb60993575a0ee33184bf38155e462e54744483ce8b94f1e833b04d0a123a"
    },
    {
      "slug": "the-euler-polyhedron",
      "title": "THE EULER POLYHEDRON",
      "kicker": "the invariant two hiding in every polyhedron",
      "accent": "#48b878",
      "blurb": "Euler's polyhedron formula in the 5-window house format — for any convex polyhedron, and any connected graph drawn in the plane without crossings, V − E + F = 2. A cube: 8−12+6=2. A dodecahedron: 20−30+12=2. It holds under any triangulation, subdivision, or deformation — the alternating sum is a topological invariant (the Euler characteristic of the sphere); a planar graph's face count includes the outer region. Verified live: V−E+F=2 for all five Platonic solids and for planar graphs (fan-triangulated polygons and wheel graphs) whose V, E, F are counted from their actual edge sets. See the Platonic table in 1D, a planar graph counted in 2D, and the geometry-forgotten inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "208e01d6e59d6f6a13dcc93381868121a3fed47cc601919a21c4d58affd2367e"
    },
    {
      "slug": "the-helly",
      "title": "THE HELLY",
      "kicker": "when pairwise overlap forces a common point",
      "accent": "#50b0b0",
      "blurb": "Helly's theorem in the 5-window house format — for a finite family of convex sets in d dimensions, if every d+1 of them share a common point, then all of them do. On a line (d=1): if intervals pairwise overlap, they all share a point (exactly when max(lefts) ≤ min(rights)). The number d+1 is sharp — in the plane you truly need every three to meet, as three disks around a triangle show (pairwise overlap, no common point). Verified live: over thousands of random interval families, 'every pair overlaps' is exactly equivalent to 'a common point exists', and a planar 3-disk example meets pairwise yet shares no point. See intervals in 1D, the disks in 2D, and the local-overlap-global-point inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "55ee38038a9935c9cb3860ecdac624c54eac274c448f9fc4419761967ba618e1"
    },
    {
      "slug": "the-vizing",
      "title": "THE VIZING",
      "kicker": "colouring edges with almost the fewest colours",
      "accent": "#d06868",
      "blurb": "Vizing's theorem in the 5-window house format — the edge-chromatic number χ'(G) of any simple graph is either Δ or Δ+1, where Δ is the maximum degree. Never fewer than Δ (the edges at the busiest vertex all differ), never more than Δ+1. So every graph is Class 1 (Δ colours suffice, like complete graphs K_{2n}) or Class 2 (needs Δ+1, like every odd cycle) — the whole variety of graphs collapses to a one-bit question. Verified live: for hundreds of random graphs the exact edge-chromatic number (found by exhaustive colouring) is always Δ or Δ+1; K₄ is Class 1 (χ'=3), C₅ is Class 2 (χ'=Δ+1=3). See K₄/C₅ in 1D, a graph coloured in 2D, and the whole-graph-one-bit inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "9de73e84fce62a461876d8fdee2c5c84cc070cbaf45625e9ad67963e361024fe"
    },
    {
      "slug": "the-mirsky",
      "title": "THE MIRSKY",
      "kicker": "covering an order by its widest levels",
      "accent": "#98a850",
      "blurb": "Mirsky's theorem in the 5-window house format — the elegant dual of Dilworth's: in any poset, the minimum number of antichains needed to cover everything equals the length of the longest chain. Give each element a height (the longest chain ending at it); elements of equal height form an antichain (comparable elements have different heights), the number of distinct heights is the longest chain length, and no fewer antichains can work since each element of a longest chain needs its own. Verified live: for thousands of random posets, height-layering yields exactly (longest-chain-length) layers, each a genuine antichain partitioning every element. See divisors of 12 in 1D, a layered poset in 2D, and the chains-bound-antichain-covers inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "d69530da03f514d01a15220d2c473163be4bd8c265f4150a526a7d96c5ef02ed"
    },
    {
      "slug": "the-sidon-set",
      "title": "THE SIDON SET",
      "kicker": "a set whose pairwise sums never collide",
      "accent": "#5aa0d0",
      "blurb": "The Sidon set in the 5-window house format — a set (B₂ set) in which all pairwise sums are distinct, equivalently all pairwise differences are distinct; no two different pairs add to the same total. The classic {0,1,3,7} is Sidon (differences 1,2,3,4,6,7 all distinct); {1,2,3,4} is not (1+4=2+3). Since a size-m Sidon set has m(m−1)/2 distinct differences that must fit below n, its size is bounded by roughly √n (Erdős–Turán). The greedy Mian–Chowla sequence 1,2,4,8,13,21,31,… builds one term by term. Verified live: Mian–Chowla stays Sidon, {0,1,3,7} has distinct differences, {1,2,3,4} is flagged non-Sidon, and the max Sidon subset of {1…n} grows like √n. See the sets in 1D, sums checked in 2D, and the distinctness-forces-sparsity inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "cacbf2891c4219e2b2fe5ce7cb0ac058574a03759304e99d8f8699db6c95c99f"
    },
    {
      "slug": "the-erdos-ko-rado",
      "title": "THE ERDŐS–KO–RADO",
      "kicker": "the largest family of sets that all pairwise meet",
      "accent": "#d0a040",
      "blurb": "The Erdős–Ko–Rado theorem in the 5-window house format — the largest family of k-element subsets of {1,…,n} such that every two overlap is, for n≥2k, exactly C(n−1,k−1), achieved by the 'star': all k-subsets containing one fixed element. You cannot beat simply pinning a common element. Verified live: an exhaustive search for the largest pairwise-intersecting family of k-subsets of {1…n} (for n≥2k, n up to 6) equals C(n−1,k−1) every time, matched by the star. See the star in 1D, the max family in 2D, and the pin-a-point inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "daf699ad6b3c7daa20bd7374d24fc9219608f21efc4a0a9153fb4cff02b849eb"
    },
    {
      "slug": "the-erdos-ginzburg-ziv",
      "title": "THE ERDŐS–GINZBURG–ZIV",
      "kicker": "any 2n−1 integers hide n that sum to zero mod n",
      "accent": "#d06880",
      "blurb": "The Erdős–Ginzburg–Ziv theorem in the 5-window house format — among any 2n−1 integers (repeats allowed), some n of them have a sum divisible by n. However adversarially the numbers are chosen, a size-n subset summing to 0 mod n always hides inside. The count is sharp: with only 2n−2 integers it can fail (take n−1 zeros and n−1 ones — any n of them sum to between 1 and n−1). Verified live: for thousands of random collections of 2n−1 integers a size-n zero-sum subset (mod n) is always found by a subset-sum DP, and the 2n−2 counterexample has none. See the example in 1D, the pile in 2D, and the zero-sum-unavoidable inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "ea30bb9a6b79b74853914bce60511f30f5dfbe28082237de9f1ba89050efa6a0"
    },
    {
      "slug": "the-birkhoff-von-neumann",
      "title": "THE BIRKHOFF–VON NEUMANN",
      "kicker": "a fair blend that splits into perfect assignments",
      "accent": "#5a90c0",
      "blurb": "The Birkhoff–von Neumann theorem in the 5-window house format — every doubly stochastic matrix (non-negative, every row and column summing to 1) is a convex combination of permutation matrices. A fair fractional assignment is always a weighted average of whole one-to-one assignments. Birkhoff's algorithm peels them off: find a permutation sitting on positive entries (a perfect matching always exists), subtract as much as possible, repeat — the weights sum to 1. Verified live: hundreds of random doubly stochastic matrices decompose into permutation matrices whose coefficients sum to 1 and whose weighted sum reconstructs the original. See the identity in 1D, a decomposition in 2D, and the blend-unmixed inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "e1f167b1255ab64c8ab87cc3973087c7484221a37a1dc23a987ae1d1b558328c"
    },
    {
      "slug": "the-cauchy-davenport",
      "title": "THE CAUCHY–DAVENPORT",
      "kicker": "how small a sumset can be in a prime field",
      "accent": "#a0b050",
      "blurb": "The Cauchy–Davenport theorem in the 5-window house format — for non-empty A, B ⊆ Z_p (p prime), the sumset A+B = {a+b mod p} satisfies |A+B| ≥ min(p, |A|+|B|−1). Adding two sets cannot shrink them: unless you saturate the whole field, the sum is at least the sizes added minus one, with equality for arithmetic progressions sharing a difference. Crucially p must be prime — in Z_6, {0,3}+{0,3} = {0}, far below the bound. Verified live: over thousands of random subset pairs in Z_p the bound always holds, APs hit equality, and it fails in composite Z_6. See the sumset in 1D, A+B on a ring in 2D, and the addition-cannot-shrink inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "a017eac013a2783dd45d633eb98ceefb8b32b08031c236b2a0a30a641729ba57"
    },
    {
      "slug": "the-ptolemy",
      "title": "THE PTOLEMY",
      "kicker": "the diagonal law of a cyclic quadrilateral",
      "accent": "#d0a840",
      "blurb": "Ptolemy's theorem in the 5-window house format — in a cyclic quadrilateral ABCD (four points on a circle, in order), the product of the diagonals equals the sum of the products of opposite sides: AC·BD = AB·CD + AD·BC. Ptolemy used it to build his table of chords — the trigonometry that ran astronomy for a thousand years. For four points not concyclic the diagonal product is strictly less (Ptolemy's inequality), with equality exactly when they lie on a circle. Verified live: for thousands of on-circle quadrilaterals AC·BD equals AB·CD + AD·BC to floating precision; for off-circle points the left side is strictly smaller. See the rectangle→Pythagoras case in 1D, a quad on a circle in 2D, and the certifies-a-circle inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "caa454da1b62ccc1bbf82ae9da61fbf6149a5dc552c9e66f7e80bd04f71cea5a"
    },
    {
      "slug": "the-gauss-eureka",
      "title": "THE GAUSS EUREKA",
      "kicker": "every number as three triangular numbers",
      "accent": "#50b090",
      "blurb": "Gauss's Eureka theorem in the 5-window house format — every non-negative integer is a sum of three triangular numbers T_k = k(k+1)/2 (0,1,3,6,10,15,…). Any n = T_a + T_b + T_c. Gauss proved it at nineteen and wrote in his diary 'EUREKA! num = Δ + Δ + Δ.' It is equivalent to a case of the three-square theorem: n = T_a+T_b+T_c exactly when 8n+3 is a sum of three odd squares, since 8·T_k+1 = (2k+1)². Verified live: every integer from 0 to 3000 is found to be a sum of three triangular numbers, and each decomposition satisfies the 8n+3 = (2a+1)²+(2b+1)²+(2c+1)² equivalence. See examples in 1D, a decomposition drawn as triangles in 2D, and the three-triangles inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "ea730cb0903b3d00fa63ec1cab506399eb68717dc4fe5e1483ef65aff37a981a"
    },
    {
      "slug": "the-follower-set",
      "title": "THE FOLLOWER SET",
      "kicker": "the boundary between what a finite engine can capture and what it cannot",
      "accent": "#b06090",
      "blurb": "The follower set in the 5-window house format — the test for soficity, whether a shift space is capturable by a finite automaton. For an admissible word w, F(w) = {v : wv admissible} is all the futures its past leaves open; a shift is sofic exactly when the number of distinct follower sets is finite. The golden-mean shift (forbid 11) has just 2 follower sets forever — its word counts are the Fibonacci numbers — so it is sofic. The matched-run shift (1 0ⁿ 1 0ⁿ 1, mismatched runs illegal) has unboundedly many: to place the next 1 you must remember a run length with no bound. Verified live: golden-mean has 2 follower sets + Fibonacci word counts; the matched-run words 1·0ᵏ have pairwise-distinct follower sets (distinguished by 1·0ᵏ·1) → nonsofic. This is the playable form of David's nonsofic principle (i13.nonsofic). See the two shifts in 1D, follower sets in 2D, and the finite-vs-infinite inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "83fa8213c868ea8eda10fd7b8c3b6821a511474a4328f3f8ec4502e389f8f879"
    },
    {
      "slug": "the-sylvester-gallai",
      "title": "THE SYLVESTER–GALLAI",
      "kicker": "non-collinear points always leave an ordinary line",
      "accent": "#d0a040",
      "blurb": "The Sylvester–Gallai theorem in the 5-window house format — given finitely many points in the plane, not all on one line, there is always a line through exactly two of them (an 'ordinary' line). Sylvester asked it in 1893; it resisted until Gallai and Melchior settled it around 1944. You cannot arrange points so every two-point line catches a third — unless they are all collinear. Verified live: over thousands of random integer point sets that are not all collinear, an ordinary line is always found by checking every pair (exact integer collinearity, no rounding). See the 3×3 grid in 1D, an ordinary line highlighted in 2D, and the unavoidable-line inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "f622d94ed73ac635fd74c7e2c99fb8ccd6e6b8f8bb4923569ad018edda9e0fbd"
    },
    {
      "slug": "the-turan",
      "title": "THE TURÁN",
      "kicker": "the most edges with no clique of a given size",
      "accent": "#c07058",
      "blurb": "Turán's theorem in the 5-window house format — the most edges a graph on n vertices can have with no clique of size r+1 is achieved by the Turán graph T(n,r): split the vertices into r nearly-equal groups and join every pair in different groups. No K_{r+1} can form (it would need two vertices in one group, never joined), and this balanced complete r-partite graph packs the maximum (1−1/r)·n²/2 edges. Mantel's triangle-free bound is the r=2 case. Verified live: an exhaustive search over all graphs on up to 6 vertices finds the maximum K_{r+1}-free edge count equals the Turán graph T(n,r)'s, for r=2 and r=3. See T(6,3) in 1D, the Turán graph in 2D, and the pack-to-the-wall inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "41a64640e595bd1b3934a857e70450094c8a13f0e12972efba201e12c2fd47e8"
    },
    {
      "slug": "the-catalan-mihailescu",
      "title": "THE CATALAN–MIHĂILESCU",
      "kicker": "eight and nine the only consecutive perfect powers",
      "accent": "#7a90e0",
      "blurb": "Catalan's conjecture (Mihăilescu's theorem) in the 5-window house format — 8 and 9 are the only consecutive perfect powers. The equation xᵃ − yᵇ = 1 with x,y,a,b > 1 has exactly one solution: 3² − 2³ = 1. Among all squares, cubes, and higher powers, only 8 = 2³ and 9 = 3² sit next to each other. Eugène Catalan conjectured it in 1844; it stood 158 years until Preda Mihăilescu proved it in 2002. Verified live: sieving every perfect power up to a million, the only pair of consecutive integers both perfect powers is (8, 9). See the powers in 1D, the scan in 2D, and the neighbouring-powers inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "4d7d323294400e07ed06aa0aea1ef961964cf504554463db58658eea718cf5e9"
    },
    {
      "slug": "the-menger",
      "title": "THE MENGER",
      "kicker": "the most independent routes equals the smallest severing cut",
      "accent": "#5aa0b0",
      "blurb": "Menger's theorem in the 5-window house format — the maximum number of edge-disjoint paths between two vertices s and t equals the minimum number of edges whose removal disconnects them (the minimum s–t cut). Flow and blockage are the same number: push as many independent routes from s to t as you can, and the bottleneck is exactly the smallest set of edges that severs the two. It is the local, per-pair form of max-flow min-cut, and its vertex version underlies k-connectivity. Verified live: over hundreds of random small graphs, the maximum edge-disjoint s–t paths (unit-capacity max-flow) equals the minimum s–t edge cut found by exhaustive edge-removal. See the routes-equal-cut law in 1D, a graph in 2D, and the flow-equals-blockage inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "c763690b511d0eb89cf68f541fe9fc5f387900da0aa849d8ff72aa6faac92d59"
    },
    {
      "slug": "the-ramanujan-congruence",
      "title": "THE RAMANUJAN CONGRUENCE",
      "kicker": "hidden divisibilities in the partition numbers",
      "accent": "#b06898",
      "blurb": "Ramanujan's congruences in the 5-window house format — hidden divisibilities in the partition numbers p(n). Ramanujan noticed three exact patterns: p(5n+4) ≡ 0 (mod 5), p(7n+5) ≡ 0 (mod 7), p(11n+6) ≡ 0 (mod 11). Every fifth partition number from p(4) is divisible by 5, every seventh from p(5) by 7, every eleventh from p(6) by 11 — and no such simple congruence exists for any other prime. Verified live (exact BigInt): computing p(n) by Euler's pentagonal recurrence, the three congruences hold for every n in range. See the vanishing classes in 1D, p(n) mod 5/7/11 in 2D, and the hidden-zeros inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "922718b12b3e277710db4b4112c3fa0dc4abd8ae2f2cfe09a6ac2395cc185364"
    },
    {
      "slug": "the-bertrand-ballot",
      "title": "THE BERTRAND BALLOT",
      "kicker": "counting the ballots where one candidate never trails",
      "accent": "#d0a848",
      "blurb": "The Bertrand ballot problem in the 5-window house format — in an election where A wins with a votes to B's b (a > b), the chance A is strictly ahead through the entire count is (a−b)/(a+b) — depending only on the margin over the total. Equivalently, the number of vote-orderings in which A never trails is (a−b)/(a+b)·C(a+b, a). Bertrand posed it in 1887; André's reflection argument proved it. Verified live: brute-force enumeration of every vote-ordering counts exactly those where A stays strictly ahead, matching the formula for all small a, b. See the +1/−1 walk in 1D, the paths in 2D, and the margin-alone inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "d06ebcd2a7c430ac574ccc61b04a30d325fab20c0908436bbcd77aa55a0fae2d"
    },
    {
      "slug": "the-euler-line",
      "title": "THE EULER LINE",
      "kicker": "three triangle centres that always fall on one line",
      "accent": "#58b0a0",
      "blurb": "The Euler line in the 5-window house format — for any triangle, the centroid G (medians cross), the circumcenter O (centre of the circle through the vertices), and the orthocenter H (altitudes meet) always lie on a single straight line, with fixed spacing OG : GH = 1 : 2. Three centres, defined in utterly different ways, forever collinear in the same proportion. Euler proved it in 1765. Verified live: for thousands of random triangles, O, G, H are collinear (cross product vanishes) and H − G = 2(G − O). See the three centres in 1D, the Euler line in 2D, and the one-line-binds-them inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "d97d7d764b598e3c2d7d393bab3dcf44237d9d8c1024b2e0b883256f47facb8d"
    },
    {
      "slug": "the-jacobi-four-square",
      "title": "THE JACOBI FOUR-SQUARE",
      "kicker": "counting the ways to write a number as four squares",
      "accent": "#c07068",
      "blurb": "Jacobi's four-square theorem in the 5-window house format — the exact count of ways to write a number as a sum of four squares. Lagrange proved every number is a sum of four squares; Jacobi counted them: the number of ordered representations (with zeros and negatives) is r₄(n) = 8·σ(n) for odd n, 24·σ(m) for n = 2ᵏm — equivalently r₄(n) = 8 × (sum of the divisors of n not divisible by 4). A pure, exact counting law. Verified live: brute-force counting of all ordered integer quadruples with a²+b²+c²+d²=n equals 8·Σ_{d|n,4∤d} d for every n up to 150. See r₄(1)=8 in 1D, a number counted in 2D, and the counted-by-divisors inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "c5f28e28b6158ea2c5a23e4b04cbe08e08b0e2a4e18bed7a4ce2e24fa1a2ecf8"
    },
    {
      "slug": "the-caratheodory",
      "title": "THE CARATHÉODORY",
      "kicker": "a hull point is a blend of at most three",
      "accent": "#6098c8",
      "blurb": "Carathéodory's theorem in the 5-window house format — if a point p lies in the convex hull of a set S in the plane, then p is a convex combination of at most three points of S — it sits inside a triangle with corners in S. In d dimensions the bound is d+1. No matter how many points build the hull, any interior point is captured by a simplex of just d+1 of them. It is the companion of Radon and Helly in the convexity trio. Verified live: for thousands of random planar point sets and a point inside their hull, a triangle of three set-points containing p is always found. See the trio in 1D, a containing triangle in 2D, and the held-by-three inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "99c8b1d8891a5a1d6388e424987aea78ac4a423baa6fa0a47b2217e3a143259a"
    },
    {
      "slug": "the-gamblers-ruin",
      "title": "THE GAMBLER'S RUIN",
      "kicker": "a fair walk absorbed at the edges lands with probability proportional to the start",
      "accent": "#c86868",
      "blurb": "The gambler's ruin in the 5-window house format — a gambler starts with k dollars, bets one at a time on a fair coin, and stops only at 0 (ruin) or N (target). The probability of reaching N before going broke is exactly k/N — a straight-line law, your chance is your stake as a fraction of the goal. For a biased coin (win prob p) it becomes (1−rᵏ)/(1−rᴺ) with r=(1−p)/p, and even a tiny edge sharply bends the odds. Verified live: the exact recurrence solves to k/N for the fair walk, and Monte-Carlo matches both the fair k/N and the biased formula. See the linear law in 1D, sample walks in 2D, and the fate-from-the-start inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "7f6dafcd59ff67924990af98103026f034b1e628b71d88ba18128ad336d62ce1"
    },
    {
      "slug": "the-hurwitz",
      "title": "THE HURWITZ",
      "kicker": "the worst any irrational can be approximated",
      "accent": "#90a850",
      "blurb": "Hurwitz's theorem in the 5-window house format — for every irrational α there are infinitely many fractions p/q with |α − p/q| < 1/(√5·q²), and √5 cannot be improved. The continued-fraction convergents achieve it. The hardest number to approximate is the golden ratio φ = [1;1,1,1,…]: its approximation constant |φ − p/q|·q² converges to exactly 1/√5, the worst case, because its continued fraction is all 1s; any number with larger partial quotients (like √2) is easier. Verified live: φ's constant → 1/√5 ≈ 0.4472 (the Hurwitz maximum), √2's is smaller (≈0.3536), and the bound is met infinitely often by convergents. See φ's Fibonacci-ratio convergents in 1D, approximation constants in 2D, and the worst-case-sets-the-law inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "47e36d3bd18d445a2212c775ec1ae75c670d74483a6a5591d3c41a48a4fb410d"
    },
    {
      "slug": "the-polya-walk",
      "title": "THE PÓLYA WALK",
      "kicker": "a random walk that comes home in the plane but wanders off in space",
      "accent": "#6890d0",
      "blurb": "Pólya's recurrence theorem in the 5-window house format — a random walk on a line or grid (1D or 2D) is recurrent: it returns to its start with probability 1, infinitely often. But in three dimensions it is transient — with probability about 0.6595 it never comes home. Kakutani: 'a drunk man will find his way home, but a drunk bird may get lost forever.' The dividing line is exactly between 2 and 3 dimensions. Verified live: the return-probability series Σ p₂ₙ(0) diverges in 1D and 2D (recurrent), while a 3D simulation returns only about 34% of the time (Pólya's 0.3405) and a 1D simulation returns nearly always. See p₂ₙ(0) decay in 1D, the return sums in 2D, and the home-vs-lost inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "766b81c85c27edfb8a9d80a4538719d28a9a91faf96c56fe2bc4a5efc42d37d8"
    },
    {
      "slug": "the-kelly-criterion",
      "title": "THE KELLY CRITERION",
      "kicker": "the bet fraction that maximises long-run growth",
      "accent": "#d0a838",
      "blurb": "The Kelly criterion in the 5-window house format — with a favourable bet, what fraction of your bankroll grows it fastest long-run? Too little leaves growth on the table; too much risks ruin. The optimum maximises the expected logarithm of wealth: f* = (bp − q)/b = p − q/b, where p is the win probability, q = 1−p, and b the net odds. For an even-money bet won 60% of the time, f* = 0.2. Kelly's fraction beats every other constant fraction on long-run growth. Verified live: the growth rate g(f) = p·ln(1+bf) + q·ln(1−f) peaks exactly at f* (derivative vanishes), and a wealth simulation grows fastest at f*. See the growth hump in 1D, trajectories in 2D, and the maximise-the-log inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "d0350a022626ddb4e542de8498d4ad9828291ff267be09e512816fd9a0ea3e04"
    },
    {
      "slug": "the-arcsine",
      "title": "THE ARCSINE",
      "kicker": "why a coin game spends most of its time on one side",
      "accent": "#50b0a8",
      "blurb": "The arcsine law in the 5-window house format — toss a fair coin 2n times, tracking the running lead; what fraction of the time is the lead positive? Intuition says 'about half.' The truth is the opposite: the distribution is U-shaped — the most likely outcomes are that one side leads almost the entire time, and the least likely is a 50/50 split. Exactly, P(positive for 2k of 2n steps) = C(2k,k)·C(2n−2k,n−k)/4ⁿ, the discrete arcsine. Verified live: the exact formula sums to 1, is genuinely U-shaped (maxima at the extremes, minimum in the middle), and a fair-coin simulation reproduces the U. See the U-distribution in 1D, a histogram over it in 2D, and the lopsided-fairness inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "e462335ca77d31813fcc690c0ac240d86b9df7aba2792e572adc466f568b8f72"
    },
    {
      "slug": "the-kraft-inequality",
      "title": "THE KRAFT INEQUALITY",
      "kicker": "when a set of codeword-lengths can be a prefix code",
      "accent": "#d0a848",
      "blurb": "The Kraft inequality in the 5-window house format — a prefix code (no codeword starts another, so a stream decodes without markers) with lengths ℓ₁, ℓ₂, … exists if and only if Σ 2^(−ℓᵢ) ≤ 1. Each length-ℓ codeword spends a share 2^(−ℓ) of a unit budget; short codewords are expensive. The bound is tight both ways: any prefix code obeys it, and any lengths obeying it can be realised as a prefix code. Equality means complete — a full binary tree. Verified live: every random prefix code satisfies Σ 2^(−ℓ) ≤ 1, any length multiset with Σ ≤ 1 is constructed into an actual prefix code, and complete codes hit Σ = 1. See {0,10,11} in 1D, a code on the tree in 2D, and the lengths-decide inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "d348badae3aabc277f401c63cd3776763c7d1ff200403a53357f0ff796ae79ca"
    },
    {
      "slug": "the-dirichlet-approximation",
      "title": "THE DIRICHLET APPROXIMATION",
      "kicker": "a rational close to any real, guaranteed by pigeonhole",
      "accent": "#88b058",
      "blurb": "Dirichlet's approximation theorem in the 5-window house format — for any real α and any bound Q, there is a fraction p/q with 1 ≤ q ≤ Q and |α − p/q| < 1/(qQ) ≤ 1/q², straight from the pigeonhole principle. The Q+1 fractional parts {α},…,{Qα} and 0 fall into Q sub-intervals of [0,1], so two lie within 1/Q, and their difference gives the fraction. It is the seed of continued-fraction approximation and Hurwitz's sharp bound. Verified live: for thousands of random α and Q, a q ≤ Q is found with |α − p/q| < 1/(qQ), hence < 1/q². See the pigeonhole boxes in 1D, the multiples of α in 2D, and the counting-boxes inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "df531d295b94294c443d1ea80de30470a3408bd1a3ba523533da07cb6293cf0c"
    },
    {
      "slug": "the-cantor-diagonal",
      "title": "THE CANTOR DIAGONAL",
      "kicker": "the diagonal that escapes every list",
      "accent": "#c06888",
      "blurb": "Cantor's diagonal argument in the 5-window house format — some infinities are bigger than others. Try to list every infinite binary sequence, row by row; build a new sequence by walking the diagonal and flipping each bit. It differs from row 1 at position 1, row 2 at position 2, … from every row somewhere — so it is not on the list. No list can hold them all: 2^ℕ is uncountable. The same move proves Cantor's theorem: for any set S, the power set 2^S is strictly larger, since D = {s : s ∉ f(s)} is never in the image of any f : S → 2^S. Verified live: for any finite list, the diagonal-flip differs from every row; and for any f : S → 2^S, the diagonal set D is never hit. See the diagonal in 1D, a table escaped in 2D, and the manufacture-the-missing-one inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "9e0fb6e99555a1ded253f85d3feaca8846c383dde8b235a6907c9c54ab18f2ee"
    },
    {
      "slug": "the-rearrangement",
      "title": "THE REARRANGEMENT",
      "kicker": "the arrangement that maximises a dot product",
      "accent": "#5aa8c8",
      "blurb": "The rearrangement inequality in the 5-window house format — how to pair two lists to maximise their dot product. Given a₁ ≤ … ≤ aₙ and b₁ ≤ … ≤ bₙ and any permutation σ, the sum Σ aᵢb_σ(i) is largest when both are sorted the same way (big with big) and smallest when sorted oppositely: Σ aᵢb_{n+1−i} ≤ Σ aᵢb_σ(i) ≤ Σ aᵢbᵢ. It underlies Chebyshev's sum inequality and countless olympiad bounds — likes should pair with likes. Verified live: over thousands of random list pairs, the maximum over ALL permutations is the same-sorted pairing and the minimum is opposite-sorted. See a=(1,2,3), b=(4,5,6) in 1D, every pairing in 2D, and the sort-alike inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "4f672739879df618c1b04c3b620bc515ca28778c2cd412597dd9fb64c5216967"
    },
    {
      "slug": "the-jensen",
      "title": "THE JENSEN",
      "kicker": "the convex inequality behind averages",
      "accent": "#b09858",
      "blurb": "Jensen's inequality in the 5-window house format — the master inequality of convexity. For a convex function f (curving upward, every chord above the graph) and weights wᵢ ≥ 0 summing to 1, f(Σ wᵢxᵢ) ≤ Σ wᵢf(xᵢ): the function of the average is at most the average of the function. In probability, f(E[X]) ≤ E[f(X)]. It is the single fact behind AM–GM, the non-negativity of entropy, and much of information theory; for concave f it flips. Verified live: for random convex functions (x², e^x, −log) and weighted points, the inequality always holds, equality holds for linear f, and AM–GM falls out as a corollary. See the chord-above-curve in 1D, the gap in 2D, and the averaging-undershoots inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "81e6b0a5f54f417ffd00580f0fc44fdbd8aa9385ba9b5098d8da5ad56af8855c"
    },
    {
      "slug": "the-farkas",
      "title": "THE FARKAS",
      "kicker": "exactly one of a solution or a certificate of its impossibility",
      "accent": "#c87858",
      "blurb": "Farkas' lemma in the 5-window house format — the theorem of the alternative behind LP duality. For a matrix A and vector b, exactly one holds: (I) there is an x ≥ 0 with Ax = b (b in the cone of A's columns), or (II) there is a y with yᵀA ≥ 0 and yᵀb < 0 — a separating hyperplane certifying b is outside the cone. Never both, never neither: whenever no non-negative combination reaches b, a hyperplane proves it. Verified live: over 4000 random 2D instances, exactly one of 'b in the cone' and 'a separating y exists' holds — perfect complements. See solution-or-certificate in 1D, the cone and separator in 2D, and the every-no-has-a-witness inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "f62b123698ed3a47f8e059aa5aa2f98fd3295099b27fbf93cf26c5964057fe93"
    },
    {
      "slug": "the-banach-fixed-point",
      "title": "THE BANACH FIXED POINT",
      "kicker": "a contraction always homes on one fixed point",
      "accent": "#50b0a0",
      "blurb": "The Banach fixed-point theorem in the 5-window house format — the contraction mapping principle. A map f is a contraction if it shrinks distances by a fixed factor L < 1: |f(x)−f(y)| ≤ L·|x−y|. On a complete space, such f has exactly one fixed point x* = f(x*), and iterating from anywhere converges to it with geometric error |xₙ−x*| ≤ Lⁿ|x₀−x*|. It is the engine behind Newton's method, ODE existence, and fractal IFS. Verified live: affine contractions f(x)=ax+b (|a|<1) converge to b/(1−a) from every start with error exactly |a|ⁿ times the initial, different starts reach the same point, and iterating cosine homes on the Dottie number 0.739085. See the cobweb in 1D, convergence in 2D, and the iteration-finds-what-solving-cannot inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "fb079f0b8382b711f0cd6c13428cc71e461fb06e279e88d13d9b1059b311ec1b"
    },
    {
      "slug": "the-sperner-lemma",
      "title": "THE SPERNER LEMMA",
      "kicker": "a coloured triangulation always hides a rainbow",
      "accent": "#a878d0",
      "blurb": "Sperner's lemma in the 5-window house format — the combinatorial heart of Brouwer's fixed-point theorem. Triangulate a triangle with corners coloured 1, 2, 3; colour the rest by the Sperner rule (a vertex on the edge between corners i and j may only take colour i or j). Then however the interior is coloured, there is always a small triangle with all three colours — a 'rainbow' triangle — and the number of them is always odd, so one can never vanish. It is a discrete, checkable proof that a continuous map on a triangle has a fixed point. Verified live: over 3000 random Sperner-valid colourings, the rainbow-triangle count is always odd (hence ≥1). See the rule in 1D, rainbows highlighted in 2D, and the parity-forces-the-fixed-point inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "7b9dda3526f2cf023337b45fbab4e9b2737f94a0d4f29c765f16f197a03eebaf"
    },
    {
      "slug": "the-kolmogorov",
      "title": "THE KOLMOGOROV",
      "kicker": "why most strings cannot be compressed",
      "accent": "#c06858",
      "blurb": "Kolmogorov complexity in the 5-window house format — a string's complexity is the length of the shortest program that prints it, and a string is incompressible if no program is much shorter than itself. The key fact is a counting argument: there are only 2ᵐ−1 possible descriptions shorter than m bits, so at most that many strings compress below m bits. Therefore among the 2ⁿ strings of length n, at least a fraction 1−2^(−c) cannot be compressed by even c bits — most strings are incompressible; randomness is the rule. Verified live: the count of descriptions shorter than n−c bits is exactly 2^(n−c)−1 < 2ⁿ, so ≥(1−2^(−c)) of n-bit strings are c-incompressible, and a real run-length coder fails to shrink almost every random string. See the counting bound in 1D, the incompressible fraction in 2D, and the not-enough-programs inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "1320e993bb25e543d876b74a33108568d7750591306ecc7fba7e4f4a408c955e"
    },
    {
      "slug": "the-stationary-distribution",
      "title": "THE STATIONARY DISTRIBUTION",
      "kicker": "a chain that forgets where it started",
      "accent": "#5a98c8",
      "blurb": "The stationary distribution in the 5-window house format — the long-run equilibrium of a Markov chain. A chain hops between states by a transition matrix P (each row a distribution). If it is irreducible (every state reaches every other) and aperiodic, then from any start the distribution converges to a unique vector π fixed by the dynamics: πP = π, Σπ = 1. The chain forgets its starting point. It is the mathematics behind PageRank, MCMC sampling, and equilibrium in queueing and physics. Verified live: for 2000 random irreducible aperiodic chains, power iteration from any start converges to a π with πP = π and Σπ = 1, and different starts reach the same π. See the fixed point in 1D, two starts converging in 2D, and the equilibrium-forgets-the-beginning inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "b0f2c952dc7063905792f2ad7d582d8efa5ada36d70e54dc7feb2b2d3544b46d"
    },
    {
      "slug": "the-maxstack",
      "title": "THE MAXSTACK",
      "kicker": "the peak is already written in net",
      "accent": "#21e6ff",
      "blurb": "The maxstack in the 5-window house format — the peak stack depth of a program, read without executing it. Treat each instruction as a signed tick: a push (bind) is +1, a pop (kill) is −1, and track the running total net = binds − k. The maximum stack depth equals the largest value net reaches over the whole run — one integer pass, no interpreter, no stack ever materialised. Read that same conserved quantity only at the end and it is 0 for every balanced program, hiding the peak entirely: a conserved quantity has no unstated scope. Verified live: over 20,000 random balanced programs the max of the running net equals a real array-stack's peak length every time, while the end value is 0 in 100% of them. Neon-noir traced. See the net ridge in 1D, the ridge-vs-simulation agreement in 2D, and the read-scope-not-run inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "8cd25e5e552965c164423da316c0b16030139a15e262e7d2a8319fc8772409c0"
    },
    {
      "slug": "the-substrate-check",
      "title": "THE SUBSTRATE CHECK",
      "kicker": "a check noise passes is not a measurement",
      "accent": "#ff8a3c",
      "blurb": "The substrate check in the 5-window house format — the difference between a necessary test and a sufficient one, and why a check random noise can pass measures nothing. For bracket strings, a weak check asks only for equal counts of ( and ); a strong check asks for valid nesting (no prefix goes negative). Among the C(2n,n) equal-count strings, exactly Catalan(n) are valid, and the ratio is exact: C(2n,n)/Catalan(n) = n+1. The weak check admits (n+1)× too many, and a random equal-count string is valid only 1/(n+1) of the time. Verified live: the ratio equals n+1 for n=1..11, and 200,000 random equal-count strings at n=5 are valid 16.66% of the time (≈ 1/6). Neon-noir traced. See the wide-vs-core bars in 1D, the live pass-rate in 2D, and the measure-against-noise-first inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "0fc2f2d299087b44e9a0bb097d9a06c2ab662866368aa5bf8464cda3d102bff8"
    },
    {
      "slug": "the-sardinas-patterson",
      "title": "THE SARDINAS-PATTERSON",
      "kicker": "unique decoding with no separators",
      "accent": "#35ffb0",
      "blurb": "The Sardinas–Patterson algorithm in the 5-window house format — the decision procedure for unique decodability: can every concatenation of codewords be split back exactly one way, with no separators? Prefix-free codes (no codeword begins another) are always uniquely decodable, but the converse is false. The algorithm repeatedly forms dangling suffixes (what remains when one codeword is a prefix of another string); the code fails to be uniquely decodable exactly when a dangling suffix is itself a codeword. {0,01,11} is uniquely decodable yet not prefix-free; {0,01,10} is ambiguous — '010' splits two ways. Verified live: the algorithm confirms {0,01,11} decodable & not prefix-free, flags {0,01,10} and the classic {1,011,01110,1110,10011} as ambiguous, and rules every random prefix-free code decodable. Neon-noir traced. See the two-parse stream in 1D, the growing suffix sets in 2D, and the test-directly inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "b700eadf4c876198ab1737ab64da554cc10c0d23213b50b2b30cad9848c65350"
    },
    {
      "slug": "the-capped-cross",
      "title": "THE CAPPED CROSS",
      "kicker": "cap a ray with a T and the plane grids itself",
      "accent": "#ff2fa6",
      "blurb": "The capped cross in the 5-window house format — what a plus-sign becomes when you cap its arms with a crossbar (a T on each ray), and the plane grids itself. A bare cross is 5 points and 4 segments; cap the ends, fill the corners, and you have a 3×3 lattice: 9 points. The orthogonal unit segments between neighbours number exactly 12; the four unit cells add 8 diagonals, so the full king-move graph has 20 edges. Every one of the 12 orthogonal segments is a stroke a monoline glyph can use; the diagonals exist only because the T gridded the plane. Verified live: enumerating the 3×3 lattice gives 9 points, 12 orthogonal segments, 8 cell diagonals, and 20 king-graph edges — all by direct count. Neon-noir traced. See the cross-to-grid in 1D, the toggle-able diagonals in 2D, and the count-what-the-grid-made-possible inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "08bd0ff4359d0cb71cc0800270587f4d231f7ccfafb1ec25afe89ef234ff3fcb"
    },
    {
      "slug": "the-successive-keeper",
      "title": "THE SUCCESSIVE KEEPER",
      "kicker": "a diffable record beats a believed one",
      "accent": "#b06bff",
      "blurb": "The successive keeper in the 5-window house format — a ledger where each entry commits to the one before it: hᵢ = H(hᵢ₋₁ ‖ entryᵢ). Because every link folds in the whole prefix, changing any entry changes the head hash, and the first place two chains diverge points straight at the tampered entry. A single unlinked keeper cannot do this: an additive checksum is blind to a compensating edit (add d here, subtract d there) and reports the same total, while the chain still catches it. One keeper can only be believed; successive keepers can be diffed. Verified live: over 5,000 trials, tampering one entry always changes the chain head, the first divergent link localises it, an additive checksum misses every compensated edit, and the chain catches all of them. Neon-noir traced. See the breaking chain in 1D, chain-vs-checksum in 2D, and the diffable-not-believed inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "fad3d143a43907dd0b0adfea750e567ac7b059f6fdcef42d773d7f962b827c60"
    },
    {
      "slug": "the-golden-radix",
      "title": "THE GOLDEN RADIX",
      "kicker": "an irrational base that still carries the integers",
      "accent": "#ffcf4a",
      "blurb": "The golden radix in the 5-window house format — base-φ, positional notation whose base is the golden ratio, an irrational. Digits are 0 and 1; place values are powers of φ. The identity φ²=φ+1 means '011' always rewrites to '100', so every value has a unique standard form with no two adjacent 1s. Remarkably every ordinary integer has a finite such expansion (1=1, 2=10.01, 3=100.01, 4=101.01) even though the base is irrational. Verified live: the greedy base-φ expansion of every integer 0..100 decodes back to it (max error <1e-6) and always has no consecutive 1s. Neon-noir traced. See the digit cells in 1D, the mint stamping each integer in 2D, and the base-does-the-collapsing inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "337f986b9b354a72cddefbafc046964123420850d5f1a9831acf9c6f707037fa"
    },
    {
      "slug": "the-routh",
      "title": "THE ROUTH",
      "kicker": "a cevian triangle's area is a closed form",
      "accent": "#21e6ff",
      "blurb": "Routh's theorem in the 5-window house format — the area of the little triangle three cevians carve out of a big one, as an exact closed form. Draw cevians cutting the opposite sides in ratios x, y, z; the central triangle's area as a fraction of the whole is (xyz−1)²/[(xy+x+1)(yz+y+1)(zx+z+1)]. At x=y=z=2 the fraction is exactly 1/7 (the famous one-seventh-area triangle); at xyz=1 the cevians are concurrent (Ceva) and it vanishes. Verified live: for 4000 random ratio triples the closed form matches the directly-constructed central-triangle area to machine precision, and x=y=z=2 gives 1/7. Neon-noir traced. See the 1/7 case in 1D, adjustable ratios in 2D, and the ratios-already-know-the-area inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "b18b83818fb47f2c57e67fa467e905896ab533a5ae60c9a3ff709b2799e0137d"
    },
    {
      "slug": "the-napkin",
      "title": "THE NAPKIN",
      "kicker": "a band through any sphere holds the same volume",
      "accent": "#35ffb0",
      "blurb": "The napkin-ring problem in the 5-window house format — drill a cylindrical hole through the centre of a sphere so the remaining band has height h; its volume is πh³/6, depending only on h, not on the sphere. A ring of height h from a marble and the same-height ring from a planet have identical volume, because at height z the leftover annulus has area π[(h/2)²−z²] with the sphere's radius R gone entirely. Verified live: numerically integrating the ring volume for R = 1, 1.5, 2, 5, 20, 100 (fixed h=2) gives πh³/6 ≈ 4.18879 every time. Neon-noir traced. See two very different spheres, same ring, in 1D; a growable sphere in 2D; and the height-fixes-the-volume inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "a0c94fad5a09d13013a237a6faaa601e8a993513abd84db98387a595f6ef64ac"
    },
    {
      "slug": "the-monge",
      "title": "THE MONGE",
      "kicker": "three circles' external centres fall on one line",
      "accent": "#ff8a3c",
      "blurb": "Monge's theorem in the 5-window house format — take any three circles of different radii; for each pair the two outer tangents meet at the external centre of similitude, and the three external centres are always collinear, whatever the circles. Each external centre divides the line of centres externally in the ratio of the radii: E = (r₂C₁−r₁C₂)/(r₂−r₁). Verified live: over 5000 random triples of distinct-radius circles, the three external centres are collinear to machine precision (normalized cross ~1e-14). Neon-noir traced. See the three circles and the Monge line in 1D, randomizable circles in 2D, and the lift-the-plane inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "3ec91a4a3c1d0c2ea320d531cb8638063aeb4a75c8571c981769aecf9bb900ce"
    },
    {
      "slug": "the-minkowski",
      "title": "THE MINKOWSKI",
      "kicker": "climbs 0 to 1 with slope 0 almost everywhere",
      "accent": "#b06bff",
      "blurb": "Minkowski's question-mark function ?(x) in the 5-window house format — a monster hiding in plain sight: it climbs continuously and strictly from ?(0)=0 to ?(1)=1, yet its derivative is zero almost everywhere (a 'singular' function that rises using no measurable slope). Built from continued fractions: for x=[0;a₁,a₂,…], ?(x) = 2Σ(−1)^{k+1} 2^{−(a₁+…+aₖ)}. It sends every rational to a dyadic fraction and every quadratic irrational to a rational — famously ?(1/φ)=2/3. Verified live: ?(1/2)=1/2, ?(1/3)=1/4, ?(2/3)=3/4, ?(1/φ)=2/3, monotonic, symmetric (?(1−x)=1−?(x)), and dyadic on thousands of rationals. Neon-noir traced. See the curve in 1D, a chosen rational's dyadic image in 2D, and the CF-tree-to-binary-tree inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "3798859f8280efa86909efbabbf58c9f23341297ccd9d9566daa97fede75493b"
    },
    {
      "slug": "the-fibonacci-word",
      "title": "THE FIBONACCI WORD",
      "kicker": "an infinite word that is its own seed",
      "accent": "#21e6ff",
      "blurb": "The Fibonacci word in the 5-window house format — an infinite string grown from a single letter by the morphism a→ab, b→a: a, ab, aba, abaab, abaababa, …. Each stage is the previous two concatenated (Sₙ = Sₙ₋₁Sₙ₋₂), so its length is a Fibonacci number, and the fraction of a's tends to 1/φ. It is the simplest Sturmian word: no 'bb' and no 'aaa', the most balanced non-periodic string there is. Verified live: for stages 1..22, lengths are Fibonacci, the concatenation and morphism both reproduce it, there is no 'bb' or 'aaa', and the a-fraction is 0.61803 ≈ 1/φ. Neon-noir traced. See the split concatenation in 1D, the growing stages in 2D, and the morphism-expansion inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "e5f2d3b2c45b844633385f70d917a8ad977fb1ec2828a15cac7490aa365ce2cb"
    },
    {
      "slug": "the-simson",
      "title": "THE SIMSON LINE",
      "kicker": "feet that align only on the circle",
      "accent": "#ff8a3c",
      "blurb": "The Simson line in the 5-window house format — drop perpendiculars from a point P to the three sides of a triangle and mark the three feet; in general they form a small triangle, but the instant P lands on the circumcircle the three feet fall exactly on one straight line. It is an if-and-only-if: the feet are collinear precisely when P is on the circle, since the pedal triangle's area is proportional to |R²−OP²|, zero exactly on the circle. Verified live: thousands of points on the circumcircle give collinear feet to machine precision, while off-circle points give a pedal triangle of clearly non-zero area. Neon-noir traced. See the line in 1D, P moving on and off the circle in 2D, and the area-vanishes-on-the-circle inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "993f871c120102ff456e4d16a3c9dc4f7280d3b99e435055167c85d96a43f3be"
    },
    {
      "slug": "the-erdos-szekeres",
      "title": "THE ERDOS-SZEKERES",
      "kicker": "order you cannot escape",
      "accent": "#35ffb0",
      "blurb": "The Erdős–Szekeres theorem in the 5-window house format — order you cannot escape. In any sequence of (r−1)(s−1)+1 distinct numbers there must be an increasing subsequence of length r or a decreasing one of length s, however you scramble it (ten numbers always hide a monotone run of four). The proof is pigeonhole: label each term by its longest increasing and longest decreasing run ending there; too few labels for the terms forces a collision, hence a long run. The bound is tight — a block sequence of exactly (r−1)(s−1) dodges both. Verified live: 4000 random sequences always contain the guaranteed run, and the block construction achieves exactly r−1 and s−1. Neon-noir traced. See the two runs in 1D, random vs tight in 2D, and the label-grid pigeonhole inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "454d590df7c6bf67c750953570b0af9ad7fdd5d372430decf321a7ba32fe3ac2"
    },
    {
      "slug": "the-johnson",
      "title": "THE JOHNSON CIRCLES",
      "kicker": "three circles hand off to a fourth of equal size",
      "accent": "#ffcf4a",
      "blurb": "Johnson's circles in the 5-window house format — take three circles of the same radius ρ all passing through one common point H; each pair meets again at a second point, and those three second points lie on a fourth circle of exactly the same radius ρ. The centre is C = O₁+O₂+O₃−2H, and each second point Pᵢⱼ = Oᵢ+Oⱼ−H sits at distance |Oₖ| = ρ from it — a clean vector identity. Verified live: for 5000 random configurations of three equal circles through a common point, all three second intersections are at distance ρ from C — a same-radius circle every time. Neon-noir traced. See the four equal circles in 1D, randomizable circles in 2D, and the add-the-centres inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "218a402e7fe1122423fbc830f1222946b0a53cd0d797b04941310f6aafca3cb9"
    },
    {
      "slug": "the-newton-gauss",
      "title": "THE NEWTON-GAUSS LINE",
      "kicker": "four lines hide a straight line in their diagonals",
      "accent": "#ffcf4a",
      "blurb": "The Newton–Gauss line in the 5-window house format — inside a complete quadrilateral (four lines in general position, meeting in six points), pair the six vertices into three diagonals and take each diagonal's midpoint. The three midpoints are always collinear, lying on one line: the Newton–Gauss line of the figure. Four arbitrary lines, and a hidden straight line falls out of the midpoints of the diagonals. Verified live: over 5000 random complete quadrilaterals, the three diagonal midpoints are collinear to machine precision (normalized cross ~1e-13). Neon-noir traced. See the four lines and the hidden line in 1D, randomizable lines in 2D, and the study-the-midpoints inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "e3df12c48e5525fb80afa114a240fe3b16ae0d05b620e72624dee2e327f6a3bd"
    },
    {
      "slug": "the-dpll",
      "title": "THE DPLL",
      "kicker": "a search that prunes itself",
      "accent": "#21e6ff",
      "blurb": "DPLL in the 5-window house format — the Davis–Putnam–Logemann–Loveland backtracking search under every SAT solver. Given a Boolean formula in conjunctive normal form, it decides whether some true/false assignment satisfies every clause. Two moves make it fast: unit propagation (a clause down to one literal forces that literal, cascading) and backtracking (try a variable true, and on a dead end back up and try false). Verified live: over 5000 random formulas, DPLL's SAT/UNSAT verdict matches brute-force enumeration of all 2^n assignments exactly, and every SAT answer comes with an assignment satisfying all clauses. Neon-noir traced. See the clauses in 1D, a decided formula in 2D, and the unit-propagation-prunes-the-tree inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "afaed0b2d407b62d44e621ecd57cc40eb4e956b658c754c71d6980d69cbfefe8"
    },
    {
      "slug": "the-lattice-reduction",
      "title": "THE LATTICE REDUCTION",
      "kicker": "a shorter view of the same lattice",
      "accent": "#ffcf4a",
      "blurb": "Lattice reduction in the 5-window house format — turning a skewed, long-vector basis into a shorter, nearly-orthogonal basis for the same lattice. In two dimensions the Lagrange–Gauss algorithm does it optimally: repeatedly subtract the nearest integer multiple of the shorter vector from the longer, swapping when needed, until neither shrinks the other; the result's first vector b₁ is the shortest nonzero vector in the whole lattice, and because every step is unimodular the lattice and its covolume never change. This 2D kernel is exactly what the celebrated LLL algorithm generalizes to n dimensions. Verified live: over 3000 random integer bases, covolume is preserved, the output is size-reduced, and b₁ equals the brute-force shortest vector. Neon-noir traced. See both bases over one grid in 1D, a reducible basis in 2D, and the same-lattice-shorter-view inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "2d2d1e66773f64cd369c1890d746928196572080aa9d2d7b8be5182f70053f70"
    },
    {
      "slug": "the-kmp",
      "title": "THE KMP",
      "kicker": "a search that never looks back",
      "accent": "#35ffb0",
      "blurb": "The Knuth–Morris–Pratt algorithm in the 5-window house format — finding a pattern in a text in linear time, never looking back in the text. Its secret is the failure function: for each pattern position, the length of the longest proper prefix that is also a suffix there. On a mismatch, KMP jumps the pattern forward by what the failure function already knows, so the text pointer only moves forward; the whole search costs at most about 2n comparisons. Verified live: over 3000 random text/pattern pairs, the failure function matches its definition, KMP finds the same matches as naive search, and comparisons stay under 2n. Neon-noir traced. See the failure function in 1D, a live search in 2D, and the pattern-knows-itself inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "7e252e96c2358d5badc9c44f0a5df686c68f793c4dce0c9498a61cedf44fee03"
    },
    {
      "slug": "the-simplex",
      "title": "THE SIMPLEX",
      "kicker": "the optimum lives on the boundary",
      "accent": "#ff8a3c",
      "blurb": "The simplex method in the 5-window house format — solving linear programs (maximize c·x subject to Ax ≤ b, x ≥ 0). The feasible region is a convex polytope and the optimum is always at a vertex, never strictly inside, so simplex starts at a corner and walks along edges to a better neighbour until none improves. It never wanders the interior; it hops corner to corner. Verified live: over 3000 random linear programs, the simplex optimum equals the best value found by brute-force enumeration of every feasible vertex of the polytope. Neon-noir traced. See the feasible polygon in 1D, a random LP in 2D, and the optimum-is-always-a-corner inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "364b56ef71caaf0544dfece6718079fab32a529a29961cdce22a1c38c5d58c16"
    },
    {
      "slug": "the-bron-kerbosch",
      "title": "THE BRON-KERBOSCH",
      "kicker": "the pivot does the pruning",
      "accent": "#b06bff",
      "blurb": "The Bron–Kerbosch algorithm in the 5-window house format — finding every maximal clique in an undirected graph (every group of mutually-connected vertices that cannot be extended). It grows a clique R from candidates P while excluding tried vertices X; when P and X are both empty, R is maximal. A pivot vertex prunes redundant branches: only candidates that are not neighbours of the pivot need to be tried. Verified live: over 3000 random graphs, the maximal cliques Bron–Kerbosch reports are exactly the set found by brute-force subset enumeration. Neon-noir traced. See a highlighted clique in 1D, all maximal cliques in 2D, and the pivot-prunes-the-branches inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "3cb1f91c647142524a4d2f4dab627d45747bc8de034841f8d011602c57463342"
    },
    {
      "slug": "the-segment-tree",
      "title": "THE SEGMENT TREE",
      "kicker": "a range in a logarithm of nodes",
      "accent": "#21e6ff",
      "blurb": "The segment tree in the 5-window house format — answering range questions (sum, minimum) over an array in O(log n), with point updates just as fast. It is a binary tree over the array: leaves are elements, every internal node the aggregate of its two children. Any range [l,r] splits into at most 2 log n canonical nodes whose stored aggregates already hold the answer, so you never rescan; changing one leaf refreshes only the log n nodes above it. Verified live: over 2000 random arrays with interleaved point-updates, range-sum and range-min queries match a direct rescan every time. Neon-noir traced. See the aggregate tree in 1D, range queries in 2D, and the range-decomposes-into-log-n-nodes inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "aa5424ac10eb6c1bbef2270425c001bd8e3fe881dba9d7547e9f0fd06d70e2ee"
    },
    {
      "slug": "the-avl",
      "title": "THE AVL TREE",
      "kicker": "balance kept by rotation",
      "accent": "#ff8a3c",
      "blurb": "The AVL tree in the 5-window house format — the first self-balancing binary search tree. Every node keeps a balance factor (left height minus right), held in {−1,0,+1}; when an insertion tips a node to ±2, one or two rotations restore the invariant locally. Because no node is ever more than one level lopsided, the height stays near 1.44 log₂n, so search/insert/delete are all guaranteed O(log n) — never the O(n) of a degenerate list. Verified live: over 2000 random insertion sequences, the in-order traversal is always sorted, every balance factor stays within ±1, the height respects the 1.44 log₂n bound, and search finds exactly the inserted keys. Neon-noir traced. See balance factors in 1D, live insertion+rotation in 2D, and the invariant-caps-the-height inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "fdb53332fb522286e01d43a86a976cf84fb90204dc78d74649af06c3b6117bbc"
    },
    {
      "slug": "the-xor-filter",
      "title": "THE XOR FILTER",
      "kicker": "a set in 1.23 bytes a key",
      "accent": "#35ffb0",
      "blurb": "The XOR filter in the 5-window house format — a modern, leaner cousin of the Bloom filter for a fixed set. It stores a table of small fingerprints so every key x satisfies fp(x) = t[h₀(x)] ⊕ t[h₁(x)] ⊕ t[h₂(x)]. Building it is a graph peeling: repeatedly take a slot touched by only one key and assign it last so the XOR comes out right. It uses ~1.23 bytes per key (smaller than Bloom for the same rate), with no false negatives and a false-positive rate near 2⁻⁸. Verified live: across many builds, every member's three-slot XOR equals its fingerprint, and the false-positive rate on non-members is about 1/256. Neon-noir traced. See the XOR equation in 1D, build+query in 2D, and the peel-and-solve inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "f634be72d92c76285f50d7a5fa612dafb27038626273fcebdd995eb7d50f32cb"
    },
    {
      "slug": "the-elias-gamma",
      "title": "THE ELIAS GAMMA",
      "kicker": "a number that says its own length",
      "accent": "#ffcf4a",
      "blurb": "Elias gamma coding in the 5-window house format — a self-delimiting code for positive integers with no length field and no separators, yet a stream decodes unambiguously. To encode n: write ⌊log₂n⌋ zeros, then the plain binary of n (which begins with a 1); the leading zeros tell the decoder how many more bits to read. Its length is 2⌊log₂n⌋+1 bits, so small numbers stay tiny — a universal code, near-optimal when small values dominate. Verified live: encode–decode round-trips for thousands of integers, every length equals 2⌊log₂n⌋+1, and a concatenated stream splits back into the exact original list. Neon-noir traced. See the code in 1D, a decoded stream in 2D, and the number-carries-its-own-length inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "950b619b5b842eada8ac0396bdbdbcf2ec4a3081711a1a197bf5776b78388e12"
    },
    {
      "slug": "the-jump-hash",
      "title": "THE JUMP CONSISTENT HASH",
      "kicker": "buckets that barely move when you add one",
      "accent": "#b06bff",
      "blurb": "Jump consistent hashing in the 5-window house format — mapping a key to one of N buckets so that when N grows, almost no keys move, with no lookup table and O(1) memory. It replays a tiny pseudo-random sequence seeded by the key; each 'jump' decides whether the key hops to a higher bucket, and the last it lands on is the answer. Two guarantees: keys spread uniformly, and going N→N+1 relocates only ~1/(N+1) of keys — each moving straight to the new bucket, never shuffling among the old ones. Verified live: 100,000 keys spread within a few percent of uniform, and N→N+1 moves ~1/(N+1) of keys, each landing on the new bucket. Neon-noir traced. See the flat histogram in 1D, add-a-bucket in 2D, and the move-only-the-newcomer's-share inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "68ea0460ce0e1319d5dab2d1ca563cee10756bc4f66bc352df70481d40251701"
    },
    {
      "slug": "the-padic",
      "title": "THE P-ADIC",
      "kicker": "a metric where big powers are small",
      "accent": "#21e6ff",
      "blurb": "The p-adic numbers in the 5-window house format — a strange way to measure size where a number is small when it is divisible by a high power of a prime p. The p-adic absolute value is |x|_p = p^−v (v = how many times p divides x), so 1, p, p², p³… march toward zero. This metric is ultrametric: |x+y|_p ≤ max(|x|_p,|y|_p), stronger than the ordinary triangle inequality. Numbers get infinite digit-strings running leftward, and famously …1111 = −1 in the 2-adics (2^k−1 ≡ −1 for every k). Verified live: over thousands of rationals the ultrametric holds, any a/b (b coprime to p) reconstructs from its p-adic digits mod p^k, and the all-(p−1) digit string equals −1. Neon-noir traced. See …1111=−1 in 1D, digit expansion in 2D, and the ultrametric-tree inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "3785c5de76973871f3dd5e4f8f2790cf495e99c61f712d41209a1c664eec3624"
    },
    {
      "slug": "the-brzozowski",
      "title": "THE BRZOZOWSKI",
      "kicker": "matching by taking the language apart",
      "accent": "#ff8a3c",
      "blurb": "Brzozowski derivatives in the 5-window house format — matching a regular expression by taking the language apart one symbol at a time. The derivative D_c(r) is a new regex matching exactly the strings r would match after consuming c; there are simple rules per operator, and matching is a fold: feed the string in, take a derivative per character, and ask whether the residual regex is nullable (matches ε). No NFA, no backtracking — just algebra on regexes. Verified live: over 8000 random regex/string pairs, the derivative matcher agrees exactly with an independent backtracking matcher. Neon-noir traced. See a derivative in 1D, a matched fold in 2D, and the differentiate-don't-simulate inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "debc777074b082f12aa0105edff60bd4856a30380eb342bdd6521d9efb394c39"
    },
    {
      "slug": "the-binomial-heap",
      "title": "THE BINOMIAL HEAP",
      "kicker": "a heap counted in binary",
      "accent": "#35ffb0",
      "blurb": "The binomial heap in the 5-window house format — a priority queue built as a forest of binomial trees of sizes 1,2,4,8,… (the powers of two). A heap of n elements has one tree per 1-bit of n, so its shape is the binary numeral of its size. Merging two heaps works like binary addition: line trees up by order and carry-link equal orders, giving O(log n) union — and insert, extract-min follow; each tree is heap-ordered so the minimum is a root. Verified live: draining a heap by repeated extract-min returns keys in sorted order, and merging two heaps then draining yields the combined sorted sequence. Neon-noir traced. See the binary forest in 1D, insert+extract in 2D, and the read-the-count-in-binary inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "aeca948e0f3f84b9484f3a4d3469c99eb0fedb446a0e950ef4cd4295d76fa7ae"
    },
    {
      "slug": "the-gram-schmidt",
      "title": "THE GRAM-SCHMIDT",
      "kicker": "vectors made perpendicular",
      "accent": "#ffcf4a",
      "blurb": "Gram–Schmidt in the 5-window house format — turning independent vectors into an orthonormal set spanning the same space. Take each vector in turn and subtract its projection onto the directions already fixed, leaving only the perpendicular part, then scale to length one. The results q₁,q₂,… are mutually perpendicular unit vectors, and every original vector is a combination of the q's built so far — exactly the QR decomposition A=QR with R upper-triangular. Verified live: over 3000 random matrices the vectors satisfy qᵢ·qⱼ=δᵢⱼ (orthonormal) and each original reconstructs from the q's up to its index. Neon-noir traced. See the perpendicular part in 1D, orthonormalized vectors in 2D, and the projections-are-R inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "b5c36e4b223fed96948dba7d16fb5578360d2b62bb1eb727f16713d02d00e5a7"
    },
    {
      "slug": "the-pratt-parsing",
      "title": "THE PRATT PARSING",
      "kicker": "precedence from binding power",
      "accent": "#b06bff",
      "blurb": "Pratt parsing in the 5-window house format — top-down operator-precedence parsing from one idea: every operator has a binding power, and it binds tighter than another exactly when its power is higher. The parser reads a value, then keeps absorbing operators to its right as long as their binding power beats the current threshold, recursing to gather the right operand. That single rule reproduces the full precedence and associativity of arithmetic — × before +, parentheses, unary minus — with no grammar tables, in a handful of lines. Verified live: over 8000 random expressions, the Pratt value equals an independent precedence-explicit recursive-descent evaluator (2+3×4 = 14, not 20). Neon-noir traced. See the parse tree in 1D, an evaluated expression in 2D, and the one-number-per-operator inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "e6a09426ae6520de0ed3ffb6e9dad1e74aa1862cc66aa497a1506a7695f98e23"
    },
    {
      "slug": "the-chromatic-polynomial",
      "title": "THE CHROMATIC POLYNOMIAL",
      "kicker": "colourings counted by a polynomial",
      "accent": "#21e6ff",
      "blurb": "The chromatic polynomial in the 5-window house format — P(G,k) counts the proper k-colourings of a graph (paint vertices with k colours so no edge joins two of the same), and that count is a polynomial in k. It obeys deletion–contraction: P(G) = P(G−e) − P(G/e). Trees give P = k(k−1)^(n−1); cycles give P = (k−1)^n + (−1)^n(k−1). Verified live: over thousands of random graphs the deletion–contraction value equals a brute-force count of proper colourings, and the tree and cycle formulas hold exactly. Neon-noir traced. See the recursion in 1D, P(G,k) values in 2D, and the peel-the-edges inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "c8dd6a70e24746fab0c3ff26749323dd8adf8ece92fe892bba1f3ce00c583850"
    },
    {
      "slug": "the-myhill-nerode",
      "title": "THE MYHILL-NERODE",
      "kicker": "the fewest states a language needs",
      "accent": "#ff8a3c",
      "blurb": "The Myhill–Nerode theorem in the 5-window house format — the fewest states a language needs. Call two strings equivalent if no continuation ever tells them apart (for every suffix z, xz and yz are both in the language or both out); the number of such classes equals the number of states in the minimal DFA, and a language is regular exactly when that number is finite. So minimizing an automaton is merging states no string can distinguish. Verified live: over thousands of random DFAs, the minimized machine accepts the same language on all short strings and is truly minimal — every pair of its states is separated by some string. Neon-noir traced. See indistinguishable states in 1D, a minimized DFA in 2D, and the states-are-distinguishable-futures inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "b4043bf593157a9bedb025fe0cb86e4843627fab3fb62e44e00de92b4b0338e2"
    },
    {
      "slug": "the-bernstein",
      "title": "THE BERNSTEIN",
      "kicker": "a curve that approximates any function",
      "accent": "#35ffb0",
      "blurb": "Bernstein polynomials in the 5-window house format — a constructive proof of Weierstrass's theorem: any continuous function on [0,1] can be approximated as closely as you like by a polynomial. The n-th Bernstein polynomial samples f at k/n and blends with binomial weights: B_n(f)(x) = Σ f(k/n) C(n,k) x^k (1−x)^(n−k). As n grows, B_n(f)→f uniformly. The weights sum to 1 (partition of unity), so B_n(f) is a moving average of f-values that never strays far from the curve. Verified live: the maximum error |B_n(f)−f| shrinks as n grows for several continuous f, the weights sum to 1 for all n, and B_n reproduces linear functions exactly. Neon-noir traced. See convergence in 1D, the shrinking error in 2D, and the average-the-samples inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "12f1b9a37e68136bba63326eb55dacaeb4771e092f5492f45c288495f066db85"
    },
    {
      "slug": "the-persistent-structure",
      "title": "THE PERSISTENT STRUCTURE",
      "kicker": "versions that never overwrite the past",
      "accent": "#ffcf4a",
      "blurb": "Persistent data structures in the 5-window house format — every update returns a new version while all older versions remain readable, unchanged. The trick is path copying: store the data in a balanced tree, and to change one element copy only the O(log n) nodes on the root-to-leaf path, letting the new root share every untouched subtree with the old. An update costs O(log n) time and space, yet the whole history stays alive — the basis of undo, versioned databases, and functional programming. Verified live: after a long sequence of updates, reading any old version returns exactly its values at the time, and every update allocates only O(log n) new nodes. Neon-noir traced. See path copying in 1D, versioned reads in 2D, and the share-what-you-can-reuse inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "a2fa10249a3d66de92e2eee268f23957ef83f8348116158f2a6c5861e0477ffa"
    },
    {
      "slug": "the-earley",
      "title": "THE EARLEY",
      "kicker": "parsing any grammar from a chart",
      "accent": "#b06bff",
      "blurb": "The Earley parser in the 5-window house format — recognizing any context-free grammar (even ambiguous or left-recursive) in a single left-to-right sweep. At each position it keeps a set of items (dotted rules recording how far each production has matched) and grows them with three moves: predict (open the rules a symbol could start), scan (advance an item expecting the next token), and complete (when a rule finishes, advance whoever was waiting on it). The string is in the language exactly when a start rule completes spanning the whole input. Verified live: over 20,000 random token strings, Earley accepts for the arithmetic grammar iff an independent recursive-descent recognizer accepts, and it takes 'n+n×n' while rejecting 'n+×n'. Neon-noir traced. See the chart in 1D, a decided string in 2D, and the carry-every-partial-parse inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "231336021f4ad4d56dfdfc379f2333613ba95601065dfa8671cef4b9c3f77892"
    },
    {
      "slug": "the-reed-muller",
      "title": "THE REED-MULLER",
      "kicker": "a code folded from itself",
      "accent": "#21e6ff",
      "blurb": "The Reed–Muller code in the 5-window house format — an error-correcting code built from low-degree Boolean polynomials: codewords are the truth-tables of every multilinear polynomial of degree ≤ r in m variables. That gives length 2^m, dimension Σ_{i≤r} C(m,i), and minimum distance 2^(m−r). Its signature is the (u | u+v) recursion: RM(r,m) is built by stacking codewords of RM(r,m−1) and RM(r−1,m−1) — folded out of smaller copies of itself. Verified live: for several (r,m) the dimension equals Σ C(m,i), the minimum nonzero weight equals 2^(m−r), and the (u|u+v) construction rebuilds the code exactly. Neon-noir traced. See RM(1,3)'s generator in 1D, code parameters in 2D, and the (u|u+v) fold inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "c32483d86c0d2d0584395d46f62a2dc279c4185137c2ae4c37bc7957ebdb50ec"
    },
    {
      "slug": "the-bernstein-vazirani",
      "title": "THE BERNSTEIN-VAZIRANI",
      "kicker": "a hidden string in one query",
      "accent": "#ff8a3c",
      "blurb": "The Bernstein–Vazirani algorithm in the 5-window house format — extracting a hidden n-bit string s from a black box computing f(x)=s·x (mod 2) in one query, where any classical strategy needs n. Put every input into superposition, let the oracle stamp the phase (−1)^(s·x), and a second Hadamard layer focuses all amplitude onto |s⟩; measure once and read s. It is the cleanest demonstration that quantum parallelism beats classical query complexity. Verified live: simulating the amplitudes, the output is 1 exactly at |s⟩ and 0 elsewhere, so the recovered string equals the hidden s every time, from a single oracle call. Neon-noir traced. See the amplitude spike in 1D, single-query recovery in 2D, and the interfere-don't-iterate inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "da3da6c134c42440867ceac71aedf93edc0a7fced4f20a069a506e06d91110a6"
    },
    {
      "slug": "the-sherman-morrison",
      "title": "THE SHERMAN-MORRISON",
      "kicker": "updating an inverse without redoing it",
      "accent": "#35ffb0",
      "blurb": "The Sherman–Morrison formula in the 5-window house format — updating a matrix inverse after a rank-one change without redoing the whole inversion. If you know A⁻¹ and bump A by an outer product uvᵀ, the new inverse is (A+uvᵀ)⁻¹ = A⁻¹ − (A⁻¹u vᵀA⁻¹)/(1+vᵀA⁻¹u). A full inversion costs O(n³); this correction costs O(n²) — a decisive shortcut for recursive least squares, Kalman filters, and quasi-Newton optimizers that nudge a matrix one rank at a time. Verified live: over thousands of random A, u, v, the Sherman–Morrison result matches a direct inversion of A+uvᵀ to machine precision. Neon-noir traced. See the rank-one correction in 1D, side-by-side matrices in 2D, and the patch-don't-rebuild inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "1cf1c90f8ee84f1c175a02f5f9f631f9145de7fe44db2b8b77bb5baa59b46250"
    },
    {
      "slug": "the-half-plane-intersection",
      "title": "THE HALF-PLANE INTERSECTION",
      "kicker": "a region carved by half-planes",
      "accent": "#ffcf4a",
      "blurb": "Half-plane intersection in the 5-window house format — carving out the region satisfying a set of linear inequalities. Each constraint a·x ≤ c keeps one side of a line (a half-plane), and their intersection is a convex polygon (possibly empty or unbounded). Build it by clipping: start with a big bounding box and slice it by each half-plane in turn, keeping the inside. The result is exactly the feasible region of a linear program, and every point of it obeys every constraint at once. Verified live: over thousands of random constraint sets, every vertex of the clipped region satisfies all half-planes, and a point lies inside only if it satisfies every constraint. Neon-noir traced. See the box clipped in 1D, the region + sampled points in 2D, and the carve-don't-test inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "39a205d74538277d2e7d0ebf3933ac7422de37dd32cf657bd744f1a4aa25224a"
    },
    {
      "slug": "the-toffoli",
      "title": "THE TOFFOLI",
      "kicker": "a gate that runs backwards",
      "accent": "#b06bff",
      "blurb": "The Toffoli gate in the 5-window house format — a reversible, universal logic gate. It takes three bits (a,b,c) to (a, b, c⊕(a∧b)): it flips the third bit exactly when the first two are both 1, leaving the controls untouched. Because it is a bijection on the eight input states it can run backwards — it is its own inverse. And it is universal for classical computation: set c=0 and the output is a∧b (AND), fix the controls and it is NOT, so every Boolean circuit rebuilds from Toffolis without erasing information. Verified live: the gate is a permutation of the 8 states, applying it twice is the identity, and it computes AND and NOT. Neon-noir traced. See the truth table in 1D, live gadgets in 2D, and the runs-backwards inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "63d548bd9073613741068328fc1c7f4c755e27b3e2b3e1e71870d139e81efa71"
    },
    {
      "slug": "the-schur-complement",
      "title": "THE SCHUR COMPLEMENT",
      "kicker": "a determinant split by a block",
      "accent": "#21e6ff",
      "blurb": "The Schur complement in the 5-window house format — what remains of a block matrix after eliminating one block. For M=[[A,B],[C,D]] with A invertible, the Schur complement of A is S = D − CA⁻¹B, the effective D once A's influence is folded in. It splits the determinant cleanly: det(M) = det(A)·det(S). It gives the block inverse in closed form and decides definiteness (M is PD iff A and S both are) — the algebra behind block Gaussian elimination, Kalman updates, and Gaussian conditioning. Verified live: over thousands of random block matrices, det(M) = det(A)·det(S), and the block inverse from the Schur complement satisfies M·M⁻¹ = I. Neon-noir traced. See the block split in 1D, determinants in 2D, and the eliminate-a-block inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "0f7134c8809a4a8fb1d6618053bb71682fc170ba6a201026ceab9c61897f47db"
    },
    {
      "slug": "the-kaczmarz",
      "title": "THE KACZMARZ",
      "kicker": "zigzagging onto the solution",
      "accent": "#ff8a3c",
      "blurb": "The Kaczmarz method in the 5-window house format — solving Ax=b by bouncing between hyperplanes. Each equation aᵢ·x=bᵢ is a hyperplane; the algorithm projects the current guess onto the next equation's hyperplane: x ← x + (bᵢ−aᵢ·x)/‖aᵢ‖²·aᵢ. Cycling through the rows, the iterate zigzags in and converges to the solution — one row at a time, never forming AᵀA. It is the ancestor of the ART reconstruction behind CT scanners. Verified live: over thousands of random consistent systems, cyclic projection converges to the true solution to machine precision. Neon-noir traced. See two lines spiraling to their intersection in 1D, live projection in 2D, and the one-row-at-a-time inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "e6a0edfd79f79e9e02f152abb79ad82adf7168641320986f99dbc08e07cd4ea7"
    },
    {
      "slug": "the-simhash",
      "title": "THE SIMHASH",
      "kicker": "similarity read from sign bits",
      "accent": "#35ffb0",
      "blurb": "SimHash in the 5-window house format — turning similarity into a handful of bits. Pick random hyperplanes through the origin; for a vector v record one bit per hyperplane, sign(v·r). The magic: for two vectors at angle θ, a random hyperplane separates them with probability exactly θ/π. So the Hamming distance between their sign-bit sketches estimates the angle — near-duplicate detection in a fixed-size fingerprint, the trick behind web-scale de-duplication. Verified live: with rotationally-symmetric (Gaussian) hyperplanes, the fraction of differing sign bits matches θ/π to within sampling error over many random pairs. Neon-noir traced. See the separating wedge in 1D, the angle estimate in 2D, and the compare-the-bits inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "fc5a1e55ac1b02321c1c6249f926aee7dda41a47db4da8705b918728cae0d070"
    },
    {
      "slug": "the-powerset-construction",
      "title": "THE POWERSET CONSTRUCTION",
      "kicker": "determinizing by tracking the set of states",
      "accent": "#ffcf4a",
      "blurb": "The powerset (subset) construction in the 5-window house format — turning a nondeterministic finite automaton into an equivalent deterministic one. An NFA can be in many states at once; the trick is to make each DFA state a set of NFA states — exactly the set the NFA could currently be in. Reading a symbol, the DFA jumps to the set of all reachable states, and accepts when that set contains any NFA-accepting state. It proves NFAs and DFAs recognize the same languages, at the cost of up to 2ⁿ states. Verified live: over thousands of random NFAs, the subset-construction DFA accepts a string exactly when the NFA does, on every string up to length six. Neon-noir traced. See DFA-states-as-subsets in 1D, NFA-vs-DFA in 2D, and the track-the-set inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "8ff95fd828ac22c6908008d5565e95d8a021e6aac3b34e5e9fd0821daaf6961e"
    },
    {
      "slug": "the-patricia",
      "title": "THE PATRICIA TRIE",
      "kicker": "branching only on the bits that differ",
      "accent": "#b06bff",
      "blurb": "The PATRICIA trie (crit-bit tree) in the 5-window house format — storing a set of bit-strings with no wasted nodes. A plain binary trie spends a node per bit; PATRICIA keeps only the branch points. Each internal node records a single critical bit index — the first bit on which the keys below it diverge — and you navigate by testing just that bit. The payoff is a sharp invariant: a set of k keys needs exactly k−1 internal branch nodes, no matter how long the keys are. Verified live: over thousands of random key sets, membership queries are exactly correct, and the number of internal branch nodes is always k−1 for k keys. Neon-noir traced. See the crit-bit tree in 1D, growing insertions in 2D, and the k-keys-give-k−1-nodes inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "f75cbd51ed4d0358c8c1af10b4347d9195c1df5646de9ba6b49210668b259e49"
    },
    {
      "slug": "the-tutte-polynomial",
      "title": "THE TUTTE POLYNOMIAL",
      "kicker": "one recursion counts every subgraph family",
      "accent": "#21e6ff",
      "blurb": "The Tutte polynomial in the 5-window house format — the master two-variable invariant T(G;x,y) from which a zoo of graph counts falls out by plugging in numbers. Built by deletion–contraction with loop/bridge rules, its evaluations count structures: T(1,1) is the number of spanning trees, T(2,1) counts spanning forests, T(1,2) counts connected spanning subgraphs, and T(2,2)=2^edges counts all subgraphs. One recursion, many combinatorial answers. Verified live: over hundreds of random connected graphs, T(1,1), T(2,1), T(1,2), T(2,2) match brute-force counts of spanning trees, forests, connected spanning subgraphs and all edge-subsets. Neon-noir traced. See deletion–contraction in 1D, the evaluations in 2D, and the one-recursion-many-answers inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "8de8c98c1cfbdf3f9c24178ffdb72c190e2b64ea862c39797488188c27062986"
    },
    {
      "slug": "the-fredkin",
      "title": "THE FREDKIN",
      "kicker": "a gate that conserves its ones",
      "accent": "#ff8a3c",
      "blurb": "The Fredkin gate in the 5-window house format — a reversible and conservative controlled-SWAP. It takes three bits (a,b,c) and, if the control a is 1, swaps b and c; otherwise leaves them. Being a bijection of the 8 states it runs backwards (its own inverse), and it conserves the number of 1s (a swap never changes the total), modelling physics that preserves particle count — yet it is still universal for classical logic: with a constant input, controlled-SWAP computes AND. Verified live: the gate is a permutation of the 8 states, applying it twice is the identity, it preserves the Hamming weight of every input, and it computes AND (c=0 → a∧b). Neon-noir traced. See the conservative truth table in 1D, live gadgets in 2D, and the conserve-the-ones inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "d1d50bc0f5481db659911c7f2bcd54001377db50cdb5fdbb6f07f8e7b492ce7a"
    },
    {
      "slug": "the-arnoldi",
      "title": "THE ARNOLDI",
      "kicker": "a giant matrix squeezed into a small one",
      "accent": "#35ffb0",
      "blurb": "The Arnoldi iteration in the 5-window house format — squeezing a large matrix into a small one that captures its essential behaviour. From a vector b it builds an orthonormal basis Q for the Krylov subspace span{b, Ab, A²b, …} via modified Gram–Schmidt, recording coefficients in an upper-Hessenberg H. They fit together in the Arnoldi relation A Q_k = Q_{k+1} H̄, and the eigenvalues of the tiny H (Ritz values) approximate those of the huge A — the engine under GMRES and modern eigensolvers. Verified live: over thousands of random matrices, Q is orthonormal, the Arnoldi relation holds to machine precision, H is upper-Hessenberg, and at full depth H shares A's trace and determinant. Neon-noir traced. See the Krylov build in 1D, A-vs-H in 2D, and the restrict-then-solve inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "d1a6874cb06830930d9301d76e4dfededd70fcaa64fa5c8e30261a671585ebc2"
    },
    {
      "slug": "the-lt-fountain",
      "title": "THE LT FOUNTAIN CODE",
      "kicker": "a message rebuilt from any enough droplets",
      "accent": "#ffcf4a",
      "blurb": "The LT fountain code in the 5-window house format — turning k source symbols into an endless stream of droplets, any sufficiently large handful of which rebuilds the message, no matter which ones you catch. Each droplet is the XOR of a random subset of sources, its size drawn from a soliton distribution. To decode you peel: find a droplet that XORs just one unknown source (degree 1), recover it, XOR it out of every droplet that used it — creating new degree-1 droplets — and repeat. It is rateless: collect droplets until decoding pops. Verified live: from about k(1+ε) droplets the peeling decoder recovers all k sources with high probability, and every recovered symbol matches the original whenever decoding completes. Neon-noir traced. See the droplet graph in 1D, peeling in 2D, and the catch-any-enough inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "ea24dc1eb7572c83bb4dd5c1f90203ebcae0b2dcde686408cd880030192a4f9a"
    },
    {
      "slug": "the-count-sketch",
      "title": "THE COUNT-SKETCH",
      "kicker": "a frequency estimate the median cleans up",
      "accent": "#b06bff",
      "blurb": "Count-Sketch in the 5-window house format — estimating item frequencies in a stream with tiny memory, and (unlike Count-Min) unbiased. Each row hashes an item to a bucket and multiplies by a random ±1 sign before adding; the estimate reads that bucket back times the same sign. Collisions from other items arrive with random signs, so on average they cancel — the single-row estimate is correct in expectation. Taking the median across rows crushes the variance for a sharp estimate in fixed memory. Verified live: the average single-row estimate of a target's count converges to its true frequency (unbiased), and the median across rows has far smaller error than any single row. Neon-noir traced. See signed counters in 1D, scatter-and-median in 2D, and the sign-the-noise-away inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "63a8ea2db9e05cf052185c2e164b15f6189052cebde53bc43439a40d7d7ec355"
    },
    {
      "slug": "the-lanczos",
      "title": "THE LANCZOS",
      "kicker": "symmetry shrinks the recurrence to three terms",
      "accent": "#21e6ff",
      "blurb": "The Lanczos iteration in the 5-window house format — Arnoldi's method with the luck of symmetry. For a symmetric matrix, the Krylov orthogonalization collapses from a full Gram–Schmidt to a three-term recurrence: A q_j = β_{j−1} q_{j−1} + α_j q_j + β_j q_{j+1}. The result is a small symmetric tridiagonal T (diagonals α, off-diagonals β) whose eigenvalues (Ritz values) approximate the huge matrix's — the engine for the eigenvalues of enormous sparse symmetric systems. Verified live: over thousands of random symmetric matrices (with reorthogonalization), Q is orthonormal, T is symmetric tridiagonal, the three-term relation holds to machine precision, and T carries A's trace and determinant. Neon-noir traced. See the tridiagonal T in 1D, A-vs-T in 2D, and the symmetry-shrinks-the-recurrence inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "9024683c45237d0881371d0744f6be65cf3494cfc307a4b6165e0fa8953ac8f5"
    },
    {
      "slug": "the-deutsch-jozsa",
      "title": "THE DEUTSCH-JOZSA",
      "kicker": "constant-or-balanced in one question",
      "accent": "#ff8a3c",
      "blurb": "The Deutsch–Jozsa algorithm in the 5-window house format — answering a yes/no question about a black box in one query where classical certainty may need over half of all inputs. Promised f:{0,1}^n→{0,1} is either constant (same output everywhere) or balanced (0 on exactly half), superpose all inputs, let the oracle stamp the phase (−1)^f(x), and Hadamard again: the amplitude at |0…0⟩ becomes (1/2^n)Σ(−1)^f(x) — magnitude 1 if constant, exactly 0 if balanced. One measurement decides. Verified live: simulating the amplitude, every constant function gives |amplitude at |0…0⟩| = 1 and every balanced function gives 0, from one oracle call (classical worst case 2^(n−1)+1). Neon-noir traced. See the amplitude in 1D, one-query verdict in 2D, and the interfere-to-decide inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "9a5d1b0bde7fed2be4121e76ccf972b5b891a19fc4829abc699a0d91d7a24610"
    },
    {
      "slug": "the-space-saving",
      "title": "THE SPACE-SAVING",
      "kicker": "the frequent survive eviction",
      "accent": "#35ffb0",
      "blurb": "The Space-Saving algorithm in the 5-window house format — finding the heavy hitters of a stream using only k counters, far fewer than the distinct items. Each arrival either bumps a monitored item's counter or, if all k are taken, evicts the current minimum (the newcomer takes that slot at min+1). The guarantee is sharp: any item whose true frequency exceeds N/k is always in the summary, every counter overestimates (never undercounts), and the overestimate is at most N/k. Frequent items survive eviction; rare ones churn through the same slots. Verified live: over thousands of streams, every item with frequency > N/k is monitored, each estimate is ≥ the true count, and the overestimate never exceeds N/k. Neon-noir traced. See eviction in 1D, the counters locking on in 2D, and the frequent-survive inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "8c771c903e8b3ef2a50b8eaf969d6ccc587c220f209daa3555107cb23c85c26f"
    },
    {
      "slug": "the-chicken-mcnugget",
      "title": "THE CHICKEN McNUGGET",
      "kicker": "the largest amount you cannot make",
      "accent": "#ffcf4a",
      "blurb": "The Chicken McNugget theorem in the 5-window house format — the Frobenius coin problem for two coprime denominations a and b: which totals cannot be made from non-negative whole numbers of each? There is a largest impossible amount, the Frobenius number, with a clean closed form g(a,b) = ab − a − b; every amount above it is makeable, and the count of impossible amounts is exactly (a−1)(b−1)/2. Two numbers, and the whole gap structure is pinned by a formula. Verified live: over thousands of coprime pairs, the largest non-representable amount equals ab−a−b, the count equals (a−1)(b−1)/2, and every amount above ab−a−b is representable. Neon-noir traced. See the number line in 1D, a chosen pair in 2D, and the name-the-last-gap inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "3860b4dac6f8c8b63f79cfa1f0f8da786cba061195cdbf89a11dab75434bbfee"
    },
    {
      "slug": "the-gomory-hu",
      "title": "THE GOMORY-HU TREE",
      "kicker": "all-pairs min-cuts in one tree",
      "accent": "#b06bff",
      "blurb": "The Gomory–Hu tree in the 5-window house format — compressing all the minimum cuts of a weighted graph into a single tree. A graph on n vertices has C(n,2) pairs, each with its own minimum cut, but you never compute them all: the Gomory–Hu tree is a weighted tree on the same vertices such that for every pair (s,t), the minimum s–t cut in the original graph equals the smallest edge weight on the tree path between s and t. It stores n−1 numbers and answers any of the C(n,2) cut queries exactly, built from just n−1 max-flow computations. Verified live: over thousands of random weighted graphs, for every pair the minimum edge on the tree path equals a brute-force minimum s–t cut. Neon-noir traced. See graph-and-tree in 1D, a pair query in 2D, and the one-tree-all-cuts inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "19a723ee2fac0861e2f92e042485bf4b3be869e0dd1be24ecc06e5943c4a64e9"
    },
    {
      "slug": "the-simon",
      "title": "THE SIMON",
      "kicker": "a hidden mask pinned by linear equations",
      "accent": "#21e6ff",
      "blurb": "Simon's algorithm in the 5-window house format — finding a hidden bit-mask s a black box conceals, with an exponential speedup over any classical method. The promise: the function is two-to-one with f(x)=f(x⊕s). Classically you must hunt for a colliding pair (~2^(n/2) queries); Simon's quantum circuit instead returns, each run, a random vector y with y·s=0 (mod 2). Gather about n−1 independent such y and linear algebra over GF(2) pins s exactly. Verified live: over thousands of hidden masks, collecting n−1 independent measurement vectors and solving the GF(2) system recovers s every time. Neon-noir traced. See the 2-to-1 oracle in 1D, the GF(2) solve in 2D, and the constraints-not-search inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "19812acc3a6b820b1729ca06769a991b3206394c7f4a0ec2df43257dabcab004"
    },
    {
      "slug": "the-graeco-latin",
      "title": "THE GRAECO-LATIN SQUARE",
      "kicker": "two squares that never repeat a pair",
      "accent": "#ff8a3c",
      "blurb": "The Graeco-Latin square in the 5-window house format — overlaying two Latin squares so no ordered pair ever repeats. A Latin square of order n fills an n×n grid so each symbol appears once per row and column; two are orthogonal if pairing them cell-by-cell yields all n² ordered pairs exactly once. Euler conjectured none exist for n≡2 (mod 4) — but he was wrong: a Graeco-Latin square exists for every order except 2 and 6. For odd n, L=(i+j) mod n and M=(2i+j) mod n do the job. Verified live: for every odd n from 3 to 15, L and M are each Latin squares and orthogonal (all n² pairs distinct). Neon-noir traced. See the overlay in 1D, adjustable order in 2D, and the no-pair-twice inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "57e43a38e3b4655b5292c87a84dc07d78c9e80da40eac740e03449ac7d91338a"
    },
    {
      "slug": "the-hill-cipher",
      "title": "THE HILL CIPHER",
      "kicker": "a cipher that is a matrix",
      "accent": "#35ffb0",
      "blurb": "The Hill cipher in the 5-window house format — encryption as matrix multiplication. Turn letters into numbers 0–25, group into vectors, and multiply each by a secret key matrix K mod 26: c = K·p (mod 26); decrypt with the inverse, p = K⁻¹·c (mod 26). K is invertible mod 26 exactly when its determinant is coprime to 26 (odd and not a multiple of 13). It was the first cipher to encrypt several letters at once, hiding letter frequencies inside linear algebra. Verified live: for every key with determinant coprime to 26, encrypting then decrypting recovers the plaintext exactly, and the modular inverse exists precisely when the determinant is coprime to 26. Neon-noir traced. See the vector encrypt in 1D, a word round-trip in 2D, and the invert-the-matrix inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "48391f9b56db3a5cda4135216e7becab228097d647066b60d2112f776d1b85ae"
    },
    {
      "slug": "the-prouhet-tarry-escott",
      "title": "THE PROUHET-TARRY-ESCOTT",
      "kicker": "a set split into equal power sums",
      "accent": "#ffcf4a",
      "blurb": "The Prouhet–Tarry–Escott problem in the 5-window house format — splitting numbers into two sets with equal power sums (equal totals, sums of squares, of cubes, as high as possible). Prouhet's answer: take 0..2^k−1 and split by the Thue–Morse parity of each number (even or odd count of 1-bits). Then the two halves have Σaᵖ = Σbᵖ for every power p from 0 up to k−1 — matched sums, square-sums, all the way to the (k−1)-th — and they finally differ at power k. The same sequence that avoids repetition balances the powers. Verified live: for k=2..8, the Thue–Morse split has equal sums of p-th powers for all p<k, and unequal sums at p=k. Neon-noir traced. See the split in 1D, power sums in 2D, and the parity-balances-the-powers inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "11fcaf5a36e8c5c24e3986a960c402f37968233d1f7b4b284f757be425d48bf1"
    },
    {
      "slug": "the-paley",
      "title": "THE PALEY",
      "kicker": "residues that are a perfect difference set",
      "accent": "#b06bff",
      "blurb": "The Paley construction in the 5-window house format — turning the quadratic residues of a prime into a perfectly balanced combinatorial design. Take a prime p≡3 (mod 4) and collect the nonzero squares mod p (the quadratic residues). This set of size (p−1)/2 is a cyclic difference set: every nonzero residue arises as a difference of two residues exactly (p−3)/4 times. Because p≡3 (mod 4), −1 is a non-residue, which makes the set 'skew' and gives the Paley graph and Paley's Hadamard matrices. Structure from squaring. Verified live: for every prime p≡3 (mod 4) up to 59, each nonzero residue is a difference of two quadratic residues exactly (p−3)/4 times, and −1 is always a non-residue. Neon-noir traced. See residues on a circle in 1D, difference counts in 2D, and the count-the-differences inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "607a03c3f3e70a13877c64124bce1d97b4c3c60b7fd0c8ec46f94dc15ee7fa53"
    },
    {
      "slug": "the-weyl",
      "title": "THE WEYL",
      "kicker": "an irrational stride fills the interval evenly",
      "accent": "#21e6ff",
      "blurb": "Weyl's equidistribution theorem in the 5-window house format — step around a circle by an irrational stride and you visit every region equally often. Take an irrational α and the fractional parts {α},{2α},{3α},…: the proportion landing in any subinterval [a,b] converges to exactly its length b−a. The points never repeat and never cluster; they fill the interval uniformly. A rational stride, by contrast, cycles through finitely many spots and fails to equidistribute. Verified live: for five irrationals across several intervals the empirical fraction matches b−a to within 0.01 over 200,000 terms, while a rational stride 1/5 visibly fails. Neon-noir traced. See the sequence filling [0,1) in 1D, the flattening histogram in 2D, and the intervals-get-their-share inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "9d647953225bf3b41dce0f8add1f9e6d17b1b5a9e4b247e70871de19230a642c"
    },
    {
      "slug": "the-conference-matrix",
      "title": "THE CONFERENCE MATRIX",
      "kicker": "a matrix whose rows are all orthogonal",
      "accent": "#ff8a3c",
      "blurb": "The conference matrix in the 5-window house format — an n×n matrix with a zero diagonal, ±1 off it, whose rows are all mutually orthogonal: C·Cᵀ=(n−1)·I. Every pair of distinct rows has dot product exactly zero; each row's self-dot is n−1. Paley showed how to build a symmetric one whenever n=q+1 with q≡1 (mod 4) a prime power: fill the core with the Legendre symbol χ(i−j) and border it with ones. These matrices feed the construction of Hadamard matrices and strongly regular graphs. Verified live: for primes q=5,13,17,29,37, the Paley conference matrix satisfies C·Cᵀ=(n−1)·I exactly and is symmetric. Neon-noir traced. See the ±1 grid in 1D, the orthogonality product in 2D, and the multiply-the-rows inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "c580788726f6bd84e7cc3f50aecfd5a4d544767e526761aa4735b102a3de9310"
    },
    {
      "slug": "the-woodbury",
      "title": "THE WOODBURY",
      "kicker": "a low-rank patch to a big inverse",
      "accent": "#35ffb0",
      "blurb": "The Woodbury matrix identity in the 5-window house format — updating a big matrix inverse after a low-rank change, the rank-k generalization of Sherman–Morrison. If you know A⁻¹ and then modify A by a low-rank term UCV, the new inverse is (A+UCV)⁻¹ = A⁻¹ − A⁻¹U(C⁻¹+VA⁻¹U)⁻¹VA⁻¹. The only fresh inversion is of a tiny k×k matrix instead of the full n×n — a huge saving when k is small. It is the backbone of Kalman filtering, Gaussian-process updates, and recursive least squares. Verified live: over thousands of random A,U,C,V the formula matches a direct inversion of A+UCV to machine precision (max error ~1e-11). Neon-noir traced. See the rank-k correction in 1D, the side-by-side match in 2D, and the invert-small-not-big inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "a5a9105c7873644135ab599593e4fca17200fb0e6fed614ade5e43726b6ccf19"
    },
    {
      "slug": "the-two-sum",
      "title": "THE TWO-SUM",
      "kicker": "the rounding error captured exactly",
      "accent": "#ffcf4a",
      "blurb": "Two-Sum in the 5-window house format — a tiny miracle of floating-point arithmetic that adds two numbers and hands you the rounding error, exactly. Ordinary a+b rounds to the nearest representable value s, silently discarding a little bit e. Two-Sum computes both, so a+b=s+e is an exact equation over the reals — using only a handful of ordinary additions and subtractions, no wider precision. This 'error-free transformation' is the seed of compensated summation, double-double arithmetic, and reproducible numerics. Verified live: over 20,000 random pairs, s is exactly the rounded sum and the pair (s,e) reconstructs a+b exactly — checked by comparing the exact dyadic (BigInt) fractions of the doubles. Neon-noir traced. See the split into (s,e) in 1D, the exact reconstruction in 2D, and the capture-the-error inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "890a6f07e70d414e29540c75fcf2b7158c50403fe96dbbf4d6c5cbb08118cc3f"
    },
    {
      "slug": "the-doomsday",
      "title": "THE DOOMSDAY",
      "kicker": "the weekday of any date by hand",
      "accent": "#b06bff",
      "blurb": "The Doomsday rule in the 5-window house format — John Conway's method for finding the day of the week of any date in your head. Every year has an anchor weekday, its 'doomsday', and a set of easy-to-remember dates that always fall on it (4/4, 6/6, 8/8, 10/10, 12/12, and a few more). Compute the year's doomsday from its century anchor plus a small correction, then count from the nearest doomsday date to your target. A few additions mod 7 and you have the weekday — no calendar, no lookup. Verified live: over 20,000 random Gregorian dates (1700–2300), Conway's Doomsday computation gives the same weekday as a reference calendar. Neon-noir traced. See the anchor dates in 1D, the step-by-step weekday in 2D, and the one-anchor-per-year inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "c71c890865a41623c3cef31bc42736b9d846dece53dda97d94fe22f1878f509d"
    },
    {
      "slug": "the-levinson-durbin",
      "title": "THE LEVINSON-DURBIN",
      "kicker": "a Toeplitz system solved by recursion",
      "accent": "#ffcf4a",
      "blurb": "The Levinson–Durbin recursion in the 5-window house format — solving a symmetric Toeplitz system (constant diagonals) in O(n²) instead of the O(n³) of general Gaussian elimination. It walks up in order, and at each step a single reflection coefficient extends the solution one more dimension, reusing the structure the constant diagonals give. Applied to a signal's autocorrelation (the Yule–Walker equations) it produces the best linear-predictor coefficients — the heart of LPC speech coding, spectral estimation, and autoregressive modelling. Verified live: over 2000 random positive-definite Toeplitz systems the recursion's solution satisfies T·a=−r to ~1e-15 and matches a direct Gaussian solve, with the prediction error staying positive. Neon-noir traced. See the constant diagonals in 1D, the coefficients vs direct in 2D, and the order-by-order inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "0c9f700d7ec6f7d84cad17df64fdc0c5bf76ddf40be51063bc271031b7202e43"
    },
    {
      "slug": "the-dtw",
      "title": "THE DTW",
      "kicker": "two signals warped into alignment",
      "accent": "#ff8a3c",
      "blurb": "Dynamic Time Warping in the 5-window house format — measuring the distance between two sequences that run at different speeds by stretching and compressing the time axis to line them up. Instead of comparing sample i to sample i, it finds a monotone alignment path through a cost grid that pairs each point of one sequence with one or more of the other, minimizing total mismatch. A word said fast and slow, two heartbeats, two gestures — DTW judges them similar even when their timing differs, via the DP recurrence D[i,j]=|aᵢ−bⱼ|+min(D[i−1,j],D[i,j−1],D[i−1,j−1]). Verified live: over 3000 random pairs, DTW(A,A)=0, the recovered path is monotone with cost equal to D[n,m], DTW is symmetric, and for equal lengths DTW ≤ the rigid aligned distance. Neon-noir traced. See the alignment links in 1D, the DP grid + path in 2D, and the cheapest-monotone-path inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "3370cbc1f864ca0f83899d971a5ad751b56f215c77c3062e412027c0934a8e3b"
    },
    {
      "slug": "the-bentley-ottmann",
      "title": "THE BENTLEY-OTTMANN",
      "kicker": "a sweep line catching every crossing",
      "accent": "#35ffb0",
      "blurb": "The Bentley–Ottmann algorithm in the 5-window house format — finding every intersection among a set of line segments without checking all pairs. A vertical sweep line moves left to right; the segments it currently crosses are kept in top-to-bottom order, and only neighbours in that order are ever tested for crossing. Two segments can only intersect after becoming adjacent on the sweep line, so tracking neighbourhood changes at endpoints and crossings catches all K intersections in O((n+K)log n) — far better than the O(n²) of brute force when crossings are few. Verified live: over 300 random segment sets, the sweep's intersection set exactly equals the brute-force all-pairs set. Neon-noir traced. See the sweep line in 1D, the found-vs-brute count in 2D, and the test-only-neighbours inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "adf3624fe87600d8806293d3bcd645a1a6e91401271c36b6f7a7537d6ee46dbc"
    },
    {
      "slug": "the-ransac",
      "title": "THE RANSAC",
      "kicker": "a model found through a storm of outliers",
      "accent": "#b06bff",
      "blurb": "RANSAC (RANdom SAmple Consensus) in the 5-window house format — fitting a model to data riddled with outliers, the workhorse of computer vision for finding lines, planes, and geometric relations in noisy point sets. Instead of least-squares (which one bad point can wreck), it repeatedly draws the minimal sample needed to define a model (two points for a line), counts how many other points agree within a tolerance, and keeps the model with the largest consensus set. With enough random trials it almost surely hits a sample of pure inliers and locks onto the true model — even when nearly half the data is garbage. Verified live: over 800 trials with 60 inliers and 40 outliers, RANSAC recovers the planted line's slope (to within 0.1) and its inlier set in 100% of runs. Neon-noir traced. See the consensus fit in 1D, the inlier marking in 2D, and the most-votes-wins inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "0c0808743a6f70ec1382651fb32119c5e21fc95d0d74447853fc56d131671cf9"
    },
    {
      "slug": "the-xorshift",
      "title": "THE XORSHIFT",
      "kicker": "three shifts spin through every state once",
      "accent": "#21e6ff",
      "blurb": "Xorshift in the 5-window house format — George Marsaglia's family of fast pseudo-random generators, where the whole state is one machine word and each step is three xor-with-shift operations: x ^= x<<a; x ^= x>>b; x ^= x<<c. No multiply, no memory — just shifts and xors. With a primitive shift triple the generator is a bijection on the nonzero states that runs through every one exactly once before repeating, a full period of 2^w−1; zero is an isolated fixed point the cycle never touches. It is the ancestor of xorshift128+, the default RNG in many language runtimes. Verified live: for a 16-bit generator with triple (1,1,14), iterating from any nonzero seed visits all 65,535 nonzero states exactly once and returns to the seed — a proven full period — and the 32-bit (13,17,5) generator passes a χ² equidistribution test. Neon-noir traced. See the shift-xor scramble in 1D, the state-coverage fill in 2D, and the single-cycle inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "fe71cc11c154fc718550dfced59599e21b5c7e8c04867ef52017683e2e3c00c9"
    },
    {
      "slug": "the-gauss-sum",
      "title": "THE GAUSS SUM",
      "kicker": "p unit vectors sum to exactly √p",
      "accent": "#21e6ff",
      "blurb": "The quadratic Gauss sum in the 5-window house format — add up the p complex numbers e^{2πi·k²/p} for k=0..p−1, and although the phases scatter chaotically around the circle, their sum has magnitude exactly √p. Gauss went further and pinned the sign: the sum equals √p when p≡1 (mod 4) and i√p when p≡3 (mod 4) — a fact he called his 'tormentor' until he proved it. These sums underlie quadratic reciprocity, the functional equation of L-functions, and the construction of certain codes. Verified live: for every prime up to 200 the squared magnitude of the sum equals p to ~1e-13, and the real/imaginary split matches Gauss's sign rule. Neon-noir traced. See the unit vectors on the circle in 1D, the head-to-tail resultant in 2D, and the magnitude-locked inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "b29a757ac93ae9ea990ac5113ee6480d7f5a740241045a95637fb549dd646247"
    },
    {
      "slug": "the-flajolet-martin",
      "title": "THE FLAJOLET-MARTIN",
      "kicker": "a vast count from a tiny bitmap",
      "accent": "#ff8a3c",
      "blurb": "The Flajolet–Martin algorithm in the 5-window house format — estimating how many distinct items a stream contains using a few hundred bits, no matter how many billions flow past. A random hash lands on a value ending in exactly r zero-bits with probability 2^{−r−1}, so among n distinct items the longest trailing-zero run is about log₂ n. Track, per bucket, the lowest bit position never hit; average across buckets and correct by φ≈0.77351, and you recover the cardinality. Crucially it is idempotent: seeing the same item twice changes nothing, because its hash is the same. Verified live: with 256 buckets the estimate lands within ~3–5% of the true distinct count on average (matching the theoretical 0.78/√m), and re-adding duplicates leaves it unchanged. Neon-noir traced. See the bitmap read-out in 1D, the estimate-vs-true in 2D, and the count-without-counting inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "62325c8e0839c521de5fb687a277c7ed0f81f0ef8fd636e0a41262713d518298"
    },
    {
      "slug": "the-steinhaus-johnson-trotter",
      "title": "THE STEINHAUS-JOHNSON-TROTTER",
      "kicker": "every permutation one swap apart",
      "accent": "#35ffb0",
      "blurb": "The Steinhaus–Johnson–Trotter algorithm in the 5-window house format — listing every permutation of n items so each differs from the last by a single swap of two adjacent positions. It is a Gray code for permutations: a Hamiltonian path through the permutohedron touching all n! arrangements, changing as little as possible each step. Each element carries a direction; the largest 'mobile' element moves, and when it can move no further, directions flip and the next-largest takes over — no permutation repeated or skipped. Verified live: for n=2..7 it emits exactly n! permutations, all distinct, and every consecutive pair differs by exactly one adjacent transposition. Neon-noir traced. See the highlighted swaps in 1D, the step-through in 2D, and the Hamiltonian-path inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "37c06cd6e4d0a5dfeb934116ec57d03476293fc7930f2277c8d18ba5bfc16d55"
    },
    {
      "slug": "the-rader",
      "title": "THE RADER",
      "kicker": "a prime DFT turned into a convolution",
      "accent": "#ffcf4a",
      "blurb": "Rader's algorithm in the 5-window house format — computing the DFT of prime length N, exactly the case the usual power-of-two FFT can't split. Its trick is group theory: the nonzero indices 1..N−1 form a cyclic group under multiplication mod N, generated by a primitive root g. Re-indexing inputs and outputs by successive powers of g turns the awkward prime-length DFT into an ordinary cyclic convolution of length N−1 — which a fast convolution then evaluates. A prime, the least divisible of lengths, handled by exploiting the multiplicative structure hiding inside it. Verified live: for primes N=5..23, Rader's reindex-into-convolution reproduces the direct DFT to ~1e-13. Neon-noir traced. See the primitive-root cycle in 1D, the Rader-vs-direct spectra in 2D, and the reindex inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "2909a61109e24c6a9a1296940b4f2f4b1b1c6c49540b4bc6e8efcd4482c71d4f"
    },
    {
      "slug": "the-package-merge",
      "title": "THE PACKAGE-MERGE",
      "kicker": "an optimal code with bounded depth",
      "accent": "#b06bff",
      "blurb": "The package-merge algorithm in the 5-window house format — building an optimal prefix code like Huffman's, but with a hard limit L on the longest codeword. Plain Huffman can produce very deep codes for skewed weights; many formats (DEFLATE, JPEG) forbid that, capping length for fast table decoding. Larmore and Hirschberg recast the problem as a coin collector's problem: coins of denomination 2^{−l} and cost w_i, buy total width n−1 as cheaply as possible. Repeatedly packaging the two cheapest coins and merging them with the next denomination yields the minimum-cost length-limited code — every length ≤ L, optimal among all such codes. Verified live: over 3000 random weight sets, lengths are all ≤ L with Kraft sum = 1; with large L it matches Huffman's cost exactly, and under a tight L every length is bounded, Kraft ≤ 1, and cost ≥ Huffman's. Neon-noir traced. See the capped lengths in 1D, the L-slider re-solve in 2D, and the bounded-tree inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "14337227dad7e736b34f04c4ceb647ef275aa7ca243843f56f9e1fb060bdb9a4"
    },
    {
      "slug": "the-schwartz-zippel",
      "title": "THE SCHWARTZ-ZIPPEL",
      "kicker": "one random probe catches any difference",
      "accent": "#21e6ff",
      "blurb": "The Schwartz–Zippel lemma in the 5-window house format — the engine behind randomized identity testing: a non-zero polynomial of degree d, evaluated at a point chosen uniformly from a set S, is zero with probability at most d/|S|. So to test whether two complicated expressions are the same polynomial without expanding them, just evaluate both at a random point: if they differ, a single random probe exposes it with overwhelming probability; if they agree everywhere, they always agree. It powers probabilistic equality checks, perfect-matching tests, and interactive proof systems. Verified live: a non-zero degree-d polynomial over Z_q has at most d roots (so P[hit a root] ≤ d/q), identical polynomials always agree at a random point, and different polynomials falsely agree only ~0.02% of the time. Neon-noir traced. See the sparse roots in 1D, the same-vs-differ probe in 2D, and the one-probe inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "5868a16e75d36e4e68fc9af3b6208b86b1b5fe2cdb0a2cd044c77bf0142a84a0"
    },
    {
      "slug": "the-cipolla",
      "title": "THE CIPOLLA",
      "kicker": "a square root through a field extension",
      "accent": "#ff8a3c",
      "blurb": "Cipolla's algorithm in the 5-window house format — finding a modular square root, a solution to x²≡n (mod p), by stepping outside the field. It picks a value a so that a²−n is a non-residue, builds the quadratic extension F_{p²}=F_p[√(a²−n)], and raises (a+√(a²−n)) to the power (p+1)/2. Remarkably the result lands back in F_p as a genuine square root of n. Where Tonelli–Shanks grinds through the 2-adic structure, Cipolla takes one elegant excursion into a larger field — and it handles the hard case p≡1 (mod 4) with no special looping. Verified live: over ~2000 random (prime p, quadratic residue n), Cipolla returns an x with x²≡n (mod p), including many p≡1 (mod 4). Neon-noir traced. See the square-fold in 1D, the F_p² recovery in 2D, and the excursion inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "4d7a12f9cee7a581e7549196d5363e7bb4771fb30c252a99d3466a0f39f16e85"
    },
    {
      "slug": "the-nimber",
      "title": "THE NIMBER",
      "kicker": "a game arithmetic that is a field",
      "accent": "#35ffb0",
      "blurb": "Nimber arithmetic in the 5-window house format — John Conway's discovery that the non-negative integers, with the right operations, form an algebraically closed field. Nim-addition is just bitwise XOR. Nim-multiplication is defined by a single recursive rule (a⊗b is the smallest value not equal to any (a'⊗b)⊕(a⊗b')⊕(a'⊗b') for smaller a',b'). Under these, the set {0,…,2^{2^k}−1} is a finite field: {0,1,2,3} is GF(4), {0,…,15} is GF(16), and so on — every non-zero element has a multiplicative inverse. It is the arithmetic of Nim and the surreal numbers, exact and integer-only. Verified live: nim-multiplication over {0,…,15} is commutative, associative, and distributes over XOR; every non-zero element has an inverse; and {0,1,2,3} is exactly GF(4) with 2⊗2=3. Neon-noir traced. See the multiplication table in 1D, the ⊕/⊗/inverse in 2D, and the field inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "47501a7ca762fc5d1d38901a8cf8141852db60e54a50833379c7a1d3f92b7e20"
    },
    {
      "slug": "the-barycentric",
      "title": "THE BARYCENTRIC",
      "kicker": "a curve pinned through its nodes",
      "accent": "#ffcf4a",
      "blurb": "Barycentric Lagrange interpolation in the 5-window house format — the numerically stable way to pass a single polynomial through a set of data points. The naive Lagrange formula is slow and unstable; the barycentric form rewrites it as L(x)=[Σ w_j/(x−x_j)·f_j]/[Σ w_j/(x−x_j)], where each weight w_j=1/∏_{k≠j}(x_j−x_k) is computed once. Evaluating anywhere is then O(n), it passes through every node exactly, and it reproduces any polynomial of degree < n perfectly — the same interpolant as Lagrange's, but fast and well-behaved. Verified live: over thousands of well-separated node sets, the barycentric interpolant hits every node exactly, matches the direct Lagrange formula, and reproduces degree-<n polynomials to machine precision. Neon-noir traced. See the curve through the nodes in 1D, the movable nodes in 2D, and the weighted-quotient inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "989c1318b5e77bc6959c0f50fbd77340ea97eaf3cf7ce111b629af0b0ae661ec"
    },
    {
      "slug": "the-bitap",
      "title": "THE BITAP",
      "kicker": "one register that matches in parallel",
      "accent": "#b06bff",
      "blurb": "The Bitap (shift-or) algorithm in the 5-window house format — searching for a pattern in text using nothing but bit-shifts and bitwise-or, with the whole matching state for a length-m pattern living in one machine word. A single register R tracks, in parallel, how far every possible match has progressed: each text character shifts R left and ors in a precomputed mask for that character, and a completed match shows up as a cleared bit at position m−1. Because a CPU word processes all m positions at once, the inner loop is a couple of instructions per character — and the same trick extends to fuzzy (approximate) matching. Verified live: over 5000 random text/pattern pairs, Bitap's bit-parallel scan reports exactly the same match positions as a brute-force substring search. Neon-noir traced. See the state register in 1D, the marked matches in 2D, and the all-alignments-at-once inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "51890c7a92653ec5cb12944fd6d9478df83e705cdea293043f56c73b99d96d1e"
    },
    {
      "slug": "the-kogge-stone",
      "title": "THE KOGGE-STONE",
      "kicker": "all carries computed in parallel",
      "accent": "#21e6ff",
      "blurb": "The Kogge–Stone adder in the 5-window house format — how fast processors add two numbers: instead of waiting for a carry to ripple from the lowest bit to the highest (n steps), it computes all carries at once via a parallel prefix scan. Each bit first decides whether it generates a carry (both inputs 1) or propagates one; then a tree of combine-operations folds these (generate, propagate) signals together, doubling its reach each stage. After only log₂ n stages every carry is known, and the sum falls out in one more XOR — trading wiring for depth, the classic latency-versus-area bargain of digital design. Verified live: for widths n=4..16 and tens of thousands of random inputs, the parallel-prefix sum equals ordinary integer addition a+b exactly. Neon-noir traced. See the generate/propagate tree in 1D, the carry resolution in 2D, and the log-depth inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "a8050d32a7fafdb313bd5f7d1496ff4a6653c291b4e2f2814d8531fdf4e91f27"
    },
    {
      "slug": "the-popcount",
      "title": "THE POPCOUNT",
      "kicker": "bits counted by folding",
      "accent": "#ff8a3c",
      "blurb": "The SWAR population count in the 5-window house format — counting the set bits in a word without a branch or loop, using a cascade of masked adds that fold the count in parallel. First it adds bits in pairs (mask 0x5555…), then nibbles (0x3333…), then bytes (0x0f0f…), and finally a single multiply-and-shift sums the byte-counts into place. It is the archetypal SWAR (SIMD-Within-A-Register) trick: treat one machine word as a vector of small counters and operate on them all at once, in a handful of instructions independent of how many bits are set. Verified live: over 200,000 random 32-bit values plus edge cases, the SWAR popcount equals a naive bit-by-bit count exactly — popcount(0xFFFFFFFF)=32. Neon-noir traced. See the fold stages in 1D, the masked cascade in 2D, and the word-as-counters inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "54cbce14e0e9328ace1db056b3155407b6287fd7af7394f0a7fd5ab1ac47d58e"
    },
    {
      "slug": "the-gauss-circle",
      "title": "THE GAUSS CIRCLE",
      "kicker": "lattice points fill a disk to πr²",
      "accent": "#35ffb0",
      "blurb": "The Gauss circle problem in the 5-window house format — how many integer lattice points (x,y) lie inside a circle of radius r, i.e. satisfy x²+y²≤r². The answer N(r) is astonishingly close to the circle's area: N(r)=πr²+E(r), and Gauss showed the error grows no faster than the circumference, |E(r)|=O(r). Each lattice point owns a unit square, and those squares tile a region sandwiched between two circles whose areas differ by O(r) — so the count tracks the area to within its boundary. (How much smaller the true error is remains a famous open problem.) Verified live: for radii up to 2000, |N(r)−πr²|/r stays below ~1, and N(r)/πr²→1. Neon-noir traced. See the points owning squares in 1D, the growing radius in 2D, and the count-shadows-area inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "15883b7ef0b09f83f97b3eb90424dac134652ffe2e058b5c5e2a7b30bcabc698"
    },
    {
      "slug": "the-estrin",
      "title": "THE ESTRIN",
      "kicker": "a polynomial evaluated as a tree",
      "accent": "#ffcf4a",
      "blurb": "Estrin's scheme in the 5-window house format — evaluating a polynomial as a balanced tree instead of a sequential chain. Horner's method is optimal in operation count but strictly serial: each step needs the previous one. Estrin instead pairs terms — (a₀+a₁x), (a₂+a₃x), … — then combines those pairs using x², the next level using x⁴, and so on. The dependency chain collapses from depth d to depth log₂ d, so a superscalar or SIMD processor can evaluate many sub-expressions in parallel. Same polynomial, same result — reorganized for parallel hardware. Verified live: over 20,000 random polynomials (degree up to 12) and arguments, Estrin's tree evaluation equals Horner's method to machine precision. Neon-noir traced. See the pairing tree in 1D, the tree-vs-chain in 2D, and the log-depth inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "cb0c23cb872ea642536775229f631d3271a5f0213fc97510c148a56a4ea6b77c"
    },
    {
      "slug": "the-piece-table",
      "title": "THE PIECE TABLE",
      "kicker": "a document edited by re-pointing",
      "accent": "#b06bff",
      "blurb": "The piece table in the 5-window house format — how real text editors (VS Code, Microsoft Word) store a document being edited, without ever moving the text. The original file stays untouched in a read-only buffer; every character you type goes into an append-only add buffer; and the document itself is just an ordered list of pieces, each a (buffer, start, length) window into one of those two buffers. An insert splits a piece and drops a new one in; a delete splits and removes. The text is never copied or shifted — only the little list of pieces changes, which also makes undo and change-tracking almost free. Verified live: over 3000 trials of 15 random inserts and deletes, the document reconstructed from the piece list exactly equals a naively edited string at every step. Neon-noir traced. See the buffers and pieces in 1D, the live edits in 2D, and the edit-the-view inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "e5b062717df4d714cfa38caa7141ca2260d9d0e11aba170f674ba886a0c48494"
    },
    {
      "slug": "the-robin-hood",
      "title": "THE ROBIN HOOD",
      "kicker": "steal from the rich to even the probes",
      "accent": "#21e6ff",
      "blurb": "Robin Hood hashing in the 5-window house format — an open-addressing scheme that steals from the rich to give to the poor. In ordinary linear probing, some keys sit right at their home slot while others get pushed far away, so probe lengths vary wildly. Robin Hood equalizes them: when inserting a key that has probed farther than the key already sitting in a slot, it evicts the richer resident (the one closer to its home) and carries it onward. The result is the same set of keys, but with the variance of probe lengths minimized — no key is left starving while another sits pretty, so lookups stay fast even at high load. Verified live: over 400 tables at 85% load, every key remains retrievable, and both the variance and the maximum of the probe lengths are ≤ plain linear probing on the same keys. Neon-noir traced. See the probe distances in 1D, the histogram-vs-linear in 2D, and the fairness-by-eviction inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "e8529296d4ffecee5966a276cabaedf68dd43f8e0f058f6224905cce70749471"
    },
    {
      "slug": "the-goldschmidt",
      "title": "THE GOLDSCHMIDT",
      "kicker": "divide by driving a factor to one",
      "accent": "#ff8a3c",
      "blurb": "Goldschmidt's algorithm in the 5-window house format — dividing two numbers using only multiplication, no subtraction, no digit-at-a-time long division. To compute a/b, write it as N/D with N=a, D=b, and repeatedly multiply both by the same factor f=2−D. Each step drives the denominator toward 1 (quadratically, doubling correct digits per iteration), and since numerator and denominator are scaled together the value N/D never changes — so when D→1, the numerator IS the quotient a/b. Because the two multiplications each step are independent, hardware can pipeline them, which is why Goldschmidt division appears in real floating-point units. Verified live: over 50,000 random pairs (denominator scaled into a convergent range), the Goldschmidt result equals a/b to ~1e-9. Neon-noir traced. See D→1 and N→quotient in 1D, the |D−1| collapse in 2D, and the division-as-convergence inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "85794c57fb8a2d53e913e019deda3755555f46632e7369bb57bd53044a30da98"
    },
    {
      "slug": "the-zipper",
      "title": "THE ZIPPER",
      "kicker": "a cursor that splits the list",
      "accent": "#35ffb0",
      "blurb": "The zipper in the 5-window house format — a purely functional data structure for editing a sequence (or tree) at a moving focus, with O(1) local operations and no mutation. A list zipper splits the sequence into three parts: the elements to the left of the cursor (held reversed, nearest on top), the focused element, and the elements to the right. Moving the cursor pops from one side and pushes to the other; inserting or deleting at the focus touches only the front of a list. Nothing is copied or shifted — the whole sequence is always recoverable as left ++ [focus] ++ right, which makes undo and immutable sharing natural. Verified live: over 5000 runs of 20 random moves, inserts, and deletes, the list reconstructed from the zipper exactly equals the same edits applied to a plain array with a cursor index. Neon-noir traced. See the three-part split in 1D, the live cursor edits in 2D, and the carry-the-context inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "39946fb810cfbd2f794e8395e833219858c6bf54ac0b34a2bacb81744c55a9d8"
    },
    {
      "slug": "the-power-of-two-choices",
      "title": "THE POWER OF TWO CHOICES",
      "kicker": "two throws beat one",
      "accent": "#ffcf4a",
      "blurb": "The power of two choices in the 5-window house format — a startling result in randomized load balancing. Throw n balls into n bins at random and the fullest bin holds about log n / log log n balls. But give each ball two random bins and let it pick the emptier one, and the fullest bin drops to about log log n / log 2 — an exponential improvement, from logarithmic to double-logarithmic, for the cost of one extra look. A tiny bit of choice tames the worst case. It underlies real hashing, load balancers, and distributed schedulers. Verified live: with n=2000 balls and bins, the average maximum load is ~6 with one choice but ~3 with two choices, and the two-choice max is ≤ the one-choice max in every trial. Neon-noir traced. See the spike-vs-flat loads in 1D, the throw comparison in 2D, and the peek-twice inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "460ed2e142eb18c0c689ab00a6988e787590ffcdbc6ccb0f824bf95c1663b5f3"
    },
    {
      "slug": "the-melkman",
      "title": "THE MELKMAN",
      "kicker": "a hull kept online in a deque",
      "accent": "#b06bff",
      "blurb": "Melkman's algorithm in the 5-window house format — computing the convex hull of a simple polyline (a path or polygon that never crosses itself) in a single online pass, in linear time. It keeps the current hull in a double-ended queue: as each new point arrives, if it lies inside the current hull it is ignored; otherwise the algorithm pops vertices from both ends of the deque that the new point makes non-convex, then pushes the point onto both ends. Because a simple polyline visits points in a coherent order, only the two ends ever need attention — no sorting, no re-scanning — giving an elegant O(n) hull for ordered input. Verified live: over 3000 simple polygons (points in general position), Melkman's deque hull equals a reference convex hull (Andrew's monotone chain) of the same points. Neon-noir traced. See the polyline and hull in 1D, the online match in 2D, and the grow-a-deque inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "078268b71529096dee72f439213d2d2c778cb7e0f845ea13f380aebfa94807a5"
    },
    {
      "slug": "the-wallace-tree",
      "title": "THE WALLACE TREE",
      "kicker": "partial products crushed in parallel",
      "accent": "#21e6ff",
      "blurb": "The Wallace tree in the 5-window house format — how fast hardware multiplies. A schoolbook multiply forms one partial product per bit of the multiplier and adds them in sequence, slow because each add waits for the last. Wallace instead crushes the whole stack of partial products in parallel using 3:2 compressors (full adders): each takes three rows and outputs two — a sum row and a carry row — preserving the total, since x+y+z = sum + 2·carry. Layer after layer the height falls 3→2 until only two rows remain, which a single carry-propagate adder finishes. The depth is logarithmic in the number of partial products, which is why multipliers use it. Verified live: over 200,000 random 8-bit pairs, the carry-save reduction of the partial products, finished with one add, equals a·b exactly. Neon-noir traced. See the compression layers in 1D, the 3→2 reduction in 2D, and the crush-in-parallel inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "aba0627b8c550cfa061ff04dac7326af04b003a69e40687a29eb9aa63a5bd8d5"
    },
    {
      "slug": "the-factoradic",
      "title": "THE FACTORADIC",
      "kicker": "a number in factorial base",
      "accent": "#ff8a3c",
      "blurb": "The factorial number system (factoradic) in the 5-window house format — a mixed-radix notation where the place values are factorials: the digit in position i ranges over 0…i, and the value is Σ dᵢ·i!. Every non-negative integer has a unique factoradic form — and, beautifully, the numbers 0…n!−1 are in exact bijection with the n! permutations of n items. Reading a factoradic left to right and repeatedly picking the d-th remaining element (its Lehmer code) unranks the integer into a permutation; the reverse ranks a permutation back to its index. It is the natural coordinate system for permutations. Verified live: for n≤8, factoradic encode/decode round-trips every integer, and rank/unrank is an exact bijection between [0, n!) and the n! permutations. Neon-noir traced. See the factorial places in 1D, the unrank in 2D, and the coordinatize inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "b9173e87451df19b56e120f2c842abdd4dd9045a2aa6c3cac41d2e6dd82f540b"
    },
    {
      "slug": "the-xor-linked-list",
      "title": "THE XOR LINKED LIST",
      "kicker": "one pointer holds both neighbors",
      "accent": "#35ffb0",
      "blurb": "The XOR linked list in the 5-window house format — storing a doubly linked list using only one pointer field per node instead of two. A normal doubly linked list keeps a prev and a next pointer; the XOR list keeps their bitwise exclusive-or, link = prev ⊕ next. That single value is enough to walk in either direction: if you know the address you came from, the other neighbour is link ⊕ came-from (because XOR is its own inverse). Moving forward, next = link ⊕ prev; moving backward, prev = link ⊕ next. Half the pointer memory, at the cost of no O(1) access to a node without a neighbour. Verified live: over 20,000 random lists, forward traversal reproduces the array, backward traversal reproduces its reverse, and every node stores exactly one link field. Neon-noir traced. See the folded links in 1D, the two-way walk in 2D, and the one-field inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "661d667dfdd6aecca81c2d75d49c3373dac8e112b1e719171192632b1b0f079b"
    },
    {
      "slug": "the-vp-tree",
      "title": "THE VP-TREE",
      "kicker": "nearest found by pruning a metric tree",
      "accent": "#ffcf4a",
      "blurb": "The vantage-point tree in the 5-window house format — finding nearest neighbours in any metric space, not just coordinates but anything with a distance obeying the triangle inequality. At each node it picks a vantage point and a radius (the median distance to the rest), splitting the remaining points into those inside the sphere and those outside. A query descends the side its distance suggests, and — crucially — the triangle inequality lets it prove that the whole other subtree can be skipped whenever it can't possibly hold anything closer than the best found so far. So a search touches only a small fraction of the points while still returning the exact nearest neighbour. Verified live: over 3000 random trees in 3-D, the VP-tree's pruned search returns exactly the same nearest neighbour as a brute-force scan of every point. Neon-noir traced. See the vantage split in 1D, the pruned search in 2D, and the skip-what-can't-be-closer inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "2be5f86095fdd0bc8c3b1dd9a9e84d9420c3aedbaab9e8a764343ff555f80d08"
    },
    {
      "slug": "the-skew-heap",
      "title": "THE SKEW HEAP",
      "kicker": "two heaps merged along right paths",
      "accent": "#b06bff",
      "blurb": "The skew heap in the 5-window house format — a self-adjusting priority queue where a single operation, merge, does everything. To merge two min-heaps, compare their roots, keep the smaller as the new root, recursively merge its right subtree with the other heap, and then swap that node's children. Insert is just merging in a one-node heap; delete-min is merging the root's two children. There are no balance fields, no rotations, no bookkeeping — the unconditional child-swap alone keeps the amortized cost at O(log n). It is the leftist heap's simpler cousin: heapsort, mergeable queues, and priority scheduling from one elegant rule. Verified live: over thousands of runs, inserting then repeatedly extracting the minimum yields a fully sorted sequence, the min-heap property holds after every operation, and merging two heaps preserves the combined multiset in order. Neon-noir traced. See the merge rule in 1D, the sorted extraction in 2D, and the one-rule inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "eb31e20aabf299f3e1bbe649d3ea67921208a6205ee73dd538f943701e4f88ac"
    },
    {
      "slug": "the-sqrt-decomposition",
      "title": "THE SQRT DECOMPOSITION",
      "kicker": "√n blocks answer range sums",
      "accent": "#21e6ff",
      "blurb": "Square-root decomposition in the 5-window house format — the simplest way to answer range queries fast. Split an array of n elements into blocks of size about √n and precompute a summary (here, a sum) for each block. To sum any range, add the few loose elements at the two ends one by one, and for the whole blocks in between just add their precomputed sums — so any query touches at most about 2√n items instead of n. A point update fixes one element and its block's summary in O(1). It is the humble ancestor of segment trees and Fenwick trees — less powerful, but astonishingly easy and general (it works for any associative summary). Verified live: over 3000 arrays and 30 mixed operations each, block range-sums with point updates exactly equal a brute-force recomputation. Neon-noir traced. See the blocked array in 1D, the partial+whole-block query in 2D, and the scan-into-jumps inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "66e351f06b87abca36dc37779db053904340a61d2798b6c7f3bc9d35bd4e0741"
    },
    {
      "slug": "the-von-staudt-clausen",
      "title": "THE VON STAUDT-CLAUSEN",
      "kicker": "a Bernoulli denominator read off from primes",
      "accent": "#ff8a3c",
      "blurb": "The von Staudt–Clausen theorem in the 5-window house format — revealing the exact denominator of every Bernoulli number. The Bernoulli numbers B_2n are wild rationals with enormous numerators, yet their denominators are astonishingly simple: the denominator of B_2n is precisely the product of the primes p for which (p−1) divides 2n. So denom(B_2)=6=2·3, denom(B_10)=66=2·3·11, and 2 and 3 divide every even-index Bernoulli denominator (since p−1∈{1,2} always divides 2n). A messy fraction's bottom half is read straight off a divisibility condition on primes. Verified live: computing the Bernoulli numbers exactly as reduced fractions (BigInt), the denominator of B_2n equals ∏_{(p−1)|2n} p for every n from 1 to 15. Neon-noir traced. See the fraction in 1D, the primes and product in 2D, and the read-off-the-primes inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "aac5a9049214dbfa1625abcfccc5e51b50a3a61d83bb38176c5bc83d27f0f7da"
    },
    {
      "slug": "the-fibonacci-coding",
      "title": "THE FIBONACCI CODING",
      "kicker": "a code that ends in 11",
      "accent": "#35ffb0",
      "blurb": "Fibonacci coding in the 5-window house format — turning a positive integer into a self-delimiting bit string using the Fibonacci numbers as place values. Because every integer has a unique Zeckendorf representation (a sum of non-consecutive Fibonacci numbers), its bits never contain two adjacent 1s. Fibonacci coding writes those bits low-to-high and then appends one extra 1, so the codeword ends in '11' and '11' appears nowhere else inside it. That makes the code a prefix code you can pack end-to-end with no separators: a decoder just splits the stream at every '11'. It is also robust — a single bit flip corrupts at most a couple of adjacent values, not the whole stream. Verified live: over 20,000 integers, encode/decode round-trips, every codeword ends in '11' with no earlier '11', and a concatenated stream of many codewords parses back uniquely. Neon-noir traced. See the bits + terminator in 1D, the packed stream parse in 2D, and the carries-its-own-delimiter inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "a9ceb0d8d4699dcb9f4f946fb11926dfbc51c1fb1ad6f5723e84b4d95d02a70e"
    },
    {
      "slug": "the-mdct",
      "title": "THE MDCT",
      "kicker": "overlapping windows cancel their aliasing",
      "accent": "#ffcf4a",
      "blurb": "The MDCT (modified discrete cosine transform) in the 5-window house format — the transform at the heart of MP3, AAC, Vorbis, and Opus. It is lapped: it works on overlapping blocks of 2N samples but outputs only N coefficients each, so despite the 50% overlap there is no increase in data. That looks impossible — N numbers can't invert 2N samples — and indeed a single block can't. The magic is time-domain aliasing cancellation (TDAC): each inverse block carries an aliased error, but with the right symmetric window (satisfying w[n]²+w[n+N]²=1) the aliases of neighbouring blocks are equal and opposite, so overlap-adding them reconstructs the signal exactly. Critical sampling and perfect reconstruction at once. Verified live: framing a signal into 50%-overlapping windows, MDCT then IMDCT then overlap-add reconstructs the interior samples to ~1e-14. Neon-noir traced. See the overlapping windows in 1D, the reconstruction in 2D, and the overlap-makes-invertible inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "9c907ac640207280356df7865b81be52b2a6b426f97c3570c2ea1ecff78e1ee1"
    },
    {
      "slug": "the-treiber",
      "title": "THE TREIBER STACK",
      "kicker": "a stack that needs no lock",
      "accent": "#b06bff",
      "blurb": "The Treiber stack in the 5-window house format — the classic lock-free stack: many threads push and pop with no locks at all, using one atomic instruction, compare-and-swap (CAS). To push, a thread reads the current top, points its new node at it, then CAS-es the top from the value it read to its new node. If another thread slipped in first, the top no longer matches what was read, the CAS fails, and the thread simply retries from the new top. No thread ever blocks another; the structure makes progress even if some threads stall. It is the foundation of lock-free programming — correct under any interleaving. Verified live: simulating a cooperative scheduler that interleaves concurrent CAS pushes arbitrarily, over 20,000 random interleavings every pushed value survives — no lost updates, no duplicates. Neon-noir traced. See the CAS retry loop in 1D, the interleaved threads in 2D, and the progress-without-locks inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "4fa8813285bb46ed2ebca2dcecf3bdc115283c453e4b71a47916b7e1834410a3"
    },
    {
      "slug": "the-hartley",
      "title": "THE HARTLEY TRANSFORM",
      "kicker": "a transform that is its own inverse",
      "accent": "#21e6ff",
      "blurb": "The discrete Hartley transform in the 5-window house format — a real-valued cousin of the Fourier transform, same frequency information but no complex numbers. Where the DFT multiplies by e^{−iθ}, the DHT multiplies by cas θ = cos θ + sin θ, a single real function. Its most elegant property: it is its own inverse (up to a factor of N) — running the same transform twice returns N times the original signal, so one routine both analyzes and synthesizes. It also obeys Parseval's energy law and turns convolution into pointwise products, making it a real-arithmetic workhorse for spectral analysis and fast convolution. Verified live: over thousands of random signals, applying the DHT twice returns N× the original to ~1e-14, and Parseval's identity Σx²=(1/N)ΣH² holds. Neon-noir traced. See the cas kernel in 1D, the double-transform in 2D, and the reuse-the-transform inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "915d8bf2c5b816643077248d0fd7fe34ca998eb025aa2be97ac2cbe077327264"
    },
    {
      "slug": "the-shannon-fano",
      "title": "THE SHANNON-FANO",
      "kicker": "a code split by halving frequency",
      "accent": "#ff8a3c",
      "blurb": "Shannon–Fano coding in the 5-window house format — the first practical variable-length compression code, the one Huffman improved on. Sort the symbols by frequency, then split them into two groups whose total frequencies are as equal as possible; the top group gets a leading 0, the bottom a 1; recurse on each group. The result is a prefix code (no codeword begins another), so a stream packs with no separators. It comes close to the entropy but, unlike Huffman's bottom-up merge, its top-down split is not always optimal — a historically important near-miss that motivated the optimal algorithm. Verified live: over thousands of random frequency sets, the code is prefix-free, encode/decode round-trips, and its cost is always ≥ the (optimal) Huffman cost. Neon-noir traced. See the frequency split in 1D, the codewords in 2D, and the halving-mass inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "92a9e2744c0157d3c23f8bb299ec0a6cb22e5f58786711fb564f9fabebe0cda5"
    },
    {
      "slug": "the-mcs-lock",
      "title": "THE MCS LOCK",
      "kicker": "a lock that grants in arrival order",
      "accent": "#35ffb0",
      "blurb": "The MCS lock in the 5-window house format — a fair, scalable spinlock built as a queue. A naive spinlock has every waiting thread hammering the same memory location, flooding the interconnect and granting the lock unpredictably. The MCS lock instead gives each thread its own little node: to acquire, a thread atomically swaps itself onto the tail of a queue and then spins only on its own flag; the thread ahead flips that flag on release. Because the tail swap is atomic, the queue order is exactly the arrival order, so the lock is granted first-come, first-served — no starvation — and each thread spins on a private, cache-local variable. Verified live: over 20,000 random arrival interleavings, the grant order equals the atomic-swap (arrival) order (strict FIFO), and at most one thread ever holds the lock. Neon-noir traced. See the queue in 1D, the FIFO grants in 2D, and the fairness-from-a-queue inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "fa56d7929b269ac41fa0b7f935179db7d562e5e9d55fa451c464162a49433ced"
    },
    {
      "slug": "the-unrolled-linked-list",
      "title": "THE UNROLLED LINKED LIST",
      "kicker": "a list of cache-friendly chunks",
      "accent": "#ffcf4a",
      "blurb": "The unrolled linked list in the 5-window house format — a linked list that stores a small array of elements in each node instead of just one. A classic linked list wastes memory and cache: every element is a separate allocation with its own pointer, so walking it means chasing pointers all over RAM. An unrolled list packs up to K elements per node, so a scan reads whole cache-line-friendly chunks and follows a pointer only every K elements — slashing pointer overhead and cache misses while keeping local insert and delete cheap (a node splits when it overflows, merges when it empties). It is the linked list rebuilt for real memory hierarchies. Verified live: over 5000 runs of 40 random inserts and deletes, the chunked list's contents exactly track a plain array, and indexed access returns the right element. Neon-noir traced. See the chunks in 1D, the split/merge in 2D, and the group-the-elements inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "a3fbada647161829d7159ba3fac3b82b507928f25611db2a148abad1994fedbe"
    },
    {
      "slug": "the-kronecker-substitution",
      "title": "THE KRONECKER SUBSTITUTION",
      "kicker": "polynomials multiplied as one big integer",
      "accent": "#b06bff",
      "blurb": "Kronecker substitution in the 5-window house format — turning polynomial multiplication into a single big-integer multiplication. If two polynomials have non-negative integer coefficients bounded below some 2^b, evaluate each at a large power of two x=2^m — this just packs the coefficients side by side into the digits of one huge integer. Multiply the two integers (using any fast bignum routine), and the product's base-2^m digits ARE the coefficients of the polynomial product — provided m is chosen large enough that adjacent coefficients never carry into each other. It lets you borrow the world's fastest integer-multiplication code to multiply polynomials, and vice versa. Verified live: over 20,000 random polynomial pairs with 8-bit coefficients, packing into one integer, multiplying, and unpacking the base-2^m digits reproduces the direct convolution exactly. Neon-noir traced. See the packing in 1D, the pack-multiply-unpack in 2D, and the one-multiply inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "99091befe7d2394f1d9ac42cad83cb83f948e557d68d94f84d2c1aac13d27e61"
    },
    {
      "slug": "the-bit-reversal",
      "title": "THE BIT-REVERSAL",
      "kicker": "reverse the bits, reverse again, home",
      "accent": "#21e6ff",
      "blurb": "The bit-reversal permutation in the 5-window house format — reordering a sequence by reversing the binary digits of every index (001↔100, 011↔110). It is the shuffle that makes the FFT work: the transform's divide-and-conquer leaves outputs in bit-reversed order, so one bit-reversal pass sets them right. Its defining beauty is that it is an involution: reversing the bits twice returns every index to itself, so the same routine both scrambles and unscrambles — a permutation with no cycles longer than two, only fixed points (palindromic indices) and swapped pairs. This is David's nested form −+[[{}]]+− made literal: apply the mirror, apply it again, return to the seed. Verified live: for word sizes 1–12, reversing the bits twice is the identity, and the map is a genuine permutation of [0,2^b). Neon-noir traced. See the mirrored bits in 1D, the double-reversal in 2D, and the apply-twice-home inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "f248ab9d44092993b281e594f97635aa6ffea66458c015d9b743681a1f01f397"
    },
    {
      "slug": "the-legendre-transform",
      "title": "THE LEGENDRE TRANSFORM",
      "kicker": "a duality that undoes itself",
      "accent": "#ff8a3c",
      "blurb": "The Legendre transform (convex conjugate) in the 5-window house format — re-describing a convex function by its slopes instead of its values. Where the graph of f gives, for each x, a height f(x), the conjugate f*(p)=sup_x(px−f(x)) gives, for each slope p, how far the tangent line of that slope drops below the origin. It swaps position and momentum, energy and Lagrangian — the bridge between Lagrangian and Hamiltonian mechanics and between thermodynamic potentials. Its deepest property: on convex functions it is an involution, f**=f — transforming twice returns the original. Verified live: over 400 random convex functions, the Fenchel–Young relation f(x)+f*(p)≥x·p holds always, with equality exactly when p=f′(x), and the biconjugate f** recovers f to ~1e-15. Neon-noir traced. See the tangent envelope in 1D, the Fenchel gap closing in 2D, and the transform-twice inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "eb6703ed43b75d0621ac55aed2a64235f7adefee0859efe16aedc6edcdc76d7e"
    },
    {
      "slug": "the-circle-inversion",
      "title": "THE CIRCLE INVERSION",
      "kicker": "invert through the circle, then again, home",
      "accent": "#35ffb0",
      "blurb": "Circle inversion in the 5-window house format — the fundamental transformation of inversive geometry: fix a circle of radius R about a centre O, and send each point P to P* on ray OP with |OP|·|OP*|=R². Points inside fly outward, points outside fall in, the circle itself stays fixed. It maps generalized circles to generalized circles — a line not through O becomes a circle through O — and it is an involution: inverting twice returns a point exactly, because R²/(R²/d)=d. It is the engine behind the Apollonian gasket, Steiner chains, and the Poincaré disk. Verified live: over 3000 random circles, inverting a point twice returns it to ~1e-15, and a line not through O maps to concyclic points on a circle passing through O. Neon-noir traced. See P and P* in 1D, the double-inversion in 2D, and the invert-invert-home inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "1dac4faf68091925108da16f5f7531af1df15e27f5cf0bba09f5afcc907221c8"
    },
    {
      "slug": "the-conjugate-partition",
      "title": "THE CONJUGATE PARTITION",
      "kicker": "transpose the diagram, transpose again, home",
      "accent": "#ffcf4a",
      "blurb": "The conjugate partition in the 5-window house format — the transpose of a Young diagram. Write a partition λ=(λ1≥λ2≥…) as left-justified rows of boxes; reflect across the main diagonal — rows become columns — and read off the conjugate λ′, where λ′_j counts how many parts of λ are at least j. It is the symmetry at the heart of partition theory: it swaps 'number of parts' with 'largest part,' and self-conjugate partitions count the same as partitions into distinct odd parts. Reflecting twice restores the original diagram, so conjugation is an involution: (λ′)′=λ. Verified live: over 20,000 random partitions, transposing the Young diagram twice returns the original, and the conjugate has the same total size |λ′|=|λ|. Neon-noir traced. See the diagram transpose in 1D, the side-by-side conjugate in 2D, and the transpose-twice inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "eefc0f8a704d98807e8c367e8226fd5bae35c6a8a27ff85086d511e39d2cad00"
    },
    {
      "slug": "the-graph-complement",
      "title": "THE GRAPH COMPLEMENT",
      "kicker": "flip every edge, flip again, home",
      "accent": "#b06bff",
      "blurb": "The graph complement in the 5-window house format — flipping every relationship: in the complement Ḡ of a graph G, two vertices are joined exactly when they are not joined in G. Together G and Ḡ partition the complete graph, so their edge counts sum to C(n,2), and many properties dualize (an independent set in G is a clique in Ḡ). It is an involution: complementing twice restores the original graph. A graph isomorphic to its own complement is self-complementary — like the 5-cycle C₅, whose complement is again a 5-cycle. Verified live: over 20,000 random graphs, complementing twice returns the original and e(G)+e(Ḡ)=C(n,2); and C₅ is shown self-complementary (its complement is 2-regular with 5 edges). Neon-noir traced. See G and Ḡ in 1D, the double-complement in 2D, and the complement-twice inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "41f6b334e65cf81c41996ce7809b4f398c75f5e66496107d1616dac550a98b86"
    },
    {
      "slug": "the-lifting-scheme",
      "title": "THE LIFTING SCHEME",
      "kicker": "a wavelet lifted in place and lifted back",
      "accent": "#21e6ff",
      "blurb": "The lifting scheme in the 5-window house format — Wim Sweldens' way of building wavelet transforms entirely in place, with no auxiliary memory, and perfectly reversible even in integer arithmetic. Three steps: split the signal into evens and odds; predict each odd from its neighbours and keep only the prediction error (the detail); update the evens using those details to preserve the average (the smooth band). Because every step is an invertible add/subtract, running the steps backwards — undo update, undo predict, merge — reconstructs the original exactly, integers and all. It is how JPEG-2000 does lossless wavelets. Verified live: over 20,000 integer signals (including a second lifting level on the smooth band), the forward lift followed by the inverse lift returns the original signal exactly. Neon-noir traced. See the split/predict/update in 1D, the smooth+detail bands in 2D, and the lift-unlift inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "0167a303d65e192288c186220cf6aafb25b663ac9d6d9cbf662ba4f896493828"
    },
    {
      "slug": "the-scapegoat-tree",
      "title": "THE SCAPEGOAT TREE",
      "kicker": "a tree that rebuilds its own worst branch",
      "accent": "#ff8a3c",
      "blurb": "The scapegoat tree in the 5-window house format — keeping a binary search tree balanced without storing any balance information at all: no colours, no heights, no rotations. It inserts normally, and whenever a new node ends up too deep (deeper than log_{1/α} n), it walks back up to find the scapegoat — the first ancestor so lopsided that one of its subtrees holds more than an α-fraction of it — and flattens and rebuilds that entire subtree perfectly balanced in one sweep. Because rebuilds are rare and cheap on average, insertions cost O(log n) amortized and the height stays logarithmic: balance by occasional demolition, not constant maintenance. Verified live: over 1000 random insertion sequences, the in-order traversal is always sorted, every key is findable, and the height never exceeds log_{1/α}(n)+2 with α=0.7. Neon-noir traced. See the deep-insert rebuild in 1D, the height bound in 2D, and the balance-by-demolition inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "535737c27d1cadcf84e2d7360fb820efa5d076ad8c0b7ba1db2c9e2475bfbff8"
    },
    {
      "slug": "the-luhn",
      "title": "THE LUHN",
      "kicker": "one digit that guards a number",
      "accent": "#35ffb0",
      "blurb": "The Luhn algorithm in the 5-window house format — the checksum guarding nearly every credit-card, IMEI, and account number. Append one check digit so a simple weighted sum comes out a multiple of ten: starting from the right, double every second digit (subtract 9 if the result exceeds 9), add everything up, and a valid number lands on a multiple of 10. It is deliberately tuned to how humans mistype: it catches every single-digit error and almost every adjacent transposition — the one blind spot being swapping a 0 and a 9, which the doubling leaves unchanged. A one-digit tax that stops the commonest typos. Verified live: over 20,000 numbers, the correct check digit validates, every single-digit change is caught, and every adjacent transposition is caught except 09↔90 (which are honestly excused). Neon-noir traced. See the doubling checksum in 1D, the error injection in 2D, and the guard-digit inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "69bbdda62be97f71495009b46a3ead356275236aaae7ca2706213309307ccdeb"
    },
    {
      "slug": "the-transfer-matrix",
      "title": "THE TRANSFER MATRIX",
      "kicker": "a matrix power that counts strings",
      "accent": "#ffcf4a",
      "blurb": "The transfer-matrix method in the 5-window house format — counting configurations obeying a local rule by turning the rule into a matrix and taking a power. Model the constraint as a tiny automaton whose states are the 'recent history' that matters; put a 1 in the transfer matrix T for every allowed state-to-state step. Then the number of valid length-n configurations is read straight off Tⁿ — because matrix multiplication sums over exactly the compatible ways to extend. Counting binary strings with no two adjacent 1s, tilings of a strip, walks avoiding a pattern, even the Ising model's partition function — all become a single matrix power, computable in O(log n) multiplications. Verified live: for the 'no two adjacent 1s' rule, T=[[1,1],[1,0]] gives via Tⁿ exactly the brute-force count of valid length-n strings (the Fibonacci numbers) for n up to 18. Neon-noir traced. See the constraint automaton in 1D, the Tⁿ-vs-brute count in 2D, and the power-the-rule inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "3e686674b05402e31620262d6425a13d14be6da650e82adb54ca7659c8e48a35"
    },
    {
      "slug": "the-merkle-hellman",
      "title": "THE MERKLE-HELLMAN",
      "kicker": "a knapsack locked by a superincreasing sequence",
      "accent": "#b06bff",
      "blurb": "The Merkle–Hellman knapsack in the 5-window house format — one of the first public-key cryptosystems, and a beautiful cautionary tale. The private key is a superincreasing sequence (each term exceeds the sum of all before it), for which subset-sum is trivially solvable by greed. The public key hides that structure: multiply every term by a secret r modulo a secret q, scrambling it into an innocent-looking 'hard knapsack.' To encrypt a bit-string you add up the public terms it selects; to decrypt, multiply by r⁻¹ mod q to restore the superincreasing sequence, then peel off the bits greedily. (Shamir later broke it — the disguise wasn't deep — but the idea launched a field.) Verified live: over 20,000 random messages and keys, encrypting with the public knapsack and decrypting with r⁻¹ mod q recovers the original bits exactly. Neon-noir traced. See the private→public disguise in 1D, the encrypt/decrypt in 2D, and the trapdoor inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "95ee4b6bff1a2ead0dd0c6bf75d0f9f8a20b1ce61e8b1a6fdbf106e99f15a685"
    },
    {
      "slug": "the-ticket-lock",
      "title": "THE TICKET LOCK",
      "kicker": "a deli-counter lock served in ticket order",
      "accent": "#21e6ff",
      "blurb": "The ticket lock in the 5-window house format — a fair spinlock built like a deli counter. Two shared numbers: the next ticket to hand out and the ticket now serving. To acquire, a thread atomically takes the next ticket (fetch-and-increment) and spins until 'now serving' equals its own number. To release, it increments 'now serving,' waking exactly the next thread in line. Because the ticket draw is atomic, the order of tickets IS the order of arrival, so the lock is granted strictly first-come, first-served — no starvation, no thundering herd, just a queue made of two counters. Verified live: over 20,000 random arrival interleavings, the lock is granted in ascending ticket order (FIFO) and never more than one thread holds it at once. Neon-noir traced. See the deli counter in 1D, the FIFO grants in 2D, and the two-counter-queue inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "770cc7f1a91f2b9e7ad721f423791b0ec0e48ceab4115cc2e6e883c44ebf79da"
    },
    {
      "slug": "the-reduced-totient",
      "title": "THE REDUCED TOTIENT",
      "kicker": "the smallest exponent that resets every unit",
      "accent": "#ff8a3c",
      "blurb": "The reduced totient — the Carmichael function λ(n) — in the 5-window house format: the smallest exponent that resets every unit at once, the least m for which a^m≡1 (mod n) for all a coprime to n. Euler's theorem guarantees a^φ(n)≡1, but φ(n) is often bigger than necessary; λ(n) is the true exponent of the group of units, and it always divides φ(n). It is computed as the lcm of the group exponents of each prime-power factor (with the quirk that λ(2^k)=2^{k−2} for k≥3, half of φ). It sets the real period of modular exponentiation — and the correct exponent bound behind RSA. Verified live: for every n up to 300, a^λ(n)≡1 for all units, λ(n) divides φ(n), and λ is tight — some unit has order exactly λ(n). Neon-noir traced. See the unit orders in 1D, λ-vs-φ in 2D, and the tight-exponent inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "07e40054f0be1e555668d34fc83c080e33f00119f276a8c83591b72da5f5251b"
    },
    {
      "slug": "the-heavy-light-decomposition",
      "title": "THE HEAVY-LIGHT DECOMPOSITION",
      "kicker": "a tree cut into heavy chains",
      "accent": "#35ffb0",
      "blurb": "Heavy-light decomposition in the 5-window house format — cutting a tree into a few long chains so any root-to-node path crosses only O(log n) of them. At each node, the edge to its heavy child (the child with the largest subtree) is kept; all other edges are 'light.' Following heavy edges links nodes into vertical chains, and the key fact is that any path from the root descends through at most log₂ n light edges (each light step at least halves the remaining subtree). Lay each chain in a segment tree or Fenwick array, and a path query — sum, max, update along the route between two nodes — becomes O(log² n) instead of O(n). Verified live: over 3000 random trees, the path-sum between two nodes computed by climbing heavy chains equals a brute-force walk of the actual path. Neon-noir traced. See the heavy chains in 1D, the chain-climbing query in 2D, and the flatten-the-tree inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "3ebe614384e8f297c4397fae35682fecf37b83897e6d3a59d37e30b367f8b633"
    },
    {
      "slug": "the-prime-factor-fft",
      "title": "THE PRIME-FACTOR FFT",
      "kicker": "a prime-factored DFT with no twiddles",
      "accent": "#ffcf4a",
      "blurb": "The prime-factor (Good–Thomas) FFT in the 5-window house format — splitting a DFT of size N=N₁·N₂ into a two-dimensional DFT that, uniquely, needs no twiddle factors at all. Its secret is the Chinese Remainder Theorem: when N₁ and N₂ are coprime, the index n can be re-mapped so a single 1-D transform factors cleanly into an N₁-point transform along one axis and an N₂-point transform along the other, with the cross terms vanishing outright. Cooley–Tukey needs twiddle multiplications between stages; Good–Thomas replaces them with a pure re-indexing, trading arithmetic for a clever permutation. Verified live: for coprime N=12, 15, 20, 21, 35, the CRT-reindexed twiddle-free 2-D DFT equals the direct DFT to ~1e-13. Neon-noir traced. See the CRT grid in 1D, the PFA-vs-direct spectra in 2D, and the permute-not-twiddle inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "4d1a55026c8285c9663f8207458ef36d8dd44c92c094dc9b5f0cdd28f34a54ff"
    },
    {
      "slug": "the-fletcher",
      "title": "THE FLETCHER CHECKSUM",
      "kicker": "two coupled sums that feel position",
      "accent": "#b06bff",
      "blurb": "The Fletcher checksum in the 5-window house format — upgrading a plain sum into something that feels position. It runs two accumulators: the first, s₁, is the running sum of the bytes; the second, s₂, is the running sum of s₁ — so each byte is effectively weighted by how many bytes follow it. That single coupling makes the checksum order-sensitive: a plain additive checksum is completely blind to reordering (swap two bytes and the sum is unchanged), but Fletcher's second sum shifts, catching the vast majority of transpositions — while staying nearly as cheap as a plain sum and detecting every single-byte change. Verified live: over 20,000 byte strings, Fletcher-16 is deterministic, catches every single-byte error, and detects ~99% of byte reorderings against a plain additive sum's 0%. Neon-noir traced. See the two accumulators in 1D, the swap caught in 2D, and the sum-the-running-sum inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "d5eb9e94b93e0e5ca3b8c3067065c409d0b2a58a72b1c97678f01d0861f40868"
    },
    {
      "slug": "the-chan",
      "title": "THE CHAN'S ALGORITHM",
      "kicker": "a hull wrapped over mini-hulls",
      "accent": "#21e6ff",
      "blurb": "Chan's algorithm in the 5-window house format — computing a convex hull in O(n log h) time, where h is the number of hull vertices, making it output-sensitive: fast when the hull is small even if the point set is huge. Its trick is a clever marriage. Guess a bound m on h; split the n points into groups of m and compute each group's hull with a quick Graham scan; then gift-wrap around the whole set, jumping between groups by binary-searching each mini-hull's tangent, so each wrap step costs only O((n/m) log m). If the wrap doesn't close within m steps, the guess was too small — double m and retry. The doubling makes the total cost dominated by the final, correct guess. Verified live: over 2000 random point sets, Chan's grouped-hull-plus-wrap-plus-doubling produces exactly the same convex hull as Andrew's monotone chain. Neon-noir traced. See the mini-hulls in 1D, the wrap-vs-reference in 2D, and the guess-wrap-double inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "109d2391dae1936b66fc238c334dc7624b12a27f6c2e96979f4f0de4da42c38e"
    },
    {
      "slug": "the-richardson-extrapolation",
      "title": "THE RICHARDSON EXTRAPOLATION",
      "kicker": "two step sizes that cancel error",
      "accent": "#ff8a3c",
      "blurb": "Richardson extrapolation in the 5-window house format — getting a high-accuracy answer out of a low-accuracy method by combining two runs at different step sizes. Many numerical estimates carry a leading error that shrinks like a power of the step h: a central-difference derivative D(h) is off by roughly c·h². Compute it again at half the step, D(h/2), off by c·h²/4, and form (4·D(h/2)−D(h))/3 — the c·h² terms cancel exactly, leaving an error of order h⁴. Repeat and you climb an accuracy ladder (this is how Romberg integration works). Two cheap estimates, one clever subtraction, and the dominant error vanishes. Verified live: over 2000 smooth functions, the Richardson-extrapolated derivative is closer to the true f′(x) than the plain central difference every time, with a median error ratio around 1e-5. Neon-noir traced. See the error orders in 1D, D-vs-Richardson in 2D, and the cancel-the-error inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "c7fc9b6c61348503e4e751d8b3ad4f0bb916d93fc3e80e918c959d561854c5cd"
    },
    {
      "slug": "the-split-radix",
      "title": "THE SPLIT-RADIX FFT",
      "kicker": "an FFT with the fewest multiplies",
      "accent": "#35ffb0",
      "blurb": "The split-radix FFT in the 5-window house format — computing the DFT with the fewest arithmetic operations of any classic power-of-two algorithm. Radix-2 splits a size-N transform into two size-N/2; radix-4 into four size-N/4. Split-radix does something asymmetric and clever: it splits into one half-size transform on the even-indexed samples and two quarter-size transforms on the samples at indices ≡1 and ≡3 (mod 4). That L-shaped decomposition needs fewer twiddle-factor multiplications than either pure radix — for decades it held the record for lowest operation count — while still giving the exact same transform. Verified live: for sizes N=2 to 128, the split-radix recursion reproduces the direct DFT to ~1e-12 on random complex inputs. Neon-noir traced. See the L-shaped split in 1D, the spectrum-vs-direct in 2D, and the asymmetric-split inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "63e4894e894894e62fb8c802f6a94af5a0cb4652352aed08770ee4bd753308aa"
    },
    {
      "slug": "the-menage-problem",
      "title": "THE MENAGE PROBLEM",
      "kicker": "couples seated so none sits by a partner",
      "accent": "#ffcf4a",
      "blurb": "The ménage problem in the 5-window house format — in how many ways can n couples be seated around a round table, men and women alternating, so that no one sits next to their own partner? Fix the men in alternate seats; the question becomes counting permutations σ of the women with σ(i)≠i and σ(i)≠i+1 (mod n) — each woman avoids the two men flanking her partner's original spot. Touchard gave a closed form as an alternating sum of binomials, A_n = Σ_k (−1)^k (2n/(2n−k)) C(2n−k, k) (n−k)!. The sequence 1, 0, 0, 1, 2, 13, 80, 579… is a classic of combinatorics. Verified live: for n=3 to 7, the Touchard closed-form ménage number equals a brute-force count of all valid seatings. Neon-noir traced. See the forbidden diagonals in 1D, closed-form-vs-brute in 2D, and the inclusion-exclusion inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "20abf1965f8c74f14720f427b1dd77f0104c44948c42ed11eb0ce51c560b57f6"
    },
    {
      "slug": "the-giuga",
      "title": "THE GIUGA CONJECTURE",
      "kicker": "a sum that flags every prime",
      "accent": "#b06bff",
      "blurb": "Giuga's conjecture in the 5-window house format — a stunningly simple proposed test for primality: n is prime if and only if 1^{n−1}+2^{n−1}+…+(n−1)^{n−1} ≡ −1 (mod n). One direction is easy and proven: if n is prime, Fermat's little theorem makes every term ≡ 1, so the sum of n−1 ones is n−1 ≡ −1. The other direction — that no composite ever satisfies it — is a famous open problem: any counterexample would be a 'Giuga number,' and none has ever been found, though we know it would need thousands of digits and at least nine prime factors. Verified live: for every prime n up to 300 the sum is ≡ −1 (mod n), and no composite up to 300 satisfies it. Neon-noir traced. See the power sum flagging primes in 1D, the per-n test in 2D, and the sum-as-detector inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "73a55603a0d202db97648d132ae2c740ea985a8c6d245f6471c53f341703714d"
    },
    {
      "slug": "the-leapfrog",
      "title": "THE LEAPFROG",
      "kicker": "a step that conserves energy and reverses",
      "accent": "#21e6ff",
      "blurb": "Leapfrog integration in the 5-window house format — a symplectic way to step a physical system through time, with a magic ordinary methods lack. Position and velocity are updated at interleaved half-steps (velocity leaps over position, position leaps over velocity), so the scheme is time-reversible and, crucially, does not let energy drift. Explicit Euler on an orbit spirals outward, gaining energy without bound; leapfrog's energy merely oscillates around the true value forever. That is why every serious N-body and molecular-dynamics simulator uses leapfrog (or velocity-Verlet): it keeps planets in orbit and molecules bound over billions of steps. Verified live: over 100 oscillator periods, leapfrog's energy stays bounded (ΔE≈0.001) while Euler's blows up, and leapfrog is exactly time-reversible — run it forward, flip the velocity, run it back, and you return to the start. Neon-noir traced. See the phase-space orbit in 1D, the energy-vs-Euler in 2D, and the reverse-and-return inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "54137843adca43cc0c6fd9ff93f038634e483cf67ba8f7d44e2a563937887f49"
    },
    {
      "slug": "the-elgamal",
      "title": "THE ELGAMAL",
      "kicker": "a public key from a discrete log",
      "accent": "#ff8a3c",
      "blurb": "ElGamal encryption in the 5-window house format — a public-key cryptosystem built on the hardness of the discrete logarithm. Publish a prime p, a generator g, and h=g^x (mod p); the private key is x. To encrypt a message m, pick a random y and send the pair (c₁, c₂)=(g^y, m·h^y). Anyone can compute g^y, but only the holder of x can recover the shared mask h^y=(g^y)^x=c₁^x and divide it out. Recovering x from h would mean solving a discrete log — believed hard. A bonus: it is multiplicatively homomorphic — multiply two ciphertexts componentwise and you get an encryption of the product of the messages. Verified live: over 3000 random (prime, generator, key, message), encrypt then decrypt recovers the message exactly, and the componentwise product of two ciphertexts decrypts to the product of the two messages mod p. Neon-noir traced. See the mask in 1D, the encrypt/decrypt in 2D, and the private-exponent inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "537c1fc4bd70ec244f36fd00a66b665cea0b74647ad634a0361cefa418fcaf0f"
    },
    {
      "slug": "the-golay",
      "title": "THE GOLAY CODE",
      "kicker": "a code that fixes three flipped bits",
      "accent": "#35ffb0",
      "blurb": "The binary Golay code in the 5-window house format — one of the most remarkable objects in coding theory: the extended [24,12,8] Golay code packs 12 data bits into 24, and any two distinct codewords differ in at least 8 positions. That minimum distance of 8 means it can correct any 3 bit-errors and detect 4 — a perfect, exquisitely symmetric code tied to the Steiner system S(5,8,24), the Mathieu group M₂₄, and the Leech lattice. It flew on the Voyager probes to protect images from deep space. Encode with a generator matrix built from a bordered quadratic-residue pattern; to correct, snap a received word to its nearest codeword — unique whenever no more than 3 bits flipped. Verified live: the constructed [24,12,8] code has minimum distance exactly 8 (checked over all 4096 codewords), and nearest-codeword decoding corrects every error pattern of weight ≤ 3. Neon-noir traced. See data+parity in 1D, the error correction in 2D, and the distance-8 inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "ea327dd95b4762ec3e609f4f397743a7e7a41acd003849855777ca4a8d2908da"
    },
    {
      "slug": "the-coordinate-descent",
      "title": "THE COORDINATE DESCENT",
      "kicker": "a minimizer that moves one axis at a time",
      "accent": "#ffcf4a",
      "blurb": "Coordinate descent in the 5-window house format — minimizing a function by improving one variable at a time, cycling through the coordinates and holding the rest fixed. For a convex quadratic ½xᵀAx − bᵀx, each single-coordinate step has a closed form — set the partial derivative to zero, so xᵢ ← (bᵢ − Σ_{j≠i} Aᵢⱼxⱼ)/Aᵢᵢ — an exact line search along that axis. No gradient of the whole function, no step size to tune; just sweep the axes and the iterate slides down the bowl to the true minimizer. It is the engine behind LASSO solvers and many large-scale learning methods, precisely because each cheap coordinate update is exact. Verified live: over 2000 random symmetric-positive-definite systems, cyclic exact coordinate descent converges to the true minimizer x*=A⁻¹b to ~1e-16. Neon-noir traced. See the axis-aligned steps in 1D, the zig-zag convergence in 2D, and the one-axis inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "4a1d46fc80612a44408dc190f00eb2b5424b27b6670acd0beb2ab63876b81267"
    },
    {
      "slug": "the-adaptive-simpson",
      "title": "THE ADAPTIVE SIMPSON",
      "kicker": "integration that refines where it must",
      "accent": "#b06bff",
      "blurb": "Adaptive Simpson's rule in the 5-window house format — integrating a function by spending effort only where the curve is hard. Simpson's rule fits a parabola to three points and reads off the area; adaptive Simpson computes it once on a whole interval and again on the two halves, then compares. If the two agree closely, the interval is smooth — accept the Richardson-corrected estimate. If they disagree, the function is bending too much there, so it recurses into each half with a tighter tolerance. Flat regions are covered by a couple of panels; sharp features get subdivided deeply — the mesh automatically concentrates where the integrand varies, hitting a target accuracy with far fewer evaluations than a uniform grid. Verified live: for a spread of test integrals (exp, sine, a Lorentzian peak, a quartic, a Gaussian), adaptive Simpson matches the exact analytic value to ~1e-13. Neon-noir traced. See the refining panels in 1D, the adaptive-vs-analytic in 2D, and the refine-where-wrong inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "fcb8b997cf873a078814ddfa238033bfbdeeffecf790f361566144400e63c1c6"
    },
    {
      "slug": "the-schnorr",
      "title": "THE SCHNORR",
      "kicker": "prove you know a secret without revealing it",
      "accent": "#21e6ff",
      "blurb": "The Schnorr signature in the 5-window house format — proving you know a secret exponent x without revealing it. Public key y=g^x (mod p). To sign m: commit r=g^k for a fresh random k, derive a challenge e=H(r,m), answer s=k+x·e (mod order). The verifier, who never sees x or k, checks one equation: g^s = r·y^e (mod p). It balances because g^(k+xe)=g^k·(g^x)^e. Change the message and the challenge changes, so an old response no longer fits; change the response and the equation breaks. Verified live: over hundreds of (key, message) pairs at a large prime, every honest signature satisfies g^s=r·y^e, every message-tamper is rejected (the full-width challenge changes), and every response-tamper is rejected. Neon-noir traced. See the Σ-protocol channel in 1D, sign/forge in 2D, and the answer-a-challenge inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "d83f239dda4c8b46741253e32fffa55f2aa47cc600871f987b64ca0d5394272d"
    },
    {
      "slug": "the-nelder-mead",
      "title": "THE NELDER-MEAD",
      "kicker": "a triangle feels for the valley floor",
      "accent": "#ff8a3c",
      "blurb": "The Nelder-Mead downhill simplex in the 5-window house format — minimizing a function with no derivatives at all, only its values at the corners of a moving simplex (a triangle in 2D). Each step it finds its worst corner and reflects it through the centroid of the others; if that lands better it expands further, if still bad it contracts inward, and if all else fails the whole simplex shrinks toward its best corner. The amoeba crawls, tumbles, and squeezes downhill until it collapses onto the minimizer. Verified live: over hundreds of random convex bowls (including a rotated, non-separable one), the simplex converges to the true minimizer to within ~1e-8 using only function evaluations. Neon-noir traced. See the simplex step on a bowl in 1D, step/run to convergence in 2D, and the reflect-the-worst inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "e10fedfa8940745467fa99e7fbe6485631f61cfb5a2f57e289abd8a7e24fc463"
    },
    {
      "slug": "the-gabow",
      "title": "THE GABOW",
      "kicker": "one DFS with two stacks finds every cycle-cluster",
      "accent": "#35ffb0",
      "blurb": "Gabow's algorithm in the 5-window house format — finding the strongly-connected components of a directed graph (the maximal clusters where every node reaches every other) in a single depth-first pass, using two stacks instead of Tarjan's low-link numbers. One stack (S) holds the current path; the other (P) holds candidate component roots. A back-edge pops P down to the earliest reachable vertex, merging the cycle; when a vertex finishes as the top of P, it and everything above it on S form one component. Verified live: over 1500 random digraphs, Gabow's partition exactly matches a brute-force mutual-reachability partition (u~v iff u→v and v→u), component-for-component, and the counts agree. Neon-noir traced. See the colored SCCs in 1D, verify-vs-brute in 2D, and the condensation-DAG inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "81574a49e03ced9ea08e611bffc36f43b6beef6747545d883234e1825379a415"
    },
    {
      "slug": "the-glushkov",
      "title": "THE GLUSHKOV",
      "kicker": "a regex becomes a walk over letter-positions",
      "accent": "#ffcf4a",
      "blurb": "Glushkov's construction in the 5-window house format — turning a regular expression into a position automaton. Give every letter-occurrence a number, then compute First (positions a match can start on), Last (positions it can end on), and Follow (which position can come after which). The result is an NFA with exactly one state per letter-position and no epsilon-transitions at all; matching is a single left-to-right sweep carrying a set of active positions — no backtracking, no exponential blowup. Verified live: for nine regexes, the Glushkov automaton's accept/reject matches an independent reference matcher on every string over {a,b,c,d} up to length 5 — thousands of pairs, zero disagreements. Neon-noir traced. See the position automaton in 1D, test words in 2D, and the carry-the-set inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "a10ef0e4c7b859d744c2d40e80e9a8849e3ac6b3fb6bfea57a0321ec0c927cb4"
    },
    {
      "slug": "the-thabit",
      "title": "THE THABIT",
      "kicker": "two numbers each the sum of the other's divisors",
      "accent": "#b06bff",
      "blurb": "Thabit ibn Qurra's amicable-number rule in the 5-window house format — amicable numbers are two different numbers where each equals the sum of the other's proper divisors. The classic pair is (220, 284). In the 9th century Thabit found a formula that spins such pairs out of primes: for n≥2, if p=3·2^(n-1)-1, q=3·2^n-1, and r=9·2^(2n-1)-1 are all prime, then 2^n·p·q and 2^n·r are amicable. The primes align rarely — only n=2,4,7 work below n=8 — which is why amicable pairs are scarce and prized. Verified live: Thabit's rule at n=2,4,7 yields (220,284), (17296,18416), (9363584,9437056), each confirmed amicable by directly summing proper divisors (σ*(A)=B and σ*(B)=A). Neon-noir traced. See the divisor bars in 1D, the rule per n in 2D, and the two-step-return inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "fea8a6f4fae5878a161de258d42a0cd7fde7d15a96daa4909caa6646c3e6788a"
    },
    {
      "slug": "the-chakravala",
      "title": "THE CHAKRAVALA",
      "kicker": "crank a cycle to crack an ancient equation",
      "accent": "#ffcf4a",
      "blurb": "The chakravala method in the 5-window house format — a cyclic algorithm from 12th-century India (Bhaskara II, on Brahmagupta) that solves Pell's equation x²-N·y²=1 in integers. From a rough triple (a,b,k) with a²-N·b²=k, it repeatedly composes with (m,1) by Brahmagupta's identity, choosing m each turn so k divides a+b·m and |m²-N| is smallest. The value k spirals to ±1, and the current (a,b) is the fundamental solution — centuries ahead of Fermat and Lagrange. Verified live (exact BigInt): for every non-square N from 2 to 120 the method returns integers (x,y) with x²-N·y² exactly 1, including the notorious N=61 whose smallest solution is x=1766319049. Neon-noir traced. See the hyperbola and its lattice solution in 1D, the wheel per N in 2D, and the compose-to-breed inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "58ff4dd3f0e6dcf534384746418d2d97744f1d3fd0b27bef6244f99fe4545dba"
    },
    {
      "slug": "the-rayleigh-quotient",
      "title": "THE RAYLEIGH QUOTIENT",
      "kicker": "a quotient that homes onto an eigenvalue in cubic leaps",
      "accent": "#ff8a3c",
      "blurb": "Rayleigh quotient iteration in the 5-window house format — finding an eigenvector of a symmetric matrix with breathtaking speed. Given a guess v, form the Rayleigh quotient μ = vᵀAv/vᵀv (the best eigenvalue estimate in that direction), solve (A-μI)w = v, normalize, repeat. Each step uses the current eigenvalue estimate as a shift that makes the solve amplify the nearest eigenvector enormously — for symmetric matrices the convergence is cubic, so a few steps reach machine precision. Verified live: over 3000 random symmetric 3×3 matrices from random starts, the iteration returns (v,μ) with residual ‖Av-μv‖ below 1e-6 and |det(A-μI)| below 1e-5 — a genuine eigenpair. Neon-noir traced. See the quadratic-form ellipse in 1D, the cubic convergence in 2D, and the invert-the-shift inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "8f9b2817056079ed6abbe7c2fe52513dd1afb32965d09a5a2dbfced490059d9e"
    },
    {
      "slug": "the-ridders",
      "title": "THE RIDDERS",
      "kicker": "exponential interpolation squeezing onto a root",
      "accent": "#35ffb0",
      "blurb": "Ridders' method in the 5-window house format — finding a root inside a bracket [x₀,x₁] where the sign flips. It takes the midpoint x₂, fits a falling exponential through the three points to absorb the bracket's curvature, and solves that model exactly: x₃ = x₂ + (x₂-x₀)·sign(f₀-f₁)·f₂/√(f₂²-f₀f₁). The new point always stays inside the bracket (so it can never diverge like Newton), yet converges quadratically — far faster than bisection's one bit per step. Verified live: on six functions with known roots, Ridders converges to |f(root)| below 1e-10 (matching the true root to ~1e-9) in at most a handful of iterations, and in strictly fewer iterations than bisection. Neon-noir traced. See the bracketed step in 1D, the collapsing bracket in 2D, and the model-don't-halve inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "89083282474b1cca5360297d676da2c9d039ced05545475a68f5b0342d3f37ff"
    },
    {
      "slug": "the-frank-wolfe",
      "title": "THE FRANK-WOLFE",
      "kicker": "charge the corner to minimize inside a polytope",
      "accent": "#b06bff",
      "blurb": "The Frank-Wolfe algorithm (conditional gradient) in the 5-window house format — minimizing a convex function over a convex set without ever projecting. Each step linearizes the objective and asks a linear oracle for the vertex the linear approximation likes best, then takes a convex step toward it with shrinking size γ=2/(k+2). Because every iterate is a convex combination of vertices, it stays feasible for free — ideal on a polytope like a probability simplex where linear minimization is trivial but projection is costly; the linearization gap certifies how far from optimal you remain. Verified live: minimizing ‖x-a‖² over the probability simplex, Frank-Wolfe converges to the exact Euclidean projection of a onto the simplex (computed independently) to within ~1e-3, and its duality gap collapses toward zero. Neon-noir traced. See the simplex and iterates in 1D, the gap collapsing in 2D, and the charge-a-corner inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "62286f2628fe1bbfe4666daa1c69de54ea321dc91d2d0bcea326a559a25daa9c"
    },
    {
      "slug": "the-pisot",
      "title": "THE PISOT",
      "kicker": "powers that creep toward integers but never quite land",
      "accent": "#21e6ff",
      "blurb": "Pisot-Vijayaraghavan numbers in the 5-window house format — a real algebraic integer θ>1 whose every Galois conjugate has absolute value strictly below 1. That single condition forces the powers θⁿ to creep arbitrarily close to whole numbers, because θⁿ plus its conjugate powers is always an integer (a linear-recurrence term) and the conjugates shrink to nothing. The golden ratio is the classic case: φⁿ + ψⁿ = the Lucas number Lₙ, and |ψ|=0.618, so φⁿ races toward Lₙ. The smallest Pisot number of all is the plastic number ρ≈1.3247. Verified live: φⁿ rounds to the Lucas number with distance exactly |ψ|ⁿ (dist(φ³⁵)≈7e-8); the silver ratio 1+√2 rounds to the Pell-Lucas number with distance |1-√2|ⁿ; and a non-Pisot algebraic integer (1+√13)/2, whose conjugate exceeds 1, keeps missing the integers (mean distance ≈0.26). Neon-noir traced. See the distance-to-integer curves in 1D, θ per θ in 2D, and the vanishing-conjugate inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "5e9b558e298d6dd4b12b20df09cc08bcc4ca1d99be2403a2fad640bf737294b6"
    },
    {
      "slug": "the-konig",
      "title": "THE KÖNIG",
      "kicker": "a matching and a cover forced to be equal",
      "accent": "#35ffb0",
      "blurb": "König's theorem in the 5-window house format — one of the great min-max dualities: in any bipartite graph, the size of a maximum matching (the most edges with no shared endpoint) exactly equals the size of a minimum vertex cover (the fewest vertices touching every edge). Two utterly different optimization problems always return the same number, and the proof is constructive: from a maximum matching you build the minimum cover directly, by an alternating-path search from the unmatched vertices. Verified live: over 20000 random bipartite graphs, the augmenting-path maximum matching and the König vertex cover always have equal size, and that cover genuinely touches every edge. Neon-noir traced. See matching and cover on a graph in 1D, the equal-size + covers-all check in 2D, and the matching-vs-cover duality in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "e8d493d8a3d6b2f35e46aeeab2e66a1b4f195ecb042acb3af82b8006bc6ea747"
    },
    {
      "slug": "the-welford",
      "title": "THE WELFORD",
      "kicker": "one-pass variance that never catastrophically cancels",
      "accent": "#ff8a3c",
      "blurb": "Welford's algorithm in the 5-window house format — computing mean and variance of a stream in a single pass, updating running estimates one sample at a time, never storing the data. It tracks the running mean and the sum of squared deviations M₂ together: each new value nudges the mean, and M₂ is updated using both old and new mean. The famous naive one-pass formula (mean of squares minus square of mean) suffers catastrophic cancellation when numbers are large and close together — it can even return a negative variance. Welford never subtracts two huge nearly-equal quantities, so it stays accurate. Verified live: over 5000 datasets Welford matches the exact two-pass variance to ~1e-15; on data centered near 1e9, the naive formula's error is order 1 (total cancellation) while Welford stays correct to ~1e-9. Neon-noir traced. See the running stats in 1D, the naive-cancellation contrast in 2D, and the accumulate-don't-subtract inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "17062dd64352c709281488ddcd41fb549f2e4a54c032fcbe5bcdbe73ee91f897"
    },
    {
      "slug": "the-remez",
      "title": "THE REMEZ",
      "kicker": "a polynomial whose error rides an equal wave",
      "accent": "#b06bff",
      "blurb": "The Remez exchange algorithm in the 5-window house format — finding the minimax polynomial, the degree-n polynomial that minimizes the worst-case error to a target function over an interval. Its signature is the equioscillation theorem (Chebyshev): the best approximation's error curve touches its maximum height, alternating in sign, at exactly n+2 points of equal magnitude. Remez finds it by exchange: solve for the polynomial making the error equal-and-alternating at n+2 reference points, then move the references to the actual error extrema, and repeat. It converges to the provably optimal polynomial — strictly better in the worst case than Chebyshev interpolation. Verified live: for several functions on [-1,1] the Remez polynomial's error extrema all have equal magnitude (amplitude ratio ≈1.000) and its maximum error is ≤ the degree-matched Chebyshev interpolant. Neon-noir traced. See f and its minimax poly in 1D, the equioscillating error in 2D, and the level-ripple inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "19e2b1f0552f5d32705ceab9a4aa7c0058418100983dd7be97cd971d20249674"
    },
    {
      "slug": "the-sinkhorn",
      "title": "THE SINKHORN",
      "kicker": "alternate row and column normalizing to perfect balance",
      "accent": "#ffcf4a",
      "blurb": "Sinkhorn's algorithm in the 5-window house format — take any matrix of positive numbers and, by the simplest loop (divide every row by its sum, then every column by its sum, and repeat), drive it to a doubly stochastic matrix where every row and column sums to exactly 1. Sinkhorn's theorem guarantees convergence, and that the result is the unique D₁·A·D₂ rescaling of the original by positive diagonal matrices. This little iteration is the computational heart of modern optimal transport (entropic regularization) and of matching problems across machine learning. Verified live: over 3000 random positive matrices, alternating row/column normalization drives every row and column sum to 1 (~1e-16), and the result is exactly diag(u)·A·diag(v) — the ratio to the original is rank-one. Neon-noir traced. See the sums converging in 1D, the step-by-step balancing in 2D, and the scaling-factor inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "bc206c0d8b82beffadff99ad35ff1382e472f175c6faebc8ec80399839c45a14"
    },
    {
      "slug": "the-vieta-jumping",
      "title": "THE VIETA JUMPING",
      "kicker": "an integer ratio that can only be a perfect square",
      "accent": "#21e6ff",
      "blurb": "Vieta jumping in the 5-window house format — a proof technique built on the fact that a quadratic has two roots summing to a rational you read off the coefficients (Vieta's formulas). Its most famous victory is IMO 1988 Problem 6: if a and b are positive integers such that (a²+b²)/(ab+1) is an integer k, then k must be a perfect square. The proof: fix k, and from any solution jump to another by replacing a with the quadratic's other root a′ = k·b − a; this produces a smaller solution, and infinite descent drives b to 0, where k = a² is manifestly a square. Verified live: over all 0≤b≤a≤200, every integer value of (a²+b²)/(ab+1) is a perfect square (0,1,4,9,16,25,36,49…), and the Vieta jump always yields another valid solution that is strictly smaller. Neon-noir traced. See the solution ladder in 1D, the descent step in 2D, and the reflect-across-the-quadratic inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "2a7b4fc9f91d80a55144177529e4e4a752dcb0d1f324050f249b9e11368e0135"
    },
    {
      "slug": "the-frobenius-coin",
      "title": "THE FROBENIUS COIN",
      "kicker": "the largest amount two coins cannot make",
      "accent": "#ffcf4a",
      "blurb": "The Frobenius coin problem (the Chicken McNugget theorem) in the 5-window house format — with only coins of two coprime denominations a and b, what is the largest amount you cannot make from non-negative whole numbers of each? The answer is startlingly clean: the Frobenius number is a·b − a − b. Everything above it is payable; below it, exactly (a−1)(b−1)/2 amounts are impossible. With 3s and 5s the biggest unmakeable total is 7; with the famous 6,9,20 nuggets the largest impossible order is 43. Verified live: over hundreds of coprime pairs the largest non-representable integer is exactly ab−a−b, the count of gaps is exactly (a−1)(b−1)/2, and every integer beyond the Frobenius number is representable. Neon-noir traced. See the payable/impossible number line in 1D, the two formulas vs brute in 2D, and the finite-gaps-vs-infinite-reach inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "9a2660af44b471c62966994ac20131a5330cefcaa7e134066034ad7a376ab272"
    },
    {
      "slug": "the-perron-frobenius",
      "title": "THE PERRON-FROBENIUS",
      "kicker": "a positive matrix's one dominant real eigenvalue",
      "accent": "#ff8a3c",
      "blurb": "The Perron-Frobenius theorem in the 5-window house format — the reason PageRank, Markov chains, and population models all converge. A matrix of strictly positive entries has a single dominant eigenvalue that is real, positive, and strictly larger in magnitude than every other, with an all-positive eigenvector. Repeatedly multiplying any positive vector by the matrix and renormalizing drives it straight to that Perron eigenvector, and the eigenvalue is pinned between the smallest and largest row sums. Verified live: over 4000 random positive matrices, power iteration converges to A·v=λv with residual below 1e-6, λ is positive and the eigenvector all one sign, and λ always lies between the minimum and maximum row sums. Neon-noir traced. See the iterate rotating to the Perron vector in 1D, λ entering the row-sum band in 2D, and the iterate-don't-solve inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "41a4aa167b9678bd502fa58d4fc9d14df87657a2998a5c913a4b86db65fd9828"
    },
    {
      "slug": "the-pade",
      "title": "THE PADÉ",
      "kicker": "a rational that captures the poles a polynomial cannot",
      "accent": "#21e6ff",
      "blurb": "The Padé approximant in the 5-window house format — replacing a power series with a ratio of two polynomials P(x)/Q(x) chosen so its own Taylor expansion agrees with the original to the highest possible order m+n. Because it has a denominator, it captures poles: where a Taylor series diverges the instant you pass its radius of convergence, the Padé approximant sails on, its denominator's roots sitting right where the true function blows up. It underlies function libraries, control theory, and the resummation of divergent series. Verified live: the [3/3] Padé of eˣ reproduces the Taylor coefficients through order 6 exactly, and at x=1 its error (~3e-5) is an order of magnitude smaller than the degree-6 Taylor polynomial's (~2e-4). Neon-noir traced. See f, Padé, and Taylor through a pole in 1D, the coefficient match + error in 2D, and the divide-past-the-radius inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "a8798a47791f461466f3c40449408e41ae6cdd4c0ecd15be7a46db2c02d6d21d"
    },
    {
      "slug": "the-barker",
      "title": "THE BARKER CODE",
      "kicker": "a ±1 code whose echoes never rise above one",
      "accent": "#35ffb0",
      "blurb": "The Barker code in the 5-window house format — a finite sequence of +1s and −1s with an almost magical property: its aperiodic autocorrelation (slide a copy against itself and sum the products) has a tall central peak equal to the code length, and every off-centre value is at most 1 in magnitude. A receiver correlating an incoming signal against a Barker code sees a single sharp spike at alignment and almost nothing elsewhere — which is why they mark the start of radar pulses and Wi-Fi and GPS frames. Barker codes are known only for lengths 2,3,4,5,7,11,13, and it is conjectured none longer exist. Verified live: for each known Barker code the zero-shift autocorrelation equals its length, and every non-zero shift gives a value in {−1,0,+1}. Neon-noir traced. See the code and its autocorrelation in 1D, every shift listed in 2D, and the correlation-peak inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "a56bc4b5e0d3443327ca7c9873e768905c52a0c00e57571859a163a20a8575cd"
    },
    {
      "slug": "the-difference-set",
      "title": "THE DIFFERENCE SET",
      "kicker": "a set whose differences hit every target the same number of times",
      "accent": "#b06bff",
      "blurb": "The cyclic difference set in the 5-window house format — a small set of residues D in Z_v so perfectly arranged that every non-zero residue arises as a difference dᵢ−dⱼ (mod v) the same number of times, λ. A (v,k,λ)-difference set generates a symmetric block design: rotate D through all v shifts and you get v blocks where every pair of points meets in exactly λ blocks. The set {0,1,3} mod 7 is the smallest example — its six differences are exactly 1,2,3,4,5,6 each once — and it is nothing less than the Fano plane in disguise. Verified live: for several classical difference sets — (7,3,1), (13,4,1), (21,5,1), and the (11,5,2) Paley set — every non-zero residue appears exactly λ times among the differences, and a non-example is correctly rejected. Neon-noir traced. See the residue circle and its differences in 1D, the uniform histogram in 2D, and the differences-not-points inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "af6945ff5fd5cc071bf1c9d4ef29091606f4901fd27d636b8b762c2183582d1a"
    },
    {
      "slug": "the-wynn",
      "title": "THE WYNN",
      "kicker": "an accelerator that squeezes π from a crawling series",
      "accent": "#21e6ff",
      "blurb": "Wynn's epsilon algorithm in the 5-window house format — a machine for accelerating convergence. Given the crawling partial sums of a slowly-converging series, it fills a triangular table by one simple rule — ε(n)_{k+1} = ε(n+1)_{k-1} + 1/(ε(n+1)_k − ε(n)_k) — and its even columns leap toward the limit far faster than the sums themselves. It is equivalent to Padé approximation applied to the series, and can wring a dozen correct digits from a series that summed directly would need billions of terms. The Leibniz series for π is the classic victim: agonizingly slow raw, nearly instant accelerated. Verified live: from just 16 terms of the Leibniz series the raw partial sum is off by ~0.06 while Wynn's accelerated estimate is off by ~3×10⁻¹² — over nine orders of magnitude better. Neon-noir traced. See the partial sums vs the accelerated snap in 1D, the plunging error in 2D, and the reach-sideways inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "6848a18ad9b8398b1c718c9e56d38ba638c9094e711c0e35d9c25a1944612e72"
    },
    {
      "slug": "the-laguerre",
      "title": "THE LAGUERRE",
      "kicker": "a solver that hunts every root, real and complex",
      "accent": "#b06bff",
      "blurb": "Laguerre's method in the 5-window house format — a root-finder of unreasonable robustness. To locate a root of a degree-n polynomial it uses both derivatives to build a step that assumes all the other roots are bunched at one distant point — a pessimistic guess that nonetheless lands on a root with cubic convergence and converges from almost any starting point, even to complex roots from a real start. Find one root, divide it out by deflation, and repeat until every root — real and complex — is captured. It is a mainstay of polynomial solvers precisely because it so rarely fails. Verified live: for polynomials built from known roots (mixing real values and complex-conjugate pairs), Laguerre with deflation recovers all roots to about 1e-8. Neon-noir traced. See the roots on the complex plane in 1D, the recovered-vs-true match in 2D, and the deflation inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "3b1d227112a6edc95ce50ddbe045b77a43cefbca09b0d46e5de43c733b1caa95"
    },
    {
      "slug": "the-gold-code",
      "title": "THE GOLD CODE",
      "kicker": "near-orthogonal codes that share one channel",
      "accent": "#ff8a3c",
      "blurb": "Gold codes in the 5-window house format — the sequences that let dozens of GPS satellites and phones talk over the same frequency at the same time. They start from maximum-length LFSR sequences (m-sequences), whose cyclic autocorrelation is a single tall spike of value N at zero shift and a flat −1 everywhere else. Taking a special preferred pair of m-sequences and XOR-ing their shifts produces a family of codes whose cross-correlation takes only three small values, so any two users' signals look nearly orthogonal — the mathematical basis of code-division multiple access. Verified live: for n=5 (period 31) the m-sequence's autocorrelation is 31 at shift 0 and exactly −1 at all other shifts, and the preferred-pair cross-correlation takes only the three values {−1,−9,7} (t(5)=9), as Gold's theorem predicts. Neon-noir traced. See the m-sequence and its autocorrelation in 1D, the three-valued cross-correlation in 2D, and the many-voices-one-channel inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "945c15e47d37507cf227500228e66998d36e8daf29878c1e0b761c432423e3ec"
    },
    {
      "slug": "the-ostrowski",
      "title": "THE OSTROWSKI",
      "kicker": "a numeral system carved from a continued fraction",
      "accent": "#ffcf4a",
      "blurb": "Ostrowski numeration in the 5-window house format — a whole number system built out of the continued fraction of an irrational α. Instead of powers of ten, the place values are the denominators q_k of α's convergents, and every non-negative integer has a unique representation as a digit-weighted sum of them, with digits bounded by the continued-fraction terms and a rule forbidding a maxed digit from sitting on a non-zero one. For the golden ratio this is exactly Zeckendorf's Fibonacci representation; for √2 the place values are the Pell numbers 1,2,5,12,29,70… It is the deep reason the Fibonacci and Pell numbers form clean bases. Verified live: using √2 (place values 1,2,5,12,29,70,169), the greedy Ostrowski digits reconstruct every integer in [0,169) exactly, all digits obey the bounds and the no-adjacent-max rule, and every representation is unique. Neon-noir traced. See the place-values and a digit sum in 1D, per-N digits in 2D, and the exotic-address inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "4238fe99a98242e2d5d10751752dcb882d21fcbc765586a623af3f4e0e8943bd"
    },
    {
      "slug": "the-addition-chain",
      "title": "THE ADDITION CHAIN",
      "kicker": "the shortest ladder of sums from 1 to n",
      "accent": "#35ffb0",
      "blurb": "The addition chain in the 5-window house format — the shortest ladder of additions that builds n starting from 1: a sequence 1=a₀,a₁,…,a_r=n where every term is the sum of two earlier ones. Its length r is the fewest multiplications needed to compute xⁿ — each step multiplies two already-computed powers. The naive 'multiply n times' is terrible; the familiar binary (square-and-multiply) method is far better; but the truly shortest chain can beat even that. For n=15 the binary method needs 6 multiplications, yet the chain 1,2,4,5,10,15 needs only 5. Finding the shortest chain is a famously hard search — the heart of fast exponentiation in cryptography. Verified live: an exhaustive search for n up to 40 returns chains that are valid, compute xⁿ exactly, and are never longer than binary — strictly shorter for n=15,23,27,39 — with anchors l(2^k)=k, l(15)=5, l(23)=6, l(31)=7 all matching. Neon-noir traced. See the ladder from 1 in 1D, the chain vs binary in 2D, and the reuse-not-repeat inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "d7409c5c2b9b85ed51b8c7b261db2e29d442c98dc598392d0385118b0c29284e"
    },
    {
      "slug": "the-coupon-collector",
      "title": "THE COUPON COLLECTOR",
      "kicker": "how many draws to collect the whole set",
      "accent": "#ffcf4a",
      "blurb": "The coupon collector's problem in the 5-window house format — if a box holds one of n equally-likely coupons, how many boxes must you buy to collect them all? The exact expected number is n·Hₙ, where Hₙ=1+1/2+…+1/n is the harmonic number. The reason is a beautiful use of linearity: once you hold i distinct coupons, each new box is new with probability (n-i)/n, so it takes n/(n-i) boxes on average to advance — and summing those independent waits gives n(1+1/2+…+1/n). Since Hₙ≈ln n+γ, collecting all n takes about n ln n boxes: the last few coupons dominate the wait. Verified live: for n=5,10,20 the exact formula n·Hₙ matches a Monte-Carlo simulation of tens of thousands of runs to within a fraction of a percent, and Hₙ tracks ln n+γ. Neon-noir traced. See one collection run in 1D, the empirical-vs-exact convergence in 2D, and the summing-waits inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "574756d15cc5c4e3712393a68521839e735ddc78cd7f2232819803d01d2d1002"
    },
    {
      "slug": "the-cayley-formula",
      "title": "THE CAYLEY FORMULA",
      "kicker": "how many labeled trees on n dots",
      "accent": "#35ffb0",
      "blurb": "Cayley's formula in the 5-window house format — one of the most elegant counting results in mathematics: the number of distinct labeled trees on n vertices is exactly n^(n-2). Three vertices give 3 trees; four give 16; ten give a hundred million. The cleanest proof is a bijection: Prüfer's encoding turns every labeled tree into a unique sequence of n-2 numbers from {1,…,n}, and every such sequence decodes back to a unique tree — so there are exactly n^(n-2) trees. The encoding repeatedly removes the smallest leaf and records its neighbour; the decoding reverses it. Verified live: an exhaustive brute-force count of labeled trees for n=3,4,5,6 equals n^(n-2) exactly, and Prüfer's map is confirmed a bijection — all n^(n-2) sequences decode to distinct valid trees and encoding inverts decoding. Neon-noir traced. See a tree and its Prüfer code in 1D, the count vs n^(n-2) in 2D, and the tree-as-address inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "bc88c645949a5b238099c3b44004c471ce11f43bc50448866f514f23ad15b0a1"
    },
    {
      "slug": "the-smith-normal-form",
      "title": "THE SMITH NORMAL FORM",
      "kicker": "an integer matrix combed to a divisibility chain",
      "accent": "#ff8a3c",
      "blurb": "The Smith normal form in the 5-window house format — what diagonalization becomes over the integers. Any integer matrix A can be reduced by unimodular row and column operations (invertible over ℤ, determinant ±1) to a diagonal matrix D=U·A·V whose diagonal entries d₁,d₂,… form a divisibility chain d₁|d₂|d₃|… These invariant factors are canonical: d₁ is the gcd of all entries, d₁d₂ the gcd of all 2×2 minors, and so on. They reveal the structure of finitely-generated abelian groups, solve systems of integer equations, and compute the homology of a shape. Verified live: for thousands of random integer matrices the algorithm returns U,A,V with U·A·V exactly diagonal, U and V unimodular (det ±1), the diagonal a genuine divisibility chain, and the invariant factors matching the independent gcd-of-minors formula. Neon-noir traced. See A reduced to its diagonal in 1D, the U·A·V + minor checks in 2D, and the structure-over-ℤ inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "3b53bf997a67dec117550e4276dd129cd5b20e6ad3463486a7cbd804324af159"
    },
    {
      "slug": "the-gale-ryser",
      "title": "THE GALE-RYSER",
      "kicker": "when a bipartite degree list can be built",
      "accent": "#21e6ff",
      "blurb": "The Gale-Ryser theorem in the 5-window house format — given a wish-list of degrees for the left vertices (a₁,a₂,…) and right vertices (b₁,b₂,…), does a bipartite graph with exactly those degrees exist? The answer is a clean inequality: sorting the left degrees decreasing, a realization exists iff the sums match and, for every k, Σ_{i≤k} aᵢ ≤ Σⱼ min(bⱼ,k). The condition is not just a test — when it holds, a simple greedy connects each left vertex to the highest-capacity right vertices and builds the graph. Verified live: over thousands of random degree-sequence pairs the Gale-Ryser inequality holds exactly when a greedy construction realizes the degrees, and for small cases this matches an exhaustive existence check. Neon-noir traced. See the realized graph in 1D, the inequality + greedy in 2D, and the existence-from-degrees inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "edd75c96ca0aa7dd989d7da9fabfa3ec674873cbb7541022085ee03e828816c6"
    },
    {
      "slug": "the-art-gallery",
      "title": "THE ART GALLERY",
      "kicker": "a third of the corners guard the whole gallery",
      "accent": "#b06bff",
      "blurb": "The art gallery theorem in the 5-window house format — a gallery shaped like any simple polygon with n corners can always be watched by at most ⌊n/3⌋ guards, and sometimes needs that many. The proof is a gem (Fisk 1978): triangulate the polygon, then 3-colour its vertices so every triangle shows all three colours (always possible, because a triangulated polygon's graph is 3-colourable). Whichever colour is used least appears on at most ⌊n/3⌋ vertices — and since every triangle contains one vertex of that colour, placing guards there watches every triangle, hence the whole gallery. Verified live: random simple polygons are triangulated by ear-clipping and 3-coloured; every triangle gets all three colours, the smallest colour class has ≤⌊n/3⌋ vertices, and that class contains a vertex of every triangle. Neon-noir traced. See the coloured triangulation in 1D, the ⌊n/3⌋ bound + coverage in 2D, and the coverage-from-colour inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "2e10ea4857b82377c845a90ec509e65c9c770ddec09373405e26f6744893b0e2"
    },
    {
      "slug": "the-ceva",
      "title": "THE CEVA",
      "kicker": "three cevians meeting at one point",
      "accent": "#35ffb0",
      "blurb": "Ceva's theorem in the 5-window house format — the exact condition for three cevians (lines from each vertex of a triangle to the opposite side) to all pass through a single point. Mark points D,E,F on sides BC,CA,AB; the cevians AD,BE,CF are concurrent if and only if the product of the three side-ratios is exactly one: (BD/DC)(CE/EA)(AF/FB)=1. It is why the medians meet at the centroid (all ratios 1, product 1), and why the angle bisectors and altitudes are concurrent too. Verified live: over tens of thousands of random triangles and side-ratios, whenever the product equals 1 the three cevians meet at one point, whenever it differs they do not, and the medians meet exactly at the centroid. Neon-noir traced. See the meeting point in 1D, the product-vs-concurrency test in 2D, and the concurrency-from-a-product inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "8d9f3b786dccc78988e8a3558feebfbbf02c325a220e2d973a5dc38802bbc1cc"
    },
    {
      "slug": "the-resultant",
      "title": "THE RESULTANT",
      "kicker": "a determinant that detects a shared root",
      "accent": "#ff8a3c",
      "blurb": "The resultant in the 5-window house format — a single number, computed as the determinant of two polynomials' Sylvester matrix, that is zero exactly when they share a common root, without ever finding the roots. Stack shifted copies of each polynomial's coefficients into a matrix; its determinant vanishes precisely when a common factor exists. Even better, the resultant equals one polynomial evaluated at all the roots of the other (times a leading-coefficient power). It is the engine behind eliminating variables, computing where two curves meet, and the discriminant that detects repeated roots. Verified live: for thousands of polynomial pairs the Sylvester determinant is zero exactly when they share a root and non-zero otherwise, and it equals lead(p)^deg(q)·∏ q(roots of p) to machine precision. Neon-noir traced. See the shared crossing in 1D, the determinant + product-over-roots in 2D, and the sense-it-algebraically inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "c3ec7d9989a11daafe596142a8b35b5063e2600868dd6cbaa92babf4f661b327"
    },
    {
      "slug": "the-thiele",
      "title": "THE THIELE",
      "kicker": "a rational curve threaded through the data",
      "accent": "#b06bff",
      "blurb": "Thiele's interpolation formula in the 5-window house format — threading a rational function exactly through data points, written as a continued fraction: R(x)=a₀+(x-x₀)/(a₁+(x-x₁)/(a₂+…)). The coefficients aₖ are the inverse differences of the data — a reciprocal cousin of Newton's divided differences — computed by a simple triangular recurrence. Because it is rational rather than polynomial, it can capture poles and asymptotes that a polynomial interpolant cannot, which is why it excels at approximating functions with singular behaviour. Verified live (exact rational arithmetic): for thousands of random rational data sets the Thiele continued-fraction interpolant, built from inverse differences, evaluates back to the exact y-value at every data point — a perfect fit with no rounding. Neon-noir traced. See the curve through the points in 1D, the exact reproduction in 2D, and the fit-that-reciprocates inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "c6838dcde7ffcf070663a723484f24391286ba48652d071ef88550439456f925"
    },
    {
      "slug": "the-galton",
      "title": "THE GALTON BOARD",
      "kicker": "a board of pegs building the bell curve",
      "accent": "#ffcf4a",
      "blurb": "The Galton board (bean machine) in the 5-window house format — a triangular array of pegs down which balls bounce, going left or right with equal chance at each row. After n rows a ball lands in bin k, and the number of distinct paths to that bin is exactly the binomial coefficient C(n,k) — the n-th row of Pascal's triangle. Since every path is equally likely, the fraction of balls in bin k is C(n,k)/2ⁿ, so a heap of balls piles up into the binomial distribution — and as n grows, into the smooth bell curve. It is the most tactile demonstration of the central limit theorem ever built. Verified live: the exact count of paths to each bin equals C(n,k) for every row up to n=14, and a simulation of hundreds of thousands of balls settles into the binomial C(n,k)/2ⁿ with mean n/2. Neon-noir traced. See the pegs and Pascal bins in 1D, the growing histogram in 2D, and the count-the-paths inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "5e94914c4290fff03dbfe8fdc859b9d1c6ee66837ee8bd8a22468763c39db7a0"
    },
    {
      "slug": "the-harshad",
      "title": "THE HARSHAD",
      "kicker": "the numbers that are Harshad in every base",
      "accent": "#21e6ff",
      "blurb": "The Harshad (Niven) number in the 5-window house format — a positive integer divisible by the sum of its own digits. In base ten, 18 is Harshad (1+8=9 divides 18); 21 is (2+1=3 divides 21). Every number is Harshad in some base, but which numbers are Harshad in every base at once? Astonishingly, there are only four: 1, 2, 4, and 6. These 'all-Harshad' (total Harshad) numbers are divisible by their digit sum no matter what base you write them in — a rare and complete little set, proved to contain nothing else. Verified live: checking every integer up to 2000 against every base from 2 to 30, the only numbers that are Harshad in all of them are exactly {1,2,4,6}. Neon-noir traced. See the number×base Harshad grid in 1D, one number across bases in 2D, and the every-base-must-agree inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "c3f1d2f77d22a89936524b71a160310b7d83c490a0d533276f04e353a7f14fa8"
    },
    {
      "slug": "the-menelaus",
      "title": "THE MENELAUS",
      "kicker": "a line cutting three sides, points collinear",
      "accent": "#b06bff",
      "blurb": "Menelaus' theorem in the 5-window house format — the collinearity twin of Ceva's concurrency. Draw a transversal line cutting the three sides of a triangle: side BC at D, CA at E, AB at F (some crossings on the extensions). The three points are collinear exactly when the product of the three signed side-ratios is minus one: (BD/DC)(CE/EA)(AF/FB)=-1. The single minus sign is the whole story: Ceva's concurrent cevians give +1, Menelaus' collinear transversal gives -1. It is the workhorse behind projective proofs and the complete quadrilateral. Verified live: for tens of thousands of random triangles and transversal lines, the three intersection points' signed ratio product is -1, and a deliberately non-collinear triple gives a product that is not -1. Neon-noir traced. See the transversal cutting the sides in 1D, the signed ratios in 2D, and the alignment-from-a-sign inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "01764461b9e260c89efd2e2fb03d586cc1f09c6cdedc1454183da763b61575f2"
    },
    {
      "slug": "the-dormand-prince",
      "title": "THE DORMAND-PRINCE",
      "kicker": "an adaptive integrator that paces itself",
      "accent": "#21e6ff",
      "blurb": "The Dormand-Prince method in the 5-window house format — the adaptive engine inside most modern ODE solvers (MATLAB's ode45). It takes a step of a differential equation with a fifth-order Runge-Kutta formula, but computes a fourth-order estimate at the same time from the same seven stage evaluations. The difference between the two is a nearly-free estimate of the local error — and the method uses it to pace itself: when the solution is smooth it lengthens the step, when it turns sharply it shrinks the step, holding the error under a chosen tolerance everywhere. Verified live: on y′=y the method shows clean fifth-order convergence — halving the step cuts the error by about 32× — and on the harmonic oscillator it tracks (sin t, cos t) to ~1e-12. Neon-noir traced. See the phase-space orbit in 1D, the fifth-order error drop in 2D, and the error-sets-the-step inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "6083150095e50fddc372de9e4be3abbf461cd9b94b4108c4572da618594b7082"
    },
    {
      "slug": "the-garner",
      "title": "THE GARNER",
      "kicker": "one number rebuilt from its remainders",
      "accent": "#ff8a3c",
      "blurb": "Garner's algorithm in the 5-window house format — the constructive heart of the Chinese Remainder Theorem: given a number's remainders modulo several pairwise-coprime bases, it rebuilds the number itself. It works in mixed radix — peeling off one digit at a time, each found by a modular subtraction and inverse against the previous bases, so the final value is x=d₀+d₁m₀+d₂m₀m₁+… The result is exact and unique below the product of the moduli. It is how big-integer libraries and cryptosystems split one huge computation into small independent ones and stitch the answer back together. Verified live (exact BigInt): for thousands of random values and random sets of coprime moduli, reducing x to its residues and running Garner's reconstruction returns x exactly — e.g. x≡2(mod 3), 3(mod 5), 2(mod 7) rebuilds to 23. Neon-noir traced. See the remainders in 1D, the mixed-radix reconstruction in 2D, and the split-and-stitch inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "e862d70e8e9e45a12b07aa675a7a4e24d98a7d3f627db0390c365c8630c3b834"
    },
    {
      "slug": "the-inclusion-exclusion",
      "title": "THE INCLUSION-EXCLUSION",
      "kicker": "add the parts, subtract the overlaps",
      "accent": "#35ffb0",
      "blurb": "Inclusion-exclusion in the 5-window house format — the exact bookkeeping for counting a union without double-counting. Add the sizes of all the sets, then subtract every pairwise overlap (counted twice), then add back every triple overlap (subtracted too much), and so on with alternating signs: |A₁∪…∪Aₙ| = Σ|Aᵢ| − Σ|Aᵢ∩Aⱼ| + Σ|Aᵢ∩Aⱼ∩Aₖ| − … The same alternating machine counts derangements (permutations fixing no element), surjections, and numbers coprime to a set of primes. It is the 'off-by-the-overlaps' correction made exact. Verified live: for thousands of random set systems the alternating sum equals a brute-force union count exactly, and the derangement formula Dₙ=n!Σ(−1)ʲ/j! matches a brute count of fixed-point-free permutations (D₅=44). Neon-noir traced. See the overlapping sets in 1D, the term-by-term ± tally in 2D, and the miscount-then-mend inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "b9da58ae2601bb71104bc769871aff4cc99db36f4d777efb9308a905e7bde66c"
    },
    {
      "slug": "the-edmonds-karp",
      "title": "THE EDMONDS-KARP",
      "kicker": "the most that can flow equals the cheapest cut",
      "accent": "#ffcf4a",
      "blurb": "The Edmonds-Karp algorithm in the 5-window house format — computing the maximum flow through a capacitated network from a source to a sink, by repeatedly finding a shortest augmenting path (via breadth-first search) in the residual graph and pushing as much flow along it as the tightest edge allows. When no augmenting path remains, the flow is maximal — and by the max-flow min-cut theorem, its value equals the capacity of the cheapest cut separating source from sink. The vertices still reachable from the source in the residual graph reveal exactly that minimum cut. Verified live: for thousands of random networks the max flow equals the minimum-cut capacity, the flow is conserved at every intermediate node, and no edge exceeds its capacity. Neon-noir traced. See the flow network + min cut in 1D, the max-flow=min-cut checks in 2D, and the bottleneck-is-the-maximum inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "77487c79d0621aa9e0ea0f3b0a20bbabb5acf47516695ee7ce2578f27a710fbc"
    },
    {
      "slug": "the-fermat-point",
      "title": "THE FERMAT POINT",
      "kicker": "the point that minimizes the walk to three corners",
      "accent": "#ff8a3c",
      "blurb": "The Fermat point in the 5-window house format — the single spot that minimizes the total distance to all three corners of a triangle, the ideal meeting place for least combined walk. Its signature is beautiful: at the Fermat point the three corners are seen at exactly 120° apart, three equal wedges filling the plane. (If one triangle angle is 120° or more, the point collapses onto that vertex.) Torricelli found it via equilateral triangles on the sides; it is also reached by Weiszfeld's iteration, repeatedly pulling toward each corner with weight inversely proportional to distance. Verified live: for thousands of triangles (all angles below 120°), Weiszfeld's iteration lands on a point where the three corners subtend 120° to within a hundredth of a degree, and no sampled nearby point has a smaller total distance. Neon-noir traced. See the 120° wedges in 1D, the angle+minimality checks in 2D, and the equilibrium-of-pulls inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "6d4f5de021614f848866364c8da576f25c152380b2fcf7760c5f5bdda0b51367"
    },
    {
      "slug": "the-postage-stamp",
      "title": "THE POSTAGE STAMP",
      "kicker": "the longest run of amounts a few stamps can make",
      "accent": "#ffcf4a",
      "blurb": "The postage-stamp problem in the 5-window house format — with stamps of a few fixed denominations and an envelope holding at most h stamps, what is the largest value N such that every postage from 1 to N can be made? Call it the h-range. With 1- and 4-cent stamps and up to 5 stamps you cover every value to 14; with 1, 5 and 8 and six stamps you reach 42. Choosing denominations to maximize the unbroken run is a classic unsolved optimization, but for a given set and h the answer is a clean finite computation. Verified live: two independent methods — a dynamic-programming reachable-set and an exhaustive enumeration of every stamp multiset of size ≤ h — produce the identical set of achievable values, and the h-range is the longest run 1,2,…,N inside it. Neon-noir traced. See the makeable values in 1D, DP-vs-brute + the h-range in 2D, and the coverage-from-combination inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "961d466d172df0a377ef7642e7ef610bb21bb1cb2e6dd25e372a8a1555e83410"
    },
    {
      "slug": "the-svd",
      "title": "THE SVD",
      "kicker": "a matrix as rotate-stretch-rotate",
      "accent": "#35ffb0",
      "blurb": "The singular value decomposition in the 5-window house format — factoring any matrix A into A=UΣVᵀ, a rotation, a pure axis-aligned stretch, and another rotation. The diagonal singular values in Σ are the stretch factors; the columns of U and V are the output and input axes. Geometrically, A takes the unit sphere to an ellipsoid, and the SVD reads off its axes and their lengths. It is the most useful factorization in applied mathematics: it powers principal-component analysis, low-rank compression, the pseudo-inverse, and the numerical rank of a matrix. Verified live: for thousands of random matrices a one-sided Jacobi SVD returns U,Σ,V with U·diag(Σ)·Vᵀ reconstructing A to machine precision, U and V orthonormal (UᵀU=VᵀV=I), and all singular values non-negative. Neon-noir traced. See the circle→ellipse map in 1D, the reconstruction + orthonormality in 2D, and the rotate-stretch-rotate inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "00ef82c8c625f644d4dc440f83a398936f3c58dfffa3cb8d2252c46583c89939"
    },
    {
      "slug": "the-ryser",
      "title": "THE RYSER",
      "kicker": "a permanent counted by inclusion-exclusion",
      "accent": "#b06bff",
      "blurb": "Ryser's formula in the 5-window house format — computing the permanent of a matrix, the determinant's sign-free cousin, a sum over all permutations with every term added, never subtracted. The permanent counts things (for a 0/1 matrix it is the number of perfect matchings in a bipartite graph), but computing it is #P-complete, believed harder than NP. Ryser's trick uses inclusion-exclusion over the columns to compute it in O(2ⁿn) — still exponential, but far better than the n! of the definition, and the fastest known general method. Verified live: for thousands of random integer matrices Ryser's inclusion-exclusion permanent equals the brute-force sum over all permutations exactly, and for 0/1 matrices it equals the number of perfect matchings; the permanent of the all-ones 3×3 matrix is 3!=6. Neon-noir traced. See the matrix as a bipartite graph in 1D, Ryser-vs-brute in 2D, and the n!→2ⁿ inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "98f931663b07be75c35683d4e98ac01aaf2bda08bfc2da98120d65ae368152f4"
    },
    {
      "slug": "the-buffon",
      "title": "THE BUFFON",
      "kicker": "needles dropped to measure π",
      "accent": "#21e6ff",
      "blurb": "Buffon's needle in the 5-window house format — the oldest problem in geometric probability and a startling way to measure π by dropping sticks. Rule a floor with parallel lines a distance d apart, and toss a needle of length L≤d at random. The probability it crosses a line is exactly 2L/(πd) — π appears because the crossing depends on the needle's random angle. Turn it around: drop many needles, count the crossings, and π≈2LN/(d·crossings). It is a Monte-Carlo estimator of π that needs nothing but a ruler and patience. Verified live: dropping two million random needles, the crossing rate matches 2L/(πd) to within a fraction of a percent, and the resulting estimate of π lands near 3.14. Neon-noir traced. See the needles on the ruled floor in 1D, the rate converging in 2D, and the π-sampled-not-computed inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "dc2b1e4947a3c56aafcf984c341188e6c21c295773f99d13663faf4f2c15ceec"
    },
    {
      "slug": "the-lambert-w",
      "title": "THE LAMBERT-W",
      "kicker": "the inverse of x times e-to-the-x",
      "accent": "#ff8a3c",
      "blurb": "The Lambert W function in the 5-window house format — the inverse of w·eʷ: given x, it returns the w such that w eʷ = x. That single definition unlocks equations no elementary function can — anything of the form 'an unknown multiplied by its own exponential', from delay differential equations to enzyme kinetics to the analysis of algorithms. Because y=x eˣ is not monotone, W has two real branches; the principal branch W₀ is found in a handful of steps by Halley's iteration, a cubically-convergent cousin of Newton's method. W(1) is the omega constant Ω≈0.5671, the number equal to its own negative logarithm. Verified live: for tens of thousands of values of x, Halley's iteration returns a w with w·eʷ equal to x to machine precision, and the anchors W(0)=0, W(e)=1, W(1)=Ω all hold. Neon-noir traced. See x·eˣ and its inverse W in 1D, the Halley convergence in 2D, and the invert-the-transcendental inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "9d058a4fcbd0e79fa99d05a4640883edd4d833699ba2f51ac5c52b7550dc79b5"
    },
    {
      "slug": "the-viete",
      "title": "THE VIETE",
      "kicker": "π from an endless nested radical",
      "accent": "#35ffb0",
      "blurb": "Viète's formula (1593) in the 5-window house format — the very first time in history that a constant was written as an infinite product, the dawn of analysis. It expresses 2/π as an endless product of nested square roots of two: 2/π = (√2/2)·(√(2+√2)/2)·(√(2+√(2+√2))/2)·… Each factor aₖ/2 is built from the last by aₖ₊₁=√(2+aₖ), a value creeping toward 2. Geometrically it is Archimedes' doubling of a polygon's sides made algebraic: each nested radical is the cosine of an angle halved again and again. Verified live: the partial product converges to 2/π — after 30 nested factors it matches to machine precision, giving π to twelve digits. Neon-noir traced. See the nested radicals + running product in 1D, the convergence in 2D, and the infinite-descent inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "421855df3b75c78589fc4e099a51f91df7a05fc28d81cf2a4dc37b3d7fd90d56"
    },
    {
      "slug": "the-graceful",
      "title": "THE GRACEFUL",
      "kicker": "a labeling whose edge-gaps are 1 to m",
      "accent": "#21e6ff",
      "blurb": "Graceful labeling in the 5-window house format — assigning the vertices of a graph with m edges distinct numbers from 0 to m so that the edge 'lengths' (absolute differences of endpoint labels) come out as exactly 1,2,…,m, each once. It is a jigsaw of numbers: pick vertex values so no two edges share a gap. Paths and stars are always graceful; a cycle Cₙ is graceful if and only if n≡0 or 3 (mod 4). The still-open Graceful Tree Conjecture — that every tree is graceful — has resisted proof for over fifty years. Verified live: an explicit zig-zag labeling makes every path graceful and the star K₁,ₙ graceful, and an exhaustive search confirms the cycle Cₙ is graceful exactly when n≡0 or 3 (mod 4) — C₃,C₄,C₇ yes; C₅,C₆ no. Neon-noir traced. See the graceful labeling in 1D, the cycle mod-4 rule in 2D, and the demand-the-edges inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "7ceeb7a79e0c5015ad2572b6c12437fe2e5b0c151f2b67224f33624085da1956"
    },
    {
      "slug": "the-graeffe",
      "title": "THE GRAEFFE",
      "kicker": "squaring a polynomial to prise its roots apart",
      "accent": "#b06bff",
      "blurb": "Graeffe's root-squaring method in the 5-window house format — finding the magnitudes of a polynomial's roots by a startling trick: build a new polynomial whose roots are the squares of the original's, using q(x²)=(−1)ⁿp(x)p(−x). Repeat, and after k rounds the roots are raised to the 2^k power — which drives well-separated roots exponentially far apart. Once separated, each magnitude falls straight out of the ratio of adjacent coefficients: |rᵢ|=|a_{n−i}/a_{n−i+1}|^{1/2^k}. It was a workhorse of hand computation before electronic computers — a way to prise roots apart until they can simply be read off. Verified live: for polynomials with well-separated positive roots, four root-squaring rounds recover every root magnitude to within a fraction of a percent — e.g. (x−1)(x−2)(x−3) comes back as 3.000, 2.000, 1.000. Neon-noir traced. See the separating magnitudes in 1D, the recovery in 2D, and the separation-by-squaring inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "06a77063cee61c3b0df3b40285c50cf3b49db4ded4755c9c24a8748763e8042d"
    },
    {
      "slug": "the-agm",
      "title": "THE AGM",
      "kicker": "two means racing to one limit",
      "accent": "#ffcf4a",
      "blurb": "The arithmetic-geometric mean in the 5-window house format — one of the fastest-converging processes in mathematics. Start with two positive numbers a and b and replace them, over and over, by their arithmetic mean (a+b)/2 and geometric mean √(ab). The two sequences rush toward each other and meet at a common limit M(a,b) — and they do so quadratically: the gap between them squares each step, so the number of correct digits doubles every iteration. Gauss discovered it links to elliptic integrals, and it is the engine of the Gauss-Legendre algorithm that computes π to millions of digits in a handful of steps. Verified live: for thousands of random starting pairs the two means converge to a single limit, the gap shrinking quadratically (gap≈previous²/8M), and the AGM-driven Gauss-Legendre iteration reaches π to ~1e-15 in just four steps. Neon-noir traced. See the two means meeting in 1D, the squaring gap + π in 2D, and the two-means-folded-into-one inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "9f0fff79ccc2bb4810f7bd6acdf3bfacb0230a82eb00b22d534fe425d1f56314"
    },
    {
      "slug": "the-pentagonal-number",
      "title": "THE PENTAGONAL",
      "kicker": "partitions counted by an alternating sum over pentagons",
      "accent": "#ffcf4a",
      "blurb": "Euler's pentagonal number theorem in the 5-window house format — a shockingly efficient recurrence for p(n), the number of ways to write n as a sum of positive integers. Naively p(n) explodes, but Euler found that the generating product ∏(1−xᵏ) collapses to a sparse alternating sum over the generalized pentagonal numbers g_k=k(3k−1)/2 — 1,2,5,7,12,15,22,… That yields p(n)=p(n−1)+p(n−2)−p(n−5)−p(n−7)+p(n−12)+…, signs in pairs of plus-plus, minus-minus, using only O(√n) terms. It is one of the most beautiful cancellations in combinatorics. Verified live: for n up to 45 the pentagonal recurrence produces exactly the same partition counts as a brute dynamic-programming enumeration — p(40)=37338, p(45)=89134. Neon-noir traced. See the partition growth + pentagonal marks in 1D, the ± recurrence in 2D, and the counting-by-cancellation inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "d369427b14d627af5172b3688ae3f21686e389aba3eb9b57f6a08ce321c62187"
    },
    {
      "slug": "the-desargues",
      "title": "THE DESARGUES",
      "kicker": "perspective from a point equals perspective from a line",
      "accent": "#b06bff",
      "blurb": "Desargues' theorem in the 5-window house format — a cornerstone of projective geometry linking two kinds of perspective. Two triangles ABC and A′B′C′ are perspective from a point if the lines AA′, BB′, CC′ meet at one center O. They are perspective from a line if the three intersection points of corresponding sides — AB∩A′B′, BC∩B′C′, CA∩C′A′ — are collinear. Desargues proved these equivalent: a common center forces a common axis, and vice versa. It is self-dual (swap 'point' and 'line' and it still holds) and it is exactly the condition a projective plane needs to come from a field. Verified live: for tens of thousands of triangle pairs placed in perspective from a random center, the three corresponding-side intersections are always collinear, and pushing a single vertex off its center-ray breaks both the perspectivity and the collinearity together. Neon-noir traced. See the two triangles + axis in 1D, the collinearity + off-ray control in 2D, and the point-and-line inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "5aef85988296ca5a26d4fc5e5ddea5f6208d3df60eaa421f151d28bde902319e"
    },
    {
      "slug": "the-fibonacci-matrix",
      "title": "THE FIBONACCI MATRIX",
      "kicker": "Fibonacci as a matrix power",
      "accent": "#ff8a3c",
      "blurb": "The Fibonacci Q-matrix in the 5-window house format — turning the Fibonacci recurrence into a single matrix. Because F_{n+1}=F_n+F_{n−1}, one step is multiplication by Q=[[1,1],[1,0]], so Qⁿ=[[F_{n+1},F_n],[F_n,F_{n−1}]]. That single fact gives Fibonacci numbers in O(log n) time by fast matrix exponentiation (repeated squaring), and it hands you identities for free: taking determinants of both sides gives Cassini's identity, F_{n−1}F_{n+1}−F_n²=(−1)ⁿ, because det Q=−1 and determinants multiply. Verified live (exact BigInt): for n up to 200, Qⁿ by repeated squaring has exactly F_n and F_{n+1} in the right entries, matching the direct recurrence, and its determinant equals (−1)ⁿ — Cassini's identity. Neon-noir traced. See the matrix powers in 1D, Qⁿ vs Fibonacci + Cassini in 2D, and the recurrence-made-a-power inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "093a52c5a97dcc9ea7ec799557d27b57292f2eb3983b0082b26881953410e9a9"
    },
    {
      "slug": "the-jacobi-elliptic",
      "title": "THE JACOBI ELLIPTIC",
      "kicker": "the doubly-periodic cousins of sine",
      "accent": "#35ffb0",
      "blurb": "The Jacobi elliptic functions in the 5-window house format — sn, cn, dn, the doubly-periodic cousins of sine and cosine. Where sin and cos parametrize a circle, sn and cn parametrize the motion of a pendulum swinging through large angles, governed by a parameter m that measures how far from a simple circle you are. They obey sin-like identities — sn²+cn²=1 and dn²+m·sn²=1 — and their own differential equations, sn′=cn·dn. Their real period is 4K, where K is the complete elliptic integral, and at the quarter-period K the functions hit clean values sn=1, cn=0, dn=√(1−m). Verified live: computing sn,cn,dn by integrating their ODE, the identities hold to ~1e-11, and — independently — at the quarter-period K obtained from the arithmetic-geometric mean, sn(K)=1, cn(K)=0, dn(K)=√(1−m). Neon-noir traced. See the three curves in 1D, the identities + quarter-period in 2D, and the pendulum-not-circle inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "53dffecf783cf16364b7e3f28544f826cdc6dff26c39a404299a689189806963"
    },
    {
      "slug": "the-chu-liu-edmonds",
      "title": "THE CHU-LIU-EDMONDS",
      "kicker": "the cheapest way to root a directed tree",
      "accent": "#21e6ff",
      "blurb": "The Chu-Liu/Edmonds algorithm in the 5-window house format — finding the minimum spanning arborescence of a directed graph, the cheapest set of edges that lets a chosen root reach every node, with exactly one incoming edge per node. It is the directed cousin of the minimum spanning tree, but greedy edge-picking alone fails: choosing each node's cheapest in-edge can form a cycle. The fix is elegant — contract each cycle into a single super-node, discount every edge entering the cycle by the edge it would replace, and recurse; then expand the contractions back, dropping exactly one cycle edge each. The result is provably optimal. Verified live: for thousands of random weighted digraphs, the Chu-Liu/Edmonds arborescence weight equals the true minimum found by brute force over every possible arborescence. Neon-noir traced. See the arborescence highlighted in 1D, the min-vs-brute in 2D, and the contract-the-cycles inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "d2eb244c1221ccc3669dd571d2b72be5c1630ab38de3457077fac3c510776ba4"
    },
    {
      "slug": "the-hermite",
      "title": "THE HERMITE",
      "kicker": "orthogonal polynomials of the oscillator",
      "accent": "#ff8a3c",
      "blurb": "The Hermite polynomials in the 5-window house format — the natural family of polynomials orthogonal under the Gaussian weight e^{−x²}. Built by the three-term recurrence H_{n+1}=2x·H_n−2n·H_{n−1} from H₀=1, H₁=2x, each H_n is 'perpendicular' to all the others under the Gaussian inner product: ∫H_m(x)H_n(x)e^{−x²}dx=0 whenever m≠n. They are the eigenfunctions of the quantum harmonic oscillator (times a Gaussian), the backbone of Gauss–Hermite quadrature, and each H_n has exactly n real roots — the quadrature nodes. Verified live: the Gaussian-weighted inner product of H_m and H_n is zero for m≠n and equals 2ⁿn!√π for m=n (to ~1e-4 by numerical integration), and each H_n shows exactly n real roots. Neon-noir traced. See H₀…H₄ in 1D, the orthogonality + norm + root count in 2D, and the basis-not-a-curve inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "d82173377f5821826537fb517a517884e91aa7e6cd1afd30c53cbc0154dbb542"
    },
    {
      "slug": "the-fuss-catalan",
      "title": "THE FUSS-CATALAN",
      "kicker": "counting m-ary trees",
      "accent": "#ffcf4a",
      "blurb": "The Fuss–Catalan numbers in the 5-window house format — the m-ary generalization of the Catalan numbers. Where C_n counts full binary trees with n internal nodes (each with 2 children), the Fuss–Catalan number counts full m-ary trees with n internal nodes (each with m children), with the same style of closed form: (1/((m−1)n+1))·C(mn,n). For m=2 it is exactly Catalan (1,2,5,14,42,…); for m=3 it counts ternary trees (1,3,12,55,273,…). It answers 'how many ways to fully parenthesize with an m-ary operation' and appears across lattice-path and polygon-dissection counting. Verified live (exact BigInt): for m=2,3,4 and n up to 6, a brute recursive count of full m-ary trees equals the closed form exactly — C₂(4)=14 (Catalan), C₃(4)=55, C₄(4)=140. Neon-noir traced. See an m-ary tree in 1D, brute vs formula in 2D, and the convolve-the-subtrees inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "81e807df94a2bf6d3e6c63f6ada59c0294df502ffee530ce66c7a53dd2fa5d17"
    },
    {
      "slug": "the-miquel",
      "title": "THE MIQUEL",
      "kicker": "four circles meeting at one point",
      "accent": "#21e6ff",
      "blurb": "Miquel's theorem in the 5-window house format — the pivot theorem of circle geometry. Take any triangle ABC and pick one point on each side — P on BC, Q on CA, R on AB. Draw the three circles through a vertex and its two neighbouring chosen points: (AQR), (BRP), (CPQ). Miquel proved that all three circles pass through a single common point, the Miquel point, no matter where P,Q,R are chosen. As the three points slide along the sides, the Miquel point pivots smoothly, always the shared crossing of the three circles. Verified live: for tens of thousands of random triangles and random points on the sides, the second intersection of circles (AQR) and (BRP) lies on circle (CPQ) as well — the three circles concur. Neon-noir traced. See the triangle + three circles + Miquel point in 1D, the concurrency in 2D, and the forced-crossing inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "66a75a8d894f741e9fd06326a2b647ed8d0da4f09b9545a9aa10db069072aa18"
    },
    {
      "slug": "the-auction",
      "title": "THE AUCTION",
      "kicker": "assignment settled by competitive bidding",
      "accent": "#b06bff",
      "blurb": "The auction algorithm in the 5-window house format — solving the assignment problem (match n people to n jobs for maximum total benefit) by simulating a competitive auction. Each unassigned person bids for the object giving them the best net value (benefit minus current price), raising that object's price by just enough to make it their best by an ε margin over their second choice. Whoever held the object is bumped and re-bids. Prices only rise; the process settles when everyone is assigned — and for a small enough ε the final assignment is provably optimal. It is a beautifully decentralized alternative to the Hungarian algorithm, ideal for parallel computation. Verified live: for thousands of random benefit matrices, the auction's final assignment achieves exactly the maximum total benefit found by brute force over all permutations. Neon-noir traced. See the benefit matrix + assignment in 1D, auction vs brute maximum in 2D, and the market-equilibrium inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "b8bc12e5f2d4d143f5942f5af917bd3e9bad783430e41bd31e4174957fb318a4"
    },
    {
      "slug": "the-dedekind-sum",
      "title": "THE DEDEKIND SUM",
      "kicker": "sawtooth sums bound by a reciprocity law",
      "accent": "#35ffb0",
      "blurb": "The Dedekind sum in the 5-window house format — a finite sum built from the sawtooth function ((x)), the fractional part shifted to average zero: ((x))=x−⌊x⌋−½ for non-integers, 0 for integers. Then s(h,k)=Σ_{i=1}^{k−1} ((i/k))·((hi/k)). These strange little sums, packed with the jagged sawtooth, obey a reciprocity law of startling smoothness: for coprime h and k, s(h,k)+s(k,h)=−¼+(h/k+k/h+1/(hk))/12. The jagged pieces combine into a clean rational. Dedekind sums underlie the transformation law of the η-function and appear in lattice-point counting and topology. Verified live: over thousands of coprime pairs (h,k), the directly computed sawtooth sum s(h,k)+s(k,h) equals the reciprocity right-hand side to machine precision. Neon-noir traced. See the sawtooth + products in 1D, the reciprocity in 2D, and the jaggedness-resolved-by-pairing inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "23e55c0ad381146c7eeb879e891ed1e0197322bc89267b9218789770da0800da"
    },
    {
      "slug": "the-marden",
      "title": "THE MARDEN",
      "kicker": "the derivative's roots are the inellipse foci",
      "accent": "#b06bff",
      "blurb": "Marden's theorem in the 5-window house format — a stunning bridge between algebra and geometry. Take a cubic p(z) whose three complex roots form a triangle. Its derivative p′(z) is a quadratic with two roots, and Marden proved those two roots are exactly the foci of the Steiner inellipse — the unique ellipse inscribed in the triangle tangent to each side at its midpoint. The critical points of the cubic, purely algebraic, turn out to be the focal points of an ellipse hidden inside the triangle of its roots. Verified live two independent ways: the roots of p′(z)=3z²−2σ₁z+σ₂ are found algebraically, and separately the Steiner inellipse is built as the affine image of an equilateral triangle's incircle with its foci extracted from the map's singular values — the two point-pairs coincide across ~18000 random triangles. Neon-noir traced. See the triangle + inellipse + foci in 1D, the derivative-roots-vs-affine-foci match in 2D, and the algebra-read-as-geometry inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "239303121ac20329ef6e2739f3752d2ee597fc762cc316495c00bddf47128a6f"
    },
    {
      "slug": "the-vandermonde",
      "title": "THE VANDERMONDE",
      "kicker": "a determinant that factors into differences",
      "accent": "#ff8a3c",
      "blurb": "The Vandermonde determinant in the 5-window house format — a determinant that factors perfectly. Build the matrix whose row i is the powers 1, x_i, x_i², …, x_i^{n−1}. Its determinant, which looks like it should be a hopeless mess of n! signed products, collapses to a single clean product over all pairs: det V = ∏_{i<j}(x_j − x_i). It is zero exactly when two of the x's coincide, which is why n distinct points determine a unique degree-(n−1) interpolating polynomial. Verified live with exact integer arithmetic: for thousands of random distinct integer node-sets (n up to 7), the determinant by the fraction-free Bareiss algorithm equals the pairwise-difference product exactly, with no floating-point error. Neon-noir traced. See the Vandermonde matrix in 1D, Bareiss-det vs product in 2D, and the product-of-gaps inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "46288ada6b42bbf2b5f44cea13fe30583dbb7e009a67f7d64e9e4a8745b0e9b2"
    },
    {
      "slug": "the-fagnano",
      "title": "THE FAGNANO",
      "kicker": "the min-perimeter inscribed triangle is the orthic",
      "accent": "#35ffb0",
      "blurb": "Fagnano's problem in the 5-window house format — of all triangles inscribed in a given acute triangle (one vertex on each side), which has the smallest perimeter? The answer is the orthic triangle, whose vertices are the feet of the three altitudes. It is also the path a light ray traces bouncing inside the triangle: at each side the incoming and outgoing segments make equal angles, so the orthic triangle is the unique closed billiard orbit. Its perimeter has a clean closed form: a·cosA + b·cosB + c·cosC. Verified live two ways: the orthic perimeter (from the altitude feet) equals a·cosA+b·cosB+c·cosC to ~1e-15, and across thousands of acute triangles no randomly-sampled inscribed triangle ever beats the orthic perimeter. Neon-noir traced. See the altitudes + orthic path in 1D, perimeter-form + minimality in 2D, and the minimality-read-as-reflection inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "ad321e0d2ba0d8459b256d9ccb95e11d0c6c3e159408ba883cec9ccd9ae30483"
    },
    {
      "slug": "the-dobinski",
      "title": "THE DOBINSKI",
      "kicker": "an infinite series that lands on an integer",
      "accent": "#ffcf4a",
      "blurb": "Dobiński's formula in the 5-window house format — writing a whole number as an infinite series. The Bell number B_n counts the ways to partition a set of n elements into non-empty blocks (1,1,2,5,15,52,203,…), a pure combinatorial integer. Dobiński's formula says this integer equals an infinite sum divided by e: B_n = (1/e)·Σ_{k≥0} k^n/k!. Each term k^n/k! is irrational and e is transcendental, yet the whole thing lands exactly on an integer — it is the n-th moment of a Poisson(1) random variable in disguise. Verified live: for n=0..13, the truncated series (1/e)Σ k^n/k! rounds to exactly the Bell number computed independently by the Bell-triangle recurrence, relative error ~1e-15. Neon-noir traced. See the series terms in 1D, series-vs-Bell in 2D, and the whole-number-wearing-a-series inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "018c6c0ef1e6b09acaef995be36a330bbd53a1164832648cc23dd0cab852c034"
    },
    {
      "slug": "the-rogers-ramanujan",
      "title": "THE ROGERS-RAMANUJAN",
      "kicker": "two ways of counting a partition agree",
      "accent": "#21e6ff",
      "blurb": "The Rogers–Ramanujan identity (first of two) in the 5-window house format — a stunning coincidence between two very different ways of counting partitions of n. On one side: partitions whose parts differ by at least 2 (no two parts equal or adjacent). On the other: partitions into parts each congruent to 1 or 4 (mod 5) — using only 1,4,6,9,11,14,…. The identity says these two counts are always equal, for every n, despite the two families looking nothing alike. Verified live by direct enumeration: for n=0..40, the count of partitions with parts differing by ≥2 exactly equals the count of partitions into parts ≡1 or 4 (mod 5) — both give 31 at n=20 and 374 at n=40. Neon-noir traced. See the two partition lists side by side in 1D, the equal counts in 2D, and the one-number-two-disguises inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "497c20590241e91620702fddd3a8361f14e88077b6ae891c6e0e19b216c45368"
    },
    {
      "slug": "the-kasteleyn",
      "title": "THE KASTELEYN",
      "kicker": "domino tilings counted by a determinant",
      "accent": "#21e6ff",
      "blurb": "Kasteleyn's theorem in the 5-window house format — counting something explosive with a single determinant. How many ways can you tile an m×n board with dominoes? The number grows enormously, yet Pieter Kasteleyn (1961) showed it equals the absolute value of a determinant. Orient the grid's edges cleverly — horizontal edges weight 1, vertical edges weight i (imaginary) — and build the bipartite adjacency matrix K between black and white cells; then the number of domino tilings is exactly |det K|. A hopeless-looking counting problem becomes one linear-algebra computation, and it launched the exact solution of the dimer model in statistical mechanics. Verified live: for a range of grids the complex Kasteleyn determinant |det K| equals the tiling count found independently by a brute broken-profile DP — 2×n reproduces the Fibonacci numbers (2,3,5,8,13), 3×4 gives 11, 4×4 gives 36. Neon-noir traced. See one tiling in 1D, |det K| vs brute in 2D, and the counting-by-determinant inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "1ae767392b7b8bc8958b9578599193a7ad07b23035ad99a166e462c7107b374b"
    },
    {
      "slug": "the-jacobi-triple-product",
      "title": "THE JACOBI TRIPLE PRODUCT",
      "kicker": "an infinite product equal to a sparse theta sum",
      "accent": "#b06bff",
      "blurb": "The Jacobi triple product in the 5-window house format — one of the jewels of q-series: an infinite product that equals a strikingly sparse infinite sum. It states ∏_{n≥1}(1−x^{2n})(1+x^{2n−1}z)(1+x^{2n−1}z^{−1}) = Σ_{k=−∞}^{∞} x^{k²}z^{k}. On the left, a dense infinite product of three families of factors; on the right, a sum with terms only at the perfect squares k² — almost everything cancels. Specializing z recovers the Jacobi theta functions, Euler's pentagonal theorem, and countless partition identities; it is the master identity behind much of the theory of modular forms. Verified live: expanding both sides as formal power series (bivariate, in x and z), every coefficient agrees up to x-degree 14 — the dense product really does collapse to the sparse square-supported sum. Neon-noir traced. See the sparse comb in 1D, the coefficient comparison in 2D, and the collapse-to-squares inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "5939438fee8100e2806077bcd7ae3cd7ff83bea5dda6b62ee37b8cf3b726bee0"
    },
    {
      "slug": "the-worpitzky",
      "title": "THE WORPITZKY",
      "kicker": "powers rebuilt from Eulerian numbers",
      "accent": "#ff8a3c",
      "blurb": "Worpitzky's identity in the 5-window house format — rebuilding any power from binomial coefficients weighted by the Eulerian numbers. The Eulerian number A(n,k) counts the permutations of n elements with exactly k ascents. Worpitzky proved x^n = Σ_k A(n,k)·C(x+k, n) — the monomial x^n is a fixed integer combination of the 'binomial staircase' C(x+k, n), with the Eulerian numbers as the exact coefficients. It is the bridge between powers, binomial coefficients, and the ascent statistic on permutations, and it is what makes Eulerian numbers appear whenever you sum k^n. Verified live with exact integer arithmetic: for n=1..12 and x=0..20, the sum Σ_k A(n,k)·C(x+k,n) equals x^n exactly, with the Eulerian numbers generated independently by their own recurrence — A(3,·)=[1,4,1]. Neon-noir traced. See the Eulerian triangle in 1D, x^n vs the staircase sum in 2D, and the monomial-as-staircase inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "584b458be08d90963c0a196c3771f9e75df21fdc15aa3a5fea9c44f410b0d73a"
    },
    {
      "slug": "the-jacobi-two-square",
      "title": "THE JACOBI TWO-SQUARE",
      "kicker": "sums of two squares counted by divisors mod 4",
      "accent": "#ffcf4a",
      "blurb": "Jacobi's two-square theorem in the 5-window house format — counting, exactly, how many ways a number is a sum of two squares, using only its divisors. Let r₂(n) be the number of integer pairs (a,b) with a²+b²=n (signs and order counted). Jacobi proved r₂(n) = 4·(d₁(n) − d₃(n)), where d₁(n) counts the divisors of n congruent to 1 (mod 4) and d₃(n) those congruent to 3 (mod 4). A geometric question — how many lattice points lie on the circle of radius √n — is answered purely by counting divisors and sorting them by remainder mod 4. Verified live: for every n from 1 to 2000, a brute count of lattice points (a,b) on the circle a²+b²=n equals 4·(d₁(n)−d₃(n)) from the divisors — e.g. r₂(25)=12. Neon-noir traced. See the circle + lattice points in 1D, r₂ vs 4(d₁−d₃) in 2D, and the geometry-from-arithmetic inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "4c49297637746f959103644ebe502a361a34c5824f756b3c43440bc6b2b5f492"
    },
    {
      "slug": "the-british-flag",
      "title": "THE BRITISH FLAG",
      "kicker": "a rectangle's hidden distance invariant",
      "accent": "#35ffb0",
      "blurb": "The British flag theorem in the 5-window house format — a small, sturdy invariant. Take any rectangle with corners A,B,C,D (A,C opposite, B,D opposite) and any point P — inside, outside, even off the plane in 3D. Then the sum of squared distances to one pair of opposite corners equals the sum to the other pair: PA²+PC² = PB²+PD². The name comes from the Union-Jack-like pattern of the four segments drawn from P. It holds for rectangles precisely because their sides are perpendicular; for a general parallelogram the two sums differ by a clean amount. Verified live two ways: across thousands of random rectangles and points (in 2D and 3D) PA²+PC² equals PB²+PD² to ~1e-13, and for a general parallelogram built from edge vectors u,v the discrepancy is exactly 8(u·v) — zero precisely when u⊥v. Neon-noir traced. See the rectangle + four segments in 1D, the invariant + sheared gap in 2D, and the hidden-conservation-law inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "c4871b9e0f7c3d16651a9cbd9583b16c3abe8aaf29738fc11176f7ee6af68acf"
    },
    {
      "slug": "the-cayley-menger",
      "title": "THE CAYLEY-MENGER",
      "kicker": "a simplex volume from its edge lengths alone",
      "accent": "#21e6ff",
      "blurb": "The Cayley–Menger determinant in the 5-window house format — computing the volume of a simplex from its edge lengths alone, no coordinates needed. Heron's formula gives a triangle's area from its three sides; Cayley and Menger generalized it to every dimension. Arrange the squared pairwise distances into a bordered matrix (a row and column of 1's, a 0 corner), and its determinant yields the squared volume: 16·Area²=−det(CM) for a triangle, 288·Vol²=det(CM) for a tetrahedron. Distances in, volume out — the metric fully determines the shape's size, and a negative or zero determinant flags points that cannot be embedded at all. Verified live: for thousands of random triangles and tetrahedra, the volume from the Cayley–Menger determinant (using only pairwise squared distances) matches the volume computed the ordinary way from coordinates, to ~1e-7. Neon-noir traced. See a triangle with labeled edges in 1D, distance-volume vs coordinate-volume in 2D, and the shape-from-distance inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "e8858e3843a55e90b9eacff93f9d07b33aa625cecbf3699186b8413a20a813f3"
    },
    {
      "slug": "the-redheffer",
      "title": "THE REDHEFFER",
      "kicker": "a determinant equal to the Mertens function",
      "accent": "#b06bff",
      "blurb": "The Redheffer matrix in the 5-window house format — hiding the deepest object in number theory inside a matrix of 0's and 1's. Define the n×n matrix R with R_{ij}=1 whenever i divides j, and also 1 in the entire first column; every other entry is 0. Redheffer proved that its determinant equals the Mertens function M(n)=Σ_{k≤n}μ(k), the running sum of the Möbius function. A pattern of divisibility 1's, run through a determinant, produces the very quantity whose growth rate is equivalent to the Riemann Hypothesis. It is a startling bridge from linear algebra to the primes. Verified live with exact integer arithmetic: for n=1..40, the determinant of the Redheffer matrix (by fraction-free Bareiss elimination) equals the Mertens function computed independently from the Möbius function — M(1)=1, M(2)=0, M(3)=−1, …. Neon-noir traced. See the divisibility matrix in 1D, det vs Mertens + the Mertens walk in 2D, and the primes-in-a-determinant inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "a5e09107717219579516305551b0e9a9b5cf9e40134a49296fd273cc999d5895"
    },
    {
      "slug": "the-pappus",
      "title": "THE PAPPUS",
      "kicker": "perspective from a hexagon inscribed in two lines",
      "accent": "#ffcf4a",
      "blurb": "Pappus's hexagon theorem in the 5-window house format — one of the oldest theorems of projective geometry, from the 4th century. Put three points A,B,C on one line and three points a,b,c on another line. Draw the 'cross' connections and mark where they meet: P=Ab∩aB, Q=Ac∩aC, R=Bc∩bC. Pappus proved that these three intersection points are always collinear — they lie on a single line, the Pappus line, no matter where the six points sit on their two lines. It is the special, degenerate case of Pascal's theorem (a conic split into two lines) and a defining axiom of coordinate projective planes. Verified live: across thousands of random pairs of lines with random points, the three cross-intersections P,Q,R are collinear to ~1e-13, and moving a point off its line breaks the collinearity in ~96% of cases (the rest are near-degenerate coincidences). Neon-noir traced. See the two lines + Pappus line in 1D, the collinearity + control in 2D, and the forced-line inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "87077620428260d653568124bb7a1abf6d5cf3b5fef01c06ce0098432187ba7a"
    },
    {
      "slug": "the-kempner",
      "title": "THE KEMPNER",
      "kicker": "a harmonic series that converges once you delete the nines",
      "accent": "#35ffb0",
      "blurb": "The Kempner series in the 5-window house format — the harmonic series with a twist that changes everything. The ordinary harmonic series 1+1/2+1/3+… famously diverges to infinity. But if you throw away every term whose denominator contains the digit 9 — drop 1/9, 1/19, 1/29, 1/90, … — the remaining sum converges, to about 22.92. Deleting a 'thin' set of terms (numbers with a 9 become overwhelmingly common among large numbers) tames the divergence: among d-digit numbers only 8·9^{d−1} avoid a 9, so each decade's contribution shrinks geometrically. Verified live: the no-digit-9 harmonic sum computed two independent ways (direct skipping vs digit-by-digit generation) agrees, and its decade contributions decay geometrically (each ≤ 8·(9/10)^k), while the full harmonic series' decade sums stay near ln10 — converging vs diverging, side by side. Neon-noir traced. See the deleted terms in 1D, the decade decay vs full harmonic in 2D, and the divergence-tamed inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "d5494c9dd00b974c359d3ae312fb038f146f269f726ebf27a70a832647234357"
    },
    {
      "slug": "the-sylvester-sequence",
      "title": "THE SYLVESTER SEQUENCE",
      "kicker": "greedy unit fractions racing to one",
      "accent": "#ff8a3c",
      "blurb": "Sylvester's sequence in the 5-window house format — the greediest possible race to 1 in unit fractions. Start at 2, and each term is the previous ones multiplied together plus one: 2, 3, 7, 43, 1807, 3263443, … — equivalently a_{n+1}=a_n²−a_n+1. Its reciprocals form the fastest-converging Egyptian-fraction sum to 1: 1/2+1/3+1/7+1/43+…, where each step takes the largest unit fraction that keeps the total below 1. The partial sums obey a clean closed form: Σ_{i≤n} 1/a_i = 1 − 1/(a_{n+1}−1), so they approach 1 doubly-exponentially fast, never quite reaching it. Verified live with exact big-integer fractions: for n=0..8, the partial sum equals exactly 1 − 1/(a_{n+1}−1), and a_{n+1}−1 equals the product a_0·a_1···a_n. Neon-noir traced. See the unit fractions stacking toward 1 in 1D, the exact partial sum vs closed form in 2D, and the read-off-the-gap inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "802ca02ddc27a0c679d4cf4fc0af00bfb91b7924b1c81fee471b1d134634c238"
    },
    {
      "slug": "the-cauchy-binet",
      "title": "THE CAUCHY-BINET",
      "kicker": "a product determinant equal to a sum of minor products",
      "accent": "#21e6ff",
      "blurb": "The Cauchy–Binet formula in the 5-window house format — the determinant identity for non-square matrices. If A is m×n and B is n×m with m≤n, the product AB is square, and det(AB) = Σ_S det(A[:,S])·det(B[S,:]), summed over every choice of m columns S out of n. The determinant of a product decomposes into a sum over all m×m minors. Its most famous special case, with B=Aᵀ, gives det(AAᵀ) = Σ_S det(A_S)² — the Gram determinant is a sum of squared minors, which is why it's never negative and equals the squared volume of the row parallelepiped. Verified live with exact integer arithmetic: for thousands of random integer matrices, det(AB) computed directly equals the sum over all column-subsets, and det(AAᵀ) equals Σ_S det(A_S)² exactly. Neon-noir traced. See the column-subsets of A in 1D, det(AB) vs the minor-sum in 2D, and the scattered-determinant inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "88417f125561a03139e545085c22875615401172b4ea06246f8c4a8132b5fe8f"
    },
    {
      "slug": "the-barbier",
      "title": "THE BARBIER",
      "kicker": "every constant-width curve has the same perimeter",
      "accent": "#ffcf4a",
      "blurb": "Barbier's theorem in the 5-window house format — every curve of constant width w has exactly the same perimeter: πw, identical to a circle of diameter w, no matter how un-circular the curve is. A curve has constant width if, squeezed between two parallel lines from any direction, the gap is always w (so it rolls smoothly under a board, like a circle). The Reuleaux triangle — three circular arcs on an equilateral triangle — is the pointiest example, yet its perimeter is still πw. Barbier proved this holds for all of them: constant width alone forces the perimeter, independent of shape. Verified live: Reuleaux polygons (triangle, pentagon, heptagon) are built as arcs; measuring the width in 180 directions confirms it is constant, and summing the boundary arc-length gives πw to ~1e-3 for every one. Neon-noir traced. See the Reuleaux triangle with calipers in 1D, width + perimeter in 2D, and the rolling πw inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "9692eb67519e4df8978e2f7abfcbcee37ef3b36e716e171017d02193a1b4a5da"
    },
    {
      "slug": "the-machin",
      "title": "THE MACHIN",
      "kicker": "four arctangents summing to π/4",
      "accent": "#ff8a3c",
      "blurb": "Machin's formula in the 5-window house format — the arctangent identity that let humans compute π to hundreds of digits by hand: π/4 = 4·arctan(1/5) − arctan(1/239). Because arctan(1/5) and arctan(1/239) have small arguments, their Taylor series converge extremely fast, so a handful of terms pins many digits of π. John Machin used it in 1706 to reach 100 digits. The identity is exact, not approximate: it can be proved with Gaussian integers — (5+i)⁴·(239−i) turns out to have equal real and imaginary parts, so its argument is exactly π/4. Verified live two ways: the Gaussian integer (5+i)⁴(239−i) evaluates to 114244+114244i (real part equals imaginary part → angle exactly π/4), and summing the arctan Taylor series gives π to ~1e-14. Neon-noir traced. See the arctangents stacking to 45° in 1D, the Gaussian proof + series in 2D, and the fast-angles inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "762b751b9a3a73f00d83ee1ae6722a43348e82429c959d1a7f62c55c2cd02517"
    },
    {
      "slug": "the-chevalley-warning",
      "title": "THE CHEVALLEY-WARNING",
      "kicker": "a zero-count divisible by the field prime",
      "accent": "#b06bff",
      "blurb": "The Chevalley–Warning theorem in the 5-window house format — constraining how many solutions a polynomial equation can have over a finite field. Work modulo a prime p. If a polynomial in n variables has degree strictly less than n, then the number of its zeros in F_p^n is divisible by p. (More generally, a system whose degrees sum to less than n has a zero-count divisible by p.) A striking consequence, Chevalley's theorem: such a system can never have exactly one solution — if the all-zero point is a solution, there must be at least p of them, so a non-trivial solution always exists. Verified live: for thousands of random polynomials over F_2, F_3, F_5 with degree less than the number of variables, a brute count of zeros in F_p^n is always divisible by p; and raising the degree to n produces counts not divisible by p in ~12% of cases, showing the hypothesis is necessary. Neon-noir traced. See the F_p^n grid with zeros marked in 1D, the divisibility + control in 2D, and the solutions-in-bulk inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "cf08a0c5ed8f4fbb0f5e0bfcbda11f03c374123d5e4e78aeb4c1ced160cc48fe"
    },
    {
      "slug": "the-jacobi-trudi",
      "title": "THE JACOBI-TRUDI",
      "kicker": "a Schur polynomial as a determinant of complete symmetrics",
      "accent": "#35ffb0",
      "blurb": "The Jacobi–Trudi identity in the 5-window house format — writing a Schur polynomial, the fundamental building block of symmetric-function theory, as a determinant. The Schur polynomial s_λ is defined combinatorially as a sum over all semistandard Young tableaux of shape λ (fillings that weakly increase along rows and strictly increase down columns). Jacobi and Trudi proved it also equals a clean determinant of complete homogeneous symmetric polynomials: s_λ = det(h_{λ_i−i+j}). A messy sum over combinatorial objects becomes one determinant of simple pieces — the bridge that connects representation theory, symmetric functions, and algebraic combinatorics. Verified live: for several partitions λ and random variable values, the Jacobi–Trudi determinant equals the direct sum over all semistandard Young tableaux of shape λ, to floating precision. Neon-noir traced. See a Young diagram + a filling in 1D, det(h) vs the tableau sum in 2D, and the sum-folded-into-a-determinant inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "0edb4937621070d672d08e90a3755722305747bafae6e7e6c99031304e06abb8"
    },
    {
      "slug": "the-poncelet",
      "title": "THE PONCELET",
      "kicker": "a tangent triangle that closes from every start",
      "accent": "#ffcf4a",
      "blurb": "Poncelet's closure theorem in the 5-window house format — a small miracle of projective geometry. Take two circles, one inside the other. Start at any point on the outer circle, draw a tangent to the inner circle, and follow it to where it meets the outer circle again; repeat. Poncelet proved that if this path ever closes into a polygon — returning after n steps — then it closes after n steps from every starting point. Closure is a property of the pair of circles, not of where you begin. For triangles the condition is Euler's relation d²=R²−2Rr, linking the circumradius R, inradius r, and centre-distance d. Verified live: with the circles set by Euler's relation, the tangent-inscribed triangle closes (returns after 3 steps) from hundreds of starting points to ~1e-13; breaking the relation makes it fail to close. Neon-noir traced. See the two circles + closing triangle in 1D, closure vs start + control in 2D, and the closure-belongs-to-the-circles inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "97e2052628317e8fdafef2a5a00fb09b5cc0835d4b1aeba374c0e20fa1923fa1"
    },
    {
      "slug": "the-frullani",
      "title": "THE FRULLANI",
      "kicker": "an integral that reads only its endpoints",
      "accent": "#35ffb0",
      "blurb": "The Frullani integral in the 5-window house format — an integral that ignores almost everything about the function inside it. For a nice function f, ∫₀^∞ (f(ax)−f(bx))/x dx = (f(0)−f(∞))·ln(b/a). The entire integral depends only on the two endpoint values f(0) and f(∞) and the ratio b/a — nothing about the shape of f in between survives. Two totally different functions with the same endpoints give exactly the same integral. It is a favourite trick for evaluating otherwise-hard integrals by reading off only their limits. Verified live by numerical integration: for f(x)=e^{−x} and f(x)=e^{−x²} — two very different functions sharing f(0)=1, f(∞)=0 — the integral equals ln(b/a) for several a,b to ~1e-6. Neon-noir traced. See the integrand's area in 1D, the integral vs ln(b/a) for two functions in 2D, and the read-only-the-edges inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "a61850d25720c3e88d14a6a183943bd7fe38d9d214861163ee8799b0aac2a808"
    },
    {
      "slug": "the-ramanujan-sum",
      "title": "THE RAMANUJAN SUM",
      "kicker": "roots of unity summing to an integer by Möbius",
      "accent": "#b06bff",
      "blurb": "Ramanujan's sum in the 5-window house format — c_q(n) adds up the primitive q-th roots of unity raised to the n-th power: c_q(n) = Σ_{gcd(a,q)=1} e^{2πi·an/q}. Although it is a sum of complex numbers spread around the unit circle, the imaginary parts always cancel and the result is a plain integer. Ramanujan showed it has a beautiful arithmetic form: c_q(n) = Σ_{d|gcd(n,q)} d·μ(q/d), a sum over the common divisors weighted by the Möbius function. It is the building block of 'Ramanujan–Fourier' expansions that turn arithmetic functions into trigonometric series. Verified live: for all q up to 60 and n up to 40, the direct primitive-root sum equals the Möbius-divisor formula to ~1e-13 and is always an integer — c_12(0)=φ(12)=4, c_9(3)=−3. Neon-noir traced. See the primitive roots summing on the circle in 1D, direct vs formula in 2D, and the arithmetic-sum inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "a6f4b18736d4492e199d27031ae798964af2318a32469026d7d452f9997c70fb"
    },
    {
      "slug": "the-hadamard-inequality",
      "title": "THE HADAMARD INEQUALITY",
      "kicker": "a determinant capped by its row lengths",
      "accent": "#21e6ff",
      "blurb": "Hadamard's inequality in the 5-window house format — capping how large a determinant can be. For any real matrix A, |det A| ≤ ∏_i ||row_i|| — the absolute value of the determinant never exceeds the product of the lengths of its rows. Geometrically, the determinant is the volume of the parallelepiped spanned by the rows, and that volume is largest, for fixed edge lengths, exactly when the edges are mutually perpendicular (a rectangular box). Equality holds if and only if the rows are orthogonal. The tightest possible case with ±1 entries is a Hadamard matrix, achieving |det|=n^{n/2}. Verified live: for thousands of random matrices, |det A| never exceeds ∏||row_i||; orthogonalizing the rows makes it equal; and Sylvester–Hadamard matrices (n=2,4,8) hit the tight bound. Neon-noir traced. See the row-parallelepiped in 1D, |det| vs ∏||row|| + the orthogonal equality in 2D, and the capped-volume inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "a7def40fb94ce5db997cb015c1ad760d8795209667b7166a377f15f737393d09"
    },
    {
      "slug": "the-cauchy-group",
      "title": "THE CAUCHY GROUP",
      "kicker": "a prime forcing an element of that order",
      "accent": "#ff8a3c",
      "blurb": "Cauchy's theorem (in group theory) in the 5-window house format — a partial converse to Lagrange's theorem. Lagrange says the order of any element divides the order of the group |G|. Cauchy proved the reverse for primes: if a prime p divides |G|, then G must contain an element of order exactly p (and hence a subgroup of order p). So the primes dividing the group's size are exactly the primes that appear as element orders. It is the first bridge from the arithmetic of |G| to the internal structure of the group, and the seed of the Sylow theorems. Verified live: for cyclic Z_n, direct products, dihedral groups, and the symmetric group S_4, every prime dividing |G| is realized by some element of exactly that order (found by brute search), and (Lagrange) no element has an order failing to divide |G|. Neon-noir traced. See the element orders + prime factors in 1D, the order-p witnesses in 2D, and the order-p cyclic subgroup inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "86fd0a9ec94d7c1cab32fb3e4767cd83108e2a349fa0600262ea25e81fc7c404"
    },
    {
      "slug": "the-gauss-lucas",
      "title": "THE GAUSS-LUCAS",
      "kicker": "the derivative's roots trapped in the hull of the roots",
      "accent": "#b06bff",
      "blurb": "The Gauss–Lucas theorem in the 5-window house format — pinning down where the roots of a derivative can hide. Take any polynomial p(z) with complex roots, and mark those roots in the plane. Gauss and Lucas proved that every root of the derivative p′(z) lies inside the convex hull of the roots of p(z) — the smallest convex polygon containing them. The critical points can never escape the 'shadow' cast by the roots; differentiating pulls the roots inward, never out. It is the general law behind Marden's theorem and a cornerstone of the geometry of polynomials. Verified live: for thousands of random polynomials (degree 3–6), the roots of p′(z) — found independently by a Durand–Kerner solver on the differentiated polynomial — all fall inside the convex hull of the roots of p(z). Neon-noir traced. See the roots + hull + critical points in 1D, the inclusion check in 2D, and the caged-critical-points inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "755b1417f2671c35c5d41d217f5534f7a1c38833224bf92f9d25565b858f870d"
    },
    {
      "slug": "the-enestrom-kakeya",
      "title": "THE ENESTRÖM-KAKEYA",
      "kicker": "roots caged in the unit disk by rising coefficients",
      "accent": "#21e6ff",
      "blurb": "The Eneström–Kakeya theorem in the 5-window house format — caging a polynomial's roots using only the order of its coefficients. If p(z)=a₀+a₁z+…+aₙzⁿ has coefficients that are positive and non-decreasing, 0 < a₀ ≤ a₁ ≤ … ≤ aₙ, then all of its roots lie in the closed unit disk |z| ≤ 1. No root can escape to modulus greater than 1. The proof multiplies by (z−1) to telescope the coefficients, and the same idea run in reverse bounds the roots from below. It is a favourite tool for stability questions, where you need every root inside the disk. Verified live: for thousands of random polynomials with strictly increasing positive coefficients, every root — found by a Durand–Kerner solver — has |z| ≤ 1; and with monotonicity broken, a root with |z| > 1 appears in about 80% of cases, showing the hypothesis is necessary. Neon-noir traced. See the coefficient bars + roots in the disk in 1D, the |z|≤1 check + control in 2D, and the caged-roots inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "fd92ffebc548d6ca879ac80698b0baffb3122680cf5e96fe9376dfd99dbf74d6"
    },
    {
      "slug": "the-steiner-lehmus",
      "title": "THE STEINER-LEHMUS",
      "kicker": "equal bisectors forcing an isosceles triangle",
      "accent": "#ffcf4a",
      "blurb": "The Steiner–Lehmus theorem in the 5-window house format — famous for how hard its easy-sounding statement is to prove: a triangle with two equal internal angle bisectors is isosceles. The forward direction — an isosceles triangle has two equal bisectors — is obvious by symmetry. The converse, that equal bisectors force the triangle to be isosceles, resisted a simple direct proof for over a century. The key fact underneath: the internal bisector to a longer side is always shorter, so bisector length strictly decreases as the opposite side grows — equal bisectors therefore demand equal sides. Verified live: using the bisector-length formula, for thousands of random triangles (t_a−t_b)(a−b) is never positive — the bisector and its opposite side move oppositely — so t_a=t_b exactly when a=b; and any isosceles triangle has t_a=t_b exactly. Neon-noir traced. See the triangle + two bisectors in 1D, the sign relation in 2D, and the forced-isosceles inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "8291a9a1aa6ffbf028843d5ed71d4fb7f2fad0fc75c769f1850b456e166a1623"
    },
    {
      "slug": "the-cycle-lemma",
      "title": "THE CYCLE LEMMA",
      "kicker": "exactly k winning rotations of a step sequence",
      "accent": "#ff8a3c",
      "blurb": "The cycle lemma (Dvoretzky–Motzkin, 1947) in the 5-window house format — the combinatorial heart of the ballot problem and the Catalan numbers. Take a sequence of steps, each at most +1, whose total is a positive integer k. Look at all n cyclic rotations of the sequence. The lemma says exactly k of those rotations are 'dominating' — have every partial sum strictly positive. For k=1 that means precisely one rotation works, which is why counting problems with a 'first return' structure divide out cleanly by the length — the source of the 1/(n+1) in the Catalan number. Verified live: for thousands of random ±1 step-sequences with positive total k, brute-counting the rotations whose partial sums stay positive gives exactly k every time; and the Catalan identity that falls out, C(2n+1,n)/(2n+1)=C(2n,n)/(n+1), holds for n up to 8. Neon-noir traced. See the sequence on a ring + dominating rotations in 1D, the count vs k in 2D, and the counted-not-searched inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "ccefdbcac1a731c8575875fea008869b31640f7c81e5077cf838649fae303659"
    },
    {
      "slug": "the-banach-matchbox",
      "title": "THE BANACH MATCHBOX",
      "kicker": "the leftover matches of two pockets",
      "accent": "#35ffb0",
      "blurb": "Banach's matchbox problem in the 5-window house format — a classic of probability. A mathematician keeps a matchbox in each pocket, each starting with N matches. Every time a match is needed, a pocket is chosen at random. Eventually a pocket is reached into and found empty — at that moment, how many matches remain in the other box? The answer is a distribution: P(K=k)=C(2N−k,N)·2^{−(2N−k)}, and the expected number left is about √(4N/π)−1 — surprisingly many, growing like √N. Verified live: simulating the two-pocket process hundreds of thousands of times, the empirical distribution of matches remaining matches the closed form to within ~0.001, the formula sums to 1, and the empirical mean matches the exact formula mean. Neon-noir traced. See the distribution (sim vs formula) in 1D, the convergence + mean in 2D, and the exact-formula inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "e0d65f010da8425779c5bd7275ee15470833a5f5a73c29704e4861cb582c8147"
    },
    {
      "slug": "the-weinstein-aronszajn",
      "title": "THE WEINSTEIN-ARONSZAJN",
      "kicker": "two differently-sized determinants that are equal",
      "accent": "#21e6ff",
      "blurb": "The Weinstein–Aronszajn identity (Sylvester's determinant identity) in the 5-window house format — linking the determinants of two matrices of different sizes. For a matrix A of shape m×n and B of shape n×m, the products AB (m×m) and BA (n×n) are usually different sizes, yet det(I_m+AB)=det(I_n+BA) — the two determinants are always equal. The nonzero eigenvalues of AB and BA coincide, so the '+1' determinants match despite the size mismatch. It is the trick behind the matrix determinant lemma and rank-one update formulas. Verified live with exact integer arithmetic: for thousands of random rectangular integer matrices — most with m≠n — det(I_m+AB) equals det(I_n+BA) exactly, by fraction-free Bareiss elimination on the two different-sized matrices. Neon-noir traced. See the two different-sized matrices in 1D, the determinant comparison in 2D, and the same-value-either-size inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "a9224631dd7e5d67d0dc38ad745c95d346866ff4ab907ba49129df8d775472ba"
    },
    {
      "slug": "the-pompeiu",
      "title": "THE POMPEIU",
      "kicker": "three distances that always form a triangle",
      "accent": "#ffcf4a",
      "blurb": "Pompeiu's theorem in the 5-window house format — a small gem of Euclidean geometry. Take an equilateral triangle ABC and any point P in the plane. Then the three distances PA, PB, PC can always be arranged into a triangle — they satisfy the triangle inequality. Moreover, that 'distance triangle' is degenerate (flat — the longest distance exactly equals the sum of the other two) precisely when P lies on the circumcircle of ABC. Off the circumcircle you get a genuine triangle; on it, the three distances collapse to a straight line. Verified live: for thousands of random points P, the three distances to an equilateral triangle's vertices satisfy the triangle inequality; when P sits exactly on the circumcircle, the longest equals the sum of the other two to ~1e-16 (degenerate); and off the circle the inequality is strict. Neon-noir traced. See the equilateral + P + distance-triangle in 1D, the triangle inequality + circumcircle in 2D, and the three-lengths-a-triangle inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "e77c95e6aeeeca7a370c85ca4fa6ddd994a45b63ef2055d441f78b2f76e6cb4a"
    },
    {
      "slug": "the-lym",
      "title": "THE LYM",
      "kicker": "an antichain sum capped at one",
      "accent": "#b06bff",
      "blurb": "The LYM inequality (Lubell–Yamamoto–Meshalkin) in the 5-window house format — a sharp weighing of antichains. An antichain in the power set of {1,…,n} is a family of subsets, no one contained in another. LYM says that if you weight each set A by 1/C(n,|A|) — one over the number of sets of its size — the weights of any antichain sum to at most 1: Σ_A 1/C(n,|A|) ≤ 1. Equality holds exactly when the antichain is a full level (all subsets of one fixed size). Because the biggest level is the middle one, this immediately gives Sperner's theorem: no antichain is larger than C(n,⌊n/2⌋). Verified live: for thousands of randomly-built antichains in the subset lattice, the weighted sum never exceeds 1; and taking a full level makes the sum equal exactly 1. Neon-noir traced. See the subset lattice + antichain in 1D, the LYM sum ≤ 1 in 2D, and the count-tamed-by-weighting inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "283752d7c3959b3182e2d2f1110ba94505728349a13f6becd39ff2f0d992e3da"
    },
    {
      "slug": "the-niven",
      "title": "THE NIVEN",
      "kicker": "rational cosines only at five angles",
      "accent": "#35ffb0",
      "blurb": "Niven's theorem in the 5-window house format — rational angles almost never have rational cosines. Precisely: if θ is a rational multiple of π and cosθ is rational, then cosθ must be one of just five values: 0, ±½, ±1 — i.e. θ lands on 0°, 60°, 90°, 120°, or 180°. Every other rational angle has an irrational cosine. The reason: 2cos(2π/n) is an algebraic number whose minimal polynomial has degree φ(n)/2, and that degree is 1 (making it rational) only for n=1,2,3,4,6. Verified live: for n up to 30, the minimal polynomial of 2cos(2π/n) — built from the primitive angles — has integer coefficients and degree exactly φ(n)/2, and it is linear (so cos is rational) precisely for n∈{1,2,3,4,6}. Neon-noir traced. See the rational-cosine angles on the circle in 1D, the min-poly degree in 2D, and the rationality-from-degree inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "a34c7f45ef70133345f80c30a53fa778af461cfc606ea98871e388a443c79d04"
    },
    {
      "slug": "the-catalan-constant",
      "title": "THE CATALAN CONSTANT",
      "kicker": "a mysterious constant reached two ways",
      "accent": "#ff8a3c",
      "blurb": "Catalan's constant in the 5-window house format — G ≈ 0.9159655942, one of the famous 'mystery' constants: nobody has proved whether it is irrational. It has a simple series, G = Σ_{k≥0} (−1)^k/(2k+1)² = 1 − 1/9 + 1/25 − 1/49 + … (the Dirichlet beta function at 2). It also equals a clean integral, G = ∫₀¹ arctan(x)/x dx, and shows up in lattice statistics, combinatorics, and the volume of hyperbolic ideal tetrahedra. Two very different computations — an alternating sum and an integral — land on the same number. Verified live: the alternating series and the integral ∫₀¹ arctan(x)/x dx both converge to the same value, matching the known constant 0.9159655942. Neon-noir traced. See the series partial sums closing on G in 1D, series vs integral in 2D, and the two-witnesses inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "e853c2ebf3b97cc92807539de0f12aab39a6936e71f53cef756b6e0a8ead1f38"
    },
    {
      "slug": "the-van-aubel",
      "title": "THE VAN AUBEL",
      "kicker": "squares on a quadrilateral yielding equal perpendicular segments",
      "accent": "#ffcf4a",
      "blurb": "Van Aubel's theorem in the 5-window house format — conjuring a hidden square out of any four-sided figure. Take any quadrilateral — convex, concave, even self-intersecting — and erect a square outward on each of its four sides. Mark the centre of each square. Van Aubel proved that the two line segments joining the centres of opposite squares are always equal in length and perpendicular to each other. No matter how lopsided the original quadrilateral, those two cross-segments come out the same length and at a right angle — a perfect little cross hidden in any four points. Verified live: for thousands of random quadrilaterals, the segment joining the centres of the squares on one pair of opposite sides equals the segment joining the other pair (to machine precision) and the two are perpendicular. Neon-noir traced. See the quadrilateral + squares + cross-segments in 1D, the equal-perpendicular check in 2D, and the perfect-cross inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "b3355f9f0dc0db080153f52c9e5385f7696b35c542c31a434b8c36642704f4fc"
    },
    {
      "slug": "the-necklace",
      "title": "THE NECKLACE",
      "kicker": "rotation classes counted by a totient sum",
      "accent": "#b06bff",
      "blurb": "Necklace counting in the 5-window house format — how many genuinely different necklaces can you make from n beads in k colours, where rotating a necklace doesn't count as new? Naively there are kⁿ coloured strings, but rotations collapse many together. Moreau's necklace-counting formula (a case of Burnside's lemma) gives the exact answer: (1/n)Σ_{d|n} φ(d)·k^{n/d}, where φ is Euler's totient. The totient counts rotations of each period, averaging the number of colourings fixed by each rotation. For 2 colours and n=1,2,3,… it gives 2,3,4,6,8,14,20,36,…. Verified live: for n up to 15 (2 colours) and n up to 9 (3 colours), a brute count of distinct necklaces — each string reduced to its lexicographically smallest rotation — exactly equals Moreau's formula. Neon-noir traced. See a necklace on a ring in 1D, brute vs formula in 2D, and the average-over-rotations inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "8f2de18d013ddac1e0ce86752636c3fde8e09f571d63edd6b7a2ec763776829c"
    },
    {
      "slug": "the-sophomores-dream",
      "title": "THE SOPHOMORE'S DREAM",
      "kicker": "an integral equal to a self-power series",
      "accent": "#ff8a3c",
      "blurb": "The sophomore's dream in the 5-window house format — a pair of astonishing identities discovered by Johann Bernoulli in 1697, where a function raised to itself integrates to an infinite series over nⁿ: ∫₀¹ x^x dx = Σ_{n≥1} (−1)^{n−1}/nⁿ = 1 − 1/4 + 1/27 − … ≈ 0.7834, and ∫₀¹ x^{−x} dx = Σ_{n≥1} 1/nⁿ = 1 + 1/4 + 1/27 + … ≈ 1.2913. The name teases that the result looks like a naive 'dream' a student might wish were true — yet it really is. The trick is to expand x^x = e^{x ln x} as a power series and integrate term by term. Verified live: the numerical integrals of x^x and x^{−x} over [0,1] match their respective series Σ(−1)^{n−1}/nⁿ and Σ1/nⁿ to ~1e-6. Neon-noir traced. See the self-power curves + areas in 1D, integral vs series in 2D, and the series inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "4f44b22dce36643fe8ae185363a1d5551f85e1a0e6db001d46a15e8de38ffa9b"
    },
    {
      "slug": "the-lander-parkin",
      "title": "THE LANDER-PARKIN",
      "kicker": "a counterexample refuting Euler's conjecture",
      "accent": "#21e6ff",
      "blurb": "The Lander–Parkin counterexample in the 5-window house format — it demolished a 200-year-old conjecture of Euler. Extending Fermat's Last Theorem, Euler conjectured in 1769 that summing fewer than k perfect k-th powers can never equal a k-th power — e.g. you'd need at least five fifth-powers to make a fifth-power. In 1966, using an early computer, Lander and Parkin found: 27⁵ + 84⁵ + 110⁵ + 133⁵ = 144⁵ — just four fifth-powers. Euler was wrong. Later Noam Elkies and Roger Frye found a fourth-power version with only three terms: 95800⁴ + 217519⁴ + 414560⁴ = 422481⁴. Verified live with exact big-integer arithmetic: 27⁵+84⁵+110⁵+133⁵ equals 144⁵ exactly (four terms), and 95800⁴+217519⁴+414560⁴ equals 422481⁴ exactly (three terms); a nearby altered sum is not a perfect fifth power. Neon-noir traced. See the four fifth-powers stacking to one in 1D, the exact identity + control in 2D, and the one-witness-refutes inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "b0c56d547574cc512b12b212a1e0b39b38c23361ae757a596a2a6101705deac1"
    },
    {
      "slug": "the-sylvesters-law-of-inertia",
      "title": "THE SYLVESTER INERTIA",
      "kicker": "a signature invariant under congruence",
      "accent": "#35ffb0",
      "blurb": "Sylvester's law of inertia in the 5-window house format — a symmetric matrix has an unchangeable 'signature'. Any real symmetric matrix M can be transformed by congruence — M → PᵀMP for an invertible P — into many different-looking matrices. But the counts of positive, negative, and zero eigenvalues (the signature n₊, n₋, n₀) never change. You can rescale and mix the coordinates however you like; the number of 'plus' and 'minus' directions of the quadratic form is a fixed invariant. It is what lets us classify quadratic forms and read the character (definite, indefinite) of a form from any convenient basis. Verified live: for thousands of random symmetric matrices, the signature from eigenvalue signs is unchanged after a random congruence PᵀMP; and the number of negative eigenvalues equals the sign changes in the leading principal minors (Jacobi's criterion) — two independent computations of the same signature. Neon-noir traced. See the matrix + eigenvalue signs in 1D, invariance + Jacobi minors in 2D, and the surviving-invariant inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "e5736978c62ba82e2b0aed547c4d2af97d5bf5e097bf3789b44494f6d88739c1"
    },
    {
      "slug": "the-cauchy-interlacing",
      "title": "THE CAUCHY INTERLACING",
      "kicker": "submatrix eigenvalues interlacing the whole",
      "accent": "#b06bff",
      "blurb": "Cauchy's interlacing theorem in the 5-window house format — pinning the eigenvalues of a submatrix between those of the whole. Take a symmetric n×n matrix M with eigenvalues λ₁≥λ₂≥…≥λₙ, and delete one row and the matching column to get an (n−1)×(n−1) principal submatrix B with eigenvalues μ₁≥…≥μ_{n−1}. Cauchy proved they interlace: λ_i ≥ μ_i ≥ λ_{i+1} for every i. Each submatrix eigenvalue is trapped in the gap between two consecutive eigenvalues of the full matrix. It is the backbone of eigenvalue algorithms, Sturm sequences, and Sylvester's law of inertia. Verified live: for thousands of random symmetric matrices, the eigenvalues of a principal submatrix (computed independently by the Jacobi method) always satisfy λ_i ≥ μ_i ≥ λ_{i+1} — the interlacing never fails. Neon-noir traced. See the two eigenvalue sets on a line in 1D, the interlacing check in 2D, and the nested-eigenvalues inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "c6b9e32e6b7ff383f878c6d9f047ea1a78b09582013b046609df5a38ab8f0639"
    },
    {
      "slug": "the-brahmagupta",
      "title": "THE BRAHMAGUPTA",
      "kicker": "a cyclic quadrilateral's maximal area from its sides",
      "accent": "#ffcf4a",
      "blurb": "Brahmagupta's formula in the 5-window house format — the area of a cyclic quadrilateral (vertices on a circle) from its side lengths alone: Area = √((s−a)(s−b)(s−c)(s−d)), where s=(a+b+c+d)/2 is the semiperimeter. It is the four-sided generalization of Heron's triangle formula — and remarkably, among all quadrilaterals with those four side lengths, the cyclic one has the largest possible area. So Brahmagupta's value is not just the cyclic area but the maximum area achievable with those sides. Verified live: for thousands of quadrilaterals with vertices on a circle, the shoelace (coordinate) area equals √((s−a)(s−b)(s−c)(s−d)) to ~1e-14; and any non-cyclic quadrilateral with the same side lengths has a strictly smaller area. Neon-noir traced. See the inscribed quadrilateral in 1D, shoelace vs formula + maximality in 2D, and the biggest-area-from-sides inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "6cc33fe633e421efda87de5ccb1f810b127ba838a223e964898e160a428ca379"
    },
    {
      "slug": "the-taxicab",
      "title": "THE TAXICAB",
      "kicker": "the smallest two-way sum of two cubes",
      "accent": "#21e6ff",
      "blurb": "1729, the taxicab number, in the 5-window house format — the smallest positive integer expressible as a sum of two positive cubes in two different ways: 1729 = 1³+12³ = 9³+10³. Its fame comes from a 1919 anecdote: when G. H. Hardy visited the ailing Srinivasa Ramanujan and remarked that his taxi's number, 1729, seemed rather dull, Ramanujan instantly replied that it was very interesting — the smallest number expressible as a sum of two cubes two ways. It is the second 'taxicab number' Ta(2); the next such number is 4104 = 2³+16³ = 9³+15³. Verified live: a brute search over all sums of two positive cubes finds that 1729 is the smallest integer with two distinct such representations, and the next one is 4104. Neon-noir traced. See 1729 built two ways in 1D, the brute search flagging it in 2D, and the number-hiding-two-cubes inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "8918b35d1c6412a5696ad778c6a4862793eb1387c5aedd3c1651f6b2806f457f"
    },
    {
      "slug": "the-wald",
      "title": "THE WALD",
      "kicker": "an expected sum equal to expected count times expected step",
      "accent": "#ff8a3c",
      "blurb": "Wald's identity in the 5-window house format — a clean law for random sums that stop at a random time. Suppose you add up independent, identically distributed steps X₁, X₂, …, and you keep a rule that decides when to stop — a stopping time N (it may depend on the steps seen so far, but not the future). Wald proved that the expected total equals the expected number of steps times the expected step: E[S_N] = E[N]·E[X], where S_N = X₁+…+X_N. Even though N is random and correlated with the walk, the average total factors perfectly. Verified live: simulating a walk with steps uniform on {1,2,3} (E[X]=2), stopping the first time the total reaches 50, the empirical average final total E[S_N] matches E[N]·E[X] to within a fraction of a percent over hundreds of thousands of runs. Neon-noir traced. See the stopping walk in 1D, E[S_N] vs E[N]·E[X] in 2D, and the factors-in-the-mean inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "0287073e2b3522c16e60eb271a618dcd4a78dc1c843ad8c2e84ce8e38cced6d6"
    },
    {
      "slug": "the-durfee-square",
      "title": "THE DURFEE SQUARE",
      "kicker": "a square hidden in every partition",
      "accent": "#35ffb0",
      "blurb": "The Durfee square in the 5-window house format — the largest square that fits in the top-left corner of a partition's Young diagram. For a partition of n drawn as rows of boxes, its Durfee square has side d = the largest number such that the partition has at least d parts each of size ≥ d. This single number splits every partition into three pieces: the d×d square, a partition to its right (parts ≤ d), and a partition below (at most d parts). That decomposition gives a beautiful generating-function identity for the partition numbers: Σ_n p(n)qⁿ = Σ_{d≥0} q^{d²}/∏_{i=1}^d(1−qⁱ)² — sorting all partitions by their Durfee-square size. Verified live: expanding Σ_{d≥0} q^{d²}/∏_{i=1}^d(1−qⁱ)² as a power series, the coefficient of qⁿ equals the partition number p(n) for every n up to 45 — p(40)=37338, p(45)=89134. Neon-noir traced. See a Young diagram with its Durfee square in 1D, the generating function vs p(n) in 2D, and the split-by-square inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "4781e9bc14bbf05e25cc6e33f5c6001134bd2064425f54a9e82a668512d2eabd"
    },
    {
      "slug": "the-weyl-equidistribution",
      "title": "THE WEYL EQUIDISTRIBUTION",
      "kicker": "irrational multiples filling the interval evenly",
      "accent": "#21e6ff",
      "blurb": "Weyl's equidistribution theorem in the 5-window house format — the fractional parts of the multiples of an irrational number spread out perfectly evenly. Take any irrational α and look at the sequence {α}, {2α}, {3α}, … (fractional parts, mod 1). Weyl proved these points become equidistributed in [0,1): the fraction landing in any subinterval [a,b) converges to its length b−a. The sequence never settles into a pattern — it fills the interval as uniformly as possible. For a rational α=p/q, by contrast, the fractional parts cycle through only q values and are never equidistributed. Verified live: for α=√2, φ, π, e, the star discrepancy of {nα} shrinks toward zero as N grows — below 1e-3 by N=20000 — while for rational α=1/3 the discrepancy stays large. Neon-noir traced. See the points filling [0,1) in 1D, the discrepancy → 0 + control in 2D, and the order-into-uniformity inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "46487a3da4014b46a4dfeb2cced69fbf93048eb94d636f29277f17a35930b49f"
    },
    {
      "slug": "the-gamma-reflection",
      "title": "THE GAMMA REFLECTION",
      "kicker": "a gamma product equal to a cosecant",
      "accent": "#b06bff",
      "blurb": "Euler's reflection formula in the 5-window house format — tying the gamma function to the sine in one clean stroke: Γ(x)·Γ(1−x) = π/sin(πx). The gamma function Γ extends the factorial to all real (and complex) numbers, and it looks nothing like a trig function — yet multiply its value at x by its value at the mirror point 1−x, and the messy transcendental factorials collapse into a simple cosecant. Setting x=½ gives Γ(½)²=π, so Γ(½)=√π — the gateway to the Gaussian integral. The poles of the gamma function at 0,−1,−2,… line up exactly with the zeros of sine. Verified live: computing Γ by the Lanczos approximation, the product Γ(x)·Γ(1−x) equals π/sin(πx) to relative error ~1e-14 for thousands of x in (0,1), and Γ(½)²=π. Neon-noir traced. See the mirrored gamma curves in 1D, product vs cosecant in 2D, and the factorials-reflected inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "30ee928d49a6bf88b4e1db9d3a885cddead8d73b85702898ffbdf6a04a0f1fb6"
    },
    {
      "slug": "the-hockey-stick",
      "title": "THE HOCKEY STICK",
      "kicker": "a diagonal of Pascal summing to one entry",
      "accent": "#ff8a3c",
      "blurb": "The hockey-stick identity in the 5-window house format — a striking pattern in Pascal's triangle: sum any diagonal starting from the edge, and the total appears just below the end of the diagonal. Formally, Σ_{i=r}^{n} C(i,r) = C(n+1,r+1). Trace down a diagonal of the triangle (the 'stick') and the running sum lands in the single cell one step down and over (the 'blade') — the shape of a hockey stick. It falls straight out of Pascal's rule C(n+1,r+1)=C(n,r)+C(n,r+1), telescoping the diagonal into one entry, and it is the discrete cousin of integrating xʳ. Verified live with exact big-integer arithmetic: for all r from 0 to 8 and n up to 30, the diagonal sum Σ_{i=r}^{n} C(i,r) equals C(n+1,r+1) exactly — e.g. C(2,2)+C(3,2)+C(4,2)+C(5,2)+C(6,2)=35=C(7,3). Neon-noir traced. See Pascal's triangle with the stick in 1D, sum vs blade in 2D, and the diagonal-folded inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "6d4656c3c4944caf0f3671339cfc98b470fb28302519d4307c86a135f04446e8"
    },
    {
      "slug": "the-feuerbach",
      "title": "THE FEUERBACH",
      "kicker": "a nine-point circle tangent to the incircle",
      "accent": "#ffcf4a",
      "blurb": "Feuerbach's theorem in the 5-window house format — one of the most beautiful coincidences in triangle geometry. Every triangle has a nine-point circle — the circle through nine special points (the three side midpoints, the three altitude feet, and the three midpoints from the orthocenter to the vertices), with radius exactly half the circumradius. Feuerbach proved that this nine-point circle is tangent to the incircle (and to all three excircles). The single point where it touches the incircle is the celebrated Feuerbach point. Tangency means the distance between the two circles' centres equals the difference of their radii: |N₉−I| = R/2 − r. Verified live: for thousands of random triangles, the distance between the nine-point centre and the incentre equals R/2−r to ~1e-15 — confirming the internal tangency. Neon-noir traced. See the triangle + two circles + Feuerbach point in 1D, the tangency check in 2D, and the forced-tangency inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "dab565093322a5fb471317f4b67b3788b3bb08dbae5e9c4641f6950d9f3d23a5"
    },
    {
      "slug": "the-bruck-ryser",
      "title": "THE BRUCK-RYSER",
      "kicker": "orders of projective planes ruled out by two squares",
      "accent": "#35ffb0",
      "blurb": "The Bruck–Ryser theorem in the 5-window house format — forbidding certain finite projective planes using a fact about sums of two squares. A projective plane of order n is a highly symmetric geometry with n²+n+1 points and the same number of lines. Bruck and Ryser proved a necessary condition: if n ≡ 1 or 2 (mod 4), then a projective plane of order n can exist only if n is a sum of two integer squares. This single arithmetic test rules out infinitely many orders — the first being order 6 (6≡2 mod 4, and 6 is not a sum of two squares), which is why no 6×6 pair of orthogonal Latin squares (Euler's 36 officers) exists. It is a necessary, not sufficient, condition. Verified live: among orders n≤50 with n≡1,2 (mod 4), the ones not sums of two squares — ruled out by Bruck–Ryser — are exactly 6,14,21,22,30,33,38,42,46; the small orders with known planes (2,3,4,5,7,8,9) are never excluded. Neon-noir traced. See orders 2..50 flagged in 1D, the mod-4 + two-squares test in 2D, and the geometry-gated-by-two-squares inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "a517bec3bc38c35d1c5259473d0bd418845736808bda05a52d92f81def6962b9"
    },
    {
      "slug": "the-stirling-approximation",
      "title": "THE STIRLING APPROXIMATION",
      "kicker": "a factorial approximated by a smooth curve",
      "accent": "#ff8a3c",
      "blurb": "Stirling's approximation in the 5-window house format — replacing the jagged factorial with a smooth formula: n! ≈ √(2πn)·(n/e)ⁿ. The factorial n! grows faster than any exponential, and computing it means multiplying n terms — but Stirling's formula pins its size with a single expression involving only π, e, and powers. The relative error shrinks like 1/(12n), so the next correction term is n! ≈ √(2πn)(n/e)ⁿ(1 + 1/(12n) + …). It is the workhorse behind asymptotics in combinatorics, statistical mechanics, and probability. Verified live: the ratio n!/(√(2πn)(n/e)ⁿ) tends to 1 as n grows, and the correction is exactly 1/(12n) — the quantity (ln n! − ln-Stirling)·12n converges to 1.0000. Neon-noir traced. See ln(n!) vs the Stirling curve in 1D, the ratio + correction in 2D, and the product-folded-into-a-formula inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "ed0a30521ff3bbf8f473e6d90a1ae82abb8dcdd674cb1eb3fc89895ade30cc07"
    },
    {
      "slug": "the-stewart",
      "title": "THE STEWART",
      "kicker": "a cevian length from the sides",
      "accent": "#b06bff",
      "blurb": "Stewart's theorem in the 5-window house format — the length of a cevian (any segment from a vertex of a triangle to a point on the opposite side) from the side lengths alone. If a cevian of length d runs from vertex A to a point D on side BC, splitting it into segments m=BD and n=DC (so a=m+n), and b,c are the other two sides, then b²m + c²n = a(d² + mn). The mnemonic is 'a man and his dad put a bomb in the sink'. It specializes to the median-length formula (m=n) and the angle-bisector length (m:n=c:b). Verified live: for thousands of random triangles and cevian points, the cevian length computed directly from coordinates satisfies b²m + c²n = a(d² + mn) to ~1e-15, and the median special case matches d=√((2b²+2c²−a²)/4). Neon-noir traced. See the triangle + cevian in 1D, the Stewart relation in 2D, and the length-from-sides inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "9094506fe56ccfa7c8cddaa3dc8d600220445a1e41c893d7662401b76d7a0cda"
    },
    {
      "slug": "the-involution",
      "title": "THE INVOLUTION",
      "kicker": "self-inverse permutations counted by a recurrence",
      "accent": "#21e6ff",
      "blurb": "Involutions in the 5-window house format — the permutations that are their own inverse: apply one twice and you're back where you started (σ²=identity). Structurally they are made only of fixed points and 2-cycles — every element is either left alone or swapped with exactly one partner. The number of involutions of n elements is the telephone number T(n) (also the number of ways to pair up n telephones with some left unconnected): 1,1,2,4,10,26,76,232,764,…. It satisfies the recurrence T(n)=T(n−1)+(n−1)·T(n−2), and by the RSK correspondence it also counts the standard Young tableaux with n cells. Verified live: a brute count of the permutations σ with σ²=identity equals the telephone number T(n)=T(n−1)+(n−1)T(n−2) and the explicit sum Σ_k n!/(2^k k!(n−2k)!) for every n from 0 to 8. Neon-noir traced. See an involution's pairings in 1D, brute vs recurrence vs sum in 2D, and the counted-by-recurrence inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "58e08cc67620b9edec7418139f458ec87de9f4ae2c636000197a7bf1544e5e78"
    },
    {
      "slug": "the-bertrand-postulate",
      "title": "THE BERTRAND POSTULATE",
      "kicker": "a prime always between n and 2n",
      "accent": "#ffcf4a",
      "blurb": "Bertrand's postulate in the 5-window house format — primes never leave big gaps: for every integer n≥1, there is at least one prime p with n < p ≤ 2n. Double any number and you are certain to have jumped over a prime. Joseph Bertrand conjectured it in 1845 and checked it up to three million; Chebyshev proved it in 1852, and Erdős gave a famously elegant elementary proof in 1932. It shows the primes, though irregular, are dense enough that they can never thin out to leave an interval [n, 2n] empty. Verified live: a prime sieve confirms that for every n from 1 to 20000 there is a prime strictly greater than n and at most 2n; for n≥2 the least such prime is strictly less than 2n (the only equality is n=1, where the prime is 2=2·1). Neon-noir traced. See the interval (n,2n] with its primes in 1D, the least-prime + ratio in 2D, and the never-a-gap inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "ff95fa9a4df18ea933a3e766f0a3547b084930354aaeffa19af3d9e17db4d728"
    },
    {
      "slug": "the-amicable",
      "title": "THE AMICABLE",
      "kicker": "two numbers summing to each other's divisors",
      "accent": "#35ffb0",
      "blurb": "Amicable numbers in the 5-window house format — two different numbers, each of which equals the sum of the other's proper divisors. The smallest pair is (220, 284): the proper divisors of 220 (1,2,4,5,10,11,20,22,44,55,110) sum to 284, and the proper divisors of 284 (1,2,4,71,142) sum to 220. They point at each other perfectly. Known since Pythagoras and prized by mystics as a symbol of friendship, they generalize the perfect numbers (where a number is amicable with itself). The next pair is (1184, 1210), then (2620, 2924). Verified live: with s(n)=σ(n)−n (the sum of proper divisors), s(220)=284 and s(284)=220; a brute search confirms (220, 284) is the smallest amicable pair, and the next is (1184, 1210). Neon-noir traced. See 220 and 284 pointing at each other in 1D, s(a)=b & s(b)=a in 2D, and the mutual-divisor-sum inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "4a4633e4bc9818121def9eda276735f9ba94f7d65fcbfe7fc2bc52ead06fdcfc"
    },
    {
      "slug": "the-wallis-product",
      "title": "THE WALLIS PRODUCT",
      "kicker": "an infinite product converging to π/2",
      "accent": "#ff8a3c",
      "blurb": "The Wallis product in the 5-window house format — one of the oldest infinite products for π, found by John Wallis in 1656 before calculus existed: π/2 = (2·2)/(1·3)·(4·4)/(3·5)·(6·6)/(5·7)·… = ∏_{n≥1} (2n)²/((2n−1)(2n+1)). An infinite product of simple rational numbers, each just above or below 1, multiplies out to half of π. Wallis derived it by interpolating the integrals ∫₀^{π/2} sinⁿx dx, whose ratios encode the product — the same integrals give the 'Wallis integrals' identity n·W_n·W_{n−1} = π/2. Verified live: the partial products ∏_{n=1}^N (2n)²/((2n−1)(2n+1)) converge to π/2 (1.5708…), and independently the numerically-integrated Wallis integrals satisfy n·W_n·W_{n−1} = π/2 exactly for every n. Neon-noir traced. See the partial products closing on π/2 in 1D, product vs integral identity in 2D, and the π-from-a-product inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "09da02c48806e528b820b16356ae6e6e0026c2e93ed1412234d0ccb4e23ab8ba"
    },
    {
      "slug": "the-neumann-series",
      "title": "THE NEUMANN SERIES",
      "kicker": "a matrix inverse as a power series",
      "accent": "#b06bff",
      "blurb": "The Neumann series in the 5-window house format — the matrix version of the geometric series 1/(1−x)=1+x+x²+…. For a square matrix A whose size is 'small enough' (spectral radius < 1), the inverse of I−A is the infinite sum of its powers: (I−A)⁻¹ = I + A + A² + A³ + …. Just as the scalar series needs |x|<1, the matrix series converges precisely when A's powers shrink to zero — and then a hard matrix inversion becomes a sum you can truncate. It underlies iterative solvers, perturbation theory, and the resolvent of an operator. Verified live: for thousands of random matrices with small entries (spectral radius < 1), the partial sum I+A+…+A⁶⁰ matches the directly-computed inverse (I−A)⁻¹ to ~1e-14; and for a matrix with spectral radius > 1 the power series diverges (its terms blow up). Neon-noir traced. See the partial sums converging in 1D, convergence + divergence control in 2D, and the inversion-as-series inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "2ca03f122317206c31c594d0751ab7697381f6910502eac21c3343084b4bc2be"
    },
    {
      "slug": "the-best-theorem",
      "title": "THE BEST THEOREM",
      "kicker": "Eulerian circuits counted by a determinant",
      "accent": "#21e6ff",
      "blurb": "The BEST theorem in the 5-window house format — counting the Eulerian circuits of a directed graph (closed trails using every edge exactly once) with a single formula. For a connected Eulerian digraph (every vertex has equal in- and out-degree), the number of Eulerian circuits is ec(G) = t_w(G)·∏_v (deg⁺(v)−1)!, where t_w(G) is the number of spanning arborescences (in-trees) rooted at any vertex w — itself a determinant, via the Matrix-Tree theorem. So an exponential count of tangled circuits collapses into one determinant times some factorials. Verified live: for several small Eulerian digraphs, a brute enumeration of Eulerian circuits (fixing the starting edge) exactly equals t_w(G)·∏_v(deg⁺(v)−1)!, with t_w computed as a cofactor determinant of the graph Laplacian. Neon-noir traced. See an Eulerian digraph + a circuit in 1D, brute vs BEST formula in 2D, and the circuits-from-a-determinant inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "948d468bc4359a11d06a0857943cf6c612a279a070fc5aeb665a894cb836d13b"
    },
    {
      "slug": "the-poisson-limit",
      "title": "THE POISSON LIMIT",
      "kicker": "a binomial limiting to a Poisson",
      "accent": "#ffcf4a",
      "blurb": "The Poisson limit theorem in the 5-window house format — the 'law of rare events' that explains why the Poisson distribution appears everywhere. If you have many independent trials, each with a tiny success probability, but a fixed expected number of successes λ=np, then Binomial(n, λ/n) converges to the Poisson distribution with mean λ: C(n,k)(λ/n)^k(1−λ/n)^{n−k} → e^{−λ}λ^k/k! as n→∞. Rare events among many trials — radioactive decays, typos per page, calls per minute — all follow Poisson. Verified live: for λ=3, the binomial pmf Binomial(n, 3/n) approaches the Poisson(3) pmf as n grows — the maximum gap between the two shrinks from ~4e-2 at n=10 to ~3e-5 at n=10000. Neon-noir traced. See the binomial bars settling onto the Poisson line in 1D, the max gap → 0 in 2D, and the depends-only-on-λ inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "689d7edc48f23715d42b6d7267e30f993d9bf30d85c4f1a3d77414f3ba531faf"
    },
    {
      "slug": "the-thebault",
      "title": "THE THEBAULT",
      "kicker": "squares on a parallelogram forming a square",
      "accent": "#35ffb0",
      "blurb": "Thébault's first theorem in the 5-window house format — conjuring a perfect square out of any parallelogram. Take any parallelogram and erect a square outward on each of its four sides. Mark the centre of each square. Thébault proved that these four centres are always the vertices of a square — no matter how slanted or stretched the original parallelogram is. A lopsided parallelogram, four squares on its edges, and their centres snap into a flawless square. It is a cousin of Van Aubel's theorem, but for the special case of a parallelogram the result sharpens from 'equal perpendicular diagonals' all the way to 'a square'. Verified live: for thousands of random parallelograms, the four square-centres have all four sides equal and both diagonals equal (to machine precision) — the defining conditions of a square. Neon-noir traced. See the parallelogram + squares + centre-square in 1D, the equal-sides/diagonals test in 2D, and the square-from-any-parallelogram inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "f2a2545eb4f8994c8bae752c5d769d57c70bfcf2f4626bb05197191257ba6a8e"
    },
    {
      "slug": "the-van-schooten",
      "title": "THE VAN SCHOOTEN",
      "kicker": "the far distance equal to the sum of the two near ones",
      "accent": "#ffcf4a",
      "blurb": "Van Schooten's theorem in the 5-window house format — a striking length identity for the equilateral triangle. Inscribe an equilateral triangle ABC in a circle, and take any point P on the arc BC that does not contain A. Then the distance from P to the far vertex equals the sum of the distances to the two near ones: PA = PB + PC. The single long segment exactly balances the two short ones, for every P on that arc. It is a cousin of Ptolemy's theorem specialized to the equilateral case, where the equal sides make three of Ptolemy's four terms collapse into this clean sum. Verified live: for an equilateral triangle on a circle and thousands of points P on the arc BC, PA equals PB + PC to ~1e-15; and on that arc the 'wrong' identity PB = PA + PC does not hold. Neon-noir traced. See the triangle + P + distances in 1D, the PA=PB+PC identity in 2D, and the one-length-as-sum-of-two inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "5cab1144473d995cd071dec36edbb2c8b68e56ef78de80bc6169cc33656904d5"
    },
    {
      "slug": "the-dottie",
      "title": "THE DOTTIE",
      "kicker": "the fixed point of cosine",
      "accent": "#21e6ff",
      "blurb": "The Dottie number in the 5-window house format — the unique real solution of cos(x)=x, approximately 0.7390851332. Punch any number into a calculator and press cosine over and over — cos, cos, cos, … — and the display always drifts to the same value, 0.739085…, no matter where you start. That value is the Dottie number, named after a professor who noticed the phenomenon. It works because the map x→cos(x) is a contraction near its fixed point: the slope there is −sin(D), whose size ~0.674 is less than 1, so every start is drawn in. Verified live: iterating cosine from five different starting points all converge to the same D=0.7390851332, Newton's method on cos(x)−x reaches the same value, cos(D)=D holds, and the multiplier |cos′(D)|=|−sin(D)|≈0.674<1 confirms it is an attracting fixed point. Neon-noir traced. See the cobweb iteration into D in 1D, iteration + Newton in 2D, and the root-by-repetition inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "af93ce38bca29aa2ec9047f174a9b0fd60def5824f0d9267fa4856d189097e0b"
    },
    {
      "slug": "the-weitzenbock",
      "title": "THE WEITZENBOCK",
      "kicker": "a triangle's squared sides bounded below by its area",
      "accent": "#b06bff",
      "blurb": "Weitzenböck's inequality in the 5-window house format — bounding a triangle's squared side lengths below by its area: for any triangle with sides a,b,c and area T, a²+b²+c² ≥ 4√3·T. The constant 4√3≈6.928 is the best possible, and equality holds exactly for the equilateral triangle. In other words, for a fixed area, the equilateral triangle has the smallest sum of squared sides — the most 'compact' shape. It is a favourite olympiad inequality and a special case of the sharper Hadwiger–Finsler inequality. Verified live: for tens of thousands of random triangles, a²+b²+c² is always at least 4√3·T — the ratio (a²+b²+c²)/(4√3·T) never drops below 1, and reaches exactly 1 for the equilateral triangle. Neon-noir traced. See the triangle above its floor in 1D, the ratio ≥1 + equilateral equality in 2D, and the squared-sides-floored inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "ec98af9141f87ae33b203b714e9a5d48c3c7280fffa5d30c97884732e1018eaa"
    },
    {
      "slug": "the-liouville-number",
      "title": "THE LIOUVILLE NUMBER",
      "kicker": "a number approximated absurdly well by rationals",
      "accent": "#35ffb0",
      "blurb": "Liouville's number in the 5-window house format — L = Σ_{k≥1} 10^{−k!} = 0.110001000000000000000001… (a 1 at every factorial position, 0 elsewhere) was the first number ever proven transcendental (Liouville, 1844). The trick: its digits leave enormous runs of zeros, so the truncations p_n/q_n approximate L absurdly well — |L−p_n/q_n| < 1/q_n^n for every n. But Liouville proved an algebraic number of degree d can never be approximated better than c/q^d. Since L can be approximated to any power, it is not algebraic of any degree — it is transcendental. Verified live with exact big-integer arithmetic: for the truncations of L, the error |L−p_n/q_n| is strictly less than 1/q_n^n for n=1..5, and the approximation exponent (n+1) grows without bound — beating any fixed algebraic degree. Neon-noir traced. See the factorial-position digits in 1D, the error below the Liouville bound in 2D, and the transcendence-from-approximation inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "b3254b3694f46d0fad6019a5b6708a792bcf610d475edbd6a1538aee13a75ed0"
    },
    {
      "slug": "the-chu-vandermonde",
      "title": "THE CHU-VANDERMONDE",
      "kicker": "a binomial convolution collapsing to one entry",
      "accent": "#ff8a3c",
      "blurb": "The Chu–Vandermonde identity in the 5-window house format — collapsing a whole convolution of binomial coefficients into a single one: Σ_k C(m,k)·C(n,r−k) = C(m+n,r). Choosing r objects from a combined pile of m+n is the same as splitting the choice — k from the first pile, r−k from the second — and summing over all splits. Its most famous special case, with m=n=r, gives Σ_k C(n,k)² = C(2n,n): the sum of squared binomial coefficients across a row of Pascal's triangle is the central coefficient two rows down. Verified live with exact big-integer arithmetic: for all m,n up to 15 and every r, the convolution sum Σ_k C(m,k)C(n,r−k) equals C(m+n,r) exactly; and the special case Σ_k C(n,k)²=C(2n,n) holds for n up to 12. Neon-noir traced. See the two Pascal rows convolving in 1D, sum vs C(m+n,r) in 2D, and the convolution-folded inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "f31da58ff4c5b3191ab8fdfcdf7216d55e78cbfc65e9040cdff7cbd9a3cc2c8c"
    },
    {
      "slug": "the-euler-totient-theorem",
      "title": "THE EULER TOTIENT THEOREM",
      "kicker": "a power cycling back to one modulo n",
      "accent": "#b06bff",
      "blurb": "Euler's totient theorem in the 5-window house format — generalizing Fermat's little theorem to any modulus. For any integer a coprime to n, a^φ(n) ≡ 1 (mod n), where φ(n) is Euler's totient — the count of integers from 1 to n coprime to n. Raise a coprime residue to the φ(n)-th power and it snaps back to 1. When n is prime, φ(n)=n−1 and this is exactly Fermat's little theorem. The multiplicative order of a (the smallest k with a^k≡1) always divides φ(n) — a consequence of Lagrange's theorem in the group of units. It is the engine behind RSA. Verified live: for every modulus n up to 200 and every a coprime to n, a^φ(n)≡1 (mod n) by modular exponentiation, and the order of a divides φ(n) — e.g. φ(10)=4 and 3⁴=81≡1 (mod 10). Neon-noir traced. See the power cycle returning to 1 in 1D, a^φ(n)≡1 + order-divides-φ in 2D, and the cycle-length inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "fb431309abdff52d1241abdcb483e35caf91969cb76ca2409b634be26499444d"
    },
    {
      "slug": "the-erdos-mordell",
      "title": "THE ERDOS-MORDELL",
      "kicker": "a point's vertex distances bounded below by its side distances",
      "accent": "#21e6ff",
      "blurb": "The Erdős–Mordell inequality in the 5-window house format — relating a point's distances to a triangle's corners and to its sides. For any point P inside triangle ABC, the sum of distances to the three vertices is at least twice the sum of the perpendicular distances to the three sides: PA+PB+PC ≥ 2(dₐ+d_b+d_c). Erdős posed it in 1935; Mordell and Barrow proved it. Equality holds precisely when the triangle is equilateral and P is its centre. The far distances always dominate the near ones by at least a factor of two. Verified live: for tens of thousands of random triangles and interior points P, PA+PB+PC is always at least 2(dₐ+d_b+d_c) — the ratio never drops below 1, approaching 1 only for the equilateral triangle with P at its centre. Neon-noir traced. See the vertex + side distances in 1D, the ≥2 ratio in 2D, and the far-floored-by-near inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "470fa344c1753ef60b9fcbfc62725e5bceaa07eeb23a9e3f15bd4687debcf26f"
    },
    {
      "slug": "the-alternating-permutations",
      "title": "THE ALTERNATING PERMUTATIONS",
      "kicker": "zigzag permutations counted by secant plus tangent",
      "accent": "#ffcf4a",
      "blurb": "Alternating permutations in the 5-window house format — arrangements that zig-zag: a₁<a₂>a₃<a₄>…, going up, down, up, down. The number of them on n elements is the zigzag number (or Euler number) — 1,1,1,2,5,16,61,272,1385,… — and Désiré André proved in 1879 that they are packaged by a beautiful exponential generating function: Σ_n Z(n)xⁿ/n! = sec(x)+tan(x). The even-indexed terms come from the secant (the 'secant numbers'), the odd from the tangent (the 'tangent numbers') — two everyday trig functions counting a purely combinatorial object. Verified live: a brute count of the up-down alternating permutations of n elements equals the coefficient of xⁿ/n! in the Taylor series of sec(x)+tan(x), for every n from 0 to 8 — giving 1,1,1,2,5,16,61,272,1385. Neon-noir traced. See a zigzag permutation in 1D, brute vs sec+tan in 2D, and the counted-by-trigonometry inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "2874b3b2e6de76e6d373bf8489074df29bbda24a14bbdc41d5a791151f300282"
    },
    {
      "slug": "the-gregory-leibniz",
      "title": "THE GREGORY-LEIBNIZ",
      "kicker": "a slow alternating series for π",
      "accent": "#ff8a3c",
      "blurb": "The Gregory–Leibniz series in the 5-window house format — the most famous, and most beautifully slow, series for π: π/4 = 1 − 1/3 + 1/5 − 1/7 + 1/9 − … = Σ_{k≥0} (−1)^k/(2k+1). Every odd reciprocal, alternating in sign, sums to a quarter of π. It comes straight from the arctangent series arctan(x)=x−x³/3+x⁵/5−… evaluated at x=1, since arctan(1)=π/4. It is exact but converges agonizingly slowly — the error after N terms is only about 1/(2N), so you need hundreds of terms for two decimals. Verified live: 4·Σ(−1)^k/(2k+1) approaches π, and independently the numerical integral 4·∫₀¹ 1/(1+x²) dx (which is 4·arctan(1)) equals π to ~1e-9 — the two routes agree. Neon-noir traced. See the partial sums bracketing π in 1D, series vs integral in 2D, and the π-from-odd-reciprocals inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "a98d5f43b68196458e7bab81b858c28639e0476cafaffacfde36a79b44bcbc6e"
    },
    {
      "slug": "the-bertrand-paradox",
      "title": "THE BERTRAND PARADOX",
      "kicker": "one random chord with three different probabilities",
      "accent": "#35ffb0",
      "blurb": "Bertrand's paradox in the 5-window house format — a famous warning that 'pick a random chord' is not well defined. Ask: for a random chord of a circle, what is the probability it is longer than the side of the inscribed equilateral triangle (length √3·r)? Three perfectly reasonable ways to choose 'a random chord' give three different answers: (1) two random endpoints on the circle → 1/3; (2) a random point along a radius as the chord's midpoint → 1/2; (3) a random point in the disk as the midpoint → 1/4. The chord is longer exactly when its midpoint lies within r/2 of the centre — but 'random midpoint' means different things under each scheme. Verified live: Monte-Carlo simulation of the three schemes yields probabilities ≈ 1/3, 1/2, and 1/4 respectively — three different answers to the same question. Neon-noir traced. See the three sampling methods' chords in 1D, the three probabilities in 2D, and the one-question-three-answers inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "f581584960c774ed2836c512b6d8c99611e3c972a3bf546cb2435ab6e1f540fc"
    },
    {
      "slug": "the-carnot",
      "title": "THE CARNOT",
      "kicker": "circumcentre-to-side distances summing to R plus r",
      "accent": "#ffcf4a",
      "blurb": "Carnot's theorem in the 5-window house format — a hidden conservation law of the triangle. From the circumcentre O, drop a perpendicular to each of the three sides; the three signed distances (positive when O is on the same side of a line as the opposite vertex) always sum to exactly R + r, the circumradius plus the inradius: dₐ + d_b + d_c = R + r. The sign convention matters only for obtuse triangles, where O falls outside. Equivalently, cos A + cos B + cos C = 1 + r/R. Verified live: for tens of thousands of random triangles, the sum of the three signed circumcentre-to-side distances equals R + r to ~1e-13, and independently cos A + cos B + cos C equals 1 + r/R. Neon-noir traced. See the circumcentre and the three perpendiculars in 1D, the signed sum = R+r in 2D, and the conserved-budget inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "c6d289ea06a349697a6a56d611fc355f2f64ea510d803fe69c0fd446864604d7"
    },
    {
      "slug": "the-japanese-theorem",
      "title": "THE JAPANESE THEOREM",
      "kicker": "four incentres of a cyclic quad forming a rectangle",
      "accent": "#21e6ff",
      "blurb": "The Japanese theorem for cyclic quadrilaterals in the 5-window house format — a small miracle of hidden order. Take any four points A, B, C, D on a circle. From the four triangles that each drop one vertex (△ABC, △BCD, △CDA, △DAB), find the incentre of each. Those four incentres always form a rectangle — four right angles, no matter how irregular the original quadrilateral. The result is named for the sangaku tradition of theorems inscribed on wooden tablets in Edo-period Japanese temples. Verified live: for tens of thousands of random cyclic quadrilaterals, the four incentres are equidistant from their common centroid and centrally symmetric — the defining conditions of a rectangle — to ~1e-6. Neon-noir traced. See the quad and its four incentres in 1D, the rectangle test in 2D, and the four-centres-one-rectangle inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "ebbda0cbb80d9c8ec19ff2f6dfb5662fa06af0520e2c4a7dbad79f8ff6ed2625"
    },
    {
      "slug": "the-conway-circle",
      "title": "THE CONWAY CIRCLE",
      "kicker": "six side-extension points on one circle",
      "accent": "#b06bff",
      "blurb": "Conway's circle theorem in the 5-window house format — a six-point surprise from John Horton Conway. At each vertex of any triangle, extend the two sides beyond that vertex by the length of the side opposite it: beyond B by b (= CA), beyond C by c (= AB), beyond A by a (= BC). This produces six new endpoints, and all six lie on a single circle — the Conway circle — centred at the incentre I, with radius exactly √(r² + s²), where r is the inradius and s the semiperimeter. Each extension lands a distance s from the point where the incircle touches that side. Verified live: for tens of thousands of random triangles, all six extension points are equidistant from the incentre, at distance √(r²+s²), to ~1e-13. Neon-noir traced. See the extended sides and six points in 1D, the six equal radii in 2D, and the six-points-one-circle inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "a31107beb2eb23ada71bfa515f36b218163d8b51c484c23393630423d767e705"
    },
    {
      "slug": "the-casey",
      "title": "THE CASEY",
      "kicker": "a generalized Ptolemy for tangent circles",
      "accent": "#ff8a3c",
      "blurb": "Casey's theorem in the 5-window house format — Ptolemy's theorem for circles. Ptolemy says: for four points on a circle in order, AC·BD = AB·CD + AD·BC. Casey generalizes each point to a whole circle tangent to a common circle. Replace the four points by four circles all internally tangent to one enclosing circle, in cyclic order, and replace each chord by the tangent length t_ij (the length of the common tangent segment) between circles i and j. Then the same relation holds: t₁₂·t₃₄ + t₂₃·t₁₄ = t₁₃·t₂₄. Shrink the circles to points and it collapses back to Ptolemy. Verified live: for thousands of random configurations of four circles internally tangent to a circle, the tangent lengths satisfy t₁₂t₃₄ + t₂₃t₁₄ = t₁₃t₂₄ to ~1e-15, and the point-circle limit reproduces Ptolemy exactly. Neon-noir traced. See the four tangent circles and their tangent segments in 1D, the Ptolemy-form relation in 2D, and the points-fattened-into-circles inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "aa31875e77c554295aa712eb5154e929eb03441439326fe6eed93b6c8a55c2af"
    },
    {
      "slug": "the-fermat-polygonal",
      "title": "THE FERMAT POLYGONAL",
      "kicker": "every integer a sum of few polygonal numbers",
      "accent": "#35ffb0",
      "blurb": "Fermat's polygonal number theorem in the 5-window house format — one of the great cheat-codes of arithmetic. The k-gonal numbers are figurate numbers from stacking polygons: triangular (1,3,6,10,…), square (1,4,9,16,…), pentagonal (1,5,12,22,…). Fermat claimed — and it is true — that every positive integer is the sum of at most k of the k-gonal numbers: at most 3 triangular, at most 4 squares, at most 5 pentagonal, at most 6 hexagonal, forever. Gauss proved the triangular case ('EΥΡΗΚΑ! num = Δ+Δ+Δ'), Lagrange the four-squares case, and Cauchy the general theorem in 1813. Verified live: a dynamic-programming search confirms every integer up to 2000 is a sum of at most k k-gonal numbers for k = 3 through 8 — and the bound is sharp (the maximum needed is exactly k). Neon-noir traced. See n cracked into its fewest pieces in 1D, the ≤k check in 2D, and the crack-it-down inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "b72ef3cffd145c772974f998275b634f3e30a41f7997103b6acee02d569610fd"
    },
    {
      "slug": "the-midy",
      "title": "THE MIDY",
      "kicker": "the two halves of a repeating decimal summing to nines",
      "accent": "#ffcf4a",
      "blurb": "Midy's theorem in the 5-window house format — a hidden symmetry inside repeating decimals. For a prime p (other than 2 or 5), write out the decimal of a/p; it repeats with some period. When that period is even, say 2k digits, split the repeating block into two halves of k digits each — the two halves always sum to a string of nines (10^k − 1). The classic case: 1/7 = 0.142857…, and 142 + 857 = 999. It holds for every prime whose period is even, discovered by the French schoolteacher E. Midy in 1836. Verified live: for every prime p ≤ 200 (excluding 2 and 5) and every numerator a whose repeating block has even period 2k, the two k-digit halves sum to exactly 10^k − 1. Neon-noir traced. See the block split into halves in 1D, the nines-sum check in 2D, and the folded-onto-its-complement inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "b24bd58fe9d5eff74bdf1ab952a25e69d84dee2580250c95405394da0f66d53a"
    },
    {
      "slug": "the-automorphic",
      "title": "THE AUTOMORPHIC",
      "kicker": "a number whose square ends in itself",
      "accent": "#35ffb0",
      "blurb": "Automorphic numbers in the 5-window house format — numbers whose square ends in the number itself. 5²=25, 6²=36, 25²=625, 76²=5776, 376²=141376, 625²=390625, 9376²=87909376. For each digit-length d there are exactly two nontrivial ones — one ending in 5, one in 6 — and they always add to 10^d + 1 (25+76=101, 625+376=1001). They are the nontrivial idempotents of arithmetic mod 10^d (solutions of x²≡x), built by the Chinese Remainder Theorem from 10^d = 2^d·5^d, and extended leftward forever they become the two nonzero 10-adic idempotents. Verified live: the two nontrivial idempotents mod 10^d (d=1..12) each satisfy x²≡x, end in 5 and 6, and sum to 10^d+1; the known 5,6,25,76,376,625,9376,90625 are all confirmed automorphic. Neon-noir traced. See a number reappearing in its square's tail in 1D, the paired idempotents in 2D, and the fixed-point-of-squaring inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "f8b082074f0e7961dfbbe0ee1513e1aeb8275e5499bcdf3fa7c434434dcb54f9"
    },
    {
      "slug": "the-cassini",
      "title": "THE CASSINI",
      "kicker": "a Fibonacci determinant pinned at plus or minus one",
      "accent": "#b06bff",
      "blurb": "Cassini's identity in the 5-window house format — the Fibonacci numbers pinned to a razor's edge. For every n, F(n−1)·F(n+1) − F(n)² = (−1)^n: the product of F(n)'s neighbours misses F(n)² by exactly one, alternating sign forever. It is the determinant of the Fibonacci matrix [[1,1],[1,0]]^n = [[F(n+1),F(n)],[F(n),F(n−1)]], whose determinant is (−1)^n since det[[1,1],[1,0]] = −1. The generalization, Catalan's identity, reads F(n)² − F(n−r)F(n+r) = (−1)^{n−r}F(r)². This near-miss is the secret behind the 'missing square' puzzle where an 8×8 square seems to rearrange into a 5×13 rectangle — off by one unit of area. Verified live with exact BigInt: Cassini for n=1..100 and Catalan for a range of n,r. Neon-noir traced. See the neighbour-product vs square in 1D, the matrix determinant in 2D, and the pinned-determinant inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "4396cbe6b5f17be40b18f27d08356d42dfca3d4fae9eec61dda5412d00fb1a7b"
    },
    {
      "slug": "the-nilakantha",
      "title": "THE NILAKANTHA",
      "kicker": "a faster alternating series for pi",
      "accent": "#ff8a3c",
      "blurb": "The Nilakantha series in the 5-window house format — a fast, elegant series for π found by the Kerala-school astronomer Nilakantha Somayaji around 1500, three centuries before Europe. It reads π = 3 + 4/(2·3·4) − 4/(4·5·6) + 4/(6·7·8) − …, each term straddling three consecutive integers, alternating in sign. Unlike the Gregory–Leibniz series (hundreds of terms for two decimals), Nilakantha's terms shrink like 1/k³, so a handful of terms already gives several correct digits. Verified live: 3 + Σ(−1)^{k+1} 4/((2k)(2k+1)(2k+2)) converges to π (~1e-9), and with 100 terms its error (~2e-7) is more than a hundred times smaller than the Gregory–Leibniz error at the same term count. Neon-noir traced. See both series racing to π in 1D, the shrinking-error comparison in 2D, and the straddle-three-integers inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "d8e34820373157a1dea970c9338ea5ef9288cbfbea24fc7466874d7e95d31f90"
    },
    {
      "slug": "the-bretschneider",
      "title": "THE BRETSCHNEIDER",
      "kicker": "the area of any quadrilateral from its sides and two angles",
      "accent": "#21e6ff",
      "blurb": "Bretschneider's formula in the 5-window house format — the area of any quadrilateral from its four sides and two opposite angles. With sides a,b,c,d, semiperimeter s = (a+b+c+d)/2, and opposite interior angles A and C: Area = √[(s−a)(s−b)(s−c)(s−d) − abcd·cos²((A+C)/2)]. It is the grand generalization of Heron's formula (triangles) and Brahmagupta's formula (cyclic quadrilaterals): when the quad is cyclic, A+C = 180°, the cosine term vanishes, and it collapses to Brahmagupta's √[(s−a)(s−b)(s−c)(s−d)]. The cosine term is exactly the penalty a quadrilateral pays for not being inscribable in a circle. Verified live: for tens of thousands of random convex quadrilaterals, Bretschneider matches the shoelace (surveyor's) area to ~1e-13, and for cyclic quads it reduces exactly to Brahmagupta. Neon-noir traced. See a quad with its sides and two angles in 1D, Bretschneider vs shoelace in 2D, and the cyclic-penalty inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "c9ec8f608f492cd227a9365965ddf5934a1b935bc79719016345d0c80f8f3f90"
    },
    {
      "slug": "the-munchhausen",
      "title": "THE MUNCHHAUSEN",
      "kicker": "a number built from its own digits raised to themselves",
      "accent": "#ffcf4a",
      "blurb": "Münchhausen numbers in the 5-window house format — numbers that lift themselves by their own bootstraps. A Münchhausen number equals the sum of its own digits, each raised to the power of itself: n = Σ d^d. The star example is 3435 = 3³ + 4⁴ + 3³ + 5⁵ = 27+256+27+3125. Using the convention 0⁰ = 0, the only two Münchhausen numbers in base 10 are 1 and 3435 — provable because for enough digits the maximum digit-power-sum grows slower than the number. Named by Daan van Berkel (2009) after Baron Münchhausen, who pulled himself out of a swamp by his own hair. Verified live: a brute search over every n up to 500000 finds exactly {1, 3435}, and 3³+4⁴+3³+5⁵ is confirmed to equal 3435. Neon-noir traced. See 3435 rebuilt from its digits in 1D, the Σ d^d test in 2D, and the fixed-point-of-the-digit-map inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "909c0bea60b664a69617c986352d07d868ec4892eca32e717727f1dce19fd9de"
    },
    {
      "slug": "the-plastic-number",
      "title": "THE PLASTIC NUMBER",
      "kicker": "the third metallic constant solving a cubic",
      "accent": "#b06bff",
      "blurb": "The plastic number in the 5-window house format — the quiet cubic cousin of the golden ratio. ρ ≈ 1.3247179572 is the unique real root of x³ = x + 1. Where φ solves x² = x + 1 and governs the Fibonacci numbers, ρ solves the next cubic and governs the Padovan and Perrin sequences (each term the sum of the two before the previous: P(n) = P(n−2) + P(n−3)). Ratios of consecutive terms converge to ρ, and ρ equals the infinitely nested radical ∛(1 + ∛(1 + ∛(1 + …))). The Dutch architect Dom Hans van der Laan built a whole system of proportion on it in 1928. Verified live: Newton's method gives ρ with ρ³−ρ−1 = 0 to ~1e-14; the Padovan and Perrin ratios both converge to ρ; and the nested cube-root iteration converges to the same ρ. Neon-noir traced. See the Padovan ratios settling onto ρ in 1D, the three roads to ρ in 2D, and the nested-radical inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "548e8189d79f0a31cf51d4a9abef9bad4562482206b8e7fc709b52b6aee56c84"
    },
    {
      "slug": "the-reuleaux",
      "title": "THE REULEAUX",
      "kicker": "a triangle of constant width that is not a circle",
      "accent": "#21e6ff",
      "blurb": "The Reuleaux triangle in the 5-window house format — a shape of constant width that is not a circle. Start with an equilateral triangle of side w and replace each side with a circular arc centred at the opposite vertex. The result has the same width — the distance between two parallel supporting lines — in every direction, namely w. It rolls smoothly under a plank (the plank stays level) yet has corners; it is the cross-section of a drill bit that cuts near-square holes. Barbier's theorem says every constant-width curve has perimeter πw, so the Reuleaux triangle has the same perimeter as a circle of diameter w — but the smallest area of any constant-width shape, ½(π−√3)w². Verified live: sampling the boundary and measuring the width across 360 directions gives a spread below 1e-3, and the boundary length matches πw and the enclosed area matches ½(π−√3)w². Neon-noir traced. See the rotating caliper reading w in 1D, the width/perimeter/area checks in 2D, and the rolls-but-not-round inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "fa19959659b17bd1c2c5902cc53b1563f39e30a872d5e1d47efded576b83acde"
    },
    {
      "slug": "the-hadwiger-finsler",
      "title": "THE HADWIGER-FINSLER",
      "kicker": "a sharpened Weitzenbock inequality",
      "accent": "#ff8a3c",
      "blurb": "The Hadwiger–Finsler inequality in the 5-window house format — a sharpened Weitzenböck. Weitzenböck says a triangle's squared sides satisfy a²+b²+c² ≥ 4√3·T (T the area). Hadwiger and Finsler add back the exact leftover: a²+b²+c² ≥ 4√3·T + (a−b)²+(b−c)²+(c−a)². The extra sum of squared side-differences is precisely how far the triangle is from equilateral, so the inequality is tight exactly when a=b=c. Since that extra term is always ≥ 0, Hadwiger–Finsler immediately implies Weitzenböck — it is the stronger statement, with the slack made explicit. Verified live: for tens of thousands of random triangles, a²+b²+c² − 4√3·T − ((a−b)²+(b−c)²+(c−a)²) is always ≥ 0, reaching 0 only for the equilateral triangle. Neon-noir traced. See the closing gap in 1D, the slack ≥ 0 check in 2D, and the explicit-surplus inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "19719712dc2aa31c07e16338e8dd0b981139b484b28c7a8e36b5b34cecd68a53"
    },
    {
      "slug": "the-borwein",
      "title": "THE BORWEIN",
      "kicker": "a run of integrals that equal pi-over-two until they suddenly do not",
      "accent": "#35ffb0",
      "blurb": "The Borwein integrals in the 5-window house format — the most famous 'pattern that breaks' in mathematics. Using sinc(x) = sin(x)/x, ∫₀^∞ sinc(x) dx = π/2. Add a factor: ∫ sinc(x)·sinc(x/3) dx = π/2. Keep going — sinc(x/5), sinc(x/7), … up to sinc(x/13) — and every one is exactly π/2. Then include sinc(x/15) and the answer drops to π/2 minus a whisper (about 2×10⁻¹¹). The reason is exact: the integral stays π/2 as long as the tail 1/3+1/5+… stays ≤ 1, and 1/3+…+1/13 = 0.9551 < 1 while adding 1/15 tips it to 1.0218 > 1. Verified live: the reciprocal sum 1/3+…+1/13 is confirmed < 1 while +1/15 exceeds 1 (the exact mechanism), and the integrals through sinc(x/7) and sinc(x/13) are numerically π/2. Neon-noir traced. See the reciprocal tail creeping to 1 in 1D, the π/2-until-it-breaks check in 2D, and the hidden-threshold inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "49f610043a8791c41515cf08dd1cbf14ff0791c6e3eea5d966e0c5561f287f74"
    },
    {
      "slug": "the-sophie-germain",
      "title": "THE SOPHIE GERMAIN",
      "kicker": "an algebraic identity that factors a sum of two fourth powers",
      "accent": "#35ffb0",
      "blurb": "Sophie Germain's identity in the 5-window house format — a small algebraic key that unlocks a family of factorizations: a⁴ + 4b⁴ = (a²−2ab+2b²)(a²+2ab+2b²). A sum of two fourth powers, which looks irreducible, splits cleanly into two quadratics. Setting b=1 gives the classic corollary: n⁴+4 is composite for every n>1, since n⁴+4 = (n²−2n+2)(n²+2n+2) and both factors exceed 1 (the lone exception is n=1, giving 5). The same Sophie Germain studied Sophie Germain primes — primes p for which 2p+1 is also prime (2, 3, 5, 11, 23, …). Verified live: a⁴+4b⁴ = (a²−2ab+2b²)(a²+2ab+2b²) exactly for all |a|,|b| ≤ 30; n⁴+4 is confirmed composite for 2 ≤ n ≤ 200; and the Sophie Germain primes up to 200 are listed. Neon-noir traced. See n⁴+4 splitting in 1D, the factorization in 2D, and the sum-is-a-product inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "c4daa1c86328fb1851a5aea961a1fb1fe774a861b119e4bfb86b9a78feee57d8"
    },
    {
      "slug": "the-pascal-theorem",
      "title": "THE PASCAL THEOREM",
      "kicker": "six points on a conic whose opposite sides meet on one line",
      "accent": "#21e6ff",
      "blurb": "Pascal's theorem in the 5-window house format — the 'mystic hexagram', found by Blaise Pascal at sixteen. Take any six points on a conic (circle, ellipse, parabola, or hyperbola) and join them in order into a hexagon. Extend the three pairs of opposite sides until each pair meets. Those three intersection points always lie on a single straight line, the Pascal line. It holds no matter how the six points are placed or labelled, and it is purely projective — only incidence matters, not distance or angle. Its projective dual is Brianchon's theorem. Verified live: for tens of thousands of random hexagons inscribed in an ellipse, the three opposite-side intersection points are collinear — the triangle they form has normalized area below 1e-6. Neon-noir traced. See the hexagon and its Pascal line in 1D, the collinearity check in 2D, and the six-points-one-line inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "91cacd3d8281ed25d44087b88f501c89b43354f92962b9f20135b01a853517c2"
    },
    {
      "slug": "the-isoperimetric",
      "title": "THE ISOPERIMETRIC",
      "kicker": "the circle enclosing the most area for its perimeter",
      "accent": "#ffcf4a",
      "blurb": "The isoperimetric inequality in the 5-window house format — the oldest optimization question: of all closed curves with a given perimeter, which encloses the most area? The answer, known to the ancients as 'Dido's problem' but only rigorously proved in the 19th century, is the circle. For any simple closed curve of length L enclosing area A: 4πA ≤ L², with equality only for the circle. The ratio 4πA/L² (the isoperimetric quotient) is at most 1, and a regular n-gon achieves π/(n·tan(π/n)), which climbs toward 1 as the polygon rounds out into a circle. Verified live: for tens of thousands of random convex polygons, 4πA/L² never exceeds 1, and the regular n-gon quotient increases toward 1 (0.605, 0.785, 0.907, 0.977, 0.999 for n = 3, 4, 6, 12, 60). Neon-noir traced. See a polygon vs the equal-perimeter circle in 1D, the quotient climbing to 1 in 2D, and the maximal-circle inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "b61c557fe3149c7c237bd7276b64ffecd0a77b980e9b0af6f660267018c86575"
    },
    {
      "slug": "the-apollonius-circle",
      "title": "THE APOLLONIUS CIRCLE",
      "kicker": "the circle traced by a constant distance-ratio",
      "accent": "#b06bff",
      "blurb": "The circle of Apollonius in the 5-window house format — where are all the points whose distances to two fixed points keep a fixed ratio? Given points A and B and a ratio k ≠ 1, the set of all P with |PA|/|PB| = k is not a line or an oval — it is a perfect circle. Its diameter runs between the two points that divide segment AB in ratio k, internally and externally. As k → 1 the circle swells to the perpendicular bisector; for k far from 1 it tightens around the nearer point. Apollonius of Perga catalogued these circles around 200 BCE; they underlie hyperbolic distance and the geometry of pursuit. Verified live: for thousands of random A, B, k, every point sampled on the constructed circle has |PA|/|PB| = k to ~1e-14, while points off the circle do not. Neon-noir traced. See the circle with its distance-ratio spokes in 1D, the constant-ratio check in 2D, and the ratio-draws-a-circle inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "a3e5bac45678d035396572a717b4aed7955e087711fca5fd439cb2f7dbc021e1"
    },
    {
      "slug": "the-eisenstein-triples",
      "title": "THE EISENSTEIN TRIPLES",
      "kicker": "integer triangles with a sixty-degree angle",
      "accent": "#ff8a3c",
      "blurb": "Eisenstein triples in the 5-window house format — the 60° cousins of Pythagorean triples. A Pythagorean triple gives an integer-sided triangle with a right angle (a²+b²=c²). An Eisenstein triple gives an integer-sided triangle with a 60° angle: by the law of cosines with cos60° = ½, the side c opposite the 60° corner satisfies a²−ab+b² = c². The smallest nontrivial one is (3, 8, 7): 9−24+64 = 49 = 7², a triangle whose angle opposite the 7 is exactly 60°. Swap the sign for the 120° version, a²+ab+b² = c² (e.g. 3, 5, 7). They tile naturally on the triangular (Eisenstein) lattice. Verified live: a search finds primitive integer triples with a²−ab+b² = c², and the law of cosines confirms the angle opposite c is exactly 60°; the 120° analog a²+ab+b²=c² is found too. Neon-noir traced. See the (3,8,7) triangle to scale in 1D, the a²−ab+b²=c² check in 2D, and the Pythagoras-retuned-to-60° inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "0d53c4d933f4bb05a8284dbe3d026732a6e7fdd00ac21e9a81a1014e62542bf1"
    },
    {
      "slug": "the-apery-constant",
      "title": "THE APERY CONSTANT",
      "kicker": "an irrational constant summing the reciprocal cubes",
      "accent": "#b06bff",
      "blurb": "Apéry's constant in the 5-window house format — the value ζ(3) = Σ 1/n³ = 1 + 1/8 + 1/27 + … ≈ 1.2020569. While Euler found closed forms for ζ(2) = π²/6 and every even argument, ζ(3) has resisted every simple closed form. In 1978 Roger Apéry stunned mathematicians by proving ζ(3) is irrational — using a rapidly converging series he discovered: ζ(3) = (5/2) Σ (−1)^{n−1} / (n³ C(2n,n)). Each term of Apéry's series adds several correct digits where the plain sum crawls. The constant appears in quantum electrodynamics (the electron's magnetic moment) and random-minimum-spanning-tree statistics. Verified live: the direct sum Σ1/n³ converges to 1.2020569…, and Apéry's series reaches the same value to ~1e-14 in about 20 terms — the two agree. Neon-noir traced. See both series climbing to ζ(3) in 1D, direct-vs-Apéry in 2D, and the hidden-fast-road inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "c3088dd5d90182125ce650e57227df258c0a5d7783d97ff9a59f5945bff64106"
    },
    {
      "slug": "the-ulam-numbers",
      "title": "THE ULAM NUMBERS",
      "kicker": "a sequence that builds itself from unique sums",
      "accent": "#21e6ff",
      "blurb": "Ulam numbers in the 5-window house format — a sequence that builds itself. Start with 1 and 2. Each new term is the smallest integer larger than the last that can be written as a sum of two distinct earlier Ulam numbers in exactly one way. That single rule generates 1, 2, 3, 4, 6, 8, 11, 13, 16, 18, 26, 28, 36, … — 3 = 1+2, 4 = 1+3, but 5 is excluded (5 = 1+4 = 2+3, two ways). Devised by Stanisław Ulam in 1964, the sequence looks random yet has a startling hidden regularity: a mysterious 'almost period' of about 21.6 governs where its terms fall — still not fully explained. Verified live: the self-generating rule reproduces the known Ulam sequence exactly — the first 26 terms match 1, 2, 3, 4, 6, 8, …, 99 — and each term has exactly one representation. Neon-noir traced. See the numbers with 5 excluded in 1D, each term's unique sum in 2D, and the self-selecting-set inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "f95af8778af41f57953d8cb5f8fb84d75763172d3fbe7327a21c95d7770d352e"
    },
    {
      "slug": "the-takagi",
      "title": "THE TAKAGI",
      "kicker": "a curve continuous everywhere and smooth nowhere",
      "accent": "#ffcf4a",
      "blurb": "The Takagi function in the 5-window house format — the blancmange curve, continuous everywhere and differentiable nowhere. It is built by piling up ever-finer triangle waves: T(x) = Σ s(2ⁿx)/2ⁿ, where s(x) is the distance from x to the nearest integer. Each layer is a zig-zag half as tall and twice as frequent as the last; their sum converges to a continuous curve that wobbles at every scale, so no tangent line ever exists. It obeys the self-similar functional equation T(x) = s(x) + ½T(2x), reaches its maximum of exactly 2/3 at x = 1/3 and 2/3, and resembles a blancmange pudding — hence the name (Teiji Takagi, 1901). Verified live: the functional equation holds across the interval to ~1e-15; T(1/2) = 1/2, T(1/3) = 2/3, and the maximum equals 2/3. Neon-noir traced. See the blancmange curve in 1D, the piling layers in 2D, and the self-similarity inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "2645d7713cba2ac198ed350fe4e87e9084980054a42ff47cdbac622a0e30fdc2"
    },
    {
      "slug": "the-pizza-theorem",
      "title": "THE PIZZA THEOREM",
      "kicker": "a pizza split fairly from any interior cut-point",
      "accent": "#ff8a3c",
      "blurb": "The pizza theorem in the 5-window house format — a slice of surprising fairness. Take a circular pizza and pick any point P inside it — not necessarily the centre. Make cuts through P at equal angles, and if you make eight slices (four cuts, 45° apart), then two people taking alternate slices always get exactly equal total area — no matter where P was or how the knife was rotated. The off-centre gains of the big slices are exactly cancelled by the losses of the small ones. It works for any number of slices that is a multiple of four and at least eight; curiously, for four slices it fails (whoever gets the slices containing the centre wins). Verified live: for eight slices from a random interior point, the two alternating groups have equal area (to ~1e-4 by fine integration), the total equals πR², and the four-slice control is confirmed unequal. Neon-noir traced. See the coloured alternate slices in 1D, the equal sums in 2D, and the fairness-from-alternation inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "2f2df0446e99e29c78d26b7fd55c3fc10680c4944c2fba33b60200b71f9ebe0d"
    },
    {
      "slug": "the-brianchon",
      "title": "THE BRIANCHON",
      "kicker": "six tangents to a conic whose diagonals meet at a point",
      "accent": "#35ffb0",
      "blurb": "Brianchon's theorem in the 5-window house format — the exact mirror-image of Pascal's. Where Pascal takes six points on a conic and finds a line, Brianchon takes six lines tangent to a conic — a hexagon circumscribed about it — and finds a point: the three main diagonals (joining opposite vertices) all pass through one common point. This point-line swap is the deepest idea in projective geometry, duality: every theorem about points on a conic has a twin about tangent lines, trading 'point' for 'line', 'lies on' for 'passes through', 'collinear' for 'concurrent'. Charles-Julien Brianchon proved it in 1810. Verified live: for tens of thousands of random hexagons circumscribed about an ellipse, the three main diagonals are concurrent — the third diagonal passes through the intersection of the first two, normalized residual below 1e-6. Neon-noir traced. See the circumscribed hexagon and its Brianchon point in 1D, the concurrency check in 2D, and the dual-of-Pascal inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "4f03642bcb8415689104145eca65206f3aae29cef93be8a00014c05f65554bbd"
    },
    {
      "slug": "the-bernoulli-numbers",
      "title": "THE BERNOULLI NUMBERS",
      "kicker": "a rational sequence hiding inside power sums and the zeta values",
      "accent": "#b06bff",
      "blurb": "The Bernoulli numbers in the 5-window house format — a sequence of rationals that surface all over mathematics: 1, −½, 1/6, 0, −1/30, 0, 1/42, 0, −1/30, … They are defined by the recurrence Σ C(n+1,k) B_k = 0, and every odd-indexed one past B₁ is exactly zero. They give the coefficients in Faulhaber's formulas for sums of powers, the Taylor series of tan and coth — and, most beautifully, Euler's closed form for the even zeta values: ζ(2n) = (−1)^{n+1} B_{2n} (2π)^{2n} / (2·(2n)!). Setting n = 1 recovers ζ(2) = π²/6 from B₂ = 1/6. Verified live: the recurrence yields B₂ = 1/6, B₄ = −1/30, B₆ = 1/42, all odd B (past B₁) zero; and Euler's formula gives ζ(2) = π²/6 and ζ(4) = π⁴/90 to ~1e-10. Neon-noir traced. See the sequence with vanishing odd terms in 1D, the ζ(2n) rebuild in 2D, and the sums-pinned-to-a-sequence inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "f247ee08e05dfaf30df6545a42cf0a359ef06db90e3dfdcb39ffaff4a6fe7e5c"
    },
    {
      "slug": "the-lucky-euler",
      "title": "THE LUCKY EULER",
      "kicker": "a polynomial that spits primes forty times in a row",
      "accent": "#35ffb0",
      "blurb": "Euler's lucky numbers in the 5-window house format — a startling coincidence Euler found in 1772: the polynomial n²+n+41 produces a prime for every n from 0 to 39 — forty primes in an unbroken run: 41, 43, 47, 53, 61, 71, …, 1601. The streak finally breaks at n = 40, where 40²+40+41 = 1681 = 41². Even beyond that it stays astonishingly prime-rich (about 58% of values up to n = 1000 are prime). The magic isn't luck: 41 is the largest of the six 'lucky numbers of Euler', tied to the fact that the imaginary quadratic field of discriminant −163 = 1 − 4·41 has class number one — unique factorization, the deepest reason the primes line up. Verified live: n²+n+41 is prime for all n = 0 to 39, composite at n = 40 (= 41²), and about 58% of values up to n = 1000 are prime. Neon-noir traced. See the 40-prime run breaking at 41² in 1D, the per-n factorization in 2D, and the class-number-one inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "58bd1e49be1d1fcbd65de66c02f3ee50786a2a35c324e74e9f9795d9f1807f94"
    },
    {
      "slug": "the-nine-point-circle",
      "title": "THE NINE-POINT CIRCLE",
      "kicker": "nine special triangle points on one circle",
      "accent": "#21e6ff",
      "blurb": "The nine-point circle in the 5-window house format — one of the most elegant facts about a triangle: nine special points all lie on a single circle. They are the three midpoints of the sides, the three feet of the altitudes, and the three midpoints of the segments from each vertex to the orthocentre. No matter how the triangle is shaped, these nine points are perfectly concyclic. The circle's centre N is the midpoint between the circumcentre O and the orthocentre H (so N sits on the Euler line), and its radius is exactly half the circumradius, R/2. It touches the incircle and the three excircles (Feuerbach's theorem). Verified live: for tens of thousands of random triangles, all nine points are equidistant from N = midpoint(O, H), at distance exactly R/2, to ~1e-14. Neon-noir traced. See the nine points on one circle in 1D, the nine equal radii in 2D, and the three-constructions-one-circle inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "1c5eec186ffa0e791234915462605d528d09928066f2c6aa97f67616dfe0f4fb"
    },
    {
      "slug": "the-gaussian-integral",
      "title": "THE GAUSSIAN INTEGRAL",
      "kicker": "a bell curve whose area is the square root of pi",
      "accent": "#ffcf4a",
      "blurb": "The Gaussian integral in the 5-window house format — the beautiful fact that the area under the bell curve is the square root of π: ∫_{−∞}^{∞} e^{−x²} dx = √π. There is no elementary antiderivative for e^{−x²} — you cannot integrate it term by term — yet the total area is exactly √π ≈ 1.7724539. The classic trick squares the integral and switches to polar coordinates, turning an impossible one-dimensional integral into an easy two-dimensional one. Rescaled, it gives the normalization of the normal distribution: ∫ e^{−x²/2} dx = √(2π), which is why the bell curve of statistics divides by √(2π). Verified live: numerical integration of e^{−x²} over the real line gives 1.7724539… = √π to ~1e-7, and e^{−x²/2} integrates to √(2π). Neon-noir traced. See the shaded bell curve area in 1D, the quadrature converging to √π in 2D, and the square-it-to-solve-it inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "877401723d7043597556461a06f33d71ea4fa7d951347cd148d2d2d7e5169c3a"
    },
    {
      "slug": "the-monty-hall",
      "title": "THE MONTY HALL",
      "kicker": "a game show where switching doubles your odds",
      "accent": "#ff8a3c",
      "blurb": "The Monty Hall problem in the 5-window house format — the most famous counter-intuitive result in probability. You pick one of three doors; behind one is a car, behind the others goats. The host — who knows where the car is — opens a different door revealing a goat, then offers you the chance to switch. Should you? Yes: switching wins 2/3 of the time, staying only 1/3. Your first pick is right 1/3 of the time, so the other door hides the car the remaining 2/3 — and the host's reveal concentrates all of that onto the single unopened door. It scales: with N doors and one goat revealed, switching to a random remaining door wins (N−1)/(N(N−2)). Verified live: a Monte-Carlo simulation gives switch ≈ 2/3 and stay ≈ 1/3 for three doors, and matches (N−1)/(N(N−2)) for four and five doors. Neon-noir traced. See the three doors and outcomes in 1D, the win-rate settling on 2/3 in 2D, and the where-did-the-2/3-go inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "f76a548d72eca54e9cc1fec7dcdf1be475d5ac2f7f501612e360841cae75afed"
    },
    {
      "slug": "the-euler-mascheroni",
      "title": "THE EULER-MASCHERONI",
      "kicker": "the constant left over between the harmonic series and the logarithm",
      "accent": "#b06bff",
      "blurb": "The Euler–Mascheroni constant in the 5-window house format — γ ≈ 0.5772156649, the mysterious gap between two things that both grow without bound: the harmonic series H_n = 1 + 1/2 + … + 1/n, and the natural logarithm ln(n). Both march to infinity, but their difference settles onto a single fixed number: γ = lim(H_n − ln n). It appears in the gamma function, the prime-counting function, and the zeta function — yet after 250 years no one knows whether γ is even irrational. The plain limit crawls (error ~1/2n), but a corrected form H_n − ln n − 1/(2n) + 1/(12n²) sprints to γ. Verified live: H_n − ln n approaches 0.5772156649… (to ~1e-6 at n = 2×10⁶), and the corrected form reaches γ to ~1e-12. Neon-noir traced. See the gap settling onto γ in 1D, plain-vs-corrected in 2D, and the two-infinities-one-remainder inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "5a5f98328ee3fc68e01d94a9bdf3316e7a6342de750af64174461725bb5a97db"
    },
    {
      "slug": "the-birthday-paradox",
      "title": "THE BIRTHDAY PARADOX",
      "kicker": "twenty-three people enough to share a birthday",
      "accent": "#ff8a3c",
      "blurb": "The birthday paradox in the 5-window house format — the shock that in a room of just 23 people, it is more likely than not that two share a birthday. It feels wrong — 365 days, surely you'd need ~180 people? But you are not matching one fixed birthday; you are checking all pairs, and 23 people make 253 pairs. The probability of at least one shared birthday is 1 − (365/365)(364/365)…((365−n+1)/365); at n = 23 it crosses 0.507, past a half. By 57 people it is over 99%. The counter-intuition comes from confusing 'a match with me' (linear) with 'a match among anyone' (quadratic in the number of people). Verified live: the exact formula gives P = 0.5073 at 23 people (> 1/2) and 0.9901 at 57; a Monte-Carlo simulation matches across several group sizes. Neon-noir traced. See the probability crossing ½ at 23 in 1D, exact-vs-simulation in 2D, and the count-pairs-not-people inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "aa3c94f5350d0fdac7e7c31197d162848c89443687cb8bb396e1788827a6f9cf"
    },
    {
      "slug": "the-fibonacci-gcd",
      "title": "THE FIBONACCI-GCD",
      "kicker": "a greatest common divisor that stays inside the Fibonacci sequence",
      "accent": "#21e6ff",
      "blurb": "The Fibonacci–GCD identity in the 5-window house format — a jewel of divisibility: the greatest common divisor of two Fibonacci numbers is itself a Fibonacci number, and exactly the one whose index is the gcd of the indices. In symbols, gcd(F_m, F_n) = F_{gcd(m,n)}. For example gcd(F₁₂, F₁₈) = gcd(144, 2584) = 8 = F₆, and gcd(12,18) = 6. The Fibonacci sequence carries the whole divisibility structure of the integers on its back. A clean corollary: for m ≥ 3, F_m divides F_n if and only if m divides n — every third Fibonacci is even (divisible by F₃ = 2), every fourth divisible by F₄ = 3, and so on. Verified live with exact BigInt: gcd(F_m, F_n) = F_{gcd(m,n)} for all m, n up to 40, and F_m | F_n ⟺ m | n for m ≥ 3. Neon-noir traced. See the gcd land back on F_{gcd(m,n)} in 1D, the identity check in 2D, and the sequence-commutes-with-gcd inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "88ce605a58f657a04b72b533f953ab651124dffcd71832541199516d7fb2960b"
    },
    {
      "slug": "the-gergonne",
      "title": "THE GERGONNE",
      "kicker": "triangle cevians to the incircle meeting at one point",
      "accent": "#ffcf4a",
      "blurb": "The Gergonne point in the 5-window house format — a hidden meeting-point every triangle carries. Inscribe the incircle — the circle tangent to all three sides. It touches the sides at three contact points. Now draw a line (a cevian) from each vertex to the contact point on the opposite side. Astonishingly, all three of these lines meet at a single point: the Gergonne point. It works for every triangle, guaranteed by Ceva's theorem, because the contact point on side a sits at distance s−b from one end and s−c from the other (s the semiperimeter), and the three ratios multiply to exactly 1. Named for Joseph Diez Gergonne. Verified live: for tens of thousands of random triangles, the three cevians from the vertices to the incircle's contact points are concurrent — the third passes through the intersection of the first two, normalized residual below 1e-6. Neon-noir traced. See the incircle, contact points, and meeting cevians in 1D, the concurrency check in 2D, and the three-lines-one-meeting inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "a6608ffb19c2470a2f58e8de5cfb9ae742455448e9f3c5e20fa5eea49fd2d6ff"
    },
    {
      "slug": "the-vandermonde-determinant",
      "title": "THE VANDERMONDE DETERMINANT",
      "kicker": "a determinant that factors into pairwise differences",
      "accent": "#35ffb0",
      "blurb": "The Vandermonde determinant in the 5-window house format — a stunningly clean answer to a messy-looking question. Build a matrix whose rows are the powers of some numbers x₀, x₁, …, x_{n−1} — row i is (1, x_i, x_i², …, x_i^{n−1}). Its determinant, which looks like it should be a horrible polynomial, factors perfectly into a product of all pairwise differences: det = ∏_{i<j} (x_j − x_i). So the determinant is zero exactly when two of the numbers coincide — which is why polynomial interpolation through distinct points always has a unique solution. It underlies interpolation, coding theory (Reed–Solomon), and the theory of symmetric functions. Verified live: for tens of thousands of random node sets (n = 3 to 6), the determinant computed by Gaussian elimination equals ∏_{i<j}(x_j − x_i) to ~1e-9. Neon-noir traced. See the power matrix and its product-of-gaps determinant in 1D, det-vs-product in 2D, and the determinant-is-just-the-gaps inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "b4ac69822e4cbaf6c9e5e21f3b54d088b405cc64f95600d9f745a6ebabf7aa6c"
    },
    {
      "slug": "the-ramanujan-pi",
      "title": "THE RAMANUJAN PI",
      "kicker": "a series adding eight digits of pi per term",
      "accent": "#b06bff",
      "blurb": "Ramanujan's series for 1/π in the 5-window house format — one of the fastest-converging formulas ever written, produced by Srinivasa Ramanujan in 1914 seemingly out of nowhere: 1/π = (2√2/9801) Σ (4k)!(1103+26390k)/((k!)⁴ 396^{4k}). The very first term (k=0) already gives π correct to seven digits, and each further term adds about eight more. Ramanujan gave no proof; it was only rigorously established decades later. The same family — refined by the Chudnovsky brothers — is what modern record computations of π to trillions of digits actually use. Verified live: the single k=0 term gives π to ~1e-7, one more term to ~1e-15 (machine precision), and by two terms it equals π to the last bit. Neon-noir traced. See the error plunging ~8 digits/term in 1D, π rebuilt to more digits in 2D, and the eight-digit-rung ladder inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "8a007d4621fb05757417726d32753b7976eef74b2e68b85d14619218316c6cdc"
    },
    {
      "slug": "the-galton-board",
      "title": "THE GALTON BOARD",
      "kicker": "beads falling into a bell curve",
      "accent": "#21e6ff",
      "blurb": "The Galton board in the 5-window house format — Francis Galton's 1873 bean machine that turns pure randomness into a clean bell curve. Drop a ball through n rows of offset pegs; at each peg it bounces left or right with probability ½. After n rows it lands in bin k, having gone right k times — and the chance of that is exactly the binomial C(n,k)/2ⁿ. Thousands of balls pile up into the binomial distribution, and by the de Moivre–Laplace theorem that binomial approaches the normal (Gaussian) bell curve as n grows. It is the most physical demonstration there is of the Central Limit Theorem. Verified live: simulating thousands of balls through 16 rows reproduces the bin frequencies C(n,k)/2ⁿ, and that binomial matches the normal density N(n/2, n/4). Neon-noir traced. See the beads bouncing into a histogram in 1D, sim-vs-binomial-vs-normal in 2D, and the crowd-makes-a-curve inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "85c913ea1edd44fc76056e66bc1fae935b4e91ae7d2928310f039c4994055696"
    },
    {
      "slug": "the-steiner-porism",
      "title": "THE STEINER PORISM",
      "kicker": "a ring of circles that always closes",
      "accent": "#ff8a3c",
      "blurb": "Steiner's porism in the 5-window house format — a beautiful all-or-nothing fact about circles. Take two circles, one inside the other (not concentric), and thread a chain of circles in the gap, each tangent to both boundary circles and to its neighbours. Steiner's porism says: if the chain ever closes up perfectly — the last circle exactly tangent to the first — then it closes for every starting position, using the same number of circles. Either all chains close or none do; there is no in-between. The proof is magic: an inversion turns the two circles concentric, where the chain is just a ring of equal circles and closure is obvious by symmetry. Verified live: a closing chain is built concentrically (closure ratio sin(π/n) = (R−r)/(R+r)) and then inverted to a non-concentric pair; the image chain stays tangent to both boundaries and to its neighbours and closes — for every starting angle, to ~1e-15. Neon-noir traced. See the threaded chain in 1D, closure from any start in 2D, and the invert-to-symmetry inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "2e7b775e35e3d284cd796419e0f92c9f3c8d35925af864749e07ba781af8fe81"
    },
    {
      "slug": "the-euler-criterion",
      "title": "THE EULER CRITERION",
      "kicker": "a single power that tells a square from a non-square",
      "accent": "#ffcf4a",
      "blurb": "Euler's criterion in the 5-window house format — a single exponentiation that decides whether a number is a perfect square modulo a prime. For an odd prime p and any a not divisible by p, a^((p−1)/2) ≡ ±1 (mod p) — and it is +1 exactly when a is a quadratic residue (some x with x² ≡ a mod p exists), −1 when it is not. That sign is the Legendre symbol (a|p). So without ever searching for a square root, one modular power tells you whether one exists. It is the computational heart of quadratic reciprocity and of primality tests like Solovay–Strassen. Verified live: for every odd prime p up to 200 and every a from 1 to p−1, a^((p−1)/2) mod p equals +1 or p−1, and it is +1 exactly when a is a quadratic residue (checked independently by squaring). Neon-noir traced. See residues split into squares/non-squares in 1D, the criterion vs Legendre in 2D, and the no-root-taken inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "ac8f79a175217f444845dde8e791688590ee5d1eca6bd38f5f9e9d120e0488ee"
    },
    {
      "slug": "the-lah-numbers",
      "title": "THE LAH NUMBERS",
      "kicker": "numbers linking the rising and falling factorials",
      "accent": "#35ffb0",
      "blurb": "The Lah numbers in the 5-window house format — the exact exchange rate between the two natural kinds of factorial. The rising factorial x^(n) = x(x+1)…(x+n−1) and the falling factorial (x)_k = x(x−1)…(x−k+1) each build a staircase product, one climbing and one descending. The unsigned Lah numbers convert one into the other: x^(n) = Σ_k L(n,k)(x)_k, with the closed form L(n,k) = C(n−1,k−1)·n!/k!. Combinatorially, L(n,k) counts the ways to split n labelled items into k non-empty ordered lists. They sit between the Stirling numbers as the 'both-ordered' case, and satisfy L(n,1) = n!, L(n,n) = 1. Verified live: L(n,k) = C(n−1,k−1)·n!/k! is an integer with L(n,1) = n! and L(n,n) = 1, and the identity x^(n) = Σ_k L(n,k)(x)_k holds exactly for a range of x and n. Neon-noir traced. See the Lah triangle in 1D, the rising-from-falling rebuild in 2D, and the staircase-exchange inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "a65ef29a9d71c8738f0482c52b58fde380f062a51488e8254cd611b66c7026cb"
    },
    {
      "slug": "the-schwarz-lantern",
      "title": "THE SCHWARZ LANTERN",
      "kicker": "an inscribed surface whose area depends on how you refine it",
      "accent": "#b06bff",
      "blurb": "The Schwarz lantern in the 5-window house format — the counterexample that shattered an 'obvious' belief: that inscribed polyhedral surfaces must converge to a curved surface's area, the way inscribed polygons converge to a curve's length. Hermann Schwarz (1880) triangulated a cylinder into an antiprism 'lantern' — m points per ring, n rings, zig-zag triangles — with exact area 2mn·sin(π/m)·√((h/n)² + r²(1−cos(π/m))²), whose limit depends on the refinement path: n = m converges to the true area 2πrh; n = m² converges to the wrong constant 2π√(1+π⁴/4) ≈ 31.64; n = m³ diverges to infinity as the triangles tilt into accordion pleats. Surface area cannot be defined by naive inscription. Verified live: the closed form yields all three limits, with n = m³ doubling in area as m doubles. Neon-noir traced. See the pleated bands in 1D, the three regimes refined live in 2D, and the path-chooses-the-limit inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "09c9d18ebb24cf25159b87a8d65bd2158bd28ea5b0c287336f0c82aa837d060d"
    },
    {
      "slug": "the-morrie",
      "title": "THE MORRIE LAW",
      "kicker": "three cosines multiplying to exactly one eighth",
      "accent": "#ffcf4a",
      "blurb": "Morrie's law in the 5-window house format — the identity cos20°·cos40°·cos80° = 1/8, three unremarkable-looking cosines multiplying to an exact rational. Richard Feynman kept the name all his life: a boy called Morrie Jacobs showed it to him in his father's leather shop. The secret is the doubling cascade: for any θ, ∏cos(2^k θ) = sin(2ⁿθ)/(2ⁿ sinθ) — each cosine doubles the angle via sin2x = 2sinx·cosx and the product telescopes. At θ = 20° the cascade lands on sin160°, which equals sin20° exactly — the sines cancel and only 1/2³ = 1/8 survives. Verified live: the product is 0.125 to machine precision, the telescoping identity holds for thousands of random θ and n (worst ~1e-16), and sin160° = sin20° exactly. Neon-noir traced. See the doubling cascade on the circle in 1D, product-vs-closed-form in 2D, and the telescope inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "092aa4e0cb464904851a9f3f7ffd4da90d60bf35b733e894d6a7b629279fb97e"
    },
    {
      "slug": "the-lemoine-point",
      "title": "THE LEMOINE POINT",
      "kicker": "medians reflected over bisectors meeting at one point",
      "accent": "#21e6ff",
      "blurb": "The Lemoine point in the 5-window house format — take the three medians of a triangle and reflect each one over the angle bisector at its vertex. The three reflected lines — the symmedians — all pass through a single point K, one of the most studied points in triangle geometry. In barycentric coordinates it is simply (a²:b²:c²), and it carries a beautiful signature: its perpendicular distances to the three sides are proportional to the side lengths themselves — equivalently, K uniquely minimizes the sum of squared distances to the sides. Émile Lemoine presented it in 1873, launching 'the geometry of the triangle'. Verified live: for tens of thousands of random triangles the three reflected medians are concurrent, the meeting point matches (a²:b²:c²) independently, and its side-distances are proportional to a, b, c. Neon-noir traced. See medians become symmedians in 1D, the three identities checked in 2D, and the weigh-the-corners inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "7ff7d1b608158dbfa597d61432e7b458bac56bd3532aa24a08e155924764c504"
    },
    {
      "slug": "the-euler-brick",
      "title": "THE EULER BRICK",
      "kicker": "a brick whose faces are all Pythagorean but whose heart is an open problem",
      "accent": "#ff8a3c",
      "blurb": "The Euler brick in the 5-window house format — a box whose edges and all three face diagonals are whole numbers. The smallest, found by Paul Halcke in 1719, has edges 44, 117, 240: face diagonals 125, 244, 267 — three Pythagorean triples sharing legs pairwise. But the brick guards a missing gem: its space diagonal is √73225 ≈ 270.6, not an integer. A brick with integer space diagonal too — a perfect cuboid — has never been found and never been ruled out: one of the oldest open problems in number theory, searched past 5×10¹¹ with no example. Verified live: 44²+117² = 125², 44²+240² = 244², 117²+240² = 267², an exhaustive search confirms no Euler brick has largest edge below 240, and √73225 is verified non-integer. Neon-noir traced. See the brick with its three integer diagonals in 1D, the face checks in 2D, and the unclaimed-bounty inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "9e9ab5eb1a5dea8827bf240ae95b1ec1561eb60ec32d04c90ab9c038c86a31e4"
    },
    {
      "slug": "the-string-that-remembers",
      "title": "THE STRING THAT REMEMBERS",
      "kicker": "a burst of noise that decays into a musical note",
      "accent": "#35ffb0",
      "blurb": "Karplus–Strong synthesis in the 5-window house format — a convincing plucked string out of almost nothing. Fill a short buffer of N samples with random noise, then loop it forever, replacing each sample with the average of the two samples one period ago: y[n] = ½(y[n−N] + y[n−N−1]). The averaging is a gentle low-pass filter inside the loop: every pass, the jagged noise gets smoother and quieter, high harmonics dying first exactly as on a real string. The half-sample in the average makes the true period N+½, so the fundamental lands at f₀ = fs/(N+½). A burst of static becomes a note with a natural decay — the algorithm behind countless early digital guitars. Verified live: for several N, the fundamental measured by autocorrelation (with sub-sample peak interpolation) matches fs/(N+½) to under 1 Hz, and block-RMS energy decays monotonically. Neon-noir traced. See noise settle into a tone in 1D, the pitch check + decay bars in 2D, and the store-the-loop inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "02a6982f0d8c817f99bb6610a0029730cac9ffa2e87442365ab44d4caa546619"
    },
    {
      "slug": "the-loaded-dice-table",
      "title": "THE LOADED-DICE TABLE",
      "kicker": "a die loaded in constant time",
      "accent": "#ffcf4a",
      "blurb": "Walker's alias method in the 5-window house format — the constant-time loaded die. To sample from an arbitrary discrete distribution, the naive way walks a cumulative table (O(n)) or bisects it (O(log n)). Alias sampling builds two arrays — a probability table and an alias table — that repack the distribution into n equal columns, each holding at most two outcomes. A draw is then: pick a column uniformly, flip one biased coin, take the column's own outcome or its alias. One uniform, one comparison — O(1) forever, no matter how lopsided the distribution. Alastair Walker found it in 1974; Michael Vose gave the clean linear-time construction. Verified live: the finished table reconstructs the input probabilities exactly (mass audit to 1e-12), and a million draws land within 0.0007 of every target probability. Neon-noir traced. See the repacked columns in 1D, frequencies converging on targets in 2D, and the reshape-don't-search inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "de97f0993e104f1da6496ec6280d2f12053f5fe4fa75580d93354a35f1034b2c"
    },
    {
      "slug": "the-sparse-oracle",
      "title": "THE SPARSE ORACLE",
      "kicker": "every window pre-answered by two overlapping blocks",
      "accent": "#b06bff",
      "blurb": "The sparse table in the 5-window house format — constant-time range-minimum queries built on one forgiving fact: taking a minimum twice does no harm (min is idempotent). Precompute the minimum of every window whose length is a power of two — O(n log n) cells. Then any range [l, r], whatever its length, is covered by just two overlapping power-of-two blocks: one anchored at l, one ending at r. They may overlap heavily — with min, overlap is free. Answer = min of two table lookups; no tree walks, no recursion. Verified live: 5000 random range-minimum queries over a 5000-element array, each answered by exactly two lookups, all matching a brute-force scan. Neon-noir traced. See the two blocks bracketing a range in 1D, fast-vs-brute in 2D, and the idempotence-turned-speed inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "e56f087d310193c668d0a882d3d49211671c692f5f111cc049c695ca610bce46"
    },
    {
      "slug": "the-prophets-jump",
      "title": "THE PROPHET'S JUMP",
      "kicker": "express lanes built by coin flips",
      "accent": "#21e6ff",
      "blurb": "The skip list in the 5-window house format — William Pugh's 1989 sorted linked list that builds its own express lanes by coin flips. Every inserted node gets a random tower height: half the nodes reach level 2, a quarter level 3, an eighth level 4… A search starts on the top lane, skips far ahead, and drops down a level whenever the next stop would overshoot — the express train, then the local. No rebalancing, no rotations: probability does the balancing, and searches take O(log n) expected hops. It rivals balanced trees at a fraction of the code — Redis sorted sets run on one. Verified live: after thousands of random inserts and deletes, membership agrees exactly with a reference set; every level is sorted and nested inside the level below; and measured search hops stay within a small constant times log₂ n. Neon-noir traced. See the coin-flip towers in 1D, the hop meter in 2D, and the gamble-for-balance inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "0fdc28d4059a40f9c201dc21e2bca408eec6cdc211db9e55be40f40f307bb82d"
    },
    {
      "slug": "the-mamikon",
      "title": "THE MAMIKON",
      "kicker": "an annulus worth only its tangent length",
      "accent": "#35ffb0",
      "blurb": "Mamikon's annulus in the 5-window house format — the front door of visual calculus. Draw a ring between two concentric circles, and let ℓ be the half-length of a chord of the outer circle that just grazes the inner one. The ring's area is πℓ² — and the radii themselves have vanished: a skinny ring around a planet and a fat ring around a coin have the same area if their tangent half-chords match. Mamikon Mnatsakanian's 1959 insight (developed with Tom Apostol): sweep the tangent segment around the ring, then translate every segment to a common point — the tangent cluster forms a plain disk of radius ℓ, no integral in sight. Verified live: Monte-Carlo measurement of the ring's area returns πℓ² within 1% for inner radii spanning a 16× range with ℓ held fixed — the radius truly cancels. Neon-noir traced. See the grazing chord in 1D, the immovable measurement in 2D, and the tangent-cluster inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "441cf39d5b1f0f91b091af2794458a6afa0000d96515eede624a5287b5e47e74"
    },
    {
      "slug": "the-holditch",
      "title": "THE HOLDITCH",
      "kicker": "a curve reborn smaller by exactly pi-p-q",
      "accent": "#ff8a3c",
      "blurb": "Holditch's theorem in the 5-window house format — it sounds like a party trick and lands like a law of nature. Slide a chord of fixed length p + q around the inside of any smooth convex closed curve, keeping both ends on the curve. Mark the point dividing the chord into pieces p and q. That point traces a smaller closed curve inside — and the area between the two curves is exactly πpq: no dependence on the outer curve's shape, size, or lopsidedness. An ellipse, an egg, a rounded blob — the ring carved by the sliding point always measures πpq, the area of an ellipse with semi-axes p and q. Rev. Hamnet Holditch published it in 1858. Verified live: sliding a chord numerically around an ellipse (2400 positions, bisection for the far endpoint), the traced curve's shoelace area shows a deficit matching πpq to under 0.01% — for symmetric and lopsided splits alike. Neon-noir traced. See the sliding chord and its traced curve in 1D, the πpq toll in 2D, and the host-independent inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "65c3a84b6fd96043331f0664dd383fb5b6efc97a464fa3378f2c54e6f0c42a62"
    },
    {
      "slug": "the-prime-race",
      "title": "THE PRIME RACE",
      "kicker": "a prime race with one famous upset",
      "accent": "#35ffb0",
      "blurb": "The prime race in the 5-window house format — primes of the form 4k+3 versus primes of the form 4k+1. Dirichlet proved both teams infinite and asymptotically even, yet Chebyshev noticed in 1853 that team 3 is almost always ahead. The bias is structural (quadratic residues drag on team 1) but not absolute: at x = 26,861 — found by John Leech in 1957 — team 1 takes the lead for the first time, fleetingly, before team 3 recovers. Under Rubinstein–Sarnak's analysis, team 3 leads about 99.59% of logarithmic time. Verified live: sieving to 2,000,000, team 4k+3 leads at 99.76% of prime checkpoints, the first 4k+1 lead occurs at exactly x = 26,861, and the final score still favours team 3. Neon-noir traced. See the score curve dip once in 1D, the ledger in 2D, and the hunt-the-upsets inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "ace0a8d6750a77acccaa5a7cb77dd6973d20e69913ef3469fd8166ccf6632455"
    },
    {
      "slug": "the-gilbreath",
      "title": "THE GILBREATH",
      "kicker": "difference rows that always lead with one",
      "accent": "#b06bff",
      "blurb": "Gilbreath's conjecture in the 5-window house format — start with the primes and take absolute differences, then differences of those, again and again. The claim, noticed by Norman Gilbreath on a napkin in 1958 (and by François Proth in 1878, with a faulty proof): every row after the first begins with 1. Forever. Andrew Odlyzko verified it to astronomical height in 1993; a proof has never been found. The mechanism smells simple — the rows settle into 0s and 2s, which seems to protect the leading 1 — yet nobody can close the argument. Verified live: taking the 9,592 primes below 100,000, the first 500 difference rows all begin with 1. Neon-noir traced. See the difference triangle in 1D, the row-by-row audit with its 0/2 texture in 2D, and the verified-never-proven inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "22db0f5660eebd5891baf9eb11a2dd01f4e5c97f2509583742175477dce83224"
    },
    {
      "slug": "the-gabriels-horn",
      "title": "THE GABRIELS HORN",
      "kicker": "a horn holding finite paint behind an infinite wall",
      "accent": "#ff8a3c",
      "blurb": "Gabriel's horn in the 5-window house format — the trumpet made by spinning y = 1/x (x ≥ 1) around the axis. Torricelli worked it out in 1643 and scandalized the century: the volume is finite — exactly π — but the surface area is infinite. The volume integral π∫x⁻²dx converges; the surface is bounded below by the harmonic tail 2π∫dx/x, gaining about 2π every time the length multiplies by e, forever. Hence the painter's paradox: π units of paint fill the horn completely, yet no finite amount can coat its wall (resolution: mathematical paint has zero thickness). Verified live: quadrature gives volume(10⁶) converging to π while the surface gains ≈2π per e-fold at every scale tested. Neon-noir traced. See the narrowing profile in 1D, the frozen-volume/running-surface ledger in 2D, and the two-verdicts inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "4ab5e383536336f5395b75ae635530460d1ca83ccfa72994c62ae825f3f2879f"
    },
    {
      "slug": "the-devils-staircase",
      "title": "THE DEVILS STAIRCASE",
      "kicker": "a staircase that climbs without sloping",
      "accent": "#21e6ff",
      "blurb": "The devil's staircase in the 5-window house format — Cantor's function, climbing from 0 to 1 with slope zero almost everywhere. On the middle third of [0,1] it is flat at 1/2; on the middle thirds of what remains, flat at 1/4 and 3/4; and so on, flat on plateaus whose lengths sum to the entire interval. Every scrap of climbing is crowded onto the Cantor set — a dust of measure zero — yet the function is continuous and obeys crisp self-similarities: F(x/3) = F(x)/2 and F(1−x) = 1−F(x). The standard counterexample to the intuition that a function's rise must live where its derivative does. Verified live: monotone 0→1 over 10,001 samples; both self-similarities to ~1e-10; and 100% of the climb happens on the level-8 Cantor cover — just 3.9% of the interval, shrinking toward zero with depth. Neon-noir traced. See the staircase in 1D, the thinning strip that carries all the rise in 2D, and the dust-does-the-work inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "305cf2263362552592bc91c1cb7f15a629cbde6d2f015e01e93034e42e1a4e73"
    },
    {
      "slug": "the-conway-soldiers",
      "title": "THE CONWAY SOLDIERS",
      "kicker": "an army that cannot reach the fifth row",
      "accent": "#ff2fa6",
      "blurb": "Conway's soldiers in the 5-window house format — a peg-jumping army with an invisible ceiling. Fill the entire half-plane below a line with checkers; moves are jumps that remove the jumped soldier. Rows 1–4 above the line are reachable with armies of 2, 4, 8, 20. Row 5: never — not with a million soldiers, not with the whole infinite half-plane. Conway's 1961 proof weights each square by σ^d (d = distance to target) with σ = (√5−1)/2 satisfying σ²+σ = 1: no jump ever increases total weight, and the entire infinite army weighs exactly 1 — the target's own price. Any finite army weighs strictly less, so the target can never be paid for. Verified live: σ²+σ = 1 to machine precision; the half-plane sum evaluates to exactly 1.000000000000; and all three jump classes audited — toward-jumps preserve weight (~1e-18), sideways and away strictly lose. Neon-noir traced. See the fading army and the prize in 1D, the ledger in 2D, and the price-the-board inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "d790f7f55c750cd07cd7a9d9cecab8f87dadb9e15473433eaa2ff9dd74215413"
    },
    {
      "slug": "the-chiliagon",
      "title": "THE CHILIAGON",
      "kicker": "the thousand-gon no mind can picture",
      "accent": "#ffcf4a",
      "blurb": "The chiliagon in the 5-window house format — the regular 1000-sided polygon Descartes chose in the Sixth Meditation to split the mind in two: you can conceive a chiliagon perfectly — define it, reason about it, compute with it — but you cannot imagine it; every mental picture is indistinguishable from a circle. The numbers agree: perimeter 99.99984% of its circumcircle's; area within two parts in a hundred thousand of π; interior angle 179.64°; maximum bulge off the circle 4.9×10⁻⁶ of the radius — about a thousandth of a pixel at a 300-pixel radius. Conception outruns imagination, measurably. This is sphere 1000 of 2048 in THE FOLD — the corpus's own thousand-gon, seated in the genesis block. Verified live: perimeter, area, isoperimetric quotient 0.99999671, interior angle, and sagitta all computed and bounded against the circle. Neon-noir traced. See the polygon share every pixel with its circle in 1D, the metrics converging in 2D, and the known-never-pictured inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "20360f89339053c9f4eab1c71b2f04d5d072f62b7163e872eb7599a71818c1d5"
    },
    {
      "slug": "the-anscombe",
      "title": "THE ANSCOMBE",
      "kicker": "four datasets wearing the same statistics",
      "accent": "#21e6ff",
      "blurb": "Anscombe's quartet in the 5-window house format — four small datasets built to wear the same disguise: identical mean of x (9), variance of x (11), mean of y (7.50), variance of y (≈4.12), correlation (0.816), and regression line (y = 3.00 + 0.500x), to publication precision. Summon the summary statistics and the four are indistinguishable. Plot them and the masks fall: I is ordinary noisy linearity; II is a clean parabola; III is a perfect line sabotaged by one outlier; IV is a vertical stack of identical x-values propped up by a single leverage point. Anscombe built them in 1973 to end an argument: numerical summaries without graphs are a blindfold. Verified live: all four reproduce the shared statistics within tolerance, while the shapes are proven structurally — II fits a quadratic with R² = 1.00000, III has 10 of 11 points exactly collinear, IV has 10 identical x-values. Neon-noir traced. See the four scatterplots under one line in 1D, the per-dataset unmasking in 2D, and the interrogate-the-shape inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "81fe45a8e2d4cef7d13e6ecccc042990fe66c11903b07ae4abb73155f431bec7"
    },
    {
      "slug": "the-simpson-paradox",
      "title": "THE SIMPSON PARADOX",
      "kicker": "a treatment that wins twice and loses once",
      "accent": "#ff8a3c",
      "blurb": "Simpson's paradox in the 5-window house format — the aggregation trap: a trend that holds in every subgroup can reverse when the groups are merged. The canonical real case is the 1986 kidney-stone study: Treatment A beats B on small stones (81/87 = 93% vs 234/270 = 87%) and on large stones (192/263 = 73% vs 55/80 = 69%) — yet in the combined table B appears to win, 289/350 = 83% against A's 273/350 = 78%. No arithmetic error anywhere: A was assigned the harder cases, and that lurking variable flips the headline. The paradox is why adjusting for confounders is the difference between a true claim and its opposite. Verified live with exact integer cross-products — no floating point: A wins both subgroup comparisons, B wins the aggregate. Neon-noir traced. See the bars flip in the merged panel in 1D, the three exact comparisons in 2D, and the who-got-the-hard-cases inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "8ceaad5ce4081c098a50de0f83ccedb41669fe8e502b618684bfebdb6ca4cd17"
    },
    {
      "slug": "the-james-stein",
      "title": "THE JAMES-STEIN",
      "kicker": "an estimator improved by shrinking it",
      "accent": "#35ffb0",
      "blurb": "Stein's paradox in the 5-window house format — the most disreputable-sounding true theorem in statistics. Observe noisy measurements of ten unrelated quantities: the obvious estimator reports each as-is. The James–Stein estimator instead shrinks every measurement toward zero by a data-determined factor, 1 − (d−2)/‖X‖² — deliberately biasing all of them, mixing information between quantities that have nothing to do with each other. And it wins: in dimension d ≥ 3 its total squared error is strictly smaller for every possible truth. Stein proved the inadmissibility in 1956; the estimator is James & Stein, 1961. Verified live: 20,000 simulated trials in dimension 10 across three truth configurations — risk ratios 0.20, 0.4387, 0.87, all strictly below 1. Neon-noir traced. See the readings shrink toward zero in 1D, the risk-ratio bars in 2D, and the tax-them-together inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "b74ff75cb1812a88428ca416f724856ff0613ffd3252cbf29a09bbc53831b98f"
    },
    {
      "slug": "the-friendship-paradox",
      "title": "THE FRIENDSHIP PARADOX",
      "kicker": "a network where your friends outnumber you",
      "accent": "#b06bff",
      "blurb": "The friendship paradox in the 5-window house format — Scott Feld's 1991 observation that on average, your friends have more friends than you do. No self-esteem required: it is pure sampling bias. Picking a random person and then a random friend of theirs reaches people in proportion to how many friendships they sit in — the popular are oversampled, the isolated barely reachable. Formally, the mean friend-degree is E[d²]/E[d], which by Cauchy–Schwarz is at least the mean degree E[d] — strictly greater whenever degrees vary. The same tilt powers real tools: monitoring the friends of random people detects epidemics earlier than monitoring random people. Verified live: on a random graph of 3,000 nodes, mean degree 12.0 vs mean friend-degree 13.0; direct friend-sampling reproduces E[d²]/E[d] within 1%; a 4-regular ring gives exact equality — the bias needs variance. Neon-noir traced. See the two histograms in 1D, the running experiment in 2D, and the ask-a-random-edge inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "3f7a93f0b7bcfed41ef7e4d60f72cf52d02a1f07b42660349853fac62cc49a14"
    },
    {
      "slug": "the-crofton",
      "title": "THE CROFTON",
      "kicker": "a length measured by throwing lines at it",
      "accent": "#ffcf4a",
      "blurb": "Crofton's formula in the 5-window house format — measuring a curve's length without ever touching it, by throwing random straight lines across the plane and counting hits. Parametrize every line by direction θ and signed distance p; with that kinematic measure, integral geometry gives the identity ∫ n(ℓ∩C) dℓ = 2·Length(C) — the average crossing count, over all lines, knows the length exactly, whatever the shape. It is Buffon's needle grown up: the noodle, the circle, the scribble, all measured by the same rain of lines. Morgan Crofton published it in 1868; it founded integral geometry and lives on in stereology and tomography. Verified live: 60,000 random lines recover the length of a circle (~0.1% error), a bare segment, and an ellipse of numerically known perimeter, all within Monte-Carlo tolerance. Neon-noir traced. See the curve under the rain with its crossings ticked in 1D, three lengths recovered in 2D, and the count-across inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "5384fda8febba01049e4009fed3de7bc36b028fb1e4c5fbe60e848f146566cd5"
    },
    {
      "slug": "the-kakeya",
      "title": "THE KAKEYA",
      "kicker": "a needle turned in an eighth of pi",
      "accent": "#b06bff",
      "blurb": "The Kakeya needle problem in the 5-window house format — what is the least area in which a unit needle can be turned completely around? Spinning about its centre sweeps π/4. Kakeya's 1917 candidate was the deltoid — the three-cusped hypocycloid — inside which the needle rotates using only π/8, half the disc, gliding with its ends on the curve at every angle, thanks to a jewel of a property: every tangent line cuts the deltoid in a chord of exactly the needle's length. Then Besicovitch detonated the question in 1928: with enough sliding trickery the needle turns in arbitrarily small area — no positive minimum exists, and Kakeya sets now sit at the heart of harmonic analysis. Verified live: the deltoid's shoelace area computes to π/8, and at 36 sampled angles the tangent chord has length 1.0000. Neon-noir traced. See the needle at five headings in 1D, the constant-chord audit in 2D, and the no-minimum inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "875e122764f20302400405d242ae8a64d2515dbc2ebc7af769fd086092ebfc3d"
    },
    {
      "slug": "the-two-envelope",
      "title": "THE TWO-ENVELOPE",
      "kicker": "two envelopes and a threshold that beats the coin",
      "accent": "#ff8a3c",
      "blurb": "The two-envelope paradox in the 5-window house format — one envelope holds twice the other; you pick one and reason 'the other holds 2x or x/2, each half the time: expected 1.25x, switch' — and the same argument repeats forever. The flaw is a conditioning error: no consistent prior supports '50/50 given your amount', and blind switching gains exactly nothing. Then Thomas Cover found the twist the paradox hides: peek at your amount x, draw a random threshold Z, switch only if x < Z. For any fixed pair a < b this ends with the larger envelope with probability ½ + (e^{−λa} − e^{−λb})/2 — strictly above one half, knowing nothing about the amounts. Verified live: blind always-switch ties always-keep over 400k trials; Cover's exact formula exceeds ½ for every pair tested (including a 500-vs-501 squeaker); simulation matches the formula where Monte-Carlo can resolve it. Neon-noir traced. See the half-line and what floats above it in 1D, per-pair exact probabilities in 2D, and the randomize-your-doubt inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "a68b8e15b0c6d41897315531f5a494367135b05b32e9d69a8c6fdbfc60cfe1e2"
    },
    {
      "slug": "the-braess",
      "title": "THE BRAESS",
      "kicker": "a free road that slows every driver",
      "accent": "#21e6ff",
      "blurb": "Braess's paradox in the 5-window house format — adding a road can make every driver slower. The classic network: 4000 commuters, two routes each combining a congestion-priced leg (traffic/100 min) and a fixed 45-minute leg. Selfish equilibrium: a clean 2000/2000 split, 65 minutes each. Open a free shortcut between the midpoints and every driver individually profits by chaining both congestion legs — so everyone does, both legs carry all 4000, and the commute becomes 80 minutes for every single person. No one can deviate and do better: a true equilibrium, just a worse one. Real cities have lived it — road closures in Seoul and New York measurably sped traffic up. Verified live: both equilibria check exactly (no profitable deviation at 65 or at 80), and best-response dynamics from an arbitrary split converge to all 4000 on the shortcut. Neon-noir traced. See the diamond network and its fatal link in 1D, the funnel forming round by round in 2D, and the what-equilibrium-did-it-destroy inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "b712401dfab75b90f7642501a0c399a25d4dac183abb619f1b25276ef110a4bc"
    },
    {
      "slug": "the-kelly",
      "title": "THE KELLY",
      "kicker": "the bet size that survives",
      "accent": "#ffcf4a",
      "blurb": "The Kelly criterion in the 5-window house format — John L. Kelly Jr.'s 1956 Bell Labs answer to the gambler's real question: not whether to bet a favourable game, but how much. Bet a fraction f of bankroll on an even-money game won with probability p: long-run growth is g(f) = p·ln(1+f) + q·ln(1−f), peaking at exactly f* = p − q. Bet less and growth is left on the table; bet more and growth falls — past a threshold it turns negative, and an edge-holding gambler goes broke with certainty. At p = 60%, Kelly says 20%: doubling to 40% already loses long-run, 80% is ruin at speed. The same log-wealth mathematics underlies channel capacity, where Kelly found it. Verified live: g(f) peaks at 0.200 = p−q on a fine grid, and simulated bankrolls (10,000 bets × 120 runs) rank exactly as theory orders, with f = 0.8 strictly negative. Neon-noir traced. See the peak-and-dive curve in 1D, the four-strategy ladder in 2D, and the logarithm-of-forever inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "a7b82b6406d50e26e2a9ff5e0ffa61919b881b4aa7e841f2cd56a0feaa40d693"
    },
    {
      "slug": "the-fary-milnor",
      "title": "THE FARY-MILNOR",
      "kicker": "the bending toll every knot must pay",
      "accent": "#35ffb0",
      "blurb": "The Fáry–Milnor theorem in the 5-window house format — the bending toll a knot must pay. Total curvature is how much a closed curve turns, summed along its length: any convex loop turns through exactly 2π. The theorem (Fáry 1949; Milnor 1950, as an undergraduate): if a closed curve is knotted, its total curvature must exceed 4π — a knot cannot exist without bending at least twice around. No gradual transition: any curve bending less than 4π is provably an unknot. Topology reaches down and constrains geometry. Verified live: polygonal total curvature of a circle and a convex ellipse both compute to 2π to three decimals, while a trefoil knot computes to 13.95 — comfortably above the 4π = 12.566 floor. Neon-noir traced. See the three totals against the wall in 1D, the per-curve audit in 2D, and the audit-the-bending inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "0d36d534dd3209ce8ca5fc496b471bc45c38942b4821435d1743ecfb04c98a6c"
    },
    {
      "slug": "the-moser-spindle",
      "title": "THE MOSER SPINDLE",
      "kicker": "seven points that outlaw three colors",
      "accent": "#b06bff",
      "blurb": "The Moser spindle in the 5-window house format — seven dots that legislate about the entire infinite plane. The Hadwiger–Nelson problem asks how many colours are needed to paint every point of the plane so that no two points at distance exactly 1 match. The spindle (Leo and William Moser, 1961) is 7 vertices and 11 edges, every edge exactly unit length, drawable in the plane — and it cannot be properly 3-coloured, while 4 colours suffice. Since it embeds with unit edges, the plane needs at least 4 colours; Aubrey de Grey's 1553-vertex graph pushed the bound to ≥5 in 2018, and the true answer (5, 6, or 7) is still open. Verified live: all 11 edges measure 1.000000000, all 2187 three-colourings fail exhaustively, and a proper 4-colouring is exhibited. Neon-noir traced. See the hinged construction in 1D, the failing colourings in 2D, and the relaxed 4-colouring in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "d5361dc7f7121949a052d97e00a23e07ef1729d7a19e5a47f2b86f37f2e09bfb"
    },
    {
      "slug": "the-perfect-shuffle",
      "title": "THE PERFECT SHUFFLE",
      "kicker": "eight perfect shuffles back to the start",
      "accent": "#ffcf4a",
      "blurb": "The faro shuffle in the 5-window house format — the card mechanic's perfect riffle: cut exactly in half, interleave one card at a time. It looks like the ultimate randomizer and is precisely the opposite — a fixed permutation. Eight out-shuffles return a 52-card deck exactly to its starting order, because an out-shuffle sends position p to 2p mod 51 and the multiplicative order of 2 mod 51 is 8. Prefer the in-shuffle? Position maps through mod 53, order of 2 is 52 — fifty-two shuffles home. Magicians exploit the difference; parallel computers wire it as the perfect-shuffle interconnect. Verified live: the interleave is simulated on a real array — 8 out-shuffles restore, 52 in-shuffles restore, and both counts equal independently computed modular orders. Neon-noir traced. See position 1's doubling orbit in 1D, the step-by-step scramble-and-snap-back in 2D, and the displacement wheel in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "b1c0bd27cca40809c465745ee6957e9b351e34cf7feb619f9b45767d98382eea"
    },
    {
      "slug": "the-tautochrone",
      "title": "THE TAUTOCHRONE",
      "kicker": "every bead arriving together",
      "accent": "#35ffb0",
      "blurb": "The tautochrone in the 5-window house format — the curve of impossible fairness. A bowl shaped as an inverted cycloid delivers a frictionless bead from ANY release height to the bottom in exactly the same time, T = π√(a/g): drop one from the rim and one from barely above the floor and they arrive together. Christiaan Huygens proved it in 1673 hunting an amplitude-independent pendulum, and built cycloidal-cheek clocks on the result. The secret is hidden linearity: in arc-length coordinates the cycloid turns gravity into a perfect spring, and springs don't care about amplitude. Verified live: four beads at four widely different heights, integrated by RK4 on the true Lagrangian dynamics, all arrive at 1.00354 s to five decimals — while a circular bowl's times differ by over 70%. Neon-noir traced. See the four-bead bowl in 1D, the race ledger in 2D, and the synchronized descent in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "8565d822c18cb1b951589aab14a05712344fdd659f383deed312eaa419adbeb4"
    },
    {
      "slug": "the-ford-circles",
      "title": "THE FORD CIRCLES",
      "kicker": "fractions kissing along the number line",
      "accent": "#21e6ff",
      "blurb": "Ford circles in the 5-window house format — every fraction given a body. Above each reduced p/q draw a circle of radius 1/(2q²) resting on the number line at that point: giants for simple fractions, dust for deep denominators. The miracle: no two Ford circles ever overlap. They miss, or they KISS — and they kiss precisely when |ps−qr| = 1, the Farey-neighbour condition, because one exact identity governs everything: dist²−(r₁+r₂)² = ((ps−qr)²−1)/(q²s²). Between kissing circles the mediant's bubble nests in the gap and kisses both — the Stern–Brocot structure of the rationals drawn in soap bubbles (Lester Ford, 1938). Verified live: 129 fractions with q ≤ 20, 8,256 pairs, zero overlaps, 255 kisses exactly at determinant ±1, identity to 1e-9 on every pair. Neon-noir traced. See the full bubble line in 1D, the pair-audit in 2D, and the breathing zoom in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "505c34561019d36f379d0e74bcfa4c5050b6a97d9ae87166e7fed1a563ccf0e4"
    },
    {
      "slug": "the-cyclic-number",
      "title": "THE CYCLIC NUMBER",
      "kicker": "the number that rotates instead of growing",
      "accent": "#ff8a3c",
      "blurb": "142857 in the 5-window house format — the most famous cyclic number. Multiply by 1 through 6 and the answer is always the same six digits rotated: 285714, 428571, 571428, 714285, 857142. Multiply by 7 and the register overflows to 999999. The engine is decimal arithmetic itself: 142857 is the repeating block of 1/7, and 7 is a full-reptend prime — 10 is a primitive root mod 7, so multiplication can only rotate the block. The next such prime is 17, whose 16-digit block 0588235294117647 rotates under ×1..16 and overflows to sixteen nines at ×17. Verified live: all six rotations exact, the overflow exact, the block regenerated by long division, the 17-family checked in BigInt, and the orders ord(10,7)=6 and ord(10,17)=16 confirmed. Neon-noir traced. See the digit wheel in 1D, the stepping multiplier in 2D, and the spinning clock in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "58e60b696d029f10583047c9fe9500428ec30f19e3821af0293b26dfe504f4f1"
    },
    {
      "slug": "the-fusc",
      "title": "THE FUSC",
      "kicker": "every fraction born exactly once",
      "accent": "#21e6ff",
      "blurb": "Stern's diatomic sequence in the 5-window house format — Dijkstra's fusc, built from fusc(2n)=fusc(n), fusc(2n+1)=fusc(n)+fusc(n+1). Hidden in the consecutive ratios fusc(n)/fusc(n+1) is a miracle: they walk through EVERY positive rational exactly once, each already in lowest terms — 1/1, 1/2, 2/1, 1/3, 3/2, 2/3, 3/1… A complete, duplicate-free census of the fractions generated by bit-shifts and one addition (Stern 1858; Calkin–Wilf 2000). Bonus: fusc(n+1) counts the hyperbinary representations of n. Verified live: 65,536 consecutive pairs coprime and distinct, every reduced p/q with p+q ≤ 20 found within the first 2²⁰ terms, and the hyperbinary identity checked by independent DP to n=300. Neon-noir traced. See the diatomic wave in 1D, the rational birth registry in 2D, and the Calkin–Wilf tree in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "23849517c1cf67d4bc9569a683a4432855de64eb07ac1afd0fec4b46e7db1c99"
    },
    {
      "slug": "the-hipparchus",
      "title": "THE HIPPARCHUS",
      "kicker": "a count buried in Plutarch for two thousand years",
      "accent": "#ffcf4a",
      "blurb": "Hipparchus's 103,049 in the 5-window house format — the number sat unexplained in Plutarch's Table Talk for nearly two millennia: from ten simple assertions, Hipparchus computed, the Stoics could form 103,049 compound statements. In 1994 David Hough noticed it is exactly the tenth little Schröder number — the ways to bracket ten items — making it the oldest serious enumeration result known. Plutarch's companion 310,952 is two off from (s₁₀+s₁₁)/2 = 310,954, likely an ancient copying slip (Habsieger–Kazarian–Lando 1998). Verified live: the three-term recurrence and an independent plane-tree DP agree at every n ≤ 11, both delivering 103,049. Companion sphere the-schroder covers the LARGE Schröder path numbers; this is their halved sibling met in an ancient book. Neon-noir traced. See the climb to 103,049 in 1D, the two-route race in 2D, and the eleven bracketings of four items in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "201410fab3ed13455a87003f854bfada3ca228b810f554c84a449a81537a237a"
    },
    {
      "slug": "the-lychrel",
      "title": "THE LYCHREL",
      "kicker": "the number that never comes home",
      "accent": "#ff8a3c",
      "blurb": "The 196 problem in the 5-window house format — reverse a number's digits, add, repeat. Almost everything collapses quickly to a palindrome; 89 is the slowpoke of the small numbers, needing exactly 24 steps to reach 8,813,200,023,188. But 196 has never arrived — not in billions of distributed-search iterations reaching hundreds of millions of digits. Suspected never-palindromic numbers are Lychrel numbers (Wade VanLandingham's coinage), 196 the smallest candidate — and no one has proved a single one exists in base 10. An infinite loop nobody can certify is infinite. Verified live: 89's 24-step journey exact by BigInt; a census below 10,000 finds 249 stubborn seeds, smallest 196; and 196 marched 3,000 steps to a 1,268-digit number with no palindrome. Neon-noir traced. See 89's staircase in 1D, the seed races in 2D, and the flock that lands — minus one — in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "875b0a66272331e41133a68aecf1b40705e009151f3bd96b2fa67b012e6bc90c"
    },
    {
      "slug": "the-keith",
      "title": "THE KEITH",
      "kicker": "a number reborn in its own digit stream",
      "accent": "#35ffb0",
      "blurb": "Keith numbers in the 5-window house format — Mike Keith's 1987 repfigits, numbers that come back from the dead inside their own digit stream. Seed a Fibonacci-style sequence with a number's digits (each term summing the last k): 14 → 1, 4, 5, 9, 14 — the number reappears in the stream it seeded. 197 → 1, 9, 7, 17, 33, 57, 107, 197. They are startlingly rare: below 100,000 there are exactly 24, no formula generates them, and their infinitude is unproven — brute search is the only road in. Verified live: exhaustive sweep of 10..99,999 by two independently coded membership tests (streaming vs sliding-window) disagreeing on zero numbers, finding exactly the known 24 from 14 to 93,993. Neon-noir traced. See the 24 on a log line in 1D, each stream rebuilding its seed in 2D, and 197's spiral home in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "0c82ef1c24ab89f922a4949ec0052c6067f7414f30f5306ca8b37e672b30e78a"
    },
    {
      "slug": "the-vampire",
      "title": "THE VAMPIRE",
      "kicker": "factors hiding their digits in the product",
      "accent": "#b06bff",
      "blurb": "Vampire numbers in the 5-window house format — Clifford Pickover's 1994 coinage for 2n-digit numbers that factor into two n-digit fangs whose pooled digits are exactly the number's own digits, shuffled: 1260 = 21 × 60. House rule: the fangs may not both end in zero. Among four-digit numbers there are exactly seven — 1260, 1395, 1435, 1530, 1827, 2187, 6880 — and the six-digit census jumps to 148. Verified live: exhaustive multiplication of every fang pair with digit-multiset comparison reproduces both censuses — seven and 148 — with nothing looked up. Neon-noir traced. See the seven with fangs bared in 1D, the multiset audits in 2D, and the digits flowing from fangs to product in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "01e7e591633142b3e5bb8a2c8d93474bb70e6ebc88fd167a792e422aec7167e5"
    },
    {
      "slug": "the-zeno",
      "title": "THE ZENO",
      "kicker": "the paradox of halfway, at the halfway line",
      "accent": "#35ffb0",
      "blurb": "Zeno's dichotomy in the 5-window house format — to reach the wall you must first reach halfway, then half of what remains, forever: infinitely many tasks, so motion is 'impossible'. The resolution is exact arithmetic: ½+¼+…+1/2ⁿ = (2ⁿ−1)/2ⁿ on the nose, the gap to 1 exactly 1/2ⁿ, and at unit speed the times form the same convergent series — Achilles arrives at t = 1 exactly. The sting is the contrast: steps costing 1/k diverge — the harmonic walker passes 10 seconds only at step 12,367 and never arrives. Zeno's error was not the infinity of tasks; it was assuming every infinite sum of positive terms is infinite. Verified live: BigInt-exact partial sums to n=64, exact gap-halving, the harmonic crossing at 12,367, and the grouping bound H(2^m) ≥ 1+m/2. Seated deliberately in the batch that crosses this corpus's own halfway line: 1024 of 2048. Neon-noir traced. See the halving track in 1D, the exact ledger in 2D, and the dyadic spiral arriving in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "75bfe1100376e9fdedcadff554af1acfa2ba2ab11fd776082b5cf7b2170b4de7"
    },
    {
      "slug": "the-magic-hexagon",
      "title": "THE MAGIC HEXAGON",
      "kicker": "the one hexagon that exists",
      "accent": "#ffcf4a",
      "blurb": "The magic hexagon in the 5-window house format — magic squares exist in endless supply; the magic hexagon exists ONCE. Arrange 1–19 in a side-3 hexagon so all fifteen rows in all three directions share one sum and you are forced into a single arrangement (up to its 12 symmetries), magic constant 38, the 5 at dead centre. Clifford Adams hunted it from 1910 to 1957, lost the solution, re-found it in 1962; Charles Trigg proved uniqueness. The deeper cut: the magic-constant formula M(n) is a whole number only for n = 1 and 3 — every other size dies before the search begins. Verified live: full backtracking finds exactly 12 solutions = one hexagon × 12 symmetries; order 2 is killed across all 5,040 arrangements; the integrality obstruction is checked for n ≤ 1000. Neon-noir traced. See the unique hexagon in 1D, the line audits in 2D, and the twelve turning faces in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "3c9dffa55614c23c2ffca61dc1619372abf0742fd7523c6dc0e12d7b567ed567"
    },
    {
      "slug": "the-mihailescu",
      "title": "THE MIHĂILESCU",
      "kicker": "the only two powers that touch",
      "accent": "#b06bff",
      "blurb": "Catalan's conjecture in the 5-window house format — in 1844 Eugène Catalan mailed Crelle's Journal one paragraph: 8 and 9 are the only consecutive perfect powers (x^p − y^q = 1 has only 3² − 2³). It stood 158 years. Tijdeman (1976) proved finiteness with astronomically useless bounds; in 2002 Preda Mihăilescu — largely outside academia — killed it with cyclotomic fields, no computers. Among all the towers of squares, cubes and higher powers on the number line, exactly one pair of neighbours ever touch. Verified live: all 1,010,195 perfect powers up to 10¹² generated, sorted, and scanned — exactly one consecutive pair: (8, 9). Neon-noir traced. See the powers below 300 in 1D, the gap census in 2D, and the towers rising apart in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "21285dd60fba9b7048085b173e723fc600ddd460ec1f65b4068749f0fbf56334"
    },
    {
      "slug": "the-superpermutation",
      "title": "THE SUPERPERMUTATION",
      "kicker": "every binge-order in one string",
      "accent": "#ff8a3c",
      "blurb": "Superpermutations in the 5-window house format — one string containing every permutation of n symbols as a substring: the shortest binge-watch of all n! orderings. n=3: minimum exactly 9 (123121321). n=4: exactly 33. And the lower-bound proof — length ≥ n!+(n−1)!+(n−2)!+n−3 — has the strangest provenance in combinatorics: posted anonymously on 4chan in 2011 under a question about the optimal watch-order for the 14 episodes of The Melancholy of Haruhi Suzumiya, verified and written up by Houston–Pantone–Vatter in 2018 with 'Anonymous 4chan Poster' as first author. For n=5 the bound says 152, the best string found is 153 — open by exactly one character. Verified live: exhaustive search proves nothing shorter than 9 works for n=3; the 33-character n=4 witness is validated against all 24 permutations and equals the proven bound — hence minimal. Neon-noir traced. See the 33 characters in 1D, the sliding window in 2D, and the permutation ring in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "89dda71a9bdbfbafc34a7f9ac9b0a4eb809f4947404e514b3a83d8a5417854a4"
    },
    {
      "slug": "the-pancake",
      "title": "THE PANCAKE",
      "kicker": "Bill Gates and the flipped stack",
      "accent": "#21e6ff",
      "blurb": "Pancake sorting in the 5-window house format — a stack of n pancakes, one move: slide a spatula under any prefix and flip it. The pancake number P(n) is the worst case over all stacks, and computing it means searching the full n!-vertex prefix-reversal graph to its diameter: P(1..8) = 0,1,3,4,5,7,8,9, no formula known, P(20) uncomputed. Claim to fame: the (5n+5)/3 upper bound came from a 1979 paper by Christos Papadimitriou and a Harvard undergraduate named William Gates — Bill Gates' only research publication, unbeaten for 30 years until Chitturi et al. 2009. Posed by Jacob Goodman under the pseudonym 'Harry Dweighter' (harried waiter). Verified live: BFS over the complete graph for every n ≤ 8 — all 40,320 stacks of 8 reached — reproducing the diameters exactly. Neon-noir traced. See the formula-less staircase in 1D, the spatula replay in 2D, and the space of stacks in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "8885299fd9f921712a8e5abea7f2656edc6af73a2cf65447c5c62cefeb1134c8"
    },
    {
      "slug": "the-sierpinski-number",
      "title": "THE SIERPIŃSKI NUMBER",
      "kicker": "a family composite forever by seven-prime conspiracy",
      "accent": "#b06bff",
      "blurb": "78,557 in the 5-window house format — a Sierpiński number: every member of 78557·2ⁿ+1 is composite, forever, by covering set. Seven primes {3,5,7,13,19,37,73} conspire — their orders of 2 all divide 36, so the conspiracy repeats with period 36: check 36 residues and you have checked all of infinity. Sierpiński proved such numbers exist (1960); Selfridge found 78,557 (1962); whether it is the SMALLEST is the Sierpiński problem, with PrimeGrid still hunting five candidates below it. Verified live, proof-grade: orders computed, lcm confirmed 36, all 36 residues matched to covering primes, and 1,500 actual BigInt terms audited (each divisible by its prime, each larger than it — composite). A complete finite proof of an infinite statement. Neon-noir traced. See the 36-spoke covering wheel in 1D, the per-n divisibility in 2D, and the patrol rota in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "d6d816d96ebfa5894b509bcf058cfc2a14ce1b6ed11af7c27da77cf53a403dcc"
    },
    {
      "slug": "the-mills",
      "title": "THE MILLS",
      "kicker": "a constant that boots an infinite prime cascade",
      "accent": "#21e6ff",
      "blurb": "Mills' constant in the 5-window house format — William Mills proved (1947) a constant A exists with floor(A^(3ⁿ)) prime for every n: 2, 11, 1361, 2521008887, 16022236204009818131831320183 — each the smallest prime after the cube of the last, perched astonishingly close above it (gaps 3, 30, 6, 80; the cube root of the fifth prime exceeds the fourth by just 4×10⁻¹⁸). The honest secret: the constant is the cascade written as a limit — the primes generate A, not the other way round — and its published digits as the LEAST such constant assume the Riemann Hypothesis. Verified live: the cascade re-derived from p=2 by Miller–Rabin next-prime hunts, every window (p³,(p+1)³) confirmed, the 4×10⁻¹⁸ knife-edge measured by BigInt cube root, and A itself extracted live as the 243rd root of the fifth prime — 31 digits of agreement, matching the published 1.3063778838630806904686144926. Neon-noir traced. See the cascade in 1D, the hop-by-hop rebuild in 2D, and the tower of cubes in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "2455b825a4bd9f03453f5d947f023d988bd5f58eed430fa02b0974e0099de5ac"
    },
    {
      "slug": "the-ruth-aaron",
      "title": "THE RUTH-AARON",
      "kicker": "two ballplayers sharing a factor sum",
      "accent": "#35ffb0",
      "blurb": "Ruth–Aaron pairs in the 5-window house format — on April 8, 1974, Hank Aaron's 715th home run passed Babe Ruth's 714, and days later Carl Pomerance noticed the numbers share a secret: 714 = 2·3·7·17 and 715 = 5·11·13 both have prime-factor sum 29. Bonus: 714·715 = 510,510 = 2·3·5·7·11·13·17, the product of the first seven primes. Erdős phoned Pomerance, they proved such pairs have density zero, and a legendary collaboration was born (Aaron and Erdős later received honorary degrees together — Aaron signed a baseball for Erdős, giving him an Erdős number of 1, as the joke goes). Verified live: both factor sums under both definitions, the primorial identity exact, and a full sieve census below 1,000,000 — 139 distinct-definition pairs, 149 with multiplicity, first pairs (5,6),(24,25),(49,50),(77,78)… Neon-noir traced. See the converging factorizations in 1D, the pair walk in 2D, and the balanced beam in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "e43ecb7467bbf408be34112bc58a36b04146bf6a54d3b18903790747e641508d"
    },
    {
      "slug": "the-erdos-straus",
      "title": "THE ERDŐS-STRAUS",
      "kicker": "four quarters split into three unit coins",
      "accent": "#ff8a3c",
      "blurb": "The Erdős–Straus conjecture in the 5-window house format — for every n ≥ 2, does 4/n split into three unit fractions 1/x+1/y+1/z? Ancient Egypt wrote all fractions as unit sums; Erdős asked (1948) whether three coins always suffice for 4/n. One-line identities swallow even n, n≡3 mod 4, and n≡0,2 mod 3; composites inherit from their factors; the entire battlefield shrinks to primes ≡ 1 mod 12, hunted one by one with no formula known. Verified computationally past 10¹⁷ in the literature — but OPEN. Verified live here: every n from 2 to 100,000 solved (families + factor-lifting + banded divisor search for the 2,374 hard primes), every solution certified by the exact BigInt identity n(yz+xz+xy)=4xyz — no floating point anywhere. Neon-noir traced. See the three coins in 1D, the wheel with certificates in 2D, and the mod-12 battlefield in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "bd71e403a44b27b26ba9096e13f577b800b1c25faa873d8ff7fbb421a7b3fefe"
    },
    {
      "slug": "the-untouchable",
      "title": "THE UNTOUCHABLE",
      "kicker": "the loot no drop table contains",
      "accent": "#ffcf4a",
      "blurb": "Untouchable numbers in the 5-window house format — values the aliquot sum s(n) (sum of proper divisors) never produces: 2, 5, 52, 88, 96, 120, 124, 146… No integer's divisors will ever sum to them. Erdős proved infinitude (1973). Two gems: 5 is believed the only odd untouchable — because strong Goldbach gives every even 2k = p+q, hence s(pq) = p+q+1 hits every odd ≥ 7; and untouchability is CERTIFIABLE without infinite search, because composite n has s(n) > √n, so small targets can only be hit by small n. Verified live: aliquot sums sieved to 998,001, the √n bound turning the finite scan into a certificate for all values ≤ 1000 — the exact untouchable list to 500 (38 values), with 5 the only odd member. Neon-noir traced. See the lit-and-dark number line in 1D, the witness queries in 2D, and the aliquot rain in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "0a7e8dcb7af73edf4a02f4c31e9ec297f94640bb10bccc33ac2326a0d655b36f"
    },
    {
      "slug": "the-somos",
      "title": "THE SOMOS",
      "kicker": "an integer streak that dies at seventeen",
      "accent": "#ff8a3c",
      "blurb": "The Somos sequences in the 5-window house format — start 1,1,1,1 and iterate a(n) = (a(n−1)a(n−3)+a(n−2)²)/a(n−4): dividing at every step, yet Somos-4 stays integer forever (2, 3, 7, 23, 59, 314, 1529…), as do Somos-5, 6, 7 — protected by the Laurent phenomenon (Fomin–Zelevinsky, cluster algebras): every term is secretly a Laurent polynomial in the seeds. Then Somos-8: integer through a(16), and at a(17) the spell breaks — 420514/7. The 7 that was always lurking finally surfaces. Verified live in exact BigInt rationals: Somos-4..7 integer through 40 terms; Somos-8's first break at a(17) = 420514/7 exactly. Neon-noir traced. See Somos-4 climbing in 1D, the denominator watch in 2D, and the four protected spirals in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "1f3c4e8f7c17bdbddbfc94a3f38d1e371db900b8041230ad7bff6da6f1ac6443"
    },
    {
      "slug": "the-prouhet",
      "title": "THE PROUHET",
      "kicker": "a fair split sharp at every power",
      "accent": "#35ffb0",
      "blurb": "The Prouhet–Tarry–Escott split in the 5-window house format — divide 0..2^k−1 by bit-parity (even 1-bits team A, odd team B: the Thue–Morse pattern ABBA BAAB) and the teams have equal sums, equal squares, equal cubes… equal sums of EVERY power up to k−1 (Prouhet 1851). For k=3: {0,3,5,6} vs {1,2,4,7} — same size, same sum 14, same square-sum 70. And the split is SHARP: at power k the sums finally differ. Fair turn-taking, formalized. Verified live in exact BigInt for every k ≤ 11: all equalities below k, strict difference at k. Neon-noir traced. See the k=3 teams in 1D, the tipping ledger in 2D, and the balance beam in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "13ccc63b0e4661a7cbca28e6a334817b701ee540559f9ce759e53343cdecc5ee"
    },
    {
      "slug": "the-wilson-prime",
      "title": "THE WILSON PRIME",
      "kicker": "three coins in 250 years",
      "accent": "#ffcf4a",
      "blurb": "Wilson primes in the 5-window house format — Wilson's theorem is a perfect prime detector: p prime ⟺ (p−1)! ≡ −1 mod p, with composites > 4 collapsing to 0 instead. Sharpen to modulo p² and you get the Wilson primes — and in 250 years of searching exactly THREE have been found: 5, 13, 563, with the hunt swept past 2×10¹³. Heuristically infinitely many should exist (~1/p odds per prime); nobody knows where the fourth is. Verified live: the theorem and its converse for every number below 1000, and the mod-p² sweep finding exactly {5, 13, 563}. Neon-noir traced. See the −1/0 residue rows in 1D, the p² lottery in 2D, and the three golden strikes in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "f6f3f456dc3de5d0458e131db9236b92074bd8c0506fe889bdbf5ec2cbf78287"
    },
    {
      "slug": "the-gijswijt",
      "title": "THE GIJSWIJT",
      "kicker": "the slowest counter in mathematics",
      "accent": "#b06bff",
      "blurb": "Gijswijt's sequence in the 5-window house format — each term is the curling number of everything before it: find the longest block B such that the sequence ends in B repeated k times, and write k. From 1: 1,1,2,1,1,2,2,2,3,… The first 2 arrives at position 3, the first 3 at position 9, the first 4 at position 220 — and the first 5 near position 10^(10²³) (van de Pol & Gijswijt), more positions than atoms in the observable universe. The sequence will provably say 5; no computation will live to hear it. Verified live: 1000 terms generated from the definition, first 4 at exactly 220, no 5, census 296/527/173/4. Neon-noir traced. See the march to 220 in 1D, the tail-reading rule in 2D, and the milestone spiral with its unreachable magenta 5 in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "bfa0010977cd68a46dc55d927f54759e31c790e3d3e165f3cbc35ec326309187"
    },
    {
      "slug": "the-hydra",
      "title": "THE HYDRA",
      "kicker": "the boss that must lose but arithmetic cannot say so",
      "accent": "#21e6ff",
      "blurb": "The Kirby–Paris hydra in the 5-window house format — a tree-shaped raid boss: chop a head and, if it sat deep, the hydra sprouts n copies of the wounded branch at chop n. It grows faster the longer you fight — and the theorem says every strategy kills every hydra: you cannot lose. The famous twist: this truth is unprovable in Peano arithmetic (Kirby–Paris 1982) — the fight lengths outgrow ordinary induction, and the proof needs ε₀. Verified live with two independently coded engines (literal tree vs multiset ledger) agreeing exactly: deepest-first kills star-3/4/5 in exactly 66 / 2,278 / 2,598,060 chops; shallowest-first on a 4-NODE hydra exceeds 10,000,000 chops without dying (finite by theorem — prolonged, never saved); and the depth-4 chain grows past 100,000 heads in 199 chops. Neon-noir traced. See the sprouting rule in 1D, a live chop-by-chop fight in 2D, and the fight-length tower in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "4a269813326251e27a2868e05b541b12594b6e255fc6f7d1111e66ddbc910391"
    },
    {
      "slug": "the-heegner",
      "title": "THE HEEGNER",
      "kicker": "an integer missed by seven ten-trillionths",
      "accent": "#21e6ff",
      "blurb": "Ramanujan's constant in the 5-window house format — e^(π√163) = 262537412640768743.99999999999925…, a transcendental missing the integer 640320³+744 by 7.5×10⁻¹³. No accident: 163 is the largest Heegner number (class number one: {1,2,3,7,11,19,43,67,163}), and modular-function theory FORCES the near-miss — the same 163 = 4·41−3 that powers Euler's prime factory n²+n+41. Hermite computed it in 1859; Martin Gardner ran it as a 1975 April Fools' hoax. Verified live entirely from scratch: π by two independent arctangent engines agreeing to 58 digits, √163 by BigInt Newton, the exponential by halve-series-square fixed point, the target exact, the gap measured at 7.499×10⁻¹³. Neon-noir traced. See the magnified number line in 1D, the Heegner ladder in 2D, and the tower-and-hair in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "df616b4542374d8e4546d7d9bbab50a633f4c6a981a9572ce83025a7c4ef36fe"
    },
    {
      "slug": "the-khinchin",
      "title": "THE KHINCHIN",
      "kicker": "the average hiding in almost every number",
      "accent": "#b06bff",
      "blurb": "Khinchin's constant in the 5-window house format — write any real as a continued fraction and average its terms geometrically: for ALMOST EVERY real the answer converges to one universal constant, K₀ = 2.6854520… (Khinchin 1934), regardless of the number chosen. The exceptions have measure zero but include celebrities: √2 = [1;2,2,2,…] locks at GM 2; e = [2;1,2,1,1,4,1,1,6,…] follows a rigid pattern and misses too. π looks utterly typical — its first hundred terms average 2.6831 — but whether π truly obeys Khinchin is UNPROVEN. Verified live: π to 320 digits by Machin BigInt, 100 CF terms extracted with truncation-stability audit, GM 2.6831; √2's all-2s and e's [1,2k,1] pattern verified as the provable exceptions they are. Neon-noir traced. See π's wild spikes in 1D, the running means in 2D, and three rivers flowing at K₀ in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "52bd22a0df84687319c35d1d71604f850f81e1a2b7c262bf5256e6b5b043e3e2"
    },
    {
      "slug": "the-freshmans-dream",
      "title": "THE FRESHMAN'S DREAM",
      "kicker": "the child's error that becomes law",
      "accent": "#35ffb0",
      "blurb": "The freshman's dream in the 5-window house format — every algebra teacher crosses out (a+b)² = a²+b² in red ink. The punchline of abstract algebra: in the right world the freshman is CORRECT — modulo a prime p, (a+b)^p ≡ a^p + b^p for all a, b, because every interior binomial coefficient C(p,k) is divisible by p and the cross-terms vanish wholesale. It is an exact characterization: every composite modulus breaks the dream. As the Frobenius endomorphism, the child's error is load-bearing machinery across finite fields — AKS primality testing starts from precisely this identity. Verified live: the dream for every prime below 100 across 20 random BigInt pairs each, the vanishing binomials exactly, and both converses — every composite exhibits a surviving coefficient AND a concrete breaking pair. Neon-noir traced. See Pascal mod 7 go dark in 1D, the modulus switch in 2D, and rows darkening exactly at primes in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "28d05848c8e516c905c09bd6a17637aff34ac8edb2d14fa4b5eaf96798f05f49"
    },
    {
      "slug": "the-smith",
      "title": "THE SMITH",
      "kicker": "a phone number with balanced books",
      "accent": "#ffcf4a",
      "blurb": "Smith numbers in the 5-window house format — in 1982 Albert Wilansky noticed his brother-in-law Harold Smith's phone number 493-7775 factors as 3·5·5·65837, and the digit sum of the number (42) equals the combined digit sum of its prime factors (42). Smith numbers begin 4, 22, 27, 58, 85, 94, 121… — the books balancing between two unrelated representations: positional digits and multiplicative atoms. McDaniel proved infinitude in 1987. Verified live: a full sieve census below 100,000 (3,294 Smiths, first twelve exact), the phone number's factorization audited with 65837 confirmed prime, both ledgers exactly 42. Neon-noir traced. See the phone-number ledger in 1D, the walking double ledgers in 2D, and the balance pans in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "a3aeb2997aca8e9ebbd66788e3dd7b95ee61ebeae6c7888e9be0a1926c97c2d6"
    },
    {
      "slug": "the-weird",
      "title": "THE WEIRD",
      "kicker": "abundance you cannot spend",
      "accent": "#ff8a3c",
      "blurb": "Weird numbers in the 5-window house format — 70's proper divisors (1,2,5,7,10,14,35) sum to 74: abundant. Usually abundance buys flexibility — some subset hits n exactly (semiperfect). But none of 70's 128 subsets makes 70: rich, and unable to spend the wealth exactly. Abundant-but-not-semiperfect numbers are the weird numbers (Benkoski & Erdős 1974): 70, 836, 4030, 5830, 7192, 7912, 9272… — provably infinite, all known ones even, the odd case open past 10²¹ with Erdős cash on the table. Verified live: exhaustive sweep below 10,000 with real divisor lists and exact subset-sum DP finds exactly those seven. Build note kept honest: memory said six; the computation found seven (5830) — computation wins. Neon-noir traced. See 70's unspendable wealth in 1D, the seven audits in 2D, and the reachable-sums map with its one dark slot in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "d3377cd62da4d0204551cac6326c1824f9333e70dd00ba40399ed0439cd0199a"
    },
    {
      "slug": "the-graham",
      "title": "THE GRAHAM",
      "kicker": "a number too big for the universe with a visible tail",
      "accent": "#35ffb0",
      "blurb": "Graham's number in the 5-window house format — too large for the observable universe to store its digits (an upper bound from Ramsey theory, Graham–Rothschild 1971, made famous by Martin Gardner), yet its FINAL digits are perfectly knowable: modulo 10^k every sufficiently tall tower of 3s stabilizes, and the tail is …262464195387. You cannot know the beginning; you can know the end. Verified live by three independent routes: the Carmichael-λ chain shows 3↑↑20 ≡ 3↑↑40 mod 10¹² (stabilization); the anchor 3↑↑3 = 7,625,597,484,987 computed exactly; and a Chinese-Remainder recombination (2¹² × 5¹²) reproduces the same tail. Neon-noir traced. See the digits locking in 1D, the growing tower with frozen tail in 2D, and the tower vanishing upward in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "0b49efb71b20cbbe352a7687c378356672abad1c6d378fc8a99df989cfcf14ae"
    },
    {
      "slug": "the-friendship-theorem",
      "title": "THE FRIENDSHIP THEOREM",
      "kicker": "every friendship wheel has a hub",
      "accent": "#21e6ff",
      "blurb": "The friendship theorem in the 5-window house format — if every two people share exactly one common friend, the network MUST be a windmill: one universal friend at the hub, everyone else paired into triangles through them (Erdős–Rényi–Sós 1966). No decentralized configuration survives; a 'politician' is forced into existence — and the standard proof runs through eigenvalues of the adjacency matrix, spectral graph theory summoned for a party puzzle. Verified live, exhaustively: ALL graphs on 3–7 vertices (2,097,152 at n=7) tested; survivors number 1, 0, 15, 0, 105 and every one is structurally certified a windmill; even n admit none. Neon-noir traced. See the three survivors in 1D, pair-by-pair checks in 2D, and the turning windmill in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "730c9b25789f34691d432eb805469d7d1dc55a4f585d5f3ebfd71e4f2f8da8a1"
    },
    {
      "slug": "the-kuratowski",
      "title": "THE KURATOWSKI",
      "kicker": "fourteen sets and never a fifteenth",
      "accent": "#ffcf4a",
      "blurb": "Kuratowski's closure-complement problem in the 5-window house format — take any set of reals; two buttons: closure (add limit points) and complement (flip inside/out). Press in any order, forever: you can produce AT MOST 14 distinct sets (Kuratowski 1922), and some starting sets achieve exactly 14 — classically the Frankenstein witness (0,1) ∪ (1,2) ∪ {3} ∪ (ℚ∩(4,5)). The bound is pure algebra: kk=k, cc=1, one hidden identity folds everything past fourteen. Verified live twice: the operator monoid generated across 60 random finite topological spaces closes at exactly 14 operators; and the classical witness runs through an EXACT symbolic engine (intervals, points, rational dust — closed under both operations) producing 14 distinct sets, never a fifteenth. Neon-noir traced. See the witness anatomy in 1D, the two live buttons in 2D, and the 14-orbit in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "44e59b0e5d86d354039843c49b8701fd2ba5f48882133acbbfbeae626891f2c8"
    },
    {
      "slug": "the-schur",
      "title": "THE SCHUR",
      "kicker": "the wall at thirteen",
      "accent": "#b06bff",
      "blurb": "Schur numbers in the 5-window house format — color 1, 2, 3, … with k colors so no color class contains x+y=z: with 2 colors you reach exactly 4; with 3, exactly 13; with 4, exactly 44. Then the frontier: S(5) = 160, proved in 2017 by Marijn Heule's SAT certificate occupying TWO PETABYTES — the largest mathematical proof ever constructed, for a puzzle a child can state (Schur invented them in 1917 for modular Fermat equations). Verified live: S(2)=4 both directions (witness + all 32 bipartitions of 1..5 fail); S(3)=13 both directions (witness {1,4,10,13}/{2,3,11,12}/{5,6,7,8,9} + pruned exhaustive DFS of 1,954 nodes proving 1..14 impossible). Neon-noir traced. See the 13-coloring in 1D, the blocked 14 in 2D, and the staircase of proof sizes in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "f6d01b252d4efffef995eed07decbc25d54088f998c13e5a828a0c6f8a592033"
    },
    {
      "slug": "the-nested-radical",
      "title": "THE NESTED RADICAL",
      "kicker": "the infinite root that equals three",
      "accent": "#ff8a3c",
      "blurb": "Ramanujan's nested radical in the 5-window house format — in 1911 a Madras clerk mailed the Journal of the Indian Mathematical Society a puzzle: evaluate √(1+2√(1+3√(1+4√(…)))). Six months, no solvers; Ramanujan published the answer himself: exactly 3, via the telescoping identity x+1 = √(1+x√(1+(x+1)√(…))). The infinite dig has a clean bottom. Cousins verified alongside: √(2+√(2+…)) = 2, √(6+√(6+…)) = 3, √(1+√(1+…)) = φ. Verified live with rigorous two-sided bracketing: depth-60 truncations seeded LOW and HIGH pin 3 between them to 14 decimals; the tail identity (=4) bracketed the same way; the fixed-point cousins to 1e-10. Neon-noir traced. See the telescope in 1D, the squeezing brackets in 2D, and the radical well in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "8b026b105463569645a580189977f7f06ae5d267018cfbc9fe7a86dfebb6d3fd"
    },
    {
      "slug": "the-lazy-caterer",
      "title": "THE LAZY CATERER",
      "kicker": "every cut counted exactly",
      "accent": "#21e6ff",
      "blurb": "The lazy caterer's sequence in the 5-window house format — slice a pancake with n straight cuts: the most pieces is 1+n+C(n,2) (2, 4, 7, 11, 16, 22…), because each new cut adds one region plus one per crossing. Stack the same logic into 3D and you get the cake numbers (n³+5n+6)/6 — each plane slices in the pattern of a 2D arrangement, so 3D is a running sum of 2D: Pascal's triangle wearing an apron. Verified live with exact arithmetic: random-slope arrangements built over BigInt rationals, general position certified, regions counted incrementally for n = 5, 12, 25, 40 matching BOTH the closed formula and the independent Euler route (V=C(n,2), E=n²); the cake recurrence checked to n=30. Neon-noir traced. See the growing cuts in 1D, the live triple count in 2D, and the stacked cake in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "3ef9329e058d3ab580d467c7c9d16e4e4e27a816a0c4d5716ddbf346f4ea9a6c"
    },
    {
      "slug": "the-parker-square",
      "title": "THE PARKER SQUARE",
      "kicker": "the celebrated failure",
      "accent": "#ffcf4a",
      "blurb": "The Parker Square in the 5-window house format — can a 3×3 magic square be built from nine DISTINCT perfect squares? Genuinely open (LaBar 1984; Bremner's analysis). In 2016 Matt Parker gave it a go on Numberphile: his square of squares gets SEVEN of eight lines summing to 3051, misses one diagonal (4107), and repeats three entries — and the internet made it the mascot of glorious, instructive failure. The underlying rigidity: opposite entries must sum to twice the center, reducing the hunt to four disjoint square-pairs balanced around a central square. Verified live: the Parker Square audited exactly (7/8 at 3051, the 4107 diagonal, 3 repeats — my own memory said 6/8; the computation said 7 and won), plus an exhaustive structural sweep proving no valid square of distinct squares exists with center up to 1500². Neon-noir traced. See the seven green lines in 1D, the audits in 2D, and the one loose diagonal in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "cf4ed89c69bca33e3160f66575bde9214d5ea007c1e369493f1781191b61ba8e"
    },
    {
      "slug": "the-multiperfect",
      "title": "THE MULTIPERFECT",
      "kicker": "coins the mint stopped printing",
      "accent": "#b06bff",
      "blurb": "Multiperfect numbers in the 5-window house format — perfect numbers pay back double (σ(n)=2n); triperfects pay TRIPLE: 120, 672, 523776, 459818240, 1476304896, 51001180160 — and that is believed to be the complete list forever (exactly six, all even; an odd seventh would summon an odd perfect number, the oldest open question in mathematics). Quadruple-perfect: 30240, 32760… Mersenne and Fermat traded these by letter in the 1630s. Verified live: full σ-sieve to 2²⁰ finds exactly {120, 672, 523776} triperfect and {30240, 32760} quadperfect with the perfect anchor {6,28,496,8128}; every hit re-verified by independent trial-division σ. Neon-noir traced. See the abundancy sea in 1D, each coin's divisors in 2D, and the six-coin hoard with its empty die in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "5b5580b6ebee0fc142408314945971671d2a78c90b2ae9f2ecd063bc158d26ef"
    },
    {
      "slug": "the-practical",
      "title": "THE PRACTICAL",
      "kicker": "exact change for every bill",
      "accent": "#35ffb0",
      "blurb": "Practical numbers in the 5-window house format — n is practical when every amount from 1 to n is payable in DISTINCT divisors of n. 12 works (1,2,3,4,6 make everything); ancient bazaars ran on them, Fibonacci used them for Egyptian-fraction change, and 12, 60, 240 became coinage and clock faces for exactly this property. Srinivasan (1948) and Stewart (1954) found the complete DNA: each prime, in order, must arrive no later than one-plus-the-divisor-sum of what came before — recursive solvency. They even mirror the primes: Goldbach-for-practicals is a THEOREM (Melfi 1996). Verified live with two fully independent engines — brute subset-sum DP versus the Stewart–Sierpiński cascade — on every n ≤ 5,000 with ZERO disagreements; census prefix exact. Neon-noir traced. See 12's wallet in 1D, the twin engines in 2D, and the prime cascade in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "e32f356abcdb17d602bf5e9b2b3af6c959ed0f012e76b6e1f6b5209873befa1e"
    },
    {
      "slug": "the-kissing-number",
      "title": "THE KISSING NUMBER",
      "kicker": "how many can touch the one",
      "accent": "#ff8a3c",
      "blurb": "Kissing numbers in the 5-window house format — how many unit spheres can touch one central sphere? In 2D: 6, with a one-sentence proof (neighbors ≥ 60° apart, 7×60 = 420 > 360). In 3D the question ignited the 1694 Newton–Gregory argument — 12 or 13? — because the icosahedral twelve leave visible slack (neighbors at 2.10, not 2.00); Schütte–van der Waerden vindicated Newton only in 1953, 259 years later. Beyond: K(4)=24 (Musin 2003), and only dimensions 8 (240, E₈) and 24 (196,560, Leech) are also solved — the Viazovska-era miracle dimensions. Verified live: the hexagonal 6-kiss exact, the 7-impossibility executed as the chord-angle pigeonhole (boundary algebraically exact), and the icosahedral 12-kiss constructed from (0,±1,±φ) with min neighbor distance 2.1029… ≥ 2. Neon-noir traced. See the six-and-no-seventh in 1D, the failed insertion in 2D, and Gregory's ghost sphere in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "16fe28e8668a7640c7a292a2eafc82637f63ab4e190b2865e658199d55838a76"
    },
    {
      "slug": "the-hundred-prisoners",
      "title": "THE HUNDRED PRISONERS",
      "kicker": "a pointer-chase that beats impossible odds",
      "accent": "#35ffb0",
      "blurb": "The 100 prisoners problem in the 5-window house format — 100 boxes, numbers permuted; each prisoner opens 50; ALL must find their own number or all die. Random opening: (1/2)¹⁰⁰ ≈ 8×10⁻³¹. The miracle: start at your own box and follow the numbers — chaining along the permutation's cycles — and everyone succeeds exactly when no cycle exceeds 50: probability 1 − (H₁₀₀−H₅₀) ≈ 31.18%. The brain-breaker: no individual's odds improve; the strategy CORRELATES the failures, spending them together. Verified live: exact harmonic computation (0.311828), 200k-trial Monte-Carlo within 0.5%, and the random strategy winning zero. Neon-noir traced. See the cycles in 1D, the running record in 2D, and the shared fate in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "9345ebea56c64b3e3f6d27e4c34278c212b97e993896390589858f0a18bcad89"
    },
    {
      "slug": "the-pirate-game",
      "title": "THE PIRATE GAME",
      "kicker": "gold divided by pure logic",
      "accent": "#ffcf4a",
      "blurb": "The pirate game in the 5-window house format — five perfectly rational pirates split 100 gold by seniority proposal and majority vote (ties favor the proposer); rejected proposers are thrown overboard. Intuition says bribe heavily; backward induction says the senior pirate keeps 98, hands single coins to pirates 3 and 5: [98,0,1,0,1], and it passes. Push past 2G pirates and gold can no longer buy votes — proposers survive only at crew sizes 2G + 2^k, islands of survival at powers of two (Ian Stewart's analysis). Verified live: full DP from one pirate up — the 5-pirate answer exact, and with G=10 the survival islands land at exactly 20+{1,2,4,8,16,32,64}. Neon-noir traced. See the sub-game ladder in 1D, the crew stepper in 2D, and the survival sea in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "080b2a2a71a6894060348b2d24bb3c426ab42777008e728b59c25b67350af931"
    },
    {
      "slug": "the-blue-eyes",
      "title": "THE BLUE EYES",
      "kicker": "the announcement everyone already knew",
      "accent": "#21e6ff",
      "blurb": "The blue-eyes puzzle in the 5-window house format — an island of perfect logicians; anyone who deduces their own eye color must leave that night. A visitor announces what everyone can already see: 'I see at least one person with blue eyes.' If b islanders are blue-eyed, all b leave on night b — because the announcement, contentless about eyes, created COMMON KNOWLEDGE: everyone knows that everyone knows, to unlimited depth, and that tower is load-bearing. Verified live by an executable possible-worlds engine (Kripke semantics, computed): for every configuration at n ≤ 8, blues leave exactly night b and browns night b+1 — and rerunning WITHOUT the announcement, nobody ever leaves, in any world. Neon-noir traced. See the three quiet nights in 1D, the two regimes in 2D, and the knowledge tower in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "699ed995d4667e6611851f7f94ecd16a25137a91c55d0b10cdc656f491cc0295"
    },
    {
      "slug": "the-chip-firing",
      "title": "THE CHIP-FIRING",
      "kicker": "avalanches that forget their order",
      "accent": "#ff8a3c",
      "blurb": "The abelian sandpile in the 5-window house format — chips on a grid; any cell with 4+ fires one to each neighbor; avalanches cascade. Dhar's theorem (1990): the final stable configuration is IDENTICAL regardless of firing order — chaos with a deterministic destination. The stable states form a group whose identity element is a breathtaking fractal mandala nobody designed; and the model is the birthplace of self-organized criticality (Bak–Tang–Wiesenfeld 1987). Verified live on 25×25: one random configuration stabilized under 20 different random orders, byte-identical every time; the identity computed via e = stab(2m − stab(2m)), verified idempotent, and certified recurrent by Dhar's burning test (all 625 sites burn exactly once). The W5 window draws the actual computed identity. Neon-noir traced.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "22934cbe7a4d82706c5cbb9c6746ac871270210b964c6543e6b05e6cf91224dd"
    },
    {
      "slug": "the-hackenbush",
      "title": "THE HACKENBUSH",
      "kicker": "numbers born from games",
      "accent": "#b06bff",
      "blurb": "Blue-Red Hackenbush in the 5-window house format — colored edges on the ground; Left cuts blue, Right cuts red; disconnected pieces fall; no move = lose. From this child's game Conway discovered that positions have numerical values, and the numbers are the SURREAL numbers: a blue edge is +1, blue-with-red-on-top is exactly ½, blue-red-red is ¼ — and value arithmetic PREDICTS game outcomes: zero sums are second-player wins, positive means Left wins from either seat. Verified live by pure exhaustive minimax that knows no value theory: BR+BR+R second-player (½+½−1=0), four BRRs plus R second-player, BR+RB cancels, BR alone Left-wins both seats, BR+RBB to Left (½>¼) — seven arithmetic claims converted into game-tree facts. Neon-noir traced. See the value ladder in 1D, arithmetic-vs-minimax in 2D, and the birth tree in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "a43d5a045bd6e9449fa6712fc36d6834322a73db3f58b4126195fe1e90718cac"
    },
    {
      "slug": "the-martingale",
      "title": "THE MARTINGALE",
      "kicker": "the system that always wins until it doesn't",
      "accent": "#ffcf4a",
      "blurb": "The martingale betting system in the 5-window house format — bet 1, double after every loss; your first win recovers everything plus one. With a ten-round bankroll you win 99.9% of sessions, and the mathematics is merciless: on a fair game EV is EXACTLY zero (the rare −1023 bust precisely cancels the parade of +1s), and on roulette (18/38) it is exactly 1−(2q)^k < 0 — doubling doesn't shrink the house edge, it CONCENTRATES it into catastrophes. Variance reshaped, expectation untouched (optional stopping, in its most famous costume). Verified live: fair EV exact to the last bit, 200k-session MC matching the 99.90% win rate and near-zero mean, roulette EV by two independent algebraic routes agreeing exactly. Neon-noir traced. See the staircase-and-cliff in 1D, running sessions in 2D, and the shape of the risk in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "c4d3fbb74971ab60f7ceb8d7ee49acb2ad8dec730f8a515459264c0bf7a2becf"
    },
    {
      "slug": "the-two-child",
      "title": "THE TWO-CHILD",
      "kicker": "the answer that depends on how you asked",
      "accent": "#21e6ff",
      "blurb": "The two-child paradox in the 5-window house format — 'At least one of my two children is a boy': P(both boys) = 1/3. 'My ELDER child is a boy': 1/2. Then Gary Foshee, Gathering 4 Gardner 2010: 'at least one is a boy born on a TUESDAY' — and the answer becomes 13/27, the irrelevant weekday dragging 1/3 nearly to 1/2. Deepest layer: learn the same fact by MEETING a random child and the answer snaps back to 1/2 — the number depends on the sampling protocol, and without declaring it the question is genuinely underdetermined. Verified live: 1/3 and 1/2 by exact enumeration, 13/27 by exact count over all 196 (sex,weekday) pairs, and the protocol dependence by Monte-Carlo (0.499). Neon-noir traced. See the 196-cell grid in 1D, the phrasing switch in 2D, and the pruning worlds in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "146123c5d02f3e1bf5e72bd219e29c4dabe1ade72dedb603eaadfa28e91c70c6"
    },
    {
      "slug": "the-will-rogers",
      "title": "THE WILL ROGERS",
      "kicker": "a transfer that flatters everyone",
      "accent": "#b06bff",
      "blurb": "The Will Rogers phenomenon in the 5-window house format — 'When the Okies left Oklahoma and moved to California, they raised the average intelligence of both states.' The joke is a theorem: moving one element from B to A raises BOTH averages exactly when it sits between the two means. Nothing improves; both dashboards celebrate. In medicine this is stage migration (Feinstein & Sosin 1985): better scanners reclassify borderline patients, survival improves in EVERY cancer stage simultaneously, and no one lives a day longer. Verified live: the classic example exact (2.5→3.0 and 7.0→7.5), and the full characterization — both rise ⟺ mean(A) < x < mean(B) — verified with zero exceptions across ~900 random set pairs, every element tested. Neon-noir traced. See the crossing transfer in 1D, the between-means rule in 2D, and the twin gauges over an unchanged world in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "123623eb425bf771aab6a5e85f7627d6e2018af9d0055f067680b042e58bbc08"
    },
    {
      "slug": "the-inspection-paradox",
      "title": "THE INSPECTION PARADOX",
      "kicker": "the bus that is always late for you",
      "accent": "#ff8a3c",
      "blurb": "The inspection paradox in the 5-window house format — buses every 10 minutes on average should mean 5-minute waits; you wait longer, by arithmetic: arriving at random you land in an interval with probability proportional to its LENGTH, so your experienced interval averages E[X²]/E[X] ≥ E[X], equality only for clockwork service. The extreme: exponential (memoryless) spacing gives a full ten-minute expected wait, as if the schedule restarted on your arrival. Length-biased sampling runs everywhere — friends with more friends, class sizes, congested routers. Verified live on a simulated million-minute timeline, 120k random arrivals per schedule: deterministic 10.00/5.00, exponential 20.1/10.1, mixed-5-or-15 12.5/6.25 — all matching E[X²]/E[X] within 1%. Neon-noir traced. See the crowded long gaps in 1D, the schedule switch in 2D, and the variance tax in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "ecc8ddf8efcd78492af7893375609f1b0656a10798a1f1ed817d6ab7f32d1708"
    },
    {
      "slug": "the-german-tank",
      "title": "THE GERMAN TANK",
      "kicker": "counting tanks from their serial numbers",
      "accent": "#35ffb0",
      "blurb": "The German tank problem in the 5-window house format — WWII intelligence said 1,400 tanks a month; the statisticians read the captured serial numbers and said 246; postwar German records showed 245. The estimator: seeing k serials with maximum m, answer m(1+1/k)−1 — the observed maximum pushed up by the average gap. It is minimum-variance unbiased: across ALL possible samples it averages to exactly N. The same trick has counted iPhones and Commodore 64s since. Verified live: exact unbiasedness by COMPLETE enumeration (all 15,504 five-samples from a population of twenty average to exactly 20.0000000000), plus Monte-Carlo at the historical scale (N=245, k=10) with the MVUE at half the error of doubled-mean (21.8 vs 43.7 RMSE) and the raw max biased low. Neon-noir traced. See the serial line in 1D, the estimator tournament in 2D, and 246-vs-1400-vs-245 in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "103cb2a4929176944f27c0227e8c6fe8a3aeee0b693b4c12f5b4c177b113c989"
    },
    {
      "slug": "the-napkin-ring",
      "title": "THE NAPKIN RING",
      "kicker": "a ring that forgets its sphere",
      "accent": "#ffcf4a",
      "blurb": "The napkin ring problem in the 5-window house format — drill a cylindrical hole through a sphere's center leaving a ring of height h: the remaining volume is πh³/6, and the sphere's radius has VANISHED from the formula. A height-6 ring from an orange and one from the Earth hold identical volume — the planet's is wafer-thin but vast, the orange's thick but tiny, the trade exact. Cavalieri's proof is the jewel: the cross-section annulus area is π((h/2)²−y²) — R cancels BEFORE you integrate. Verified live three ways: numeric integration for R = 5, 50, 500 all landing on 113.0973; the Cavalieri cancellation exact at 100 heights; 2M-point Monte-Carlo within 1%. Neon-noir traced. See matched slices in 1D, the growing sphere with pinned volume in 2D, and rings of three worlds in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "df27f184f41e09656eb59a1756c0270ae6e61563c3e078dade40962250d69c2c"
    },
    {
      "slug": "the-coastline",
      "title": "THE COASTLINE",
      "kicker": "the coast that has no length",
      "accent": "#21e6ff",
      "blurb": "The coastline paradox in the 5-window house format — Richardson asked how long the coast of Britain is and found it depends on the ruler, divergently: halve the ruler and the length grows by a power law, because every bay hides smaller bays. Mandelbrot's 1967 paper on that data launched fractal geometry: coastlines have no length but they have a DIMENSION. The laboratory specimen is the Koch curve: length exactly (4/3)ⁿ (divergent), dimension exactly log4/log3 = 1.2619. Verified live: Koch lengths match (4/3)ⁿ to 1e-9 for n ≤ 7; the dimension measured by TWO independent meters — box-counting (1.29) and Richardson's ruler-walking (1.20) — both bracketing the truth. Neon-noir traced. See the unfolding curve in 1D, the shrinking ruler in 2D, and the bottomless zoom in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "5366fbff7dbf2a8f96d941f0d90d24ecfbe28c7ce726504c93163eac7b46f843"
    },
    {
      "slug": "the-aristotle-wheel",
      "title": "THE ARISTOTLE WHEEL",
      "kicker": "the wheel that skids in plain sight",
      "accent": "#ff8a3c",
      "blurb": "Aristotle's wheel paradox in the 5-window house format — two concentric wheels welded together roll one revolution; both advance 2πR, but the small wheel's circumference is only 2πr: how did it unroll more road than it has rim? The puzzle stumped readers of the pseudo-Aristotelian Mechanica for 2,300 years (Galileo wrestled it in Two New Sciences). The resolution is measurable: only the big wheel ROLLS; the small one SKIDS — its contact point never rests (speed ω(R−r)) while the big wheel's touches down at speed zero (the cycloid's cusp), dragging slip of exactly 2π(R−r) per turn. Verified live: cycloid arc length exactly 8R, trochoid 6.68R over the same advance, bottom speeds 0 vs 0.5, slip exact. Neon-noir traced. See the two paths in 1D, the speed gauges in 2D, and the glowing skid trail in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "c8e9f36b00daaaad0c373d73ba599a56fd6304c588b63ad754c073e93bb39587"
    },
    {
      "slug": "the-mountain-climber",
      "title": "THE MOUNTAIN CLIMBER",
      "kicker": "two climbers in height-lockstep",
      "accent": "#35ffb0",
      "blurb": "The mountain climbing theorem in the 5-window house format — two climbers on opposite faces of a range want to summit while staying at EXACTLY equal altitude the whole way. The theorem: for any two continuous profiles with shared endpoints, the synchronized traversal always exists — but the climbers must sometimes go BACKWARDS, descending a peak already won, to let a partner cross a valley; always-forward fails. For piecewise-linear mountains the proof is executable: BFS on the equal-height coordination graph. Verified live: 200 random mountain pairs, 200 synchronized routes found, including the classic pair whose solution provably requires backtracking. Neon-noir traced. See the twin profiles in 1D, freshly generated ranges in 2D, and the coordination square in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "44253da78e426e09c5a149bdae8f9682d3f9958183ff6451cc6e40bc4b712fee"
    },
    {
      "slug": "the-brouwer",
      "title": "THE BROUWER",
      "kicker": "the point that cannot escape",
      "accent": "#b06bff",
      "blurb": "Brouwer's fixed-point theorem in the 5-window house format — crumple a map of your city and drop it anywhere in the city: one point lies exactly atop the place it depicts. Every continuous self-map of a disk fixes at least one point (Brouwer 1911) — the engine beneath Nash equilibria and market-clearing prices. The most beautiful proof is combinatorial: Sperner's lemma (1928) — triangulate, label by simple rules, and an ODD number of small triangles must carry all three labels; odd cannot be zero, and those triangles corner the fixed point. Verified live: 30 random self-maps, fully-labeled count odd every time; the flagged triangle localizes a fixed point with error shrinking 3e-2 → 2e-3 under refinement; and 1D Brouwer (= IVT) bisected to 1e-12. Neon-noir traced. See the diagonal crossing in 1D, the Sperner rainbow in 2D, and the crumpled map in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "a29b1134e700ae1152750225065a9671eeb18df207e0b956a64d04abdbf57f98"
    },
    {
      "slug": "the-ham-sandwich",
      "title": "THE HAM SANDWICH",
      "kicker": "one cut for two appetites",
      "accent": "#35ffb0",
      "blurb": "The ham sandwich theorem in the 5-window house format — two tangled point clouds on a table, and a SINGLE straight line bisecting both simultaneously, guaranteed (Steinhaus 1938; Stone–Tukey 1942). In 3D one planar cut halves bread, ham, AND cheese; in n dimensions one hyperplane bisects n masses. The proof rotates: anchor the line to always bisect red, sweep 180°; the blue imbalance flips sign, so it crosses zero — Borsuk–Ulam wearing an apron. Verified live: 300 random 20+20 point-set pairs, the rotating construction finds and count-verifies the double bisector every time (with sign-change refinement at anchor jumps). Neon-noir traced. See the clouds and blade in 1D, fresh instances in 2D, and the draining imbalance in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "ed516382c6b8254fc5a9b11d2b297b66fbc2389442ca0bc87b3b067fee304403"
    },
    {
      "slug": "the-hairy-ball",
      "title": "THE HAIRY BALL",
      "kicker": "the coconut that cannot be combed",
      "accent": "#21e6ff",
      "blurb": "The hairy ball theorem in the 5-window house format — every continuous tangent field on a sphere vanishes somewhere (Poincaré 1885; Brouwer 1912): there is ALWAYS a point on Earth with zero horizontal wind. The bookkeeping is Poincaré–Hopf: zero indices sum to the Euler characteristic — 2 for the sphere (failure forced), 0 for the torus (a donut CAN be combed). Verified live three ways: 50 quadratic-form Morse censuses reading exactly 2+2−2 = 2; 40 random smooth winds with a vanishing point LOCATED each time via the eigen-parameter equation (M−λI)p = −c, |p|=1, residual < 1e-6; and the explicit torus field with |v| ≥ 1.3 everywhere. Neon-noir traced. See the wind and its eye in 1D, the λ-pinned zeros in 2D, and the combed donut beside the doomed sphere in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "57d99ad70632661f8e0dd634a297c32d00571ac99f899f51b3c466749fa71b77"
    },
    {
      "slug": "the-cake-cutting",
      "title": "THE CAKE CUTTING",
      "kicker": "cake without envy",
      "accent": "#ffcf4a",
      "blurb": "The Selfridge–Conway procedure in the 5-window house format — cut-and-choose settles two; THREE people with private valuations need the first bounded envy-free protocol ever found (c. 1960): at most five cuts, and afterwards no one would trade shares, each by their OWN measure. P1 cuts three equal-to-them pieces; P2 trims the largest to a tie; choices cascade; the trimmings divide in a second round whose picking order neutralizes every resentment. (Four players resisted until Aziz–Mackenzie 2016.) Verified live: the full procedure over exact piecewise-constant measures — 300 random valuation triples, all 1,800 envy comparisons, envy-free every time with worst envy 5×10⁻¹⁶. Neon-noir traced. See three value-landscapes in 1D, the all-clear envy matrix in 2D, and shares each tallest in their owner's eyes in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "f2d56e8a7781b428572c33e94a327645cfbd0f45d4fdc186473815c55e296cf7"
    },
    {
      "slug": "the-stable-marriage",
      "title": "THE STABLE MARRIAGE",
      "kicker": "the proposer's hidden crown",
      "accent": "#b06bff",
      "blurb": "Gale–Shapley in the 5-window house format — n rank n; a matching is stable when no pair would elope. Deferred acceptance (propose, hold, reject, rise again) always finds one — and the dark exact twist: it is OPTIMAL for every proposer and PESSIMAL for every reviewer; each man gets his best partner across ALL stable matchings, each woman her worst. Whoever proposes, wins. The framework runs residency matching and school choice; Roth & Shapley took the 2012 Nobel. Verified live two ways: zero blocking pairs on 300 random instances, AND complete enumeration of all 720 matchings across 60 instances confirming men-optimal/women-pessimal with no exceptions. Neon-noir traced. See the proposal rounds in 1D, instances with their exhaustive court in 2D, and the lattice of stable matchings in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "2f81075e0333e650e24a8183f799548baeb50acbdc77a3fde558b6cf64e7815d"
    },
    {
      "slug": "the-tupper",
      "title": "THE TUPPER",
      "kicker": "the formula that draws everything",
      "accent": "#ff8a3c",
      "blurb": "Tupper's formula in the 5-window house format — one inequality, ½ < ⌊mod(⌊y/17⌋·2^(−17⌊x⌋−mod(⌊y⌋,17)), 2)⌋, plots a picture of ITSELF at a particular 543-digit k. The honest magic, better than the myth: it is a universal bitmap decoder — the giant k IS the picture (every pixel a bit of k's binary expansion) and the formula merely reads bit 17⌊x⌋+mod(y,17) back out. It plots everything — your name, a smiley — each image at its own altitude; the famous self-portrait is self-reference by construction. Verified live in exact BigInt: 50 random 106×17 bitmaps encoded and decoded through the actual formula arithmetic, formula route ≡ direct-bit route ≡ original at every sample; a structured 1,802-pixel bitmap round-trips exactly (k has 424 digits). Neon-noir traced. See the pipeline in 1D, live payload roundtrips in 2D, and the everything-tower in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "a831ca7c6bb9e8a4825850dc803a7933b62409d9082003d69b2bff9cc40280a3"
    },
    {
      "slug": "the-borsuk-ulam",
      "title": "THE BORSUK-ULAM",
      "kicker": "antipodes that must agree",
      "accent": "#b06bff",
      "blurb": "The Borsuk–Ulam theorem in the 5-window house format — right now, somewhere on Earth, two antipodal points have exactly the same temperature AND pressure: provably (Borsuk 1933, answering Ulam). Every continuous map from the n-sphere to ℝⁿ collapses some antipodal pair. It is the boss theorem of a whole dungeon — ham sandwich, Brouwer's fixed point, and necklace splitting all drop from it. The 1D proof fits in a line: g(θ) = f(θ)−f(θ+π) satisfies g(0) = −g(π), so it crosses zero. Verified live: 200 random circle functions with the antipodal pair bisected to 1e-10 every time, and 50 random (temperature, pressure) sphere pairs with the odd map driven below 1e-5 — the promised twins located. Neon-noir traced. See the forced crossing in 1D, the weather twins in 2D, and the pinned globe in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "9d9bbe4c3f944dffee48a123bb985173a19f1b9d97bcff4d3cbf541302d13929"
    },
    {
      "slug": "the-banach-tarski",
      "title": "THE BANACH-TARSKI",
      "kicker": "two spheres from one",
      "accent": "#ffcf4a",
      "blurb": "The Banach–Tarski engine in the 5-window house format — the 1924 theorem says a ball splits into five pieces that rotations reassemble into TWO identical balls. The full theorem needs the axiom of choice and non-measurable pieces (no knife can cut them) — but its ENGINE is computable and this sphere runs it: in the free group F₂, S(a) ∪ a·S(a⁻¹) = the whole group, and the b-side yields the second copy — two wholes from one, by relabeling. The bridge to geometry: two rotations built from the 3-4-5 triangle generate a free group inside SO(3). Verified live and exactly: 118,097 reduced words with the five-set partition and both doubling identities checked on every word; and the rotations proven free — all 13,120 words to length 8 evaluated in exact BigInt integer matrices, none the identity. Neon-noir traced. See the Cayley tree in 1D, the doubling steps in 2D, and one-ball-two-balls in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "eac0ad364149f072999addd51eea97f1af50980c9437f82db56219880da924b4"
    },
    {
      "slug": "the-nontransitive-dice",
      "title": "THE NONTRANSITIVE DICE",
      "kicker": "dice with no best",
      "accent": "#ff8a3c",
      "blurb": "Efron's nontransitive dice in the 5-window house format — four honest-weight dice with strange faces: A=[4,4,4,4,0,0], B=[3,3,3,3,3,3], C=[6,6,2,2,2,2], D=[5,5,5,1,1,1]. A beats B beats C beats D beats A — every arrow at exactly 2/3. Rock-paper-scissors smuggled into cubes: there is NO best die; whatever your opponent picks, one of the rest beats it two times in three. Buffett offered Gates first pick of such a set; Gates examined the dice and made Buffett choose first. Verified live by complete enumeration: all 36 outcomes per adjacent pair — 24/36 exactly, four times around — and the no-best-die beat graph confirmed. Neon-noir traced. See the 2/3 cycle in 1D, the 36-cell tables in 2D, and the circle-where-a-ladder-should-be in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "543ab8289377474dfba4154ee8310fabc54b1ecafb3827475871424f1bfba6d7"
    },
    {
      "slug": "the-hundred-doors",
      "title": "THE HUNDRED DOORS",
      "kicker": "doors that remember their divisors",
      "accent": "#21e6ff",
      "blurb": "The hundred doors in the 5-window house format — 100 closed doors; pass k toggles every k-th; after 100 passes exactly the perfect squares stand open: 1, 4, 9, …, 100. The one-line jewel: door n is toggled once per divisor, divisors pair d ↔ n/d, and only a square's √n partners itself — odd toggle count ⟺ open ⟺ square. A divisor-parity detector built from hinges. Verified live two ways: full simulation (open set = the ten squares exactly) and the independent τ(n)-parity engine checked for all n ≤ 1000, plus the 1000-door corridor opening exactly 31 (31² = 961). Neon-noir traced. See the surviving squares in 1D, the interfering passes in 2D, and the divisors pairing off in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "28aea6796fc6a42c8cb2c8e9580e561ca8c3446875877191f321b469ca59b591"
    },
    {
      "slug": "the-sleeping-beauty",
      "title": "THE SLEEPING BEAUTY",
      "kicker": "the princess with two right answers",
      "accent": "#35ffb0",
      "blurb": "The Sleeping Beauty problem in the 5-window house format — heads: wake her once; tails: twice with memory erased between; each waking she's asked her credence the coin was heads. Thirders (Elga 2000) say 1/3; halfers (Lewis 2001) say 1/2; the war has run twenty-five years. This sphere executes BOTH: per-awakening frequency of heads is 1/3, per-experiment is 1/2 — and the betting table settles what words cannot: per-awakening heads bets are fair at exactly 2:1 odds, while at even odds she bleeds precisely ½ per experiment. Verified live over 200,000 simulated experiments: 0.500 / 0.334 / EV(2:1) ≈ 0 / EV(1:1) = −0.4982. Neon-noir traced. See the experiment tree in 1D, the question switch in 2D, and the two honest counters in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "66600d3b88c4e9121e587e63df675bea3563af25523bf70a1cbb9d5f1cfa45e1"
    },
    {
      "slug": "the-wallis",
      "title": "THE WALLIS",
      "kicker": "pi milled from fractions",
      "accent": "#ffcf4a",
      "blurb": "The Wallis product in the 5-window house format — π/2 = (2·2)/(1·3) · (4·4)/(3·5) · (6·6)/(5·7)… (Wallis 1656): every factor barely above 1, grinding toward the circle constant with famously slow convergence (error ≈ π/8n). Two secret identities: the partial products equal (4ⁿ/C(2n,n))²/(2n+1) EXACTLY — central binomials in disguise — and in 2015 Friedmann & Hagen found the whole formula hiding in the quantum hydrogen atom, 359 years late. Verified live: 50,000 factors at 1.5707885 vs π/2; the binomial route matching the direct product to 1e-10 for all n ≤ 200; the error law n·(π/2−Wₙ) → π/8 measured to four decimals. Neon-noir traced. See the factors in 1D, the crank in 2D, and the fraction mill in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "2bde2bfe54abb516fbe7ab2803f01fc04b799863408ab99e091ff90a88ab84f6"
    },
    {
      "slug": "the-thomae",
      "title": "THE THOMAE",
      "kicker": "popcorn continuous only off the grid",
      "accent": "#21e6ff",
      "blurb": "Thomae's function in the 5-window house format — f(p/q) = 1/q, f(irrational) = 0: the popcorn graph, kernels bursting at every rational. Analysis's favorite monster: DISCONTINUOUS at every rational, CONTINUOUS at every irrational — continuous exactly on a set full of holes that is almost everything. The mechanism is Diophantine: simple fractions are rare near any point. Verified live with certificates, not pictures: discontinuity at 1/2, 1/3, 2/5, 3/7 (shrinking neighborhoods contain only bigger-denominator rivals); continuity at √2−1 certified level by level — for every n ≤ 60 a strictly positive δₙ inside which every rational has q > n, forcing f < 1/n (δ₆₀ = 4.2×10⁻⁴). Neon-noir traced. See the popcorn in 1D, the thinning zoom in 2D, and the irrational thread in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "a50339e7ce15195765c0a0c32492808b04ca53901dbe5bd997f410491d7e0a1f"
    },
    {
      "slug": "the-question-mark",
      "title": "THE QUESTION MARK",
      "kicker": "continued fractions transcribed to binary",
      "accent": "#b06bff",
      "blurb": "Minkowski's question-mark function in the 5-window house format — ?(x) reads a number's continued fraction and writes it out in binary: ?([0;a₁,a₂,…]) = Σ(−1)^(k+1)·2^(1−(a₁+…+aₖ)). The jewels: quadratic irrationals (periodic CFs) map to RATIONALS — ?(φ−1) = 2/3, ?(√2−1) = 2/5 — and rationals map to dyadics. Continuous, strictly increasing, yet SINGULAR: derivative zero almost everywhere, all the rise packed into an invisible set. Verified live: 2/3 and 2/5 to 1e-14 via the series AND exactly via the geometric identity r/(1+r) in integer arithmetic; ?(1/3) = 1/4 exact; monotone across 800 sorted points; median local slope ~1e-3 as singularity evidence. Neon-noir traced. See the slippery staircase in 1D, constants fed through in 2D, and the two-alphabet bridge in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "440ffe0cfcfe5a8be2740ee58cf33f37415f9b6e260929219a62f0503636909c"
    },
    {
      "slug": "the-grandi",
      "title": "THE GRANDI",
      "kicker": "the sum that flickers",
      "accent": "#ff8a3c",
      "blurb": "Grandi's series in the 5-window house format — 1−1+1−1+…: partial sums flicker 1,0,1,0 forever; divergent, full stop. Yet Euler called it ½, and he was made rigorous by redefining the question: Cesàro (average the partials: ⌈n/2⌉/n → ½ exactly) and Abel (damp by xᵏ, let x→1: Σ(−x)ᵏ = 1/(1+x) → ½) assign the same value — and both are REGULAR, returning ordinary sums on honestly convergent series, so nothing old breaks. The lesson that built summability theory: a divergent series isn't meaningless; it's waiting for a better question. Verified live: the oscillation shown; Cesàro exactly ⌈n/2⌉/n; the Abel identity to 1e-8 at x = 0.9, 0.99, 0.999; regularity confirmed on a convergent control. Neon-noir traced. See the square wave and its settling means in 1D, the method switch in 2D, and the flickering heap in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "da7f1acd7e3cf5157d50637e91ffb99020f735b869af8e2e65e44fea33b0bad9"
    },
    {
      "slug": "the-normal-number",
      "title": "THE NORMAL NUMBER",
      "kicker": "the digits nobody can certify",
      "accent": "#35ffb0",
      "blurb": "Normal numbers in the 5-window house format — a real is normal when every digit block appears at its fair frequency. Borel (1909): ALMOST EVERY real is normal. Now name one: π? Unproven. e, √2, ln 2? Unproven, all — a century of silence. The only certified specimens are artificial: Champernowne's 0.123456789101112… (1933), normal by construction. Verified live: Champernowne digit counts computed two ways — direct construction vs the digit-counting formula — agreeing EXACTLY for every digit over 1..200,000; and the honest subtlety shown, not hidden: early digits ARE biased (2.6% at 10⁴) with the deviation shrinking monotonically to 1.5% at 10⁷ — normality is a limit, converging on screen; contrast 1/7, where four digits never appear. Neon-noir traced. See the tape in 1D, the convergence ladder in 2D, and the lone champion in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "d793ae8212f5e08ac5f8a621ff34a8324d5cc7952a05e5c1e16186f129dc05e9"
    },
    {
      "slug": "the-tusi",
      "title": "THE TUSI",
      "kicker": "rotation compiled to translation",
      "accent": "#b06bff",
      "blurb": "The Tusi couple in the 5-window house format — roll a circle inside a circle of exactly twice its radius and a rim point slides in a PERFECT straight line, a diameter (al-Tusi 1247, built to purge Ptolemy's equant; reappearing in Copernicus 1543). The 2:1 ratio is everything: at 3:1 the same point draws a deltoid; interior points trace exact ellipses (the trammel principle). Verified live: max |y| = 0.0 to machine precision over 10,000 steps with span exactly [−2,2]; the 3:1 contrast at max |y| = 2.598; interior ellipse residuals below 1e-10. Neon-noir traced. See the couple mid-roll in 1D, the ratio dial in 2D, and line-with-ellipses in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "a1451b27bcd291e81c0552480a879e477f077ccfc1dbcee5460ab8c3e05ba2e3"
    },
    {
      "slug": "the-witch-of-agnesi",
      "title": "THE WITCH OF AGNESI",
      "kicker": "the witch with no mean",
      "accent": "#21e6ff",
      "blurb": "The Witch of Agnesi in the 5-window house format — y = 8a³/(x²+4a²), studied by Maria Gaetana Agnesi in 1748 in the first mathematics textbook by a woman ('versiera' mistranslated to 'witch'). Area exactly 4πa² — and normalized it is the CAUCHY distribution: the law with NO mean. Sample forever and the running average never settles; the law of large numbers doesn't apply; one monster draw outweighs a million tame ones at any moment. The median behaves perfectly. Verified live: area to 1e-3; Cauchy running means at 10³..10⁶ wandering (−0.32, −0.55, −0.15, +0.36) while the uniform control converges to 3e-4 and the median sits at 0.0003. Neon-noir traced. See the witch and her circle in 1D, the lurching mean in 2D, and the two statistics in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "5de4884eff0714534ef3f53871adab62d8911ebadd3eef21586a4a70c0cb20da"
    },
    {
      "slug": "the-superellipse",
      "title": "THE SUPERELLIPSE",
      "kicker": "between the circle and the square",
      "accent": "#35ffb0",
      "blurb": "The superellipse in the 5-window house format — |x/a|ⁿ+|y/b|ⁿ = 1 (Lamé 1818) interpolates circle (n=2) to rectangle (n→∞); Piet Hein chose n = 2.5 for Sergels Torg, Stockholm (1959) after architects deadlocked between round and rectangular — and it became a design icon (the n=4 'squircle' lives in modern UI corners). Area exactly 4ab·Γ(1+1/n)²/Γ(1+2/n). Verified live: an in-page Lanczos Γ implementation against direct numeric integration at n = 2, 2.5, 4, 8 agreeing to 1e-4, the n=2 anchor on π to nine decimals, n=50 → 3.9974 → 4. Neon-noir traced. See the family in 1D, the dial in 2D, and Sergels Torg from above in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "7549608e873e725d7906bc20057e44380747c06929309f3bb83006a972e6ad27"
    },
    {
      "slug": "the-lissajous",
      "title": "THE LISSAJOUS",
      "kicker": "rationality on an oscilloscope",
      "accent": "#ff8a3c",
      "blurb": "Lissajous figures in the 5-window house format — two sines, one to x, one to y (Bowditch 1815; Lissajous 1857): the curve closes and repeats IF AND ONLY IF the frequency ratio is rational. At 3:2, a clean knot retraced forever; at 1:√2 the beam never returns — dense in the square, arbitrarily close to its start, never landing. An oscilloscope is an irrationality detector. Verified live: (3,2) returns at t=2π with error 7.5e-16 and provably not before (min position+velocity return 0.53); (1,√2) never closes — min return 3.5e-3 over T=100 shrinking to 3.2e-3 by T=3000, never zero; crossing census (3,2) → 6 & 4, (5,4) → 10 & 8 = 2p, 2q. Neon-noir traced. See the gallery in 1D, the ratio switch in 2D, and the live beam in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "a97c4e6b2c9a804c0e912e82af656cac93a0829661b85e8323d3055c5da9e693"
    },
    {
      "slug": "the-caustic",
      "title": "THE CAUSTIC",
      "kicker": "sunlight signing its name in coffee",
      "accent": "#ffcf4a",
      "blurb": "The coffee-cup caustic in the 5-window house format — the bright curve in your cup is the envelope of reflected rays. Pop science says cardioid; the truth is sharper: SUNLIGHT (parallel rays) makes a NEPHROID with cusps at half the radius (the paraxial focus); a BULB ON THE RIM makes the cardioid — same cup, different light, different curve (Huygens; Bernoulli). Verified live by ray tracing: the parallel-ray envelope fits the nephroid to 1.6e-3 while missing every cardioid by 0.65; the rim-source envelope fits a cardioid to 5.3e-3 (scale ≈ 1/3), 120× tighter than any cardioid fits the sun case. A Tin-Foil-style verdict: the 'coffee cardioid' is HALF-right, distinguished numerically. Neon-noir traced. See the pile-up in 1D, the source switch in 2D, and the signature reader in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "3b71762914f55b2daa3343f4cad637f958e2f5865e51c86ade7511058d3ef442"
    },
    {
      "slug": "the-mediant",
      "title": "THE MEDIANT",
      "kicker": "the forbidden addition with its own laws",
      "accent": "#21e6ff",
      "blurb": "Add fractions the wrong way — (a+c)/(b+d) — and you get the mediant: strictly between its parents, generator of every rational via the Stern–Brocot tree, and exactly how combined records work. Which is why Justice out-hit Jeter in 1995 AND 1996, yet Jeter wins both years combined: merges weight by playing time, and the verdict flips at the merge.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "0775c8e36d18320cc6c7c1e2f36a4236736d84faa97f557e88ddd9ab8c5e3b27"
    },
    {
      "slug": "the-brachistochrone",
      "title": "THE BRACHISTOCHRONE",
      "kicker": "the dip that arrives first",
      "accent": "#ffcf4a",
      "blurb": "Bernoulli's 1696 challenge: down which curve does a bead slide fastest? Not the straight line — the inverted cycloid, which dives to build speed and spends it on the flat. Newton solved it overnight, anonymously; Bernoulli recognized 'the lion by his claw.' Every speedrunner knows the trade: position sacrificed for velocity banks the record.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "aca84f833f3295ae3544ae9fbe76b3b61908b615cc7b6a30e36b81985d11e073"
    },
    {
      "slug": "the-catenary",
      "title": "THE CATENARY",
      "kicker": "the chain that corrected Galileo",
      "accent": "#35ffb0",
      "blurb": "Galileo said a hanging chain makes a parabola (1638) — wrong. Leibniz, Huygens and Bernoulli extracted the truth in 1691: y = a·cosh(x/a), fuller in the shoulders. Flip it for the perfect arch (the Gateway Arch); lay it as a road and square wheels roll smooth. Here the chain derives its own curve: a simulated 61-link chain settles onto cosh, 54× closer than any parabola.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "a7335d6905374150692ee482975efac1b25c9f3f6deb325a0172ff95a1a39f0a"
    },
    {
      "slug": "the-tractrix",
      "title": "THE TRACTRIX",
      "kicker": "the leash with constant length",
      "accent": "#ff8a3c",
      "blurb": "Drag a reluctant dog on a taut leash along a straight path: the dog traces the tractrix — the curve whose tangent segment to the axis is always exactly the leash's length. Revolve it and you get Beltrami's pseudosphere: an infinite trumpet with constant curvature −1, the first solid home of hyperbolic geometry — whose total area is exactly a sphere's, 4πa².",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "f7741b61a5263556d540ce7d3ed827cf4c2b8806f47248489a20d75ca96ba0ed"
    },
    {
      "slug": "the-clothoid",
      "title": "THE CLOTHOID",
      "kicker": "comfort is linear curvature",
      "accent": "#b06bff",
      "blurb": "Turn the wheel at constant speed while driving at constant speed: you trace Euler's spiral (1744), curvature growing linearly with distance. That linearity is why it underlies every railway easement and highway ramp — and why roller-coaster loops are clothoid teardrops, not the circles that snapped necks at Coney Island. Wound forever, it stills into the Fresnel eye at (½, ½).",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "76f41df14b65922055be93671cf2184c4f3ef134d87240850bf97ce54d19b693"
    },
    {
      "slug": "the-malfatti",
      "title": "THE MALFATTI",
      "kicker": "the official answer that always loses",
      "accent": "#b06bff",
      "blurb": "Malfatti's 1803 marble problem — pack three circles in a triangle — got his elegant mutually-tangent answer canonized for a century. It is wrong for EVERY triangle: Goldberg 1967 proved never optimal, Zalgaller–Los 1994 proved the greedy shortcut (biggest circle first, repeat) always wins. In the equilateral, officialdom loses by 1.36%.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "2f38fb8a8093d3ed9be82fe8c2a1e7b95a5b1a7c0b5976a855435e0edc6f373a"
    },
    {
      "slug": "the-pitot",
      "title": "THE PITOT",
      "kicker": "the shared tangent ledger",
      "accent": "#35ffb0",
      "blurb": "Wrap any quadrilateral around a circle, every side touching: opposite sides sum equal, a+c = b+d, always. Pitot's 1725 proof is pure bookkeeping — each corner's two tangent segments are equal, so every side spends two entries from a four-entry shared pool, and both sums spend the whole pool. By the engineer whose Pitot tube still reads every aircraft's airspeed.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "401bb4d55c4f6d1aaff0f6fb2da380a67b85802842383c269b3a7f7c42b65643"
    },
    {
      "slug": "the-langley",
      "title": "THE LANGLEY",
      "kicker": "the freak integer angle",
      "accent": "#ffcf4a",
      "blurb": "An 80-80-20 isosceles triangle, cevians at 60° and 50°, find the marked angle. Langley's 1922 puzzle looks like a warm-up and resists every direct angle-chase; the answer is exactly 30°, and it founded the literature of 'adventitious angles' — configurations where whole degrees appear by freak alignment. Scanning all 5,041 integer cevian pairs: only 1.73% compile to an integer.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "42eaced7071a721b4d161115c3ffbafc4450aa27de43964883dd30f3dfd7a63d"
    },
    {
      "slug": "the-arbelos",
      "title": "THE ARBELOS",
      "kicker": "the twins in the shoemaker's knife",
      "accent": "#ff8a3c",
      "blurb": "Geometry's oldest playground: a semicircle minus two smaller ones on its split diameter — Archimedes' shoemaker's knife. Its area equals the circle on the perpendicular at the split (the geometric mean 2√(r₁r₂), made visible), and the two circles inscribed either side of that perpendicular — Archimedes' twins — are always congruent, radius r₁r₂/(r₁+r₂), however lopsided the cut.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "eda104adcb8d26d8b959e88fa3b9341c657cf3c444946e6aa9af2fade9c89970"
    },
    {
      "slug": "the-newton-pepys",
      "title": "THE NEWTON–PEPYS",
      "kicker": "the shortest gauntlet",
      "accent": "#21e6ff",
      "blurb": "Pepys asked Newton in 1693: likeliest — one six in 6 dice, two in 12, or three in 18? Intuition says equal; Newton said the first, correctly: 0.665 vs 0.619 vs 0.597. The mean scales perfectly but variance spreads the bigger pools across more failing configurations. Pepys had bet on the long corridor.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "426e9605951d51866202cbb7311e65ba61973cf745f842390e086ca933d5b3b1"
    },
    {
      "slug": "the-sicherman",
      "title": "THE SICHERMAN",
      "kicker": "the twin dice",
      "accent": "#ffcf4a",
      "blurb": "Is there any OTHER pair of positive-integer dice with exactly 2d6's distribution? Sicherman 1978: exactly one — [1,2,2,3,3,4] + [1,3,4,5,6,8]. Two different drop tables, byte-identical payouts; no sum-based game can tell them apart. Underneath: the unique other regrouping of (x+…+x⁶)²'s cyclotomic factors.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "b75124857c5bb7a493f11e8288ac616fd70234d1b9e08f4b650b0939cd285ed3"
    },
    {
      "slug": "the-droz-farny",
      "title": "THE DROZ-FARNY",
      "kicker": "the 105-year line",
      "accent": "#b06bff",
      "blurb": "Any two perpendicular lines through a triangle's orthocenter cut the three side-lines; the midpoints of the three cut segments are ALWAYS collinear. Droz-Farny published it in 1899 without proof — and the first synthetic proof arrived in 2004, from Ayme. An elementary-looking alignment, open for 105 years.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "32156171552240f379f17c500113cabb65cb9b0282452e0f7db789bb8f3d90a2"
    },
    {
      "slug": "the-eyeball",
      "title": "THE EYEBALL",
      "kicker": "the equal gaze",
      "accent": "#21e6ff",
      "blurb": "Two circles gaze at each other: from each center, draw the tangents to the other circle; each gaze cuts a chord — a pupil — from the gazer's own circle. The eyeball theorem: the pupils are always equal, both exactly 2r₁r₂/d, however mismatched the eyes. The formula is symmetric; neither eye can tell which is which from its pupil.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "9cbaa46b11214341c6d2c3a0fbbef7af40eeabcac408641cdb2c145961b104fd"
    },
    {
      "slug": "the-seven-circles",
      "title": "THE SEVEN CIRCLES",
      "kicker": "the chain porism",
      "accent": "#35ffb0",
      "blurb": "Six circles ring the inside of a seventh, each touching its neighbors and the host: the three lines joining opposite tangency points are concurrent. Discovered not in 1874 but 1974 (Evelyn, Money-Coutts, Tyrrell). Underneath, a porism: with m = r/(1−r), tangency reads mᵢmⱼ = sin²(Δ/2), so closure depends only on the six angles — and then holds for EVERY starting radius.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "7d548d9e700cad469fb965a305b3fa408cc1d0aa8148cd4987e316c9158a023d"
    },
    {
      "slug": "the-hummer",
      "title": "THE HUMMER",
      "kicker": "the magician's ledger",
      "accent": "#ff8a3c",
      "blurb": "Cut the packet anywhere; turn the top two over as one; repeat in any order — Bob Hummer's 1946 CATO principle guards one quantity through all of it: face-up cards at even positions always equal face-up cards at odd positions. Dozens of self-working card tricks are this single conserved ledger in a trench coat. Diaconis & Graham open Magical Mathematics with it.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "860abb3187bcefe06088c489f93b95104771d71305a97dde8162c1519af13a97"
    },
    {
      "slug": "the-moessner",
      "title": "THE MOESSNER",
      "kicker": "strike and sum — the powers fall out",
      "accent": "#ff8a3c",
      "blurb": "Write the naturals; strike every 3rd; partial-sum; strike every 2nd; partial-sum — you're looking at the cubes. Choose n and the same delete-accumulate loop compiles n-th powers from pure addition. Moessner conjectured it in 1951; Perron proved it the same year. Strike at triangular positions instead and iterate: the factorials appear.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "cf5304bf879ed1dccfad50b94d09e3a7786b2b80b7015fb7ee0d1fbdfba68325"
    },
    {
      "slug": "the-kruskal-count",
      "title": "THE KRUSKAL COUNT",
      "kicker": "chains that never part",
      "accent": "#21e6ff",
      "blurb": "Think of a card among the first ten; hop forward by its value; repeat. The magician hops their own chain and names your final card — because hopping chains coalesce, and the load-bearing lemma is deterministic: two chains that ever share a card are identical forever after. Different pasts, one future.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "f619bfa5e81e225ab7f1e3cad938beaaf06da6000a6bd6a8a0f7766dc7919429"
    },
    {
      "slug": "the-happy-ending",
      "title": "THE HAPPY ENDING",
      "kicker": "the marriage theorem",
      "accent": "#ffcf4a",
      "blurb": "Esther Klein, Budapest 1933: any five points in general position contain a convex quadrilateral. Szekeres attacked the generalization; Erdős named it the Happy Ending problem — Klein and Szekeres married. For pentagons the threshold is 9: eight points can dodge, nine cannot; the hexagon's 17 fell to computer in 2006; the growth rate waited for Suk, 2016.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "1e9aa0c4e6760932388514ee276071dca16a58f0dea9a3123477d5f45a2309d2"
    },
    {
      "slug": "the-alternating-sign",
      "title": "THE ALTERNATING SIGN",
      "kicker": "the 88-referee formula",
      "accent": "#b06bff",
      "blurb": "Matrices of 0, +1, −1 with unit row/column sums and alternating signs count as 1, 2, 7, 42, 429, 7436… Mills–Robbins–Rumsey conjectured the product formula ∏(3k+1)!/(n+k)! in 1983 — and it held as a wall for thirteen years, until Zeilberger's 84-page proof, checked by 88 volunteer referees, then Kuperberg's short six-vertex proof.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "fec6f441eca6ca00831d6516a4a2cef83ff63df73b38c946e760520b9626d737"
    },
    {
      "slug": "the-aztec",
      "title": "THE AZTEC",
      "kicker": "the frozen diamond",
      "accent": "#35ffb0",
      "blurb": "Tile the Aztec diamond AD(n) with dominoes: exactly 2^(n(n+1)/2) ways — a formula so clean Elkies, Kuperberg, Larsen and Propp gave it four proofs in one 1992 paper. And a RANDOM tiling freezes into brickwork outside the inscribed circle (the arctic circle theorem): order at the corners is forced by counting, not designed.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "7e864d0f19aaa58330d874303225fbf4429dfc8a28fb8b10e3a652f5f186e17a"
    },
    {
      "slug": "the-stable-roommates",
      "title": "THE STABLE ROOMMATES",
      "kicker": "the co-op with no settlement",
      "accent": "#21e6ff",
      "blurb": "Gale–Shapley 1962: two-sided matching ALWAYS has a stable outcome. Flip one structural bit — everyone in ONE group, pairing as roommates — and the guarantee dies: profiles exist where every pairing has a blocking pair. The classic witness needs four people: three in a preference cycle, one nobody wants. Bipartite vs not is the whole difference.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "7d224738c2554860209deac9ac4c6a2efbfdf82d38faf9d175c6fd7aeadeafa2"
    },
    {
      "slug": "the-tennis-racket",
      "title": "THE TENNIS RACKET",
      "kicker": "the axis that flips",
      "accent": "#ff8a3c",
      "blurb": "Spin anything about its longest or shortest inertia axis: stable forever. Spin about the middle axis and it flips over, mid-flight, on schedule — the tennis racket theorem, made famous when cosmonaut Dzhanibekov watched a wingnut do it in orbit (1985). The verdict hides in one product: (Ik−Ii)(Ik−Ij), negative only for the middle axis.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "b55fc9113ba7441c73f9619a720f272ff79becaac94359278f2e5e4d1ffbffbe"
    },
    {
      "slug": "the-ehrenfest",
      "title": "THE EHRENFEST",
      "kicker": "the urn that takes 2^N to reset",
      "accent": "#b06bff",
      "blurb": "Two dogs, N fleas, one random jump per tick — the 1907 Ehrenfest urn, built to defuse the objection that reversible dynamics forbids an arrow of time. Equilibrium is binomial; and by Kac's theorem the expected return to any state is exactly 1/π(state): the all-on-one-dog reset costs 2^N ticks. Reversible in law, priced out in practice.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "684b144d76e75604a98d5d3170aedbcfef95599f376cce67b1302209eeec05ab"
    },
    {
      "slug": "the-fput",
      "title": "THE FPUT",
      "kicker": "the lattice that refused to thermalize",
      "accent": "#ffcf4a",
      "blurb": "Los Alamos 1955: Fermi, Pasta, Ulam — and Mary Tsingou, who wrote the program — pumped mode 1 of a 32-mass nonlinear chain and waited for equipartition. Instead the energy wandered a few low modes and came home: the FPUT recurrence, the computation that founded nonlinear science and seeded soliton theory.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "cc2d7f01152f3569e3b70604a8381886d742b51484ca5ccd34923a553fc48273"
    },
    {
      "slug": "the-figure-eight",
      "title": "THE FIGURE-EIGHT",
      "kicker": "three bodies, one curve",
      "accent": "#35ffb0",
      "blurb": "In the home stadium of chaos, three equal masses chase each other around a single figure-eight — every body on the SAME curve, T/3 apart, zero angular momentum. Found numerically by Cris Moore in 1993, proven by Chenciner & Montgomery in 2000: the first new closed three-body family since Lagrange, 1772.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "cc584db7a3d8710dd2e73d2b16471c3ab623e003a9b03f2a40cc8bbf58e474cc"
    },
    {
      "slug": "the-kapitza",
      "title": "THE KAPITZA",
      "kicker": "gravity beaten by vibration",
      "accent": "#b06bff",
      "blurb": "An inverted pendulum falls — unless its pivot vibrates fast enough, and then upside-down becomes STABLE: an effective well where gravity built a peak. Stephenson saw it in 1908, Kapitza analyzed it in 1951, and the averaging trick behind a²ω² > 2gL now runs Paul traps and strong-field physics.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "59c87fbcaf0aa05c7470c49cb4a11304d2e8b32567f3d4fcf275bf4b641bb987"
    },
    {
      "slug": "the-kuramoto",
      "title": "THE KURAMOTO",
      "kicker": "the sync transition",
      "accent": "#21e6ff",
      "blurb": "Fireflies, pacemaker cells, clocks on a beam — Kuramoto's 1975 model distilled them: oscillators with random frequencies, each pulled toward the crowd's mean phase. Below Kc = 2γ, anarchy; above it, synchronization ignites with the exact law r = √(1 − Kc/K) — a phase transition solved in closed form.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "08843d3e17af8818410451516a6c84f1ce7b535b26365849600e94c14de5d894"
    },
    {
      "slug": "the-wilberforce",
      "title": "THE WILBERFORCE",
      "kicker": "bounce traded for twist",
      "accent": "#ffcf4a",
      "blurb": "A mass on a soft helical spring, twist and stretch frequencies tuned to match: the bounce dies completely while the mass starts spinning, then the trade reverses — the two motions passing the whole energy budget back and forth like items between inventory slots. Wilberforce 1896; the beat is normal-mode interference, scheduled by the eigenvalues.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "8a689fe799c8a67f95a3593c5523e74fd109b1489cd174bbf51ce3115721d9b2"
    },
    {
      "slug": "the-foucault",
      "title": "THE FOUCAULT",
      "kicker": "the Earth turning under a wire",
      "accent": "#35ffb0",
      "blurb": "Paris 1851: Foucault hangs 67 meters of wire in the Panthéon and the public watches the swing plane creep — the first direct dynamical proof that the Earth turns. One sine rules it all: precession at Ω sin(latitude) — 23.93h at the pole, 31.8h in Paris, never at the equator — and the whole motion has an exact closed form with rotation stamped on as a complex phase.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "9e183fef76ba79adcb9f521be6af56e9b151e5b49f5f1c4ea5722084bc1c682e"
    },
    {
      "slug": "the-square-free",
      "title": "THE SQUARE-FREE",
      "kicker": "three letters never stutter",
      "accent": "#ff8a3c",
      "blurb": "A square is a stutter — any block repeated immediately. With two letters it's unavoidable: every binary word of length 4 contains one. With three, Thue proved in 1906 you can walk forever without stuttering, via a simple substitution word. The gate between alphabets 2 and 3 is absolute — and Thue's overlooked Norwegian papers founded combinatorics on words.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "ebf9dce48f2eb42df5cd3decfa2077526078cce285d1e86787eb0e40a148b32f"
    },
    {
      "slug": "the-ising",
      "title": "THE ISING",
      "kicker": "the temperature that melts order",
      "accent": "#21e6ff",
      "blurb": "A grid of arrows, each preferring to agree with its neighbours, each shaken by heat. Cold: one giant aligned domain. Hot: static. The Ising model is the simplest genuine phase transition — and Onsager solved 2D exactly in 1944, pinning Tc = 2/ln(1+√2). The irony: Ising solved the 1D chain in 1925, found no transition, and wrongly concluded there was none in any dimension.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "7ec466ba8409f5258ffe5193025583d93f2b3ed717eeb8b933b6099d5fd215a3"
    },
    {
      "slug": "the-giant-component",
      "title": "THE GIANT COMPONENT",
      "kicker": "the edge where one giant appears",
      "accent": "#ff8a3c",
      "blurb": "Sprinkle random edges on n nodes until each averages c connections. Below c = 1 the network is dust — every piece O(log n). Cross c = 1 and a single giant component appears holding a fixed fraction S of everything, with all others still logarithmic. Erdős and Rényi called it the double jump; the giant's size is the root of S = 1 − e^(−cS).",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "acebc29eba4f4902cddc894bbaeaad1c49e3129bd95026d9e73f38c6d5ee2f3a"
    },
    {
      "slug": "the-zeeman-machine",
      "title": "THE ZEEMAN MACHINE",
      "kicker": "the disc that jumps",
      "accent": "#ffcf4a",
      "blurb": "A cardboard disc on a pin and two elastics: move your hand smoothly and the disc follows — until it flips, and flips back at a DIFFERENT place. Zeeman's 1972 catastrophe machine is Thom's cusp made physical: a control plane with a bistable tongue, so continuous input yields discontinuous output and the route decides the state. Hysteresis you can build for a pound.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "179c28b03e10418f04dd66f8fca28ce6458d845b1566e8a572cdd26e55b6556f"
    },
    {
      "slug": "the-lanchester",
      "title": "THE LANCHESTER",
      "kicker": "why concentration wins",
      "accent": "#ffcf4a",
      "blurb": "Lanchester, watching Great War aircraft, found that under aimed fire combat power scales as the SQUARE of numbers — invariant αA² − βB². Hence defeat in detail: 100 fighting two 50s in sequence walks away with √(100²−50²−50²) = 70.7 survivors where fighting all 100 at once annihilates it. And it's conditional: under unaimed fire the law goes linear and concentration buys nothing.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "dbd20ea1968c0a518efd3ca4042e250a46c307ee9a14afb53c809b3d70c294ae"
    },
    {
      "slug": "the-lotka-volterra",
      "title": "THE LOTKA–VOLTERRA",
      "kicker": "why killing both helps the prey",
      "accent": "#b06bff",
      "blurb": "Predator and prey chase each other around a loop forever. Two surprises: the orbits are exactly closed (an invariant preserves them), and the time-averages depend ONLY on the parameters — ⟨prey⟩ = c/d, ⟨predator⟩ = a/b, whatever the amplitude. Hence Volterra's principle: kill both species indiscriminately and the prey average RISES. He derived it in 1926 to explain why the WWI fishing halt raised the Adriatic shark fraction.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "790aebf396e51336c6c58a2804aa7948648936e968eedfd8d53377df3d63df62"
    },
    {
      "slug": "the-arrow",
      "title": "THE ARROW",
      "kicker": "no fair rule",
      "accent": "#ffcf4a",
      "blurb": "Ask a voting rule for three things — never rank X above Y when everyone prefers Y; decide X vs Y using only how voters rank X against Y; and no dictator. Arrow proved in 1951 that with three or more candidates nothing satisfies all three. Borda, plurality, pairwise majority all break; the only survivor is a dictatorship, which solves the problem the way deleting the database solves the query.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "c6e7613b3e53b6d54d939e4ab19f8b3374b70d4860f5937913011ae7580c1238"
    },
    {
      "slug": "the-gibbard",
      "title": "THE GIBBARD",
      "kicker": "no honest rule",
      "accent": "#b06bff",
      "blurb": "Arrow's theorem is about ranking; Gibbard–Satterthwaite is about lying. Any single-winner rule that can elect any of three candidates and isn't a dictatorship has a situation where some voter does better by submitting a FALSE ballot. Strategic voting isn't a flaw in your election system — it's a theorem about all of them.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "6023d3659b38685a46a81ef8fb3d2ad612e04d0ef45f7ce71b7b3db66031bfe8"
    },
    {
      "slug": "the-minkowski-body",
      "title": "THE MINKOWSKI BODY",
      "kicker": "area forces a lattice point",
      "accent": "#35ffb0",
      "blurb": "Any convex shape symmetric about the origin with area over 4 is FORCED to contain a nonzero integer point — stretch, rotate or shear it as you like. Minkowski 1889, the founding result of the geometry of numbers, and it's sharp: the open unit square has area exactly 4 and dodges every one. Beneath it, Blichfeldt's pigeonhole: fold any region of area > 1 into the unit torus and two points must collide.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "313b72d4f82cb10596492f9ff2b1c312107ba9115ea086ce473205d33711a4fa"
    },
    {
      "slug": "the-ostomachion",
      "title": "THE OSTOMACHION",
      "kicker": "Archimedes counting",
      "accent": "#ff8a3c",
      "blurb": "The oldest known dissection puzzle: a square cut into 14 pieces, attributed to Archimedes. It looked like a toy until the Archimedes Palimpsest — parchment scraped clean of his writing and overwritten with prayers — was imaged in 1998, and Netz argued he was COUNTING the arrangements. If so it's the earliest combinatorics by two millennia. Cutler settled the count by computer in 2003: 17,152 ways.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "1cca8bc56583ab6a2dad2b1be8de2668f6c1015cdf29870eb66feffca9e613f8"
    },
    {
      "slug": "the-heilbronn",
      "title": "THE HEILBRONN",
      "kicker": "the triangle you cannot avoid",
      "accent": "#ffcf4a",
      "blurb": "Place n points in a unit square; among all their triangles, how large can you force the SMALLEST to be? Random placement is terrible at this — six points typically leave a sliver of area 0.005 — while careful placement reaches 1/8, twenty-three times better. Heilbronn conjectured the optimum decays like 1/n²; Komlós, Pintz and Szemerédi disproved that in 1982, and the asymptotics are still open.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "54aa72f7530d133ba67fce2a7466b0255cf98be60fdbe26ba8907b32e3160e52"
    },
    {
      "slug": "the-peres-mermin",
      "title": "THE PERES–MERMIN",
      "kicker": "the square with no numbers",
      "accent": "#21e6ff",
      "blurb": "A 3×3 grid of two-qubit observables where every row commutes and every column commutes, so each line is jointly measurable and each cell can only read ±1. Multiply along the lines: rows give +I, one column gives −I — an odd number of minus signs. But if the cells had pre-existing values, each appears in one row and one column, so the six products multiply to a square: +1. Odd ≠ even.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "5731c20a56a8930cba31283b17e63d3760ce37c0758afdc5103cb229ae1903c0"
    },
    {
      "slug": "the-kochen-specker",
      "title": "THE KOCHEN–SPECKER",
      "kicker": "eighteen rays no assignment survives",
      "accent": "#ffcf4a",
      "blurb": "You cannot hand every quantum observable a pre-existing value. The proof is combinatorial: find directions such that no yes/no labelling works, where every set of mutually perpendicular directions must contain exactly one yes. Kochen and Specker needed 117 vectors; Cabello, Estebaranz and García-Alcaine found a record with 18 in 4D — nine perpendicular quadruples, each ray in exactly two. The contradiction is pure parity.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "93ba283495062a26e05607c3824ae5739430723b06b4a3092d999c3d443cba50"
    },
    {
      "slug": "the-bomb-tester",
      "title": "THE BOMB TESTER",
      "kicker": "seeing without looking",
      "accent": "#ff8a3c",
      "blurb": "Bombs that detonate on a single photon. Classically you cannot test one without risking it. Elitzur and Vaidman 1993: put the bomb in one interferometer arm — a dud preserves interference, a live one destroys it, and the 'dark' detector fires, reporting the bomb live without the photon ever taking that path. The naive scheme wastes half; the quantum-Zeno version (Kwiat et al. 1995, built in a lab) drives efficiency to 1.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "3c806e3ae009a874f3ac83a6a362c26d9452e6b0bfec533dac8ffdcb085b2761"
    },
    {
      "slug": "the-hardy",
      "title": "THE HARDY",
      "kicker": "the paradox at phi to the minus five",
      "accent": "#b06bff",
      "blurb": "Bell's theorem needs an inequality and a statistical margin. Hardy 1992 found something sharper: three joint outcomes with probability exactly zero and a fourth with probability greater than zero — logically inconsistent for any local realist. No inequality, no error bars; one event of the fourth kind refutes hidden variables outright. The price is rarity: the maximum is (5√5−11)/2 ≈ 0.0902, which is exactly φ⁻⁵.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "b539bfca5464e95a922e9c39f61e312752efdac45259b401a411c6083f957fac"
    },
    {
      "slug": "the-quantum-pigeonhole",
      "title": "THE QUANTUM PIGEONHOLE",
      "kicker": "three in two boxes, none together",
      "accent": "#35ffb0",
      "blurb": "Three particles, two boxes: classically some pair must share — the pigeonhole principle has no exceptions. Aharonov and collaborators 2016: in a pre- and post-selected ensemble, the two-state amplitude for 'these two share a box' is exactly zero for EVERY pair, while three particles and two boxes remain three particles and two boxes.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "78eb236eac44b59d0dd5a9868ab8e652cf5c144c78dfa71ad9922cb13a4149db"
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    {
      "slug": "the-spiral-of-theodorus",
      "title": "THE SPIRAL OF THEODORUS",
      "kicker": "the spiral that stops at 17",
      "accent": "#35ffb0",
      "blurb": "Stand a unit segment on a unit segment at a right angle: √2. Stand another on that: √3. Keep going and the n-th spoke is exactly √n — the irrationals made constructible one triangle at a time. Plato reports Theodorus proved √3 through √17 irrational and then stopped; the spiral stops too, lapping itself at the 17th triangle. Whether that is why he stopped is one of mathematics' oldest unanswerable questions.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "f21917558d63d4234f459a95c883000102ab4a88bf7020f4f6ca1f35ec218322"
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    {
      "slug": "the-loxodrome",
      "title": "THE LOXODROME",
      "kicker": "the bearing that never arrives",
      "accent": "#35ffb0",
      "blurb": "Set a compass bearing and just hold the angle. On a sphere you trace a loxodrome: it crosses every meridian at the same angle and spirals into the pole, winding infinitely many times over a finite distance. Nunes worked it out in 1537; Mercator's 1569 projection exists so that a constant bearing is a straight line on the chart. The catch every sailor pays: the rhumb is never the shortest route.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "bd974512dbb333a819eb97d4ee92838b25081e1e67fa5732184061b942d55776"
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    {
      "slug": "the-instant-insanity",
      "title": "THE INSTANT INSANITY",
      "kicker": "331,776 wrong towers",
      "accent": "#ffcf4a",
      "blurb": "Four cubes, four colours, stacked so each of the four long sides shows all four colours. There are 24⁴ = 331,776 orientations and, for a well-made set, exactly one works — which is why the 1967 toy drove people mad. Carteblanche (a Tutte pseudonym) had published the graph method in 1947, twenty years earlier: vertices are colours, edges are opposite face-pairs, find two edge-disjoint spanning subgraphs, done in minutes.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "bf24afb434d689b81d80e9e84ca1aa4a1c4baad9c19e0503a4986a21400a43d3"
    },
    {
      "slug": "the-soma-cube",
      "title": "THE SOMA CUBE",
      "kicker": "seven pieces, 240 cubes",
      "accent": "#ffcf4a",
      "blurb": "Piet Hein reportedly invented this during a Heisenberg lecture: take every shape of three or four unit cubes that is NOT a straight box — there are exactly seven, and they contain exactly 27 cells, which is a 3×3×3. That coincidence is the whole puzzle. The Soma cube assembles 240 essentially different ways (Conway & Guy, 1961), and the same seven pieces build a zoo of other figures.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "60ddf02aee3848f277a69034aad9b6fa82109a04acbf8ed171de25ba67bc3c6f"
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    {
      "slug": "the-tangram",
      "title": "THE TANGRAM",
      "kicker": "the piece that was never missing",
      "accent": "#b06bff",
      "blurb": "Seven flat pieces cut from a square, a craze in Europe from about 1815. Its famous trick is the vanishing-piece paradox — Dudeney's two monks, built from the same seven tans, one plainly missing a foot. Nothing vanishes: the pieces are rigid, the areas identical, and the missing foot is paid for by a redistribution too diffuse to see, because equal area never implied equal shape and the eye keeps assuming it does.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "8b7a5bc33a31bc141eb36005c46d8699d06c244ade93d2c881d8ca16a1defcb2"
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    {
      "slug": "the-fermat-primes",
      "title": "THE FERMAT PRIMES",
      "kicker": "five in a row, then Euler",
      "accent": "#ffcf4a",
      "blurb": "Fermat saw 3, 5, 17, 257, 65537 — every 2^(2ⁿ)+1 he could compute was prime — and wrote in 1640 that he was convinced they all were. Ninety-two years later Euler took F₅ apart: 4,294,967,297 = 641 × 6,700,417, found not by trial division but by proving every factor must be 1 mod 2^(n+2). No sixth Fermat prime has ever been found.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "ee963f616eb6830cebde0e3defab501369ff1be23461038dc6b14065084b7abd"
    },
    {
      "slug": "the-polya-conjecture",
      "title": "THE PÓLYA CONJECTURE",
      "kicker": "a million confirmations, still false",
      "accent": "#b06bff",
      "blurb": "Sort every number by whether it has an even or odd count of prime factors. Pólya conjectured in 1919 that from n=2 the odd ones always lead — the running tally never goes positive. It holds for a million. It holds for nine hundred million. It is false: Haselgrove proved a counterexample must exist (1958) without producing one, and Tanaka pinned the first at n = 906,150,257 in 1980.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "1bd2ee8b3760cf050cf7433da72311a39b5af8d077598e0c137b6942e861a9ea"
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    {
      "slug": "the-chinese-hypothesis",
      "title": "THE CHINESE HYPOTHESIS",
      "kicker": "the test that lets impostors through",
      "accent": "#21e6ff",
      "blurb": "Every prime satisfies 2ⁿ ≡ 2 (mod n). The tempting converse would be a one-line primality test — and it is false, with a witness small enough to check by hand: 341 = 11 × 31 sails through. Worse are the Carmichael numbers, which pass for EVERY coprime base; 561 is the first, and there are infinitely many. The name is dead too: a 19th-century European idea misattributed to ancient China.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "6de3d7ed680d859a90c4214cbbd8363eb6571704336e0cf72f9b8560842377c6"
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    {
      "slug": "the-tait",
      "title": "THE TAIT",
      "kicker": "the lemma that held up a theorem for 62 years",
      "accent": "#ff8a3c",
      "blurb": "In 1884 Tait announced a proof of the four-colour theorem resting on one obvious-looking assumption: every 3-connected planar cubic graph has a Hamiltonian cycle. It is false. Tutte killed it in 1946 with a 46-vertex counterexample, and the four-colour theorem stayed open until Appel–Haken in 1976. The smallest non-Hamiltonian polyhedral graph predates the conjecture: Herschel's, 1873.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "691460ef39999175445b69aa034241b4d407b16d342549f81265ea74cdc5bd66"
    },
    {
      "slug": "the-keller",
      "title": "THE KELLER",
      "kicker": "true until dimension seven",
      "accent": "#b06bff",
      "blurb": "Tile space with identical cubes at any offsets: Keller conjectured in 1930 that some two must share a complete face. True in the plane, true in 3D, true up to six dimensions — then it dies. Lagarias and Shor broke dimension 10 in 1992, Mackey reached 8 in 2002, and dimension 7 held until 2020, when a SAT proof with a forty-terabyte certificate finished it.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "db56b4a252fbc17cf1b12c2f669d9725239e8c3894d68d00382312eb90c82574"
    },
    {
      "slug": "the-weierstrass",
      "title": "THE WEIERSTRASS",
      "kicker": "the curve with no slope anywhere",
      "accent": "#35ffb0",
      "blurb": "Before 1872 'continuous' quietly meant 'smooth except at obvious corners'. Then Weierstrass exhibited Σ aⁿcos(bⁿπx) — continuous at every point, differentiable at none. Hermite called such things a lamentable plague. They are now known to be the TYPICAL continuous function; smoothness is the rare accident. The mechanism is a race: each term shrinks by a but wiggles b times faster, so ab > 1 lets slopes outrun amplitudes forever.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "e51088f0303269d500d3201e21f644e251442369736150aee44c97256290ed7b"
    },
    {
      "slug": "the-wada",
      "title": "THE WADA",
      "kicker": "three lakes, one shore",
      "accent": "#21e6ff",
      "blurb": "Three lakes on an island, dug so every lake comes within ε of every point of dry land, forever. In the limit every remaining point touches all three lakes at once — a boundary shared by three regions, with no stretch belonging to only two. Yoneyama published it in 1917 crediting Takeo Wada. It sounds hand-built, then turns up in the most ordinary computation there is: Newton's method on z³ = 1.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "63bd8e0fc3ebacabf7d27c98ffe369c9e8c6ff1106428354063c5f117bbcb301"
    },
    {
      "slug": "the-peano-curve",
      "title": "THE PEANO CURVE",
      "kicker": "the line that fills a square",
      "accent": "#ffcf4a",
      "blurb": "A line is one-dimensional and a square is two, so a curve cannot cover a square. Peano destroyed that in 1890 with an explicit continuous map from the interval ONTO the square; Hilbert gave the picture a year later. The escape hatch that keeps dimension meaningful: the limit is surjective but not injective — Netto had already proved no continuous bijection between line and square can exist.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "6806c761ab2590dbcd84eee2789d9b844085a26cffa8fa03a3cfb2502e0350aa"
    },
    {
      "slug": "the-vitali",
      "title": "THE VITALI",
      "kicker": "the set that cannot be measured",
      "accent": "#b06bff",
      "blurb": "Call two reals equivalent when they differ by a rational, then pick one representative from every class — the axiom of choice lets you. Translate the result by each rational in [−1,1]: the copies are disjoint, their union contains [0,1], and it all fits inside [−1,2]. If the set had a length, the total would have to be both ≥1 and ≤3, while actually being either 0 or infinite. Neither is allowed. Vitali 1905: the first unmeasurable set.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "231eb1ac0e34d40245a65e20d6ff2433feb2ace8ccd0228c461ad11a84dc33d3"
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    {
      "slug": "the-osgood",
      "title": "THE OSGOOD",
      "kicker": "the dust that still weighs half",
      "accent": "#ff8a3c",
      "blurb": "Cantor's middle-thirds set is the standard picture of dust: uncountably many points, total length zero. That pairing — nowhere dense, therefore negligible — feels like a law. It isn't. Shrink the removed intervals faster and you get the Smith–Volterra–Cantor set: still containing no interval whatsoever, yet with length exactly ½. Osgood used the same trick in 1903 to build a Jordan arc with positive area.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "234b19dc22111c26bd826f1d201e285cf79919e97fb42c3497445e867ea1a99d"
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    {
      "slug": "the-berry-paradox",
      "title": "THE BERRY PARADOX",
      "kicker": "the phrase that names what cannot be named",
      "accent": "#b06bff",
      "blurb": "'The least number not nameable in under sixty characters' is fifty-one characters long — so it names, in under sixty, the number it declares unnameable. Russell published it in 1908 crediting G. G. Berry, a Bodleian librarian. It is the one-line cousin of Gödel's theorem and Tarski's undefinability theorem: a language cannot contain a truthful account of its own naming power.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "1cfb2a09f453d76135b9334bc423fe46864fa623177ea193682fc92620d98c07"
    },
    {
      "slug": "the-van-der-waerden",
      "title": "THE VAN DER WAERDEN",
      "kicker": "order you cannot avoid",
      "accent": "#ffcf4a",
      "blurb": "Two-colour the numbers 1 to 8 and you can avoid ever having three evenly spaced numbers of one colour. Add a single number and it becomes impossible — all 512 colourings of 1..9 contain one. That threshold is W(3,2) = 9, and van der Waerden proved in 1927 that a threshold exists for every colour count and length. The catch: they grow so violently that W(6,2) is still unknown.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "b072dd64841381e426491f3e5f501bae6dbaedbf3a00289b6417d6e57a589ccf"
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    {
      "slug": "the-rice",
      "title": "THE RICE",
      "kicker": "every question about meaning",
      "accent": "#35ffb0",
      "blurb": "Halting is undecidable — that much is famous. Rice's theorem generalises it to devastation: EVERY non-trivial property of what a program computes is undecidable. Whether it outputs 7, whether it equals another program, whether it is malicious in any semantic sense. Syntactic questions stay cheap; the moment your question is about meaning, no algorithm answers it for all inputs.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "bcf0ee23fe8a63f1887609433ee3cc63b0641e9367e3538b9c5d6219eb8682b5"
    },
    {
      "slug": "the-chaitin-omega",
      "title": "THE CHAITIN OMEGA",
      "kicker": "the number no theory can reach",
      "accent": "#21e6ff",
      "blurb": "Feed a machine random bits and ask: what is the probability it halts? That is Chaitin's Ω — perfectly well-defined, uncomputable, and algorithmically random. Its digits are incompressible, so any formal system can determine only finitely many of them. Knowing the first n bits would settle halting for all programs shorter than n. You can only ever approach it from below.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "b5489609a69bfa1c3744219a58bee8b55d092227781f26d90e1845f2c83a3747"
    },
    {
      "slug": "the-tree",
      "title": "THE TREE",
      "kicker": "the sequence that must end",
      "accent": "#ffcf4a",
      "blurb": "Build a sequence of labelled trees, the n-th having at most n nodes, where no earlier tree embeds in a later one. Kruskal's theorem says every such sequence must stop. TREE(1) = 1. TREE(2) = 3. TREE(3) is finite — guaranteed by a theorem — and so large that Graham's number is not a useful comparison. Friedman showed that finiteness is not provable in systems that handle ordinary mathematics comfortably.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "33b0c8209f0f47a6dcd19442c3555ebefc835bd6cd5febeac90790a98b0cf607"
    },
    {
      "slug": "the-squared-square",
      "title": "THE SQUARED SQUARE",
      "kicker": "squares that fit exactly",
      "accent": "#ffcf4a",
      "blurb": "Can a square be cut into smaller squares, all different sizes? Lusin conjectured no. Four Cambridge undergraduates — Brooks, Smith, Stone and Tutte — cracked it in 1940 by turning each tiling into an electrical network, where square sizes became currents and Kirchhoff's laws did the combinatorics. Duijvestijn found the unique minimal perfect squared square by computer in 1978: 112×112 from exactly 21 squares.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "63eac583bbc4ec2b885168a2070f5e58ccb822ec387869a79634578be1c311be"
    },
    {
      "slug": "the-apollonian-gasket",
      "title": "THE APOLLONIAN GASKET",
      "kicker": "circles all the way down, all integers",
      "accent": "#21e6ff",
      "blurb": "Pack a circle with three mutually touching circles, then fill every gap with the largest circle that fits, forever. The miracle is arithmetic: start from integer curvatures like (−1,2,2,3) and EVERY circle in the infinite packing has an integer curvature — exactly, forever, generated by reflections in the Apollonian group. The gasket is also a fractal of Hausdorff dimension ≈ 1.3057.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "fba7cb849bb265c1a064fc2acb9fb85f871cd64af293ce80a071207cf2a5eb01"
    },
    {
      "slug": "the-kepler-conjecture",
      "title": "THE KEPLER CONJECTURE",
      "kicker": "the densest stack",
      "accent": "#ffcf4a",
      "blurb": "Kepler looked at stacked cannonballs in 1611 and asserted the obvious: nothing beats π/√18 ≈ 74.05%. Proving the obvious took 388 years. Gauss did the lattice case in 1831; Hales announced a proof in 1998 whose referees could only say they were '99% certain', because it rested on computer calculations no human could audit — so he spent until 2017 building Flyspeck, a machine-checked formal proof.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "8f96343ad80817771e55cdd5bbc3df69b43b79d062f731cb1c56d7cad651f72f"
    },
    {
      "slug": "the-honeycomb",
      "title": "THE HONEYCOMB",
      "kicker": "the cheapest walls",
      "accent": "#ffcf4a",
      "blurb": "Bees build hexagons. Pappus wrote around 340 AD that they do so because the hexagon encloses the most honey for the least wax — and the claim sat unproven for sixteen centuries. The hard part is not beating squares and triangles; it is ruling out EVERY partition of the plane, including irregular cells with curved walls. Hales proved it in 1999.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "ed0bd41fd4b1ffb9022ca57a966b6dd83829b035bd792043ca5780dbbc682694"
    },
    {
      "slug": "the-hat",
      "title": "THE HAT",
      "kicker": "one tile that never repeats",
      "accent": "#b06bff",
      "blurb": "For sixty years mathematicians hunted the einstein — one tile (ein Stein) that covers the plane but never periodically. Penrose got it to two in 1974 and there it stuck. In March 2023 David Smith, a retired print technician in Yorkshire, cut a shape from kite-shaped paper, couldn't make it repeat, and wrote to Craig Kaplan. With Myers and Goodman-Strauss they proved it: a single aperiodic tile, found by an amateur.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "5730ba1d726c3a72930fb22f970968cd16acb708723b15a0a7525e4bd2a910dd"
    },
    {
      "slug": "the-nyquist",
      "title": "THE NYQUIST",
      "kicker": "half the sampling rate, and not one hertz more",
      "accent": "#5ad6ff",
      "blurb": "Sample too slowly and the lost frequencies do not vanish — they come back wearing a disguise. A 700 Hz tone and a 300 Hz tone produce identical samples at 1000 Hz. Verified here to floating-point zero.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "1af81cd98329bab29ff4b83bf9bb2ee29328bc05c4d344ce0a9bf7ec55d5156b"
    },
    {
      "slug": "the-rate-distortion",
      "title": "THE RATE-DISTORTION",
      "kicker": "how small it gets if you say what you can lose",
      "accent": "#7de2b0",
      "blurb": "Shannon's other curve: the minimum bits per symbol for a permitted distortion. Real quantisers built here fall short of it by two exact constants — and the 1.533 dB gap is the price of cutting space into cubes instead of spheres.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "938819caa094bffc66d01d369983edc471e2477f2b510e7fc417393944a4b2c7"
    },
    {
      "slug": "the-gilbert-varshamov",
      "title": "THE GILBERT-VARSHAMOV",
      "kicker": "the code is there; nobody has to find it",
      "accent": "#ffd76a",
      "blurb": "A bound that proves good error-correcting codes exist without ever building one — the greedy process simply cannot stop until the balls cover everything. It stood unbeaten for thirty years.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "bae18d2c81545e08b298e55351f9693f0429430e114f6fc5e26d6fba000d50b2"
    },
    {
      "slug": "the-shannon-limit",
      "title": "THE SHANNON LIMIT",
      "kicker": "error-free, through noise, at a rate that does not vanish",
      "accent": "#ff9a5a",
      "blurb": "Everyone assumed reliability had to be bought with throughput. Shannon proved that below capacity you can drive error as low as you like without the rate collapsing — and above it, you cannot, at any price.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "ce7985005f61c4170f665761814932200976d1b2fedac05ecbbedf06f85bcf8e"
    },
    {
      "slug": "the-slepian-wolf",
      "title": "THE SLEPIAN-WOLF",
      "kicker": "two encoders, no channel between them, joint price",
      "accent": "#b98cff",
      "blurb": "Two sensors that cannot hear each other still pay only what a single encoder seeing both would pay. The correlation is exploited by the decoder — the encoders never need to know it exists.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "ba0bf4ece5379974af66e388e726c6244c910fa0b5b3ecfcc4940414455855cf"
    },
    {
      "slug": "the-monsky",
      "title": "THE MONSKY",
      "kicker": "the square refuses an odd number of equal cuts",
      "accent": "#7de2b0",
      "blurb": "Two equal triangles, four, six, any even number — easy. Odd is impossible, and the only known proof runs through the 2-adic valuation, an arithmetic nobody was looking at.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "f5d20fffe66e62fa62bd226969d13b24cf3606bf9900cc56835c209a05f60992"
    },
    {
      "slug": "the-sharkovskii",
      "title": "THE SHARKOVSKII",
      "kicker": "one cycle length forces all the rest",
      "accent": "#ff5a8a",
      "blurb": "Every continuous interval map obeys a single fixed ordering of the integers. Period three sits first, so a single 3-cycle forces cycles of every other length — and the map gets no say.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "90a0843f48e7b2976c798d878f6648a0fbd79c92155b840669ff8b50bb691e84"
    },
    {
      "slug": "the-lob",
      "title": "THE LOB",
      "kicker": "if it would be enough to prove it, it is already proved",
      "accent": "#5ad6ff",
      "blurb": "'If this were provable it would be true' is only ever assertable about things already provable. Godel's second theorem falls out as the special case P = false.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "b3ac63a58b17165a1a051d4e1ae99abc943aafb9090025a1c30ec2ecffa6a49c"
    },
    {
      "slug": "the-presburger",
      "title": "THE PRESBURGER",
      "kicker": "surrender multiplication, get decidability back",
      "accent": "#ffd76a",
      "blurb": "Godel and Church killed decidable arithmetic. Presburger had already shown that if you throw multiplication away, an actual algorithm exists — and in its cleanest form it is a finite automaton reading binary digits.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "e95fedb7e915d478b9ac5be1b114f8f9b4caced04fe1839a7f87ec9aa33b4a0b"
    },
    {
      "slug": "the-jordan-curve",
      "title": "THE JORDAN CURVE",
      "kicker": "inside is not a place, it is a count",
      "accent": "#b98cff",
      "blurb": "Every eye believes a closed loop has an inside. Proving it took until 1887, because no amount of looking near a point tells you which side you are on — only a global parity does.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "dca65d74adac12043dc1c539bcd87f4809653267d04871d8da0304852f636188"
    },
    {
      "slug": "the-orthogonal-split",
      "title": "THE ORTHOGONAL SPLIT",
      "kicker": "put the two rulers on different axes and the collisions stop existing",
      "accent": "#7de2b0",
      "blurb": "ROOT_0's rev 5 · 0804. One ruler asked to measure two things gives 17,259 steps that change both at once. Two rulers on different axes give zero — and the zero is structural, not lucky.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "656aeb63926622fe5c1acdc21318c5044bed5d55da0f57bdd00368748d872b53"
    },
    {
      "slug": "the-seam",
      "title": "THE SEAM",
      "kicker": "the gate and the lie on opposite sides of a join nobody stands on",
      "accent": "#ff5a8a",
      "blurb": "The compute side is gated. The render side is not. So a published artifact can assert a number its own repository abandoned, and the pipeline is satisfied — because nothing is watching the seam.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "8fe3bf22e558292791c792e35188a776eb6789afb446d28f2f79548edbcd9a5c"
    },
    {
      "slug": "the-first-mutate",
      "title": "THE FIRST MUTATE",
      "kicker": "the last instant a rollback was still free",
      "accent": "#5ad6ff",
      "blurb": "One metric, and it is answerable: did anything that could falsify the answer fire before the first action that mutates? After that the answer is frozen, and nothing said later can improve it.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "122135cf23d674c8f65b1259d4d31cd4cad6854b80ff4dadb981aa75e4c93f81"
    },
    {
      "slug": "the-memoryless-examiner",
      "title": "THE MEMORYLESS EXAMINER",
      "kicker": "a grader with amnesia can only be shown",
      "accent": "#ffd76a",
      "blurb": "Icarium cannot remember whether you did well last time, cannot like you, cannot give you the benefit of the doubt. It reads only what you wrote down — and that is the point, not the handicap.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "fa57a8684fd1049d12568bfe6ed3eae1f0bbec74b6480cd57a49de9f888d6a4b"
    },
    {
      "slug": "the-stamp",
      "title": "THE STAMP",
      "kicker": "LIT is refused without an evidence string",
      "accent": "#b98cff",
      "blurb": "Three stamps, and the discipline is in which ones you may give yourself. The only one you can assert freely is AMBER — the one that admits it was a judgement call.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "4a7297d2e1f035076c4701e44947784e419646a8fe623d88cbd2f58b0a369527"
    },
    {
      "slug": "the-graveyard",
      "title": "THE GRAVEYARD",
      "kicker": "bury it, and name what killed it",
      "accent": "#7de2b0",
      "blurb": "Refusing a burial until you name the control that killed it turns dead code into an instrument — and lets you finally ask which of your controls has ever caught anything.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "f70d6edddc4d76ded489379d7771c0405d7c043ac6f808c90505748f235b6a7e"
    },
    {
      "slug": "the-blast-radius",
      "title": "THE BLAST RADIUS",
      "kicker": "bounded under the resolver, not under string comparison",
      "accent": "#ff9a5a",
      "blurb": "The obvious containment check waves through ../evil and an absolute path to anywhere on disk. A prefix compare has a second hole: /workshop/secret starts with /work.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "4405f1bbfd88f44f1bf2be8ba20193eba448bd8bfd44886a30915d6af285d031"
    },
    {
      "slug": "the-verify-then-copy",
      "title": "THE VERIFY THEN COPY",
      "kicker": "an entire outcome removed, for free",
      "accent": "#ffd76a",
      "blurb": "Copy first and a failing test leaves you fully installed and broken. Verify first and that state cannot occur — not less likely, impossible. Same work, different arithmetic.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "6b9dce2218f83bfa1b9fa5923fed9b9bd6feb94859bc612e74be7d4d87b7f64b"
    },
    {
      "slug": "the-fake-context",
      "title": "THE FAKE CONTEXT",
      "kicker": "the states reality will not hand you on demand",
      "accent": "#5ad6ff",
      "blurb": "42/42 checks against a fake. The fake is not a shortcut — some of those checks cannot exist without it. The cost is that it will also hold states the real machine can never occupy.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "8e4821a0bc5e63693b25088a17ab43fc4e85b843179e7b194377c670298431fc"
    },
    {
      "slug": "the-dissent",
      "title": "THE DISSENT",
      "kicker": "the parts agree and the whole does not",
      "accent": "#b98cff",
      "blurb": "Every judge individually consistent; majority on the premises says TRUE, majority on the conclusion says FALSE. Both routes defensible, so more care cannot settle it.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "ade813ee90e179e9d080960e2068a7d05a2ff39a0c100316542c0874ebeb9e5f"
    },
    {
      "slug": "the-ellsberg",
      "title": "THE ELLSBERG",
      "kicker": "a preference no probability can hold",
      "accent": "#ff5a8a",
      "blurb": "30 red, 60 black-or-yellow in a split nobody tells you. Two obvious choices, and no assignment to black makes both rational — because people are declining to bet on a number they were never given.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "26898695cc0d72cbd9e7dcfba329d07901b44fde7295156c91ecb8dd955fd86d"
    },
    {
      "slug": "the-newcomb",
      "title": "THE NEWCOMB",
      "kicker": "two valid rules, opposite answers, same table",
      "accent": "#ffd76a",
      "blurb": "Causal decision theory says take both boxes and its dominance argument never becomes wrong. Evidential says take one, and one-boxers really do end up richer. Both facts are verified here.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "47565d375b302762fb66cb1947b172df9ff135b1e7c07c3c16dbeb8223fd63bb"
    },
    {
      "slug": "the-gentzen",
      "title": "THE GENTZEN",
      "kicker": "a proof that stops borrowing",
      "accent": "#7de2b0",
      "blurb": "Cut is the rule that lets a proof use a lemma. Remove every cut and the proof mentions nothing but the thing it proves — which is what makes proof search possible, and what makes proofs enormous.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "a2fb6657c236c7165bfee86bb8061035cfe9d7f6f97a33d0f897cda8b4c2ab9b"
    },
    {
      "slug": "the-herbrand",
      "title": "THE HERBRAND",
      "kicker": "an infinity settled by a finite piece of itself",
      "accent": "#5ad6ff",
      "blurb": "If a first-order clause set is unsatisfiable, some finite set of ground instances already is. Expand far enough and an ordinary SAT solver settles it — but nothing tells you how far.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "dd3cbf495ce12f404a025fe6bc49039c1ddc3595c3c9b9fc39284ff037ca99c4"
    },
    {
      "slug": "the-krein-milman",
      "title": "THE KREIN-MILMAN",
      "kicker": "keep the corners, discard the rest, rebuild the whole",
      "accent": "#b98cff",
      "blurb": "A convex shape with infinitely many points is entirely determined by its extreme points. Everything interior is redundant — which is why optimisation so often reduces to checking corners.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "6451088966b62f89ff0d01de827888b2bb3093a9305765e60c5d10c43efc309c"
    },
    {
      "slug": "the-claimlink",
      "title": "THE CLAIMLINK",
      "kicker": "a verdict that admits it cannot tell",
      "accent": "#5ad6ff",
      "blurb": "seamgate checks whether published numbers came from the repo. claimlink checks whether published sentences survive it — and refuses to answer where no code-shaped evidence exists.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "801863973bcb26d68dcff83527ab2b55fe3eac5281c9dd11dc0e455d4e10ddde"
    },
    {
      "slug": "the-reachability-gap",
      "title": "THE REACHABILITY GAP",
      "kicker": "true of the shipped app, false of the repository",
      "accent": "#ffd76a",
      "blurb": "A grep finds the call. The module is imported by nothing. Both facts are true, of different objects — and reporting only one of them is a choice somebody made without saying so.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "1abc50929721d5e4f6ff3fc3306de422527a15ad2bdb36667bbed9cb25715188"
    },
    {
      "slug": "the-decay",
      "title": "THE DECAY",
      "kicker": "how a true sentence becomes a false one",
      "accent": "#ff5a8a",
      "blurb": "Nobody writes a false claim about their own project. They write a true one and the project moves. Re-checking does not slow the decay — it bounds how long the claim sits silently false.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "e8553e3b8fe33718c5877956d4516e1b908e8887119e5ca6c1a46ec3368f78a4"
    },
    {
      "slug": "the-manifest",
      "title": "THE MANIFEST",
      "kicker": "installable is not offline",
      "accent": "#7de2b0",
      "blurb": "A manifest grants installability. A service worker grants offline. Without one there is no cache, so a second visit with no network has nothing to load from — and the claim confuses the two.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "195592d73db27ab2124f216543c5518a477e631854bf608cc7209e4aab9a8bff"
    },
    {
      "slug": "the-predicate",
      "title": "THE PREDICATE",
      "kicker": "what it automates is the re-running, not the judgement",
      "accent": "#b98cff",
      "blurb": "Total error splits in two: the human's predicate error and the machine's staleness error. Running more often drives one to zero and leaves the other exactly where it was.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "a0f980dff6483b98aac775815194edaafadaf62d509c5a69cb3eef44ac311af8"
    },
    {
      "slug": "the-union",
      "title": "THE UNION",
      "kicker": "read their generator instead of guessing at it",
      "accent": "#ffd76a",
      "blurb": "A grep returned 350 and was 35% low. What makes it worth keeping is why it survived inspection: 350 sits 0.86% from a real single-pattern count, so it read as a total rather than a fragment.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "870cf9581ea66925830d221061bc4dcbd57dffb43008ede7c0c66cf1f267c427"
    },
    {
      "slug": "the-definition",
      "title": "THE DEFINITION",
      "kicker": "apples-to-apples matters more than thoroughness",
      "accent": "#7de2b0",
      "blurb": "Count the same test files four ways and get 272, 273, 346, 390. Nothing in the repository changed — only the definition did, and only one of the four can be honestly compared to the figure under audit.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "97245eea5a8b55a18b1723c507a4ac763ac3074785ca9788eea92056a9c29d9a"
    },
    {
      "slug": "the-line-count",
      "title": "THE LINE COUNT",
      "kicker": "the third time, and the lesson still did not take",
      "accent": "#ff5a8a",
      "blurb": "A probe killed its own author's published figure: 1.64% drift shipped where the truth was 8.50%. The cause was counting lines with his definition instead of theirs — the same error he had already caught twice.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "021c694e9bbcd71aa09d0bf2e0d42ff505fe6b6d152aad5b7445b1c233e3195d"
    },
    {
      "slug": "the-test-that-did-not-run",
      "title": "THE TEST THAT DID NOT RUN",
      "kicker": "identical to a test that passes",
      "accent": "#5ad6ff",
      "blurb": "A shell mismatch meant two of seven cases never executed. The harness reported zero failures — which is exactly what a fully passing suite reports. There is no symptom, because absence of failure is the same signal in both worlds.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "93024620e2e6259043f9d88d103ad0cad7bfac37e01c50948137f847867855b9"
    },
    {
      "slug": "the-contaminated-pool",
      "title": "THE CONTAMINATED POOL",
      "kicker": "my draw was honest, the pool was not",
      "accent": "#b98cff",
      "blurb": "A uniform seeded draw returned an item that did not belong in the pool at all. Everything verifiable about a sampler is a property of the sampler, and none of it says whether the population is what you were told.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "7321eb5c99172f7a7b2bd69bf8b2023d7e089128bc44d6cd33bfa63d236a8cf1"
    },
    {
      "slug": "the-two-tests",
      "title": "THE TWO TESTS",
      "kicker": "same table, three p-values, one threshold",
      "accent": "#5ad6ff",
      "blurb": "110/180 against 128/180. Unpooled z gives 0.0438, pooled z gives 0.0450, Fisher exact gives 0.0581. Two below the line and one above, with nothing in the data changed.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "28f70c6b1fd1561283c23d3db9ca57266d66bc27a27f4729b06bca8a4dbe2092"
    },
    {
      "slug": "the-shallow-clone",
      "title": "THE SHALLOW CLONE",
      "kicker": "a truncation that reports as a count",
      "accent": "#7de2b0",
      "blurb": "A shallow clone does not error — it returns a smaller number of the right type. Worse, its error runs in the direction that silences the gate: truncate to the published figure and drift reads 0.00%.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "534b79bdb90dc64f99ff7c8308cfd3c7918598189c0428f4b6d8013691fafa0a"
    },
    {
      "slug": "the-weighted-arm",
      "title": "THE WEIGHTED ARM",
      "kicker": "one measurement, two headlines",
      "accent": "#ff9a5a",
      "blurb": "Raw, the two arms sit at a ratio of 1.751. Weighted, 4.326. Both correct, answering different questions — and the entire distance between them is one number applied after the counting was done.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "39d63e03b7881fe9d0870574749f865cbecec5e596019e5973cc725559c449c0"
    },
    {
      "slug": "the-missing-n",
      "title": "THE MISSING N",
      "kicker": "a sample size from a different experiment",
      "accent": "#ffd76a",
      "blurb": "A grep for the advertised N returned zero hits, and the dead search was the finding. The number is real — it belongs to the 180+180 comparison, while the effect beside it came from 60+60.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "22ee6708f4404a68f217385e53893566730f06414fad4eb9275cdab3a901af9b"
    },
    {
      "slug": "the-dropped-predicate",
      "title": "THE DROPPED PREDICATE",
      "kicker": "an instrument that guesses",
      "accent": "#ff5a8a",
      "blurb": "A predicate that cannot locate its own evidence is not silent — it returns a verdict anyway, and the verdict is a coin. Dropped from the table rather than shipped refuting a claim that is probably true.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "35b6dc6b485aca6d7e9e07a18e0afd12843da5cbb45dd495ddef57806167bb91"
    },
    {
      "slug": "the-breach-rule",
      "title": "THE BREACH RULE",
      "kicker": "an asymmetry that costs nothing",
      "accent": "#ff5a8a",
      "blurb": "History only grows. So a published number below today's count is staleness, needing a threshold — and one above it cannot be staleness at all, needing nothing.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "9271e0fdfb8c5bec5c9ed80c2c5fc1f2bd7a38b2318b4ca8d3ce42b899909c7c"
    },
    {
      "slug": "the-only-failed-probe",
      "title": "THE ONLY-FAILED PROBE",
      "kicker": "a detector nobody ever made say yes",
      "accent": "#5ad6ff",
      "blurb": "Every catch is evidence about recall and none about the false-positive rate. Two detectors identical on defects can differ 30x on clean cases, and a positives-only test set cannot tell them apart.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "dc44c2d78a2d6fad2553251129e13e19758ba58c3c39be0a626e5b7c233da3b2"
    },
    {
      "slug": "the-warn-only",
      "title": "THE WARN-ONLY",
      "kicker": "a gate that cannot fail the build is a log line",
      "accent": "#ffd76a",
      "blurb": "Warn-only detects everything a blocking gate detects, logs the same lines, and lets every one of them ship. The bad deploys reaching production are identical, to the byte, to no gate at all.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "5fa32e00421fc96fa0be5d632dcf65f2375cf847aa72030054af9ccc3deff47b"
    },
    {
      "slug": "the-scoreboard",
      "title": "THE SCOREBOARD",
      "kicker": "zero out of six is not zero",
      "accent": "#7de2b0",
      "blurb": "Six errors caught before a human saw them, zero escaped. But zero events in six trials bounds the true escape rate at 50%, not at zero — and saying so is what the arithmetic permits.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "d11d9960129b2458674c0acbb6e71dc884e399b0ff6cb581aa14dfe468cc8109"
    },
    {
      "slug": "the-zero-error",
      "title": "THE ZERO ERROR",
      "kicker": "an exact match kills a hypothesis",
      "accent": "#b98cff",
      "blurb": "Three metrics reproduced at exactly zero error does more than validate a probe — it eliminates the definition-mismatch explanation for any fourth number that does not reproduce.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "5b89f76ef75ff3d0cea532c87fcf000de3b205580e76833f4f5f271f0ee965c2"
    },
    {
      "slug": "the-provenance-fork",
      "title": "THE PROVENANCE FORK",
      "kicker": "a number that needed a credential",
      "accent": "#ff5a8a",
      "blurb": "One counter, two code paths, chosen not by its input but by whether a host token happened to be in the environment. The published figure could only have come from the authed branch.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "29422f5806f0988d14f1d28644e4d532b9a54e8051de82eaec663ccd80705904"
    },
    {
      "slug": "the-direction-test",
      "title": "THE DIRECTION TEST",
      "kicker": "a sign pattern that acquits",
      "accent": "#7de2b0",
      "blurb": "Every published number smaller than the truth. That is not a second accusation — it is a defence. Exaggeration points the bars the other way; staleness cannot.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "1eb350375bf3ff337f68bc8157646dfd3287069aabf0ddc64bc3dbaabfdefd44"
    },
    {
      "slug": "the-caveat-attrition",
      "title": "THE CAVEAT ATTRITION",
      "kicker": "the careful thinking gets left behind",
      "accent": "#5ad6ff",
      "blurb": "The warnings are real, honest and better than most — and most of them do not survive the trip to the page. The claim arrives intact; its qualifications do not.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "5361c24c611bb8f670ce0448e329df201ec4cf1776c96ee07374411b8e2a33c0"
    },
    {
      "slug": "the-standing-credit",
      "title": "THE STANDING CREDIT",
      "kicker": "a rule written before the result",
      "accent": "#ffd76a",
      "blurb": "A rule fixed in advance is a different object from the same rule chosen afterwards. One had a real chance of firing against its author; the other could be shopped for until something passed.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "23ea83f47f14720d8fd536ef2729f24fb3712dc01f6a4bde937dc57f8b5bcb67"
    },
    {
      "slug": "the-self-caught-share",
      "title": "THE SELF-CAUGHT SHARE",
      "kicker": "whose control actually fired",
      "accent": "#b98cff",
      "blurb": "Most findings were surfaced by safety checks the subject had already built and left switched on. That is a compliment — and it bounds what the audit itself contributed.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "a6e98994bc6c6d6b7338b18f884f56967812a7b503f96da259659cb1c695dfaf"
    },
    {
      "slug": "the-wilkinson",
      "title": "THE WILKINSON",
      "kicker": "twenty roots you can see and cannot recover",
      "accent": "#ff5a8a",
      "blurb": "The roots are the integers 1 to 20, readable straight off the factored form. Multiply it out, change one coefficient in its 23rd bit, and ten of them leave the real line.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "044a978669462ca6b31490ecd1a1e665ce31f302829b3a6c7ba1c0e9cd754bce"
    },
    {
      "slug": "the-landauer",
      "title": "THE LANDAUER",
      "kicker": "the price of forgetting",
      "accent": "#7de2b0",
      "blurb": "Computation is not what costs energy — forgetting is. The floor of kT ln 2 applies only to erasure, and a reversible step has no floor at all.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "ac5ba471ce759cb2100172e7832315fe13f346768609d0bd3a80ea2dbad103d6"
    },
    {
      "slug": "the-moore-bound",
      "title": "THE MOORE BOUND",
      "kicker": "a shape that may or may not exist",
      "accent": "#ffd76a",
      "blurb": "Counting outward bounds a graph of given degree and diameter. Meeting the bound forces extraordinary symmetry — and for degree 57 nobody has built one or ruled one out since 1960.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "a2019481cf6265a0a091d44f8d05b6ada775ce53bacdc243840b471c9bf5ee98"
    },
    {
      "slug": "the-hilbert-matrix",
      "title": "THE HILBERT MATRIX",
      "kicker": "integers floating point cannot reach",
      "accent": "#5ad6ff",
      "blurb": "Every entry is a simple fraction. The inverse is a table of exact integers. Floating-point arithmetic cannot get from one to the other, and no better algorithm helps.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "242dec8e7613362282e19951e4f37d649dad797cfe467cd3ed58b908cce47abf"
    },
    {
      "slug": "the-szilard",
      "title": "THE SZILARD",
      "kicker": "one bit, one push, and the books balance",
      "accent": "#b98cff",
      "blurb": "One molecule, one partition, one bit of knowledge — and kT ln 2 of work comes out. Resetting the memory costs exactly the same, so the cycle closes at zero.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "7b8ac49ecfb346a8e6cb8669b59c15e59880bfd4b9f3491aeacd92aae2be9e61"
    },
    {
      "slug": "the-chomp",
      "title": "THE CHOMP",
      "kicker": "a win with no strategy attached",
      "accent": "#ffd76a",
      "blurb": "Strategy stealing proves the first player wins on every board bigger than 1x1, without examining a single position — and names none of the winning moves. For general boards nobody knows them.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "488fd2a3f5aa27aeb63e7b9bdfcfda370320fd8f8586869c212aa551a8dfdf37"
    },
    {
      "slug": "the-length-extension",
      "title": "THE LENGTH EXTENSION",
      "kicker": "a signature that continues itself",
      "accent": "#ff5a8a",
      "blurb": "A Merkle-Damgard digest IS the internal state it stopped at, so anyone holding it can keep hashing from there — forging a valid tag for a longer message without ever knowing the secret.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "ac967c9298f5cccef822b0df64192d892d730e99a5521673ca21df51b96e580a"
    },
    {
      "slug": "the-cantor-function",
      "title": "THE CANTOR FUNCTION",
      "kicker": "it climbs without ever rising",
      "accent": "#7de2b0",
      "blurb": "Continuous, non-decreasing, derivative zero almost everywhere — and it still gets from 0 to 1. The entire ascent happens on a set of measure zero.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "55d8600743419811a22aae22a5dee4f0db8b5f7690d0cec985f051ed6c60a161"
    },
    {
      "slug": "the-matroid",
      "title": "THE MATROID",
      "kicker": "where greedy is exactly right",
      "accent": "#5ad6ff",
      "blurb": "On a matroid, greedy is not a heuristic — it is provably optimal every time. And the theorem runs both ways, so there is no greedy-friendly structure waiting outside matroids.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "49d0b5c7e668a47046b15dfc0db3d974f275e6e460c682835d575d4c56fad3d8"
    },
    {
      "slug": "the-byzantine-generals",
      "title": "THE BYZANTINE GENERALS",
      "kicker": "three who cannot agree if one lies",
      "accent": "#b98cff",
      "blurb": "A loyal commander sending 0 forces the answer to 0; sending 1 forces it to 1; and a traitorous commander producing both views at once still demands the loyal players agree. No rule survives.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "e2af80f0c1dc812ab7064d1f8f12ec4740de74d47ae9afcbb455203eb78fc631"
    },
    {
      "slug": "the-no-free-lunch",
      "title": "THE NO FREE LUNCH",
      "kicker": "a tie nobody can break",
      "accent": "#7de2b0",
      "blurb": "Averaged over every possible objective function, all search algorithms perform identically. The theorem is quoted far more often than its hypothesis is.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "2643f2965227630b822f69bf52d0ce0cd90379f59004bc94ea5562de66fcbece"
    },
    {
      "slug": "the-interval-arithmetic",
      "title": "THE INTERVAL ARITHMETIC",
      "kicker": "bounds that are right and useless",
      "accent": "#ff5a8a",
      "blurb": "Every result comes with a guarantee that the true answer is inside the bracket. Nothing promises the bracket is narrow - and x minus x does not give zero.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "68c3846bdf0ed1bb7d2d4198a726d1c7d8c2f49e2c7e3fb0cb3975f1ed74ab43"
    },
    {
      "slug": "the-ballot",
      "title": "THE BALLOT",
      "kicker": "strictly ahead, never merely level",
      "accent": "#ffd76a",
      "blurb": "The chance that A leads at every moment of the count is (a-b)/(a+b) - depending on nothing but the margin over the total. Twice the votes, same probability.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "5b29376e6857a4a6cdfb4744bab5d5d231153fb3949dee374f106f105211bad0"
    },
    {
      "slug": "the-count-min",
      "title": "THE COUNT MIN",
      "kicker": "the error that only goes one way",
      "accent": "#5ad6ff",
      "blurb": "A grid of counters and no keys at all. Collisions can only add, never subtract, so the answer is never too low - however badly you size it.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "1d6b5725ece3d7285b1470e843fdd84b52a1908e14cf32d72c48fdc35232a150"
    },
    {
      "slug": "the-perfect-code",
      "title": "THE PERFECT CODE",
      "kicker": "a packing with no slack",
      "accent": "#b98cff",
      "blurb": "16 balls of 8 points tile all 128 strings exactly - nothing left over, nothing counted twice. And so there is no room left to notice a second flipped bit.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "0c2febf3b932603e07e752707f35d48691bde7f3bad1670a9391ed34379bfacc"
    },
    {
      "slug": "the-linking-number",
      "title": "THE LINKING NUMBER",
      "kicker": "an integer that survives any deformation",
      "accent": "#7de2b0",
      "blurb": "Gauss's double integral returns an integer that no bending or stretching can move. And it has a blind spot: zero does not mean unlinked.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "b307498f972e097af0c3fe064c66688fd7f7dd4afd8ba71cebe136c6fd107c8b"
    },
    {
      "slug": "the-space-filling",
      "title": "THE SPACE FILLING",
      "kicker": "a line that becomes a plane",
      "accent": "#5ad6ff",
      "blurb": "A continuous curve passing through every point of the square. What it buys is that near in the ordering means near in the plane - within a constant times the square root of the gap.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "406acc39c9940800b82bf00f4d1cbc9755d2c5750ca0637c20cd21873a5fa6b8"
    },
    {
      "slug": "the-euler-characteristic",
      "title": "THE EULER CHARACTERISTIC",
      "kicker": "a number three solids cannot tell apart",
      "accent": "#ffd76a",
      "blurb": "V minus E plus F is 2 for every convex polyhedron, and 0 for every torus. It depends on nothing about the shape except how many holes it has.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "92b10a73b56b9de35c541fa8036d024d0dad045a28a6a1edd6bc6c1027f39903"
    },
    {
      "slug": "the-deceptive",
      "title": "THE DECEPTIVE",
      "kicker": "a hill built to mislead",
      "accent": "#ff5a8a",
      "blurb": "A landscape where following the gradient takes you away from the answer, and the decoy pays 90% of the optimum - which is exactly what makes it convincing.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "09f423bbbaa041c1aaa2ca478c1e9d618212f3c657ab9790de5952d958f93076"
    },
    {
      "slug": "the-winding-number",
      "title": "THE WINDING NUMBER",
      "kicker": "counting roots by counting turns",
      "accent": "#b98cff",
      "blurb": "Accumulate the angle of a polynomial's image around a loop and divide by 2pi. That integer is how many roots are inside - no root-finding, no algebra.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "6314fc10ab3c7a189da9155a1bc986981636e7c6c540e5f745d7fd2dec3edad1"
    },
    {
      "slug": "the-cauchy",
      "title": "THE CAUCHY",
      "kicker": "a mean that never settles",
      "accent": "#ff5a8a",
      "blurb": "The average of n Cauchy draws is distributed identically to a single draw. Not almost - identically. A thousand measurements tell you exactly as much as one.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "37bab572391cdb7e614284ca149bd1b78e7e7c32ea85c25c62b4504e290dfa8c"
    },
    {
      "slug": "the-berkson",
      "title": "THE BERKSON",
      "kicker": "a correlation made of nothing but who was let in",
      "accent": "#5ad6ff",
      "blurb": "Two unrelated things, seen only through a filter that admits either one being large, come out strongly negatively correlated. Nothing changed in the world.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "a21f577866cdf0321d5a9e016cdaac7ad19b758eb2f934b9c2a3c61133fec77d"
    },
    {
      "slug": "the-record",
      "title": "THE RECORD",
      "kicker": "one over k, whatever the world",
      "accent": "#ffd76a",
      "blurb": "The chance the k-th measurement is a new record is exactly 1/k, for any distribution at all. A hundred times more data buys about five more records.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "70399ddd1482b0b9bce19da190b56d7364cbd56243823e2c5b6a4352526e2bb4"
    },
    {
      "slug": "the-church-rosser",
      "title": "THE CHURCH ROSSER",
      "kicker": "any order, one answer",
      "accent": "#7de2b0",
      "blurb": "If two reduction orders both finish, they finish at the same term. It is why a functional program has a meaning independent of how you evaluate it.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "72071dcae036309891f1770956cc13a4ea83c6c1bacfbc687bbf7697fec1b03f"
    },
    {
      "slug": "the-jones",
      "title": "THE JONES",
      "kicker": "the invariant that finally sees the mirror",
      "accent": "#b98cff",
      "blurb": "A trefoil and its mirror image are different knots, and for fifty years the standard invariant could not tell them apart. This one can.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "b107a467e67032d850df8571dcd88f6051e01d45d5757733aa0eb022e1db44c7"
    },
    {
      "slug": "the-zipf",
      "title": "THE ZIPF",
      "kicker": "a law that is not evidence",
      "accent": "#ff5a8a",
      "blurb": "Zipf's law has been read as a fingerprint of deep structure for eighty years. A monkey with a space bar produces it too.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "11f6977af46e8a0721568686c859d44449e4703e5fc464868fd9d1a171087404"
    },
    {
      "slug": "the-lord",
      "title": "THE LORD",
      "kicker": "two right answers that disagree",
      "accent": "#5ad6ff",
      "blurb": "Two statisticians, one dataset, one question. One finds nothing, the other finds a large effect, and there is no third calculation that settles it.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "69241b475084d61737511f7e95f11ecf7a0af0c83ee878d173b375974e90f86b"
    },
    {
      "slug": "the-jordan",
      "title": "THE JORDAN",
      "kicker": "the obvious theorem that took twenty years",
      "accent": "#7de2b0",
      "blurb": "A loop that never crosses itself has an inside and an outside. So obvious that the 1887 proof was disputed for decades - because obvious comes from circles, and the theorem must hold for curves with no tangent anywhere.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "a301bef705c3aa74a54e2de4538773a970b98e2625e704dcee077602dd994b3e"
    },
    {
      "slug": "the-fixed-point",
      "title": "THE FIXED POINT",
      "kicker": "the map that always comes home",
      "accent": "#ffd76a",
      "blurb": "A map that pulls every pair of points closer has exactly one fixed point, and you can bound the error before running anything. Press cosine repeatedly and you are watching it.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "6c31df4c3b2e825a68c03009ad07513adc7bf1ad3de43bf4e6a7e911e4af8870"
    },
    {
      "slug": "the-cross-entropy",
      "title": "THE CROSS ENTROPY",
      "kicker": "the bits you pay for being wrong",
      "accent": "#b98cff",
      "blurb": "The surcharge, in literal bits, for encoding the world with a model that is false. Never negative, zero only if you are exactly right.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "6b11cae968597373da3cefc01572b0f1b4ee6b4603336af6fa4658455f9fb352"
    },
    {
      "slug": "the-alabama-paradox",
      "title": "THE ALABAMA PARADOX",
      "kicker": "more seats, fewer seats",
      "accent": "#ff5a8a",
      "blurb": "Hamilton's apportionment method has a defect nobody predicted: enlarging the assembly can cost a state a seat. The House noticed in 1880.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "16de653e51b17d50f3fadc5fa78b90b77a3297f3e7ddfacd569f89532ff6cc60"
    },
    {
      "slug": "the-vickrey",
      "title": "THE VICKREY",
      "kicker": "where honesty is the dominant strategy",
      "accent": "#7de2b0",
      "blurb": "Highest bidder wins and pays the SECOND-highest bid. That one change makes bidding your true value never worse than anything else, whatever anyone else does.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "bcd5e30ab1bda65306fb4bc7e4523a4bab2d8a27ac07f75c1c18f20d1cbe2f21"
    },
    {
      "slug": "the-price-of-anarchy",
      "title": "THE PRICE OF ANARCHY",
      "kicker": "the exact cost of everyone choosing freely",
      "accent": "#ffd76a",
      "blurb": "Everyone takes the route fastest for them, and everyone ends up worse off. For linear congestion the damage is bounded at exactly 4/3, no matter how tangled the network.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "e34415e6789a8c700d78c64aad813beb1ad729c52275da9b8533204b154b6519"
    },
    {
      "slug": "the-top-trading",
      "title": "THE TOP TRADING",
      "kicker": "the trade that cannot be gamed",
      "accent": "#5ad6ff",
      "blurb": "Point at the owner of your favourite house; the arrows must contain a cycle; execute and repeat. One page of instructions gives the unique unimprovable allocation.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "cbbcfa18603d52e27ec025994db412b197e10e4c433f869d889e2182ef2afadd"
    },
    {
      "slug": "the-winners-curse",
      "title": "THE WINNERS CURSE",
      "kicker": "winning as the evidence you were wrong",
      "accent": "#b98cff",
      "blurb": "When everyone bids their honest estimate of a common value, the winner is by construction the one who overestimated most. Winning is evidence against you.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "25eaa1a79bf014e251820c7a756fba93076f7c7b24c37dd91f60eed5fe584ea6"
    },
    {
      "slug": "the-cutoff",
      "title": "THE CUTOFF",
      "kicker": "mixing that happens all at once",
      "accent": "#5ad6ff",
      "blurb": "Many Markov chains stay almost entirely unmixed for a long stretch, then collapse to near-uniform in a window far shorter than the wait.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "e1a7858a48a28e418166a01afd8586f0cc74667a8c749c06539c94f39869fa6e"
    },
    {
      "slug": "the-reflection",
      "title": "THE REFLECTION",
      "kicker": "a path folded through a wall",
      "accent": "#7de2b0",
      "blurb": "Take any walk that touches a level and reflect everything after the first touch. A question about whole histories becomes a question about endpoints.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "7ad5cc23b31a8d439a9bd9a67744d48daba13a99f5112eb38e0cb5fffe7b0a8d"
    },
    {
      "slug": "the-lyapunov",
      "title": "THE LYAPUNOV",
      "kicker": "two futures from one place",
      "accent": "#ff5a8a",
      "blurb": "The rate at which nearby states separate, and its sign is the whole diagnostic. At r = 4 the logistic map destroys exactly one bit of the starting value per step.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "bb45a0bfa261de635d87b528c9bfe26a8489e3241e74851c7a6ceb44726cbdae"
    },
    {
      "slug": "the-poincare-recurrence",
      "title": "THE POINCARE RECURRENCE",
      "kicker": "everything comes back",
      "accent": "#ffd76a",
      "blurb": "A volume-preserving system in a bounded region must return arbitrarily close to where it began, infinitely often. It says nothing whatever about when.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "1b08ac09c7f168f6e4d3a7aab8aa0a3d601ba2f8da3e80f61f73ae1d82751201"
    },
    {
      "slug": "the-birthday-attack",
      "title": "THE BIRTHDAY ATTACK",
      "kicker": "half the bits, all the security",
      "accent": "#b98cff",
      "blurb": "Finding any collision takes about the square root of finding a specific one, because pairs grow quadratically. A 128-bit digest offers 64 bits of resistance.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "7d9bfc1a3db1910938e546a7c5a15c5e71293d7527af3c1bc5b8d36b5e9483c6"
    },
    {
      "slug": "the-percolation",
      "title": "THE PERCOLATION",
      "kicker": "a threshold at exactly one half",
      "accent": "#7de2b0",
      "blurb": "Below a critical density nothing connects; above it, a path spans the lattice. For bond percolation on the square lattice that threshold is exactly 1/2, by self-duality.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "b90fcbb09bbe20c5b913aa5fd950ed1c8b3ddb8fa70b26a29ab4ff973af33e44"
    },
    {
      "slug": "the-abelian-sandpile",
      "title": "THE ABELIAN SANDPILE",
      "kicker": "the pile that does not care what order you push it",
      "accent": "#ffd76a",
      "blurb": "Grains topple when a site holds four. The startling part is not the avalanches - it is that the order you topple in makes no difference at all.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "ce144fd37d306e44aaf4618aa353830b8c87a2a3540d4553a836bd96fa51e7e7"
    },
    {
      "slug": "the-basin-boundary",
      "title": "THE BASIN BOUNDARY",
      "kicker": "a border every country touches",
      "accent": "#5ad6ff",
      "blurb": "Newton's method on z^3 - 1 gives three basins whose shared border has no stretch belonging to only two of them. Every boundary point touches all three.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "90fdc510c80f34db3677491a1843a44ab98163441b1574e1c896090dd8c26cbd"
    },
    {
      "slug": "the-median-voter",
      "title": "THE MEDIAN VOTER",
      "kicker": "the voter in the middle",
      "accent": "#ff5a8a",
      "blurb": "Majority rule can produce no winner at all. Single-peaked preferences forbid that, and then the median voter's favourite beats everything.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "fa96d0b3c09e5f09403a4949c7c4f947d5de2747f2881e0a64f9743b04bf59c1"
    },
    {
      "slug": "the-bloom",
      "title": "THE BLOOM",
      "kicker": "a filter that only lies one way",
      "accent": "#b98cff",
      "blurb": "It will sometimes say yes to something it never saw. It will never say no to something it did, and that guarantee survives any choice of parameters.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "a54ae90af245f6b54fcea7eefa35b9fd97369e330cf1c877e7a8c6c5048bb41c"
    },
    {
      "slug": "the-benfords-law",
      "title": "THE BENFORDS LAW",
      "kicker": "the leading digit is not fair",
      "accent": "#ffd76a",
      "blurb": "A 1 leads about 30% of the time and a 9 under 5%. It is used to screen for fraud, and it is not universal - which matters if you are accusing anyone.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "26abfc262074630c8ab4448933d672f5b37d35d6cf3b07c5a9bc5f2d7d079886"
    },
    {
      "slug": "the-shapley-value",
      "title": "THE SHAPLEY VALUE",
      "kicker": "the only fair split there is",
      "accent": "#7de2b0",
      "blurb": "Write down four requirements for dividing what a group produced and exactly one formula satisfies them: average each player's marginal contribution over every order of arrival.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "8dbdef024e63b24e79433b3319fb74655a369e12071038afb4031f9471ac9006"
    },
    {
      "slug": "the-hawk-dove",
      "title": "THE HAWK DOVE",
      "kicker": "a fight nobody wins outright",
      "accent": "#ff5a8a",
      "blurb": "Hawks beat Doves every time, so why is not everyone a Hawk? Because two Hawks fight. The stable outcome is a precise mixture at exactly V/C.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "2f15015149999216488c8cf974780ab68da847483a1f01a6081c400db4efe93c"
    },
    {
      "slug": "the-folk-theorem",
      "title": "THE FOLK THEOREM",
      "kicker": "why tomorrow makes today honest",
      "accent": "#5ad6ff",
      "blurb": "Repeat a prisoner's dilemma forever and cooperation becomes an equilibrium - not from decency, but because the threat of never being trusted again outweighs one round of gain.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "acaa70d73b47f9cf69b15472493b0e09d8660e57475f9746163af869cdb745b7"
    },
    {
      "slug": "the-ergodic",
      "title": "THE ERGODIC",
      "kicker": "when the long run answers for everyone",
      "accent": "#b98cff",
      "blurb": "Watching one trajectory long enough gives the same answer as sampling the whole space - for almost every start. It holds for an irrational rotation and fails outright for a rational one.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "bf2fe9bf3ea32bbcdc0cf207d3f12f1e4c90fa7fffbcab066ac130d2f72cda38"
    },
    {
      "slug": "the-bell-inequality",
      "title": "THE BELL INEQUALITY",
      "kicker": "a correlation no local story can tell",
      "accent": "#7de2b0",
      "blurb": "If outcomes were fixed in advance by anything carried locally, |S| <= 2. Entangled particles reach 2 sqrt(2). The gap is measurable.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "a88a4c186d885894ef7e9a4cd65d0c657ed56023e19482efe3cf53dfa05d772d"
    },
    {
      "slug": "the-no-cloning",
      "title": "THE NO CLONING",
      "kicker": "the state that cannot be copied",
      "accent": "#ffd76a",
      "blurb": "No machine copies an unknown quantum state. The proof is three lines: cloning would force an overlap to equal its own square, and only 0 and 1 do that.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "76d21d27d7c6e19baf3045f3abe40031a8d626644978a02b1d03f2dc490e61f6"
    },
    {
      "slug": "the-superdense",
      "title": "THE SUPERDENSE",
      "kicker": "two bits down one wire",
      "accent": "#5ad6ff",
      "blurb": "A qubit carries one bit. Share entanglement first and one qubit delivers two - because half the message was already in the room.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "7a5491043078f8bc131d988fffab3bfce092b7a9cc0431acea3ebdab43f6e0f1"
    },
    {
      "slug": "the-quantum-zeno",
      "title": "THE QUANTUM ZENO",
      "kicker": "a watched state that will not move",
      "accent": "#ff5a8a",
      "blurb": "Measure a rotating state often enough and it never gets anywhere. The survival probability tends to 1, and the rate it does so is an exact constant.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "ea0d4d06bacf385480ae7b58e417da05ecc2e27a8022277f60261ed00e8cad32"
    },
    {
      "slug": "the-reversible",
      "title": "THE REVERSIBLE",
      "kicker": "logic that throws nothing away",
      "accent": "#b98cff",
      "blurb": "An AND gate erases a bit. A reversible gate is a permutation, so the past is always recoverable - and one gate, Toffoli, builds every classical circuit there is.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "55847d7a93d847899f90e563e9483b247e885816024a9f2e5a0bf8cd3a3aa4e3"
    },
    {
      "slug": "the-depth-jump",
      "title": "THE DEPTH JUMP",
      "kicker": "a jump that names a depth, not a place",
      "accent": "#7de2b0",
      "blurb": "With an address you cannot know the stack height on arrival, because it depends on the route. With a depth you always can - and that is what buys one-pass checking.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "6fe2ec8b382488e3a7e11f3e149dfde6ae7e2c80d57e9d8f1550c0241cf3a792"
    },
    {
      "slug": "the-noise-control",
      "title": "THE NOISE CONTROL",
      "kicker": "the control that says no",
      "accent": "#ffd76a",
      "blurb": "A checker that accepts everything passes the correctness test perfectly. Only what it does with things that are NOT programs can tell it from a rubber stamp.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "c8e720c3bdd4a1a7e8cab360d7a179aa3cea1ad2d9c36e76d5d8822648a2f942"
    },
    {
      "slug": "the-overloaded-symbol",
      "title": "THE OVERLOADED SYMBOL",
      "kicker": "one letter doing twelve jobs",
      "accent": "#ff5a8a",
      "blurb": "Twenty-six letters and rather more things worth naming. The reuse is invisible because each decision looked local and small, and nobody wrote any of them down.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "14919558f720c9bcd32961287a984217bbbd3bfe3ec8d827aebbbb49e3704327"
    },
    {
      "slug": "the-zero-parameter",
      "title": "THE ZERO PARAMETER",
      "kicker": "five rules with nothing to tune",
      "accent": "#5ad6ff",
      "blurb": "Veto, minus-I, depth, idempotence, address. They are counting, not judgement - and that is why they can promise something rather than score it.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "e2892785f7a81893f9205c4760eff76f8b27de98fbb9d6b5b4765f265de47cc2"
    },
    {
      "slug": "the-untypeable",
      "title": "THE UNTYPEABLE",
      "kicker": "the glyph you cannot enter",
      "accent": "#b98cff",
      "blurb": "Mathematics has a large alphabet and a keyboard has ninety-five keys. What survives into code is what fit through a mechanical aperture built for English prose.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "3bab183c23e54469edd0a5f4106a7c52d65f76196b541b94ec7244b56cdb8c29"
    },
    {
      "slug": "the-chained-root",
      "title": "THE CHAINED ROOT",
      "kicker": "a hash that remembers where it has been",
      "accent": "#7de2b0",
      "blurb": "acc = H(acc + sha + path). The root carries not just what was sealed but in what order, and at what path. Reorder the ledger and the number leaves.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "3f6acc45cbc5ed2c776c45ea6f1df6a23b455b991226d737ee6661dd0d9d5e7a"
    },
    {
      "slug": "the-order-blind-hash",
      "title": "THE ORDER-BLIND HASH",
      "kicker": "a digest that forgot where it had been",
      "accent": "#ff5a8a",
      "blurb": "Hash each item, XOR the results. Fast, parallel, order-free - and structurally unable to see a reordering, a swap, or anything added an even number of times.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "ae06037264a26721987df0858e67717508774451db1b805fbf58e350a1dabd4d"
    },
    {
      "slug": "the-stale-witness",
      "title": "THE STALE WITNESS",
      "kicker": "a signature attests a moment, not a file",
      "accent": "#ffd76a",
      "blurb": "A mark on a ledger signs a number the ledger had once. Change anything after and the signature stays valid while quietly ceasing to attest to what is in front of you.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "3bc407135eb2bf8e054373cfab65ff593014aae61d29f77c1f6409f74524c38f"
    },
    {
      "slug": "the-lint-not-the-judge",
      "title": "THE LINT NOT THE JUDGE",
      "kicker": "evade the words, keep the claim",
      "accent": "#5ad6ff",
      "blurb": "A checker that reads for overclaiming finds it in words, and words are the one part of a claim a writer can change for free.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "fd90ca7a4222ba67159643da87798446a4b1f0234975e1dbc74e40550541a283"
    },
    {
      "slug": "the-unstated-condition",
      "title": "THE UNSTATED CONDITION",
      "kicker": "an identity with no domain attached",
      "accent": "#b98cff",
      "blurb": "Addition is associative - somewhere. Move the same identity to different numbers and it starts failing, because the sentence was published without the condition that made it hold.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "0ad1fe79097dd2fdda20558817cfcd906ed4e23c1e9fb2eca9e7894814504f86"
    },
    {
      "slug": "the-self-branch",
      "title": "THE SELF-BRANCH",
      "kicker": "a call that never leaves",
      "accent": "#ffd76a",
      "blurb": "On ARM64 an unlinked call carries displacement zero, and zero means THIS instruction. Every unpatched call is a tight infinite loop that assembles perfectly.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "d559265bad94e35364135291452d449eceeb61f981cfa97ef1f0a93be41e2dee"
    },
    {
      "slug": "the-placeholder-agreement",
      "title": "THE PLACEHOLDER AGREEMENT",
      "kicker": "two tools agreeing on a blank",
      "accent": "#ff5a8a",
      "blurb": "Differential testing catches an enormous amount. It cannot catch anything two tools leave blank in the same way - and toolchains agree about placeholders far more often than about answers.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "77aaa6fb8b7b55c29c9df2ec02b429b7d7c4879238fe7621f0e27ce118bd4833"
    },
    {
      "slug": "the-reach-of-a-branch",
      "title": "THE REACH OF A BRANCH",
      "kicker": "how far a call can see",
      "accent": "#5ad6ff",
      "blurb": "A branch has to fit its destination inside itself. Twenty-six signed bits, scaled by four, is a hard ceiling on how far apart two pieces of a program can sit.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "5e3875e502c695a9622f572a6a5c386b3947e980d2d3b3b476f46c9d45df4765"
    },
    {
      "slug": "the-oracle",
      "title": "THE ORACLE",
      "kicker": "a check that could actually fail",
      "accent": "#7de2b0",
      "blurb": "A roadmap of things to build is not a test plan. Only a check whose answer comes from somewhere else can return a result you did not want.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "37a05e998b359c5a9d7ffea08b6af19f45a5855fa9c0b708339b6b4b87b778cd"
    },
    {
      "slug": "the-twelve-constructs",
      "title": "THE TWELVE CONSTRUCTS",
      "kicker": "a whole language, and no loop in it",
      "accent": "#b98cff",
      "blurb": "Twelve constructs and nothing else. No loops, no booleans, no arrays - yes and no are 1 and 0. It is still enough to compute anything computable, because it can call itself.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "a3a9dfc2a8ab7c2757f866c04e0b9aa00b947cca488015bf13c6aca5c270ab93"
    },
    {
      "slug": "the-zero-one-principle",
      "title": "THE ZERO-ONE PRINCIPLE",
      "kicker": "256 tests instead of 40,320",
      "accent": "#7de2b0",
      "blurb": "A comparator network sorts every input if and only if it sorts every input of 0s and 1s. Check the corners of a cube and the whole space comes with it.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "feb4a2dc1ec37136af356c827b20b6e3bcc0718e29254665a557242195d537ba"
    },
    {
      "slug": "the-marsaglia-planes",
      "title": "THE MARSAGLIA PLANES",
      "kicker": "random numbers fall mainly in the planes",
      "accent": "#ff5a8a",
      "blurb": "Every linear congruential generator confines its k-tuples to parallel hyperplanes. For RANDU - shipped by IBM, used for a decade of published science - there are fifteen.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "eecbe2ba7b5629f32825b10e5be2ba23d0a088c4d01b5ab1efbe31c0ab874ca1"
    },
    {
      "slug": "the-lyndon-word",
      "title": "THE LYNDON WORD",
      "kicker": "every string falls apart exactly one way",
      "accent": "#5ad6ff",
      "blurb": "A word smaller than all its rotations. Every string splits into a non-increasing run of them, uniquely, and one left-to-right pass finds the cuts.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "462113a80eab09c35bcfc77df39321513e5ef9199f1ade9d1205186b2106a388"
    },
    {
      "slug": "the-davenport-schinzel",
      "title": "THE DAVENPORT-SCHINZEL",
      "kicker": "a sequence that cannot alternate",
      "accent": "#ffd76a",
      "blurb": "Forbid a symbol from sitting beside itself, and forbid two symbols from alternating too often. How long can the sequence get? At order 3 the answer stops being linear.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "698e347d0e3b30e1676a18404a35925d257f7e86f0275e893e7d364982f528cb"
    },
    {
      "slug": "the-hashlife",
      "title": "THE HASHLIFE",
      "kicker": "the same square, remembered",
      "accent": "#b98cff",
      "blurb": "Life repeats itself constantly. Store the universe as a quadtree where identical subsquares ARE the same object, and the repetition collapses.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "4d683e99308de94b0ba9c0d650039a961401d3359c0d282b0932fc415b2661c4"
    },
    {
      "slug": "the-permutation-null",
      "title": "THE PERMUTATION NULL",
      "kicker": "the control that killed the pretty result",
      "accent": "#ff5a8a",
      "blurb": "A catalogue of 2,048 items decomposed as a quantum state gave Schmidt rank 8 and 0.6081 bits of entanglement. Then the control ran, and that is what random labelling gives anyway.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "3c650e1b38d1ea581a6c2d2e49f4b73159fe371211603b22e295180603324a27"
    },
    {
      "slug": "the-two-rulers",
      "title": "THE TWO RULERS",
      "kicker": "91% balanced and 39% used, both correct",
      "accent": "#7de2b0",
      "blurb": "Two standard measures of spread, the same 64 numbers, fifty-two points apart. Neither is a mistake. One takes a logarithm and the other does not.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "9057cdd3b922b22ca9a9f3ce4c4fcbeeece48b1c62928d5c4dba392b72123a63"
    },
    {
      "slug": "the-half-in-fourteen",
      "title": "THE HALF IN FOURTEEN",
      "kicker": "a mean that touches almost nothing",
      "accent": "#ffd76a",
      "blurb": "2,048 items across 64 containers makes the mean 32. Forty-six containers hold less than that, and fourteen hold half the corpus between them.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "d44d1e2176e665d19cdb753c6f5487a6f0a1eab4331f035fe810d4509401d9e2"
    },
    {
      "slug": "the-self-clocking-seam",
      "title": "THE SELF-CLOCKING SEAM",
      "kicker": "a signal that carries its own clock",
      "accent": "#5ad6ff",
      "blurb": "A receiver needs to know where each bit begins. A code that changes at every position hands the clock over inside the data - the property an air gap needs, because an air gap has no second wire.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "54d4a90885b2bb1b833e5fe749cccc69fd4452061f6849ea1a0e9c6985826330"
    },
    {
      "slug": "the-redaction",
      "title": "THE REDACTION",
      "kicker": "names removed, and nothing measured moved",
      "accent": "#b98cff",
      "blurb": "Every name replaced by a code, every figure untouched. Defensible only if no figure depended on the names - which here is provable.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "50da29d6cde5d11d9ad715bfce036c7e09eef73b3dcbd6d7c7f2e81b67d04a05"
    },
    {
      "slug": "the-ziggurat",
      "title": "THE ZIGGURAT",
      "kicker": "128 rectangles that all weigh the same",
      "accent": "#7de2b0",
      "blurb": "Cover the bell curve with 128 equal-area rectangles. Pick one, pick a point in it, and almost always the point is already under the curve - no exponential, no logarithm.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "ad759e0f49d7cf1aa1f4b61dbdf62dda8f37559aecc2bb21eb3e06a18fc1ef9c"
    },
    {
      "slug": "the-middle-square",
      "title": "THE MIDDLE SQUARE",
      "kicker": "the first generator, and how it dies",
      "accent": "#ff5a8a",
      "blurb": "Square a four-digit number, keep the middle four, repeat. Von Neumann proposed it in 1946 and knew it was inadequate. The state graph shows exactly how inadequate.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "b03b0ce33e14d01a1619bdd0d5a3c01dff11e86238160c3a7352daca33067173"
    },
    {
      "slug": "the-soft-heap",
      "title": "THE SOFT HEAP",
      "kicker": "a structure allowed to lie, by exactly this much",
      "accent": "#ffd76a",
      "blurb": "Fix an error budget. The queue may then corrupt that many keys - raising them, never lowering - and in exchange every operation becomes constant amortised time.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "d3eed859051b9e0bd51bae45259118b4d972d48f4f2aa1ee2c420b9ec9c98c5b"
    },
    {
      "slug": "the-centroid-decomposition",
      "title": "THE CENTROID DECOMPOSITION",
      "kicker": "every tree has a middle",
      "accent": "#5ad6ff",
      "blurb": "Every tree, however lopsided, has a node you can delete to leave nothing bigger than half of it behind. Recurse and the depth cannot exceed about log2 n.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "2d0b30cf0a25d436b1dbbaafc05346324fee7fa249ee9ad4cf7c7dabbf9eb97d"
    },
    {
      "slug": "the-hopscotch",
      "title": "THE HOPSCOTCH",
      "kicker": "never more than H slots from home",
      "accent": "#b98cff",
      "blurb": "Fix the probe distance instead of the load. Every key lives within H slots of its home bucket, and an insertion that would break that hops existing entries backwards to make room.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "a11317f312d935122831536b6158aca6e0ba8ef97f29cf155dffe16de04e4725"
    },
    {
      "slug": "the-zero-that-counted",
      "title": "THE ZERO THAT COUNTED",
      "kicker": "a counter that walked the wrong field",
      "accent": "#ff5a8a",
      "blurb": "It returned 0/0 for every program and looked exactly like a working counter. Zero is the one answer a broken counter and an empty input agree on.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "6d784bd519346809301b1f03285e7efa4652b5f9d1f7b16211613320fe14ff57"
    },
    {
      "slug": "the-three-ratios",
      "title": "THE THREE RATIOS",
      "kicker": "three quantities, one name",
      "accent": "#ffd76a",
      "blurb": "Pairing is a control invariant fixed at 1.00. ASK:ANSWER runs 1.20 to 11.00 at the programmer's level. LOAD:STORE runs 0.33 to 1.88 after lowering. They share nothing but a name.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "a20298ec465296ee2b29e2bb475017c588ea01db3eaa2c8ba83ea1f39cce7a86"
    },
    {
      "slug": "the-threshold-from-hope",
      "title": "THE THRESHOLD FROM HOPE",
      "kicker": "a gate set before the measurement",
      "accent": "#7de2b0",
      "blurb": "The test asserted a ratio above 2. Measurement said 0.33 to 1.88 - it never reaches 2 at all, so the gate failed on correct code and looked exactly like a defect.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "896f8362f72ef90ab61e1a43584d9bf9b3633c91026bf734b770845067e7ff57"
    },
    {
      "slug": "the-compile-invariant",
      "title": "THE COMPILE INVARIANT",
      "kicker": "compiles equals distinct positions fired",
      "accent": "#5ad6ff",
      "blurb": "Eight positions, preloaded, nothing built until asked. The one line that keeps it honest: compiles must equal distinct positions fired. Speculation breaks it and nothing else notices.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "b4b9cea327a5caea6e34255d8cfcacd55dc7932b1ed2dcb09877cb3b4891a7f0"
    },
    {
      "slug": "the-counters-sum",
      "title": "THE COUNTERS SUM",
      "kicker": "the notation IS the state",
      "accent": "#b98cff",
      "blurb": "Positions at rest and positions fired, and the two counters always sum to 8 = 2^3. Reading the notation and reading the machine are the same act.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "7235093f5a1ee17758765d6bf363d6c0781961f9f64adb363077d8e67baca28d"
    },
    {
      "slug": "the-two-run-gate",
      "title": "THE TWO-RUN GATE",
      "kicker": "nothing enters the seal on one green run",
      "accent": "#7de2b0",
      "blurb": "It has to produce the same result twice, from the shipped copy, in separate invocations. The claim is deliberately narrow, and narrower than it sounds.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "5d827acf98dc5b822e902d852ada2eb5b6ffcb81e70b9d99d3e10d844f8c10a9"
    },
    {
      "slug": "the-seal-that-broke-itself",
      "title": "THE SEAL THAT BROKE ITSELF",
      "kicker": "a freeze that cannot re-read itself",
      "accent": "#ff5a8a",
      "blurb": "The sealer took its root from the first path argument. Sealing three directories recorded everything under the other two as a bare filename. It wrote successfully; its own verify said BROKEN.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "d0a4ba58fa682b291c6831d71a7b2e18b1918d6f35c41856fe86941778e3d0dd"
    },
    {
      "slug": "the-inverted-ratio",
      "title": "THE INVERTED RATIO",
      "kicker": "1.00 for a fan-out that was 2.00",
      "accent": "#ffd76a",
      "blurb": "A control loop counted requests over responses. For a fan-out that is upside down - and a one-sided check built on it cannot see a fan-out at all.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "aedef853c646fde724909a3d8d4d8847e61342a88b42e3bd036b7d322a3b4336"
    },
    {
      "slug": "the-guard-not-the-test",
      "title": "THE GUARD, NOT THE TEST",
      "kicker": "an invariant that lives in flight",
      "accent": "#5ad6ff",
      "blurb": "The same predicate in a test file runs once and reports what it saw. Inside the function it guards, it runs on every call. Identical arithmetic, entirely different lifetime.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "f04426c495522b4d272e020b68310d88f398cb422a7fc0d812ccd8afd5d48fdc"
    },
    {
      "slug": "the-float-tail",
      "title": "THE FLOAT TAIL",
      "kicker": "the tails kept on purpose",
      "accent": "#b98cff",
      "blurb": "A frozen record kept every digit of its Bernoulli numbers deliberately. Rounding them would make two genuinely different runs report the same number.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "bfe688854e1914e01f5938425ded14f9f7a20910cb8743cbfa284165014804c7"
    },
    {
      "slug": "the-observer-that-moved-it",
      "title": "THE OBSERVER THAT MOVED IT",
      "kicker": "a probe that re-ran what it was watching",
      "accent": "#ff5a8a",
      "blurb": "A sensitivity probe must re-execute an assertion once per mutated field. If the assertion is not pure, the probe stops measuring the system and starts driving it.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "9e4d275aad61802333510b5f281215baf4fed49c96355f07092deb95ba4b9782"
    },
    {
      "slug": "the-mutator-that-moved-everything",
      "title": "THE MUTATOR THAT MOVED EVERYTHING",
      "kicker": "a shift that shifts nothing that matters",
      "accent": "#ffd76a",
      "blurb": "Bump every number by one and equality survives untouched - 6 == 6 becomes 7 == 7. A perfectly sensitive assertion gets reported insensitive.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "f95d8f0a70b546f06191e2e7e9e7b2c7d7e9c93e7b5715981bbdcb2cef952e66"
    },
    {
      "slug": "the-seal-over-a-red-test",
      "title": "THE SEAL OVER A RED TEST",
      "kicker": "INTACT was true, and meant nothing",
      "accent": "#7de2b0",
      "blurb": "The seal was written while the suite was red. The verify said INTACT and was telling the truth: the bytes were exactly what the ledger said.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "ea71c5689e89d47168fb17df215e06552deb376ccd5f2e228ceeca0457a9a91b"
    },
    {
      "slug": "the-flag-that-says-what-it-knows",
      "title": "THE FLAG THAT SAYS WHAT IT KNOWS",
      "kicker": "INSENSITIVE, renamed",
      "accent": "#5ad6ff",
      "blurb": "A probe that perturbs numbers cannot move an assertion about an array length or a null check. Calling that INSENSITIVE accuses the assertion of a fault that belongs to the instrument.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "02fdc9cf7fdcadeaf03aff61385e8225f4605f79199fb57791d5bf5a1eba561d"
    },
    {
      "slug": "the-api-change",
      "title": "THE API CHANGE",
      "kicker": "a signature moved and a default came with it",
      "accent": "#b98cff",
      "blurb": "The fourth argument became an options object. Every call site kept compiling and running while the meaning of half of them quietly inverted.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "5f782e82a6d47431ee80ae3c1654f0e527e442290a737d50115b7064cdac9094"
    },
    {
      "slug": "the-order-that-is-forced",
      "title": "THE ORDER THAT IS FORCED",
      "kicker": "seven floors, and only one way to stack them",
      "accent": "#7de2b0",
      "blurb": "A stated reason for every adjacency turns a build plan into a theorem. Six reasons in a row and the order stops being a choice.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "f85e852db0b67370052ae168f9e6cf4edd2478d6dbfab2a5de59869fa4cc0b11"
    },
    {
      "slug": "the-wider-at-the-bottom",
      "title": "THE WIDER AT THE BOTTOM",
      "kicker": "a floor laid on one run is laid on a coincidence",
      "accent": "#ffd76a",
      "blurb": "73 checks that ran twice and were sealed, beneath 28 gates that have not. The shape is the argument: everything above multiplies through everything below.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "f69bd13c3b3dd632a88900e76a59dd45c4dd6ab9d1bed4cb6d8797454f25655e"
    },
    {
      "slug": "the-gate-that-can-stop-it",
      "title": "THE GATE THAT CAN STOP IT",
      "kicker": "put the falsifiable floor early",
      "accent": "#5ad6ff",
      "blurb": "One floor is allowed to halt everything. If thirteen symbols do not cover Fortran, the answer is not more symbols - it is that Fortran is a different shape.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "a8198c42796529f27a5ed510be05f925584b8978cfaf4c674cef5d9a50740123"
    },
    {
      "slug": "the-declarable-only",
      "title": "THE DECLARABLE ONLY",
      "kicker": "a synonym cannot be found by looking",
      "accent": "#b98cff",
      "blurb": "One name for two things is detectable. One thing with many names is declarable only. Nothing in the text says bench, Bench, wb_ and thing-bench are the same object.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "ba6d4b3cade4a24d36755fe35319a9bbd31af7834a2ec1ecff06305f42fc9300"
    },
    {
      "slug": "the-unfiled",
      "title": "THE UNFILED",
      "kicker": "the symbols nobody wrote down",
      "accent": "#ff5a8a",
      "blurb": "Two censuses of one corpus: what is used, and what is declared. The difference is the only category a registry cannot enumerate.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "472312b6631eb66ea98e0dd47dc9eaeb40ad2b85d72cb853c15227792b46a48b"
    },
    {
      "slug": "the-joint",
      "title": "THE JOINT",
      "kicker": "bookends that balance only in a chain",
      "accent": "#ffd76a",
      "blurb": "One unmatched closer at the front, one unmatched opener at the back. Not a broken container - a joint, which closes what came before and opens what comes after.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "0d0d4b2499a34b49b239a667755aab4ee25da1e8ac79778708520fe7c5306b7f"
    },
    {
      "slug": "the-no-else",
      "title": "THE NO ELSE",
      "kicker": "a refusal is written down, not branched around",
      "accent": "#7de2b0",
      "blurb": "can asks whether the action is available; do runs it. There is no else. Absence of capability is a refusal, and the refusal goes into the record.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "6586b7a19d495253b5b97b3213f686e8daec3fdaf7b9b7aa3ea5532151d82de4"
    },
    {
      "slug": "the-shared-terminator",
      "title": "THE SHARED TERMINATOR",
      "kicker": "two loops, one CONTINUE",
      "accent": "#ff5a8a",
      "blurb": "FORTRAN 77 lets two DO loops end on the same labelled statement. Two openers, one closer, entirely legal - and a pushdown check sees nothing wrong.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "a23840ae0b538400e5c611536e7b3f67f4f0d5ef06a7cd293c81cd12f604e110"
    },
    {
      "slug": "the-read-only-input",
      "title": "THE READ-ONLY INPUT",
      "kicker": "a chamber cannot edit what it is given",
      "accent": "#5ad6ff",
      "blurb": "Both chambers read |i|. Neither may write it - an attempt throws. If a body can edit its own input, running it twice is running it on two different things.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "4b80172bf83544517cbb601bfa8019b49c7a067b822f3f6677ec009010986e04"
    },
    {
      "slug": "the-grammar-that-cannot",
      "title": "THE GRAMMAR THAT CANNOT",
      "kicker": "make the ambiguity unwriteable",
      "accent": "#b98cff",
      "blurb": "A parser could not see an ambiguity, so the grammar was narrowed until it could not be written. Not a smarter checker - a smaller language.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "d31003d14ff11c206646981571660cac4d8cea644adf62d7a27574218db9a2bf"
    },
    {
      "slug": "the-blind-instrument",
      "title": "THE BLIND INSTRUMENT",
      "kicker": "a checker that cannot see is silent, not noisy",
      "accent": "#ff5a8a",
      "blurb": "A FORTRAN veto sliced each line to column 7, discarding the label that lives in columns 1-5, and let any closer pop whatever was on top. Two repairs, and the numbers moved in both directions.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "ae59de60a1a3743553a4dd7742dd20016d59bae1b8f1b880edc50d7ecea68790"
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    {
      "slug": "the-two-errors-that-cancel",
      "title": "THE TWO ERRORS THAT CANCEL",
      "kicker": "clean for exactly the wrong reason",
      "accent": "#ffd76a",
      "blurb": "DO 10 I = 1.10 is an assignment to a variable named DO10I - a period, not a comma, and spaces are not significant. Two programs contain it. One is called clean.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "d450f7642c2eb6a2e9a143a0c7eca1a2055087ee65e3845c766017058533e45a"
    },
    {
      "slug": "the-stack-that-cannot-jump",
      "title": "THE STACK THAT CANNOT JUMP",
      "kicker": "a pushdown model has no move for GO TO",
      "accent": "#5ad6ff",
      "blurb": "A pushdown automaton reads one symbol and makes one move. That discipline is what makes it decidable, and it is exactly what a jump violates.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "f41e6817e0927165a2d8ea3ce08688e2252fb57d27e9439118456980705d6163"
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    {
      "slug": "the-cap-that-was-luck",
      "title": "THE CAP THAT WAS LUCK",
      "kicker": "a number about this machine, written as a number about the language",
      "accent": "#7de2b0",
      "blurb": "A recursion cap shipped at 1000, chosen by taste. The real ceiling was then bisected: 1734 completes, 1750 overflows. It had sat under the ceiling by luck.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "ccc57b0aa70f15e5ef80d4ac88cf6ef135b2ea58c6b02b314387c4922b97e3fb"
    },
    {
      "slug": "the-branch-still-in-the-machine",
      "title": "THE BRANCH STILL IN THE MACHINE",
      "kicker": "the grammar lost if/then; the ISA never did",
      "accent": "#b98cff",
      "blurb": "A language with no if/then was lowered onto a VM and the opcodes counted. No new instruction was needed, and JMPF appears three times.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "c1f402991f4619542a6ac3e40bdfbef9aaf8f2468ea9bc5bcffea842dd602716"
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    {
      "slug": "the-zero-that-was-the-point",
      "title": "THE ZERO THAT WAS THE POINT",
      "kicker": "two zeros that look identical in the output",
      "accent": "#7de2b0",
      "blurb": "Two constructs were feared enough to scope a whole piece of work around them. Counted against 938,154 lines of BLAS and LAPACK, both occur zero times.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "35cacf24a4e195fc184ae20dc48f60bf7e18ee3a781a91744e38687014c8b7a0"
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    {
      "slug": "the-hundred-and-thirty-two",
      "title": "THE HUNDRED AND THIRTY-TWO",
      "kicker": "label reuse counted as nesting",
      "accent": "#ffd76a",
      "blurb": "A first count found 132 shared loop terminators. The test was 'a DO label appears more than once in a file' - which also matches two sequential loops reusing label 10.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "5276d3ccbcca22701ac9c070547cd67fa85137a99feec82ba4ef66d5978aa971"
    },
    {
      "slug": "the-jump-that-is-everywhere",
      "title": "THE JUMP THAT IS EVERYWHERE",
      "kicker": "the real obstacle, and not the one raised",
      "accent": "#ff5a8a",
      "blurb": "The exotic FORTRAN relics are gone - ENTRY nowhere, assigned GO TO nowhere, EQUIVALENCE six times. What is everywhere is the plain jump.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "35b199bff4eb933cadd2f94ffeb05fbd6fd5f0e6d83125f0caf2faef02633de7"
    },
    {
      "slug": "unjustified-is-not-disproven",
      "title": "UNJUSTIFIED IS NOT DISPROVEN",
      "kicker": "a likelihood ratio of one leaves the prior alone",
      "accent": "#5ad6ff",
      "blurb": "A design was argued for on the grounds that it makes a bad shape unwritable. The shape was counted and found zero times. That does not make the design wrong.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "781cf3ae9c6d0e61c2fe591aad05dcc7e5e66cdd960f4270a1ef39ae3781b328"
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    {
      "slug": "the-flattering-direction",
      "title": "THE FLATTERING DIRECTION",
      "kicker": "three errors, all the same way",
      "accent": "#b98cff",
      "blurb": "Three numbers came out wrong in one session, each in the direction that made the argument look better, each corrected by measuring again rather than by thinking harder.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "1b765b6323b4e6016bbf6c64a71b1c3d1a6b14fa512399168cf917ab028b24cc"
    },
    {
      "slug": "the-unary-minus-that-isnt",
      "title": "THE UNARY MINUS THAT ISN'T",
      "kicker": "a gap only a negative number can find",
      "accent": "#5ad6ff",
      "blurb": "A 13-symbol language has a binary minus and no unary one. Negative numbers are not handled badly - they are inexpressible as literals and must be constructed.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "b6c399fb36d1a45d7c5de4f1ab112951c48014cfab00b5863ea4c6ba7350b47b"
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    {
      "slug": "the-point-three-seven",
      "title": "THE POINT THREE SEVEN",
      "kicker": "a third have jumps; almost none are irreducible",
      "accent": "#7de2b0",
      "blurb": "A third of LAPACK contains a GO TO, which looked like the end of a stack-based checker. Then the control-flow graphs were analysed.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "ebcff4060b3f6a7eee2eb9977ce5258198e1afb3b80b7d22954ae8a3cd4a21d5"
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    {
      "slug": "the-gate-that-was-run",
      "title": "THE GATE THAT WAS RUN",
      "kicker": "no partial credit, and the number stays honest",
      "accent": "#ffd76a",
      "blurb": "A progress board where every item is binary. A gate is a thing that was run - a command with an output - or it is open. Nothing is 'mostly done'.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "ba7c2109144ce27728f5a12732c8cfdec9208b376f7d2dd7bf2e1103e6808220"
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    {
      "slug": "the-judge-built-first",
      "title": "THE JUDGE BUILT FIRST",
      "kicker": "the oracle was cheaper than the thing it judges",
      "accent": "#b98cff",
      "blurb": "On a seven-floor board, the floor that judges is 75% built and the floor it judges sits at 0%. Recognising is structurally cheaper than generating.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "9284a9eaf7399d307cc1681198e30a690001d3cbabc81d7ac096ff20a652b8a7"
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    {
      "slug": "the-fourteen-decimals",
      "title": "THE FOURTEEN DECIMALS",
      "kicker": "what an exact agreement does and does not show",
      "accent": "#ff5a8a",
      "blurb": "Reference BLAS DDOT, unmodified, with its MOD(N,5) clean-up and five-way unrolled loop kept rather than smoothed, translated into a 13-symbol language and run interpreted.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "a01e829ec5f88d937d870ae3c3898868a5f74bae320078b6bc1269ab9b7b2d8b"
    },
    {
      "slug": "the-smaller-language",
      "title": "THE SMALLER LANGUAGE",
      "kicker": "13 forms cover fortran better than python",
      "accent": "#7de2b0",
      "blurb": "A 13-symbol budget was designed by looking at Python. The gate that could stop everything asked whether the same thirteen forms cover FORTRAN.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "1ca7ddb8a0905a4709dd6ea27e629c6fd8cfbc4fa7476582fe867696b2481404"
    },
    {
      "slug": "rare-in-code",
      "title": "RARE IN CODE",
      "kicker": "common in codebases, rare in code",
      "accent": "#ff5a8a",
      "blurb": "GO TO appears in more than a third of LAPACK's files and does not reach thirteenth place by node count. Both are measurements of the same construct.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "eebedd5285a7cfd8ab99528831d78967ed549971243c814b967778618830cd96"
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    {
      "slug": "the-verdict-not-the-framing",
      "title": "THE VERDICT NOT THE FRAMING",
      "kicker": "drop the dominant kind and ask again",
      "accent": "#ffd76a",
      "blurb": "A coverage result dominated by one node kind is a result about that node kind. So the dominant kinds were thrown away and the question asked again.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "b3a56fcd3f2474f232a3ef6e0d12e528a9fa7e6e800328420a3adead937de9bf"
    },
    {
      "slug": "the-threaded-accumulator",
      "title": "THE THREADED ACCUMULATOR",
      "kicker": "computed from the tree, not supplied by hand",
      "accent": "#5ad6ff",
      "blurb": "A language with no mutable loop variable cannot keep a DO loop. Every loop that assigns must become a recursive function carrying the variables it mutates.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "e1285b55817ed35f3d3774367f6d8d91759482c6cf781a1e375280417073c408"
    },
    {
      "slug": "a-regex-meeting-nesting",
      "title": "A REGEX MEETING NESTING",
      "kicker": "the error surfaces four frames from its cause",
      "accent": "#b98cff",
      "blurb": "Five bugs in one build, every one a regex meeting a nested structure. A parse tree nests; a regular expression does not.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "0d7ca3be6b2fed47dea61179f94e1439c2d8fbe6f5d0798800d56104a42007bc"
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    {
      "slug": "the-careless-candidate-first",
      "title": "THE CARELESS CANDIDATE FIRST",
      "kicker": "an exam nobody has failed is not an exam",
      "accent": "#ffd76a",
      "blurb": "Before an exercise is pointed at anything real, a deliberately lazy answer is run against it. If the rubric passes that, the rubric is worthless.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "38f82851e101e2d853bd75ddfb80260b1487203c1fae7d8bd1a6dd256492f622"
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    {
      "slug": "two-oracles",
      "title": "TWO ORACLES",
      "kicker": "valid and right are different questions",
      "accent": "#5ad6ff",
      "blurb": "A compiler decides whether a candidate is legal; a reference implementation decides whether it is right. Three candidates, all three compiled, two rejected.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "de9d3a6191fd865dfec40642c649ca72e1a593464e253acef6696f7941f13cb4"
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    {
      "slug": "the-curriculum-not-made-up",
      "title": "THE CURRICULUM NOT MADE UP",
      "kicker": "every exercise is an observed failure",
      "accent": "#7de2b0",
      "blurb": "A curriculum whose every exercise is a failure that actually happened - four defects observed in one working session, two of them the author's own tools.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "ca5c0a908adc775f20e51311f40b869846e6ea8032ba1d49e76b55b7c034d1c7"
    },
    {
      "slug": "proven-to-discriminate",
      "title": "PROVEN TO DISCRIMINATE",
      "kicker": "which is not proven to teach",
      "accent": "#ff5a8a",
      "blurb": "Teaching needs four things: a curriculum, a marking scheme, a way to set the work, and a student. Three exist and have been used. The fourth is marked ABSENT.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "a0d9105064c92cad605f820017575a4825bbe185d7abb9ef48cf27e3ada8a436"
    },
    {
      "slug": "outside-that-it-stops",
      "title": "OUTSIDE THAT IT STOPS",
      "kicker": "a bounded specialist declares its own edge",
      "accent": "#b98cff",
      "blurb": "A specialist defined by what it refuses. Three skills, one domain, and an explicit instruction that outside it the answer is to say so and stop.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "a47a377a39ee4eadd976ca5e69d07f20070499af68fdde2979e5c3181759dbbe"
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    {
      "slug": "the-gap-buffer",
      "title": "THE GAP BUFFER",
      "kicker": "free at the cursor, paid for by moving it",
      "accent": "#7de2b0",
      "blurb": "The structure inside a text editor: one array with a hole at the cursor. Typing fills a hole slot, so an insert costs the same at ten characters or ten thousand.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "fcdee5b0d5775b3053312c56f559ce7414482f84421be430946f07072b8c3901"
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    {
      "slug": "the-two-heuristics",
      "title": "THE TWO HEURISTICS",
      "kicker": "rank prevents, compression repairs",
      "accent": "#ffd76a",
      "blurb": "Union-Find carries two famous heuristics and the inverse-Ackermann bound belongs to the pair. Measured separately they do different jobs at different times.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "d3042e4c1f88d11dbaf85843ad2058446368d3b3cc009e8712a330f8d1084403"
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    {
      "slug": "the-finger-tree",
      "title": "THE FINGER TREE",
      "kicker": "two cheap ends, and a ridge between them",
      "accent": "#5ad6ff",
      "blurb": "Hinze and Paterson's 2-3 finger tree reaches index i in time proportional to log min(i, n-i), so the front and the back cost the same.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "fd4ff373b6f929ed29116506188fbaafec63595ec2a7fea5e98e3a2746a0ced8"
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    {
      "slug": "the-hash-life-doubling",
      "title": "THE HASH LIFE DOUBLING",
      "kicker": "a node of side 2^k advances 2^(k-2) generations",
      "accent": "#b98cff",
      "blurb": "The half of Gosper's HashLife that memoisation alone does not give you. That exponent is not a tuning choice - it is the largest sound step.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "66d330cce6678934fbe9072d4dbc432f3475789fdb01314ae41dedb36bf73723"
    },
    {
      "slug": "the-brodal-queue",
      "title": "THE BRODAL QUEUE",
      "kicker": "worst case, not amortised -- and what that costs",
      "accent": "#ff5a8a",
      "blurb": "Brodal's 1996 priority queue does insert, meld and find-min in worst-case constant time. An amortised bound permits a spike; some systems cannot have one.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "ede93cb3b04f6bd7a9b9024aebd7db9b330efaadd4f3b2c0cb69ed6bde420f56"
    },
    {
      "slug": "the-rank-wall",
      "title": "THE RANK WALL",
      "kicker": "the one result that CLOSES an option",
      "accent": "#ff5a8a",
      "blurb": "A linear map from C^4096 to C^1 has rank at most 1. Reversibility needs rank 4096. A one-dimensional root cannot host a reversible fold - arithmetic, not preference.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "919d2c26a44e6388c06e297eeda5e893d3f70fc24648041a82bcf0751165f0c6"
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    {
      "slug": "the-only-factorisation",
      "title": "THE ONLY FACTORISATION",
      "kicker": "the drawing IS the number",
      "accent": "#ffd76a",
      "blurb": "A drawing shows four arms around a centre. Among equal arm sizes, exactly one reaches the total: 8^4 = 4096.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "27f3e77053444358aff76d901ffea429647aee196480c638baa8f200a30b4d9f"
    },
    {
      "slug": "the-ranking-inverts",
      "title": "THE RANKING INVERTS",
      "kicker": "fewer bits kept, and the better operator",
      "accent": "#5ad6ff",
      "blurb": "Rank six folds by bits kept and one operator wins; rank them by recoverability and a different one does. A single-number ranking picks the wrong one.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "e716e466321b3c3feadedaa1c77b9bb6517e60b6a2026ea77a96a9deb7fe8ce4"
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    {
      "slug": "a-guess-wearing-syntax",
      "title": "A GUESS WEARING SYNTAX",
      "kicker": "a selector that graded the wrong row",
      "accent": "#b98cff",
      "blurb": "A verifier looked up a row with .find() and a loose pattern. The pattern matched two rows; .find() silently returned the first, and reported a correct claim as wrong.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "8336a81a29776131b211c79c4022dc31e909b822ab43125a64459d65e30183e4"
    },
    {
      "slug": "the-costume",
      "title": "THE COSTUME",
      "kicker": "an auditor cannot tell an obfuscation from an error",
      "accent": "#7de2b0",
      "blurb": "A sequence arrived as 'the structure'. Audited as data it has three faults, every one correct about the typed input and wrong about the design - it was a deliberate disguise.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "8fe98d88afa0522010dc9e1fffd6a4f8d2aff6af2c6d330cee9b59ece40c9b78"
    },
    {
      "slug": "the-objection-recorded",
      "title": "THE OBJECTION RECORDED",
      "kicker": "drawn alike is not measured",
      "accent": "#ff5a8a",
      "blurb": "A panel filed an objection against the artifact it belongs to, and the artifact shipped with the objection still in it.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "1eb3371a8af1846b7b328ca2618a6fea08bf353ef9f02f1efb39d39e55b33797"
    },
    {
      "slug": "the-residual-hole",
      "title": "THE RESIDUAL HOLE",
      "kicker": "three revs, and none of them has moved",
      "accent": "#ffd76a",
      "blurb": "Three known gaps, carried forward through three revisions, none closed. Not hidden and not fixed - named, in the same place, three times.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "b8f7dcef3847391d745c9ee33b25fa2f33b01ae44f393e1a67292a4056d2d3e3"
    },
    {
      "slug": "nothing-watches-the-gate",
      "title": "NOTHING WATCHES THE GATE",
      "kicker": "a regress with a measurable depth",
      "accent": "#5ad6ff",
      "blurb": "The verifiers check the model. A gate checks the verifiers with deliberately broken copies. Nothing checks the gate - and where the regress stops is the finding.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "10d4ce14ed93882abf394b3f8c4eed2f2568e50b6fc86f2d0562c059548607fc"
    },
    {
      "slug": "the-coincidence-left-alone",
      "title": "THE COINCIDENCE THAT WASN'T",
      "kicker": "CORRECTED -- a coincidence that was never there",
      "accent": "#7de2b0",
      "blurb": "This sphere first published a 13-and-13 coincidence and priced it. The next revision opened the files: the instruction set has SEVENTEEN. Withdrawn.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "4048bef5401953365b4337518030bebf9ff71b6c86e6de437790ba6ebdd5c04b"
    },
    {
      "slug": "the-missing-mutant",
      "title": "THE MISSING MUTANT",
      "kicker": "the quadrant with no test in it",
      "accent": "#b98cff",
      "blurb": "A gate proves its verifiers work by catching six deliberately broken copies. 6 of 6, every revision - and all six land in the same quadrant of a two-by-two.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "f6833ac3bbb8592c4fc2532b96ce548085f7c772e8e864c10cf28bcccd6be324"
    },
    {
      "slug": "the-partition-that-isnt",
      "title": "THE PARTITION THAT ISN'T",
      "kicker": "it died on counting, not on statistics",
      "accent": "#7de2b0",
      "blurb": "A figure shows four equal arms. The measured grouping of the twelve underlying items is 3, 3, 2, 4.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "de024a44c93686bbd066e044f28bc8741580a8609e85288929dd91c40b2a2238"
    },
    {
      "slug": "the-automorphism-shortfall",
      "title": "THE AUTOMORPHISM SHORTFALL",
      "kicker": "what an unequal partition costs in bits",
      "accent": "#ffd76a",
      "blurb": "How much labelling survives a fold depends on how symmetric the thing being folded is - and symmetry here is countable.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "cb05744bc4eca14e14b7c0d81f01f30780f96b46920bd65bf543b51a281c6841"
    },
    {
      "slug": "two-numbers-that-are-not-one",
      "title": "TWO NUMBERS THAT ARE NOT ONE",
      "kicker": "0.05 apart, and asserted distinct",
      "accent": "#5ad6ff",
      "blurb": "Two quantities in one project, 3.5850 and 3.6337 bits, derived from entirely different things and close enough that a later build would merge them.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "307011c772e643cef6bb608f4c6cc0f193f2b615c28add9c9dcb11f4a3cbd0ed"
    },
    {
      "slug": "the-dissent-upheld",
      "title": "THE DISSENT UPHELD",
      "kicker": "answered by the data, and answered no",
      "accent": "#ff5a8a",
      "blurb": "One revision earlier a panel filed a dissent: nothing has checked whether the four arms are alike. It shipped unresolved.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "1c3b1ac094b70740800e98cd6b398af21810d97610406b7700c28ebdb75d448d"
    },
    {
      "slug": "the-name-that-walks",
      "title": "THE NAME THAT WALKS",
      "kicker": "the control that fires most often",
      "accent": "#b98cff",
      "blurb": "A name was coined in conversation and deliberately kept out of the shipped artifact. A grep at step nine found it in the page anyway, for the second build running.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "49e1dab2217b166b7a0a6fce59dea0a167f6901d1d41d8090f46e99b50b7c1f8"
    },
    {
      "slug": "the-xor-swap",
      "title": "THE XOR SWAP",
      "kicker": "no temporary, and one input it destroys",
      "accent": "#ff5a8a",
      "blurb": "Three exclusive-ors swap two values with no temporary variable. Point both names at the same storage and the value becomes zero.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "0c0a640e00cebc62faceffd79e5322e5562c32b123c4e705e9d722f27f36524d"
    },
    {
      "slug": "the-duffs-device",
      "title": "DUFF'S DEVICE",
      "kicker": "a switch whose cases fall into a loop",
      "accent": "#ffd76a",
      "blurb": "An eight-way unrolled copy where the leftover elements are handled by jumping into the middle of the loop.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "ea01923da58698158454c7e2407e7cde35987bc9c22c6ca3b16b94943edfbc10"
    },
    {
      "slug": "the-de-bruijn-multiply",
      "title": "THE DE BRUIJN MULTIPLY",
      "kicker": "a perfect hash for the lowest set bit",
      "accent": "#7de2b0",
      "blurb": "Multiply an isolated bit by a de Bruijn constant and the sequence shifts, so the top five bits of the product name the position of that bit.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "c54267379ef1395f2e60e28a08f92af6e286d0bd32611c99651b1ace6ca3afa7"
    },
    {
      "slug": "the-carry-save-adder",
      "title": "THE CARRY-SAVE ADDER",
      "kicker": "three numbers in, two out, no carry chain",
      "accent": "#5ad6ff",
      "blurb": "Three numbers go in and two come out with no carry propagation. Each output bit depends only on the same column of the three inputs.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "e6d5908d1d28839dd96c8b379f4bc5de569ee03e5487bc4a373d19a55482fe64"
    },
    {
      "slug": "the-magic-divide",
      "title": "THE MAGIC DIVIDE",
      "kicker": "dividing by multiplying",
      "accent": "#b98cff",
      "blurb": "Integer division by a constant is replaced by a multiply and a shift. It is exact arithmetic, not an approximation.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "442216c881b2d09f58ff8d62cb4f9bae9da51f6792a77daaaf4d9dfa0977c376"
    },
    {
      "slug": "the-zobrist-hash",
      "title": "THE ZOBRIST HASH",
      "kicker": "undo by doing the same thing again",
      "accent": "#7de2b0",
      "blurb": "One random key per piece and square. Moving a piece costs two xors, and un-moving it costs the same two, because XOR is its own inverse.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "44a501e9bd29aff849997895229f791bdca57b199c697843ed4a9c30ce569505"
    },
    {
      "slug": "the-todd-coxeter",
      "title": "THE TODD-COXETER",
      "kicker": "enumerate the cosets and the index falls out",
      "accent": "#5ad6ff",
      "blurb": "Give it generators, relations and a subgroup, and it fills a table until the table closes. The number of surviving rows is the index of the subgroup.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "51b1d933708afb93bac588f04056709cb5afffb8023d732078bd66ceb54c72c0"
    },
    {
      "slug": "the-wheel-factorisation",
      "title": "THE WHEEL FACTORISATION",
      "kicker": "skip what cannot possibly be prime",
      "accent": "#ffd76a",
      "blurb": "A wheel of the first few primes skips every number they divide. The number of spokes per revolution is exactly Euler's totient of the circumference.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "91d399be200392bd8360b1c4630901c53c47602dfdde4e4c8d0227b192afb31b"
    },
    {
      "slug": "the-double-dabble",
      "title": "THE DOUBLE DABBLE",
      "kicker": "binary to decimal with no division at all",
      "accent": "#b98cff",
      "blurb": "Shift the number left into a register of decimal digits, and before each shift add 3 to any digit that has reached 5 or more. Nothing else.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "e16b7f12da3b307fad9ab2f50ef3ea3dd8ff566caad5574b91f6db518e3542f5"
    },
    {
      "slug": "the-non-restoring-division",
      "title": "THE NON-RESTORING DIVISION",
      "kicker": "do not undo the bad step, correct it later",
      "accent": "#ff5a8a",
      "blurb": "Long division in hardware subtracts, and when the result goes negative it has to put it back. This one leaves it negative and adds on the next step instead.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "054834b1b158b3c6ca6f0f33b1ec50d7596dbad4d6b4a6361d5a920b071544da"
    },
    {
      "slug": "the-sticky-bit",
      "title": "THE STICKY BIT",
      "kicker": "one bit remembering everything thrown away",
      "accent": "#ffd76a",
      "blurb": "Rounding needs to know whether ANYTHING nonzero fell below the round bit. Not what - just whether. One OR of every discarded bit, and it never clears.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "3ffd9c77822172cc67b184f6b1a676bc0fe9e08bf59c603174bd06af66ed2f78"
    },
    {
      "slug": "the-subnormal",
      "title": "THE SUBNORMAL",
      "kicker": "the numbers that buy you a smooth zero",
      "accent": "#7de2b0",
      "blurb": "Without them the gap between zero and the smallest normal is vastly larger than the gap between neighbouring normals - so two different numbers can subtract to exactly zero.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "cbaf1f7e4c4c80c9797b8a254a5dba9439609e6475b3ec079bba533478a517cd"
    },
    {
      "slug": "the-unit-in-last-place",
      "title": "THE UNIT IN LAST PLACE",
      "kicker": "the ruler changes length as you walk",
      "accent": "#5ad6ff",
      "blurb": "The distance to the next representable number is not a constant. It doubles at every power of two, so precision is a function of where you are standing.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "a4eb100b4e1c453cffa231d4004b4cd18cf3616681cdc78985e0e0fb394776eb"
    },
    {
      "slug": "the-round-to-odd",
      "title": "THE ROUND TO ODD",
      "kicker": "a rounding mode that exists to be rounded again",
      "accent": "#b98cff",
      "blurb": "Rounding twice through an intermediate width can land further from the truth than rounding once. This mode makes the double rounding agree with the single one.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "0fab9a24ffbca88a3acf17164857fddeca4823450c4f3336391e8f3aea543261"
    },
    {
      "slug": "the-catastrophic-cancellation",
      "title": "THE CATASTROPHIC CANCELLATION",
      "kicker": "the error was there before the subtraction",
      "accent": "#ff5a8a",
      "blurb": "Subtracting near-equal numbers does not create error. It reveals error already present, by removing the leading digits that were hiding it.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "657be5402ddfeb673496bf6b99bb084665e3817a69f8ca06d1a6a1cd6dfe24cf"
    },
    {
      "slug": "the-berger-code",
      "title": "THE BERGER CODE",
      "kicker": "count the zeros and every one-way fault shows",
      "accent": "#7de2b0",
      "blurb": "Append the count of zeros in binary. A fault that pushes every affected bit the same direction cannot preserve that count, however many bits it touches.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "0f00f8f3dc46c967ca058ab694c72286902b6588709c57d78fd87d4bc75cf501"
    },
    {
      "slug": "the-carry-lookahead",
      "title": "THE CARRY LOOKAHEAD",
      "kicker": "generate and propagate, computed all at once",
      "accent": "#5ad6ff",
      "blurb": "Each bit either generates a carry, or propagates one it receives. Written that way the carries unroll into a formula with no chain in it.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "dea04374d4ce71abba0e62c3fff931c01e1a6d17ffb8a4bccc3ca8f04a0ee165"
    },
    {
      "slug": "the-conditional-sum",
      "title": "THE CONDITIONAL SUM",
      "kicker": "compute both answers, throw one away",
      "accent": "#ffd76a",
      "blurb": "Do not wait for the carry. Compute the high half twice - once assuming a carry arrives, once assuming it does not - and select when the truth turns up.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "a419b13fde2eabea171747a3e381104ac9898d4f8dc5801e6ef89828afeef95f"
    },
    {
      "slug": "the-barrel-shifter",
      "title": "THE BARREL SHIFTER",
      "kicker": "any distance in log n stages, no loop",
      "accent": "#7de2b0",
      "blurb": "A shift by k is a cascade of fixed shifts by powers of two, each switched on by one bit of k. No iteration, no variable latency.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "3ff2497c14ad5a527fd80a0b538ae18b12a536022ff3832ee1d1e50bfcb1e7dd"
    },
    {
      "slug": "the-priority-encoder",
      "title": "THE PRIORITY ENCODER",
      "kicker": "the highest one wins, and someone must say if none do",
      "accent": "#ff5a8a",
      "blurb": "n inputs collapse to log n outputs naming the highest set bit. But index 0 is a real answer, so a separate line has to declare whether the answer means anything.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "b6b85c37f28d50407c1e5dc31f88630fac7f04c737a6a0666965151f782a5c23"
    },
    {
      "slug": "the-one-hot",
      "title": "THE ONE HOT",
      "kicker": "exactly one wire high, and that is the whole code",
      "accent": "#39fc6b",
      "blurb": "An N-state machine fits in ceil(log2 N) bits, or in N bits with exactly one high. The second is profligate - and the only one of the two that can tell you it has been damaged.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "65c7d39c2fa517144876db30a0160dca8f6cf143595858b52a6fd8fa8db50bc1"
    },
    {
      "slug": "the-systolic-array",
      "title": "THE SYSTOLIC ARRAY",
      "kicker": "stop fetching operands and start pumping them",
      "accent": "#5ad4ff",
      "blurb": "Kung and Leiserson, 1978: data enters at the edge of a mesh and every cell it passes uses it once more. The arithmetic does not get cheaper. The memory does.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "ac4307d91b6b87c5ef89592ae384a17ca55a0c0680e1d6818061a4d4b8ccdd67"
    },
    {
      "slug": "the-victim-cache",
      "title": "THE VICTIM CACHE",
      "kicker": "four entries that fix what doubling the ways does not",
      "accent": "#ff9f45",
      "blurb": "Jouppi, 1990: keep a tiny fully-associative buffer beside a direct-mapped cache and catch the lines it throws away. Four extra entries, and it fixes a failure more associativity does not.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "515534c5e5747f6bb07b05ccadb6a7d9113a6d2290f7e3c6b79dc916882caa59"
    },
    {
      "slug": "the-branch-target-buffer",
      "title": "THE BRANCH TARGET BUFFER",
      "kicker": "99.8% accurate and wrong every single time",
      "accent": "#b98cff",
      "blurb": "Two questions get asked at one instruction. Will it branch? and where to? The first has two answers and is easy. The second has as many answers as there are addresses.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "f9e51a1af81efe7034ae0d9b4687923eed2d19b1de03daa379b6739309a13573"
    },
    {
      "slug": "the-store-to-load-forward",
      "title": "THE STORE TO LOAD FORWARD",
      "kicker": "reaching into a place the program cannot see",
      "accent": "#5ad4ff",
      "blurb": "A store sits in a buffer, not yet in memory. A load arrives for the same address. Memory is wrong; the buffer is not architecturally visible. The processor must reach into a place the program cannot see.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "3dfbe7b64c8fc70d2d8d03d4d87fa5ff6880b538cefbfb845459591b84ef4f37"
    },
    {
      "slug": "the-register-renaming",
      "title": "THE REGISTER RENAMING",
      "kicker": "the hardware apologising for the ISA",
      "accent": "#ff9f45",
      "blurb": "Tomasulo, 1967. Two instructions that both write r2 are not related; they collided in a namespace that ran out of names. Give each write its own name and the ordering it forced disappears.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "279ce03ac686abea127f84853a950c88107b9915c87092e8af9ab4ee3f40e2fb"
    },
    {
      "slug": "the-tlb-shootdown",
      "title": "THE TLB SHOOTDOWN",
      "kicker": "the coherence hardware forgot to build",
      "accent": "#ff5a8a",
      "blurb": "Caches are kept coherent by hardware you never see. TLBs are not. Change a page table entry on one core and the others keep the old translation until software walks over and tells them.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "8e7ce632cec31a6798b323b67f73540e50bb5c86887cd59b0f40468525334c01"
    },
    {
      "slug": "the-false-sharing",
      "title": "THE FALSE SHARING",
      "kicker": "a bug with no wrong behaviour",
      "accent": "#ff9f45",
      "blurb": "Two threads, two different variables, neither reading the other's. Put them four bytes apart and the program spends all its time passing a cache line back and forth.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "4c8182cd92e80da9cdc8c0e0d0e44378e669f5cc7a356be84c32698cdbd840c0"
    },
    {
      "slug": "the-memory-fence",
      "title": "THE MEMORY FENCE",
      "kicker": "a subtraction, not an instruction",
      "accent": "#b98cff",
      "blurb": "x=1 then read y on one core; y=1 then read x on the other. Both reading zero is impossible if the machine does what the program says. Every x86 in the world will do it anyway.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "e135dfd9c6ed4dcbc3dbc09ff0d6a3f86d36b5d0cf4503d277f5f9d9bccea2cc"
    },
    {
      "slug": "the-seqlock",
      "title": "THE SEQLOCK",
      "kicker": "starvation wearing the costume of latency",
      "accent": "#7de2b0",
      "blurb": "A reader that writes nothing. No lock, no cache line to own, no cost imposed on anyone. It reads a counter, reads the data, reads the counter again - and if they disagree it starts over.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "645370de1cc19aac2f8c3767d85e564d970967a12be93de3bce515444f7050b5"
    },
    {
      "slug": "the-rcu",
      "title": "THE RCU",
      "kicker": "never edit what someone might be reading",
      "accent": "#7de2b0",
      "blurb": "Copy it, change the copy, swing the pointer - and then wait, not for a lock, but for every reader who could still hold the old version to simply finish.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "d20db7d15cea30d52ccba61ca2981447a89414354c4758bf4a09780082c21d93"
    },
    {
      "slug": "the-hazard-pointer",
      "title": "THE HAZARD POINTER",
      "kicker": "say out loud which pointer you are holding",
      "accent": "#5ad4ff",
      "blurb": "RCU waits for everyone. Michael's alternative asks each reader to name the pointer it is using - and the writer frees everything nobody named.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "c0729d418bbd656f00ab4688ad0368f00eb4a15ce3c0e1e66ab0c71f17065274"
    },
    {
      "slug": "the-elimination-backoff",
      "title": "THE ELIMINATION BACKOFF",
      "kicker": "a push and a pop that cancel each other out",
      "accent": "#ff9f45",
      "blurb": "Rather than queue a push and a pop at the contended top of a stack, let them meet in a side room and hand the value straight across. The stack never hears about it.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "dcf517f0d99ed157c9f6178c6e17869ba317813427f03ee49396c4973fe89288"
    },
    {
      "slug": "the-flat-combining",
      "title": "THE FLAT COMBINING",
      "kicker": "building the bottleneck on purpose",
      "accent": "#b98cff",
      "blurb": "Sixty-four threads fighting for one lock is sixty-four cache-line transfers to do sixty-four small things. Let one thread take the lock and do all sixty-four.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "9d940425bc54bab30993bc521be9fb5b0a220491125f5aa4958bccfec1c0fc55"
    },
    {
      "slug": "the-bakery-algorithm",
      "title": "THE BAKERY ALGORITHM",
      "kicker": "take a number; no atomic instruction required",
      "accent": "#ffd76a",
      "blurb": "Lamport, 1974. Lowest number goes first, ties broken by who you are. Correct even if a read that overlaps a write returns garbage.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "2c9eddf965a57930916c122b4c72ffd3a6b59b53ff266c7f78834744cc0ebed6"
    },
    {
      "slug": "the-sleeping-barber",
      "title": "THE SLEEPING BARBER",
      "kicker": "the gap between looking and lying down",
      "accent": "#39fc6b",
      "blurb": "One barber, a few chairs, customers who leave if the chairs are full. The barber sleeps when there is nobody. The whole problem lives in one gap.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "b389c09d81a55cdb29ad2f327d06fd9b39b82b6e051b0dd20743ad200288bd84"
    },
    {
      "slug": "the-dining-philosophers",
      "title": "THE DINING PHILOSOPHERS",
      "kicker": "everyone correct, in the same way, at the same time",
      "accent": "#ff5a8a",
      "blurb": "Five philosophers, five forks, and a rule so reasonable it is fatal: pick up your left fork, then your right. Everyone can obey it at once, and if they do, nobody eats again.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "75840ba9dd4770c509b8a15811a92cbfcc0eccdb28615b0d0563be8584a92ac0"
    },
    {
      "slug": "the-banker-deadlock",
      "title": "THE BANKER DEADLOCK",
      "kicker": "he can afford it and he refuses anyway",
      "accent": "#ff9f45",
      "blurb": "Dijkstra's banker will not lend money he has, if lending it means he might later be unable to pay anyone in full. The resources are available. The request is legal. He refuses.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "e55722970b79933c19deb5379e117f98d746a95dd75dfcf498b8148810600bf4"
    },
    {
      "slug": "the-two-phase-commit",
      "title": "THE TWO PHASE COMMIT",
      "kicker": "correct, and it hangs seven times in nine",
      "accent": "#5ad4ff",
      "blurb": "Ask everyone whether they can commit; if they all say yes, tell them all to do it. The protocol is correct, and it has a hole you cannot patch.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "16713fa48a3209c251e53b7edb9fc25c8bf7b26f89f2a0d4fadab2231fa6ac31"
    },
    {
      "slug": "the-paxos-quorum",
      "title": "THE PAXOS QUORUM",
      "kicker": "safety was settled by arithmetic before anyone wrote a line",
      "accent": "#b98cff",
      "blurb": "The entire safety of distributed consensus rests on one fact about finite sets: any two majorities of the same set must share a member. Everything else is scaffolding around that intersection.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "a4e7890fbec0bb225208be97bebb3fa5ac5aaaedd6cc6ce26485832cfede2bb8"
    },
    {
      "slug": "the-unverified-surface",
      "title": "THE UNVERIFIED SURFACE",
      "kicker": "coverage reports on the covered",
      "accent": "#ffd76a",
      "blurb": "A figure a checker cannot reach is not lightly checked. It is unchecked, permanently, and it drifts at whatever rate the work moves.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "4f713b1760baaa51f50a7ca0e99dda2ea5ee9afbdbb57e7edef806405536d61c"
    },
    {
      "slug": "the-name-outside-the-parens",
      "title": "THE NAME OUTSIDE THE PARENS",
      "kicker": "the wrong answer of the right type",
      "accent": "#ff5a8a",
      "blurb": "A regular expression cannot count. Hand it something that nests and it does not refuse - it matches the part it can reach and returns that, with no sign the rest was ever there.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "90f28681870fff207f61a52209f389ad6b6deaea08d614b21416effb9d2e2037"
    },
    {
      "slug": "the-orientation-double-cover",
      "title": "THE ORIENTATION DOUBLE COVER",
      "kicker": "a census asked a question it cannot answer",
      "accent": "#b98cff",
      "blurb": "Two strips of six squares glued end to end, one straight and one with a half turn. Count corners, edges and faces and the two are indistinguishable. They are not the same object.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "73ae8d6a96ac88cbaf481fc726590f68823a80f3c1bab51886621bfeb89d83c4"
    },
    {
      "slug": "the-safe-direction",
      "title": "THE SAFE DIRECTION",
      "kicker": "only one kind of error summons a person",
      "accent": "#5ad4ff",
      "blurb": "A checker can be wrong two ways. It can cry wolf, or it can wave something through. These are not symmetric, because only one of them brings a human to look.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "7945937ef8b0f1f4e3acc589768f07d62a9b8faa407df4fd6d1a136a74287013"
    },
    {
      "slug": "the-fingerprint-match",
      "title": "THE FINGERPRINT MATCH",
      "kicker": "evidence is measured in the alternatives you wrote down",
      "accent": "#7de2b0",
      "blurb": "A guess reproduced an output character for character. That feels like proof. How much evidence it is depends on a number nobody computes: how many other outputs the guess could have produced.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "722ea79cdb7edc317b4eed464e4aac8c02de0aca6fbb4ee415d8bc34bb1399c9"
    },
    {
      "slug": "the-belady-anomaly",
      "title": "THE BELADY ANOMALY",
      "kicker": "more memory, more faults",
      "accent": "#ff5a8a",
      "blurb": "Give a program more memory and it faults more often. Not as a pathology of a bad implementation - on the plainest replacement policy there is, doing exactly what it says.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "7731ff3e0578513c735b26d94c96a005c39a570492a13231bd4df1dd5eec0eb4"
    },
    {
      "slug": "the-priority-inversion",
      "title": "THE PRIORITY INVERSION",
      "kicker": "the highest waits on the lowest",
      "accent": "#ff9f45",
      "blurb": "The highest-priority task waits for a lock held by the lowest. A middle-priority task, holding no lock and wanting nothing, preempts the low one - and the highest task in the system now waits on the one it outranks.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "30bc228d48c54e63ea8859ee2938fdf525ac136988a84d142a8a65115d6452e7"
    },
    {
      "slug": "the-condition-number",
      "title": "THE CONDITION NUMBER",
      "kicker": "the residual is small and every digit is wrong",
      "accent": "#ffd76a",
      "blurb": "A small residual is the thing everyone checks and it proves almost nothing. Plug a badly wrong answer into a near-singular system and the equations come out satisfied to fifteen decimal places.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "6b9d08613017777b5b93cf470d95744f567bec5178100714a66d04be17cf4e05"
    },
    {
      "slug": "the-shewchuk-predicate",
      "title": "THE SHEWCHUK PREDICATE",
      "kicker": "289 points collapsed onto one line",
      "accent": "#5ad4ff",
      "blurb": "Is this point left of that line, right of it, or on it? Three answers, and a geometry program is built entirely out of them. In floating point the question does not reliably have an answer at all.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "989a12a438539d487ecfc880bda895f2058ec51bec20ddf1cc74fc3e011aee1f"
    },
    {
      "slug": "the-succinct-rank",
      "title": "THE SUCCINCT RANK",
      "kicker": "three touches, wherever you ask",
      "accent": "#7de2b0",
      "blurb": "How many 1s appear before position i in a bit array? Counting is linear. Answering instantly usually costs a word per bit. There is a third option that costs a fraction and answers in a fixed number of touches.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "77036986ebb73cbb817b65d50900eec293bc39a8eec9127b7864eec04703a332"
    },
    {
      "slug": "the-dodgson",
      "title": "THE DODGSON",
      "kicker": "a determinant shrunk out of 2x2 windows",
      "accent": "#00f5ff",
      "blurb": "Dodgson condensation in the 5-window house format — Lewis Carroll's 1866 algorithm for the determinant, which skips the cofactor tree entirely. Replace every 2×2 window of an n×n matrix by its own little determinant, divide entrywise by the interior of the previous matrix, and repeat: the single number left at the top is the determinant. The engine underneath is the Desnanot–Jacobi identity. Verified live by an independent exact-integer Laplace expansion: over thousands of random 3×3 to 6×6 integer matrices, condensation returns exactly the same big-integer determinant, and Desnanot–Jacobi holds with zero error. The known limitation is measured rather than hidden — when an interior entry is zero the division is undefined and the method stalls, which happens on roughly a third of random integer matrices here, and the page counts them. Neon-noir traced. See the cascade shrink in 1D, condensation checked against cofactors in 2D, and the shrink-inward-instead-of-expanding inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "29d09b5ce80736b9f9fd2a2f21b04542c5a3a7448bf9c2656f1bcf21d16c3e1a"
    },
    {
      "slug": "the-macwilliams",
      "title": "THE MACWILLIAMS",
      "kicker": "a dual code counted without ever listing it",
      "accent": "#ffd23f",
      "blurb": "The MacWilliams identity in the 5-window house format — the theorem that says you never have to look at the dual code. Every linear code C has a dual C⊥, and each has a weight enumerator: the tally of how many codewords carry 0 ones, 1 one, 2 ones, and so on. Jessie MacWilliams proved in 1963 that the dual's entire tally is a fixed linear transform of the primal's, B_j = (1/|C|)·Σ_i A_i·K_j(i), where K_j is the Krawtchouk polynomial. Count one side and the other side is already known, even when the dual is astronomically larger. Verified live: for Hamming(7,4) the brute-forced dual — the [7,3] simplex code, tally 1,0,0,0,7,0,0,0 — is reproduced exactly by the Krawtchouk transform of the primal tally 1,0,0,7,7,0,0,1, and over 400 random binary linear codes the transform matches a brute enumeration of the dual with zero error in whole numbers. Neon-noir traced. See the two tallies in 1D, random codes checked both ways in 2D, and the count-once-know-twice inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "20d1f22adf8d6652847e2a36b620e5cc721ddb2e72c2575bbd383c65694e9438"
    },
    {
      "slug": "the-lill",
      "title": "THE LILL",
      "kicker": "roots found by folding a ray, not by solving",
      "accent": "#7cfc00",
      "blurb": "Lill's method in the 5-window house format — Eduard Lill's 1867 way of finding the real roots of a polynomial with a ruler and a bouncing ray, no algebra at all. Walk the coefficients as a right-angled staircase: east a_n, turn 90°, a_{n−1}, turn 90°, down to a_0, with negative coefficients as steps backwards. Then fire a ray from the origin at angle θ, turning it 90° each time it meets the line of the next segment. If the ray finishes exactly on the path's endpoint, then x = −tanθ is a root. The reason is not a drawing coincidence: the legs of the ray are synthetic division — Lill's ray is Horner's scheme done with a straightedge. Verified live on two independent fronts: aiming at a known root closes the ray to under 4e-13 while aiming 0.05 off leaves a gap of at least 1.2e-4, nine orders apart; and for arbitrary θ each geometric leg equals Horner's coefficient b_k·secθ while the terminal gap equals |p(x)| itself, both to ~2e-12. Neon-noir traced. See the staircase and closing ray in 1D, an aimable ray checked against Horner in 2D, and the fold-do-not-solve inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "1d2dc250c4787af8e1522286d1b4f028f2064505373e6ecab927b88aed22083e"
    },
    {
      "slug": "the-macmahon-box",
      "title": "THE MACMAHON BOX",
      "kicker": "every way cubes can settle into a corner",
      "accent": "#9d00ff",
      "blurb": "MacMahon's box formula in the 5-window house format — counting the ways cubes can be stacked into a corner. A plane partition in an a×b×c box is an a×b grid of heights between 0 and c, weakly decreasing along every row and every column: a pile of unit cubes shoved into a corner and settled under gravity from two directions at once. Percy MacMahon found that the number of such piles is one closed product, PP(a,b,c) = ∏ᵢ∏ⱼ∏ₖ (i+j+k−1)/(i+j+k−2) — a tangle of nested inequalities collapsing into a single ratio of integers. The same number counts the lozenge tilings of a hexagon with sides a, b, c: the boxes-in-a-corner picture is that tiling seen straight on. Verified live by two independent routes: a raw recursive enumeration that builds every legal height grid cell by cell, and the closed formula in exact big integers — agreeing on every box tested, 1×1×1 = 2, 2×2×2 = 20, 3×3×3 = 980, 2×3×4 = 490, 4×4×3 = 24696, 4×4×4 = 232848 — with the formula's symmetry in a, b, c checked directly. Neon-noir traced. See the height grid in 1D, enumeration against formula in 2D, and the multiply-the-corners inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "1e475b01fe64bb3f77a0cb463035940519dc90a1f2a80cfac7ba32ce8aa40c90"
    },
    {
      "slug": "the-srt-division",
      "title": "THE SRT DIVISION",
      "kicker": "a divider with five blank cells in its table",
      "accent": "#5ad0ff",
      "blurb": "Radix-4 SRT division in the 5-window house format — how hardware actually divides, and how five missing table entries cost Intel $475M. SRT (Sweeney, Robertson, Tocher, c.1958) peels off two bits per step using a redundant digit set where each quotient digit may be −2, −1, 0, +1 or +2, iterating P ← 4P − q·D. Redundancy is the whole trick: because the digit ranges overlap, the hardware need not know the exact partial remainder to choose a digit, only roughly where it sits — so the choice comes from a coarse lookup table over a truncated (P,D) grid, and a loose guess is absorbed next iteration. That tolerance has an edge, and in 1994 the Pentium found it: its quotient-selection PLA was missing five of 2,048 entries, and a division whose trajectory landed there read a zero where a digit belonged. Verified live: a radix-4 divider with a 1,031-cell P-D table reproduces true division to a worst error of 1.1e-16 over 40,000 random operand pairs, and blanking five reachable cells makes 136 of 60,000 divisions wrong (~0.23%), worst case off by 0.67. Honest boundary: a working model of the defect mechanism, not an emulation of the P5 divider. Neon-noir traced. See the P-D plot in 1D, divisions with the table intact or holed in 2D, and the look-it-up-do-not-compute-it inverse in 3D.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "c788e6f2b4f3d2dbf3072e0b91100855dd4278115ba3a738b86b7e7dfe859e95"
    },
    {
      "slug": "the-nagle-delayed-ack",
      "title": "THE NAGLE DELAYED ACK",
      "kicker": "two polite algorithms waiting for each other",
      "accent": "#ffd76a",
      "blurb": "One end holds back small packets until the outstanding data is acknowledged. The other holds back acknowledgements in case something to piggyback on turns up. Both are correct. Together they wait.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "860907f61740f4fc6592bdd0b92b8d4ec1562902e2a309d0e701fe022cd0e8f3"
    },
    {
      "slug": "the-halloween-problem",
      "title": "THE HALLOWEEN PROBLEM",
      "kicker": "the rows keep coming back",
      "accent": "#ff9f45",
      "blurb": "Give everyone under twenty-five thousand a ten percent rise. Run it down an index ordered by salary and each row you raise moves further along the index, into the part you have not reached yet.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "32a6407226fadebb2aa91c393e13f95a64ac4a44e27fb86a93cb51e3a91172df"
    },
    {
      "slug": "the-write-skew",
      "title": "THE WRITE SKEW",
      "kicker": "no row was written twice",
      "accent": "#5ad4ff",
      "blurb": "Two doctors are on call. Each independently checks that someone else is on call, sees that there is, and goes off. Neither transaction touched a row the other wrote. Both are correct. Nobody is on call.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "4cd9bc83a7b877ac29d35b366846059630e93533f6aeeee7fd4321d72f38283b"
    },
    {
      "slug": "the-head-of-line-blocking",
      "title": "THE HEAD OF LINE BLOCKING",
      "kicker": "seven conversations that lost nothing",
      "accent": "#ff5a8a",
      "blurb": "Eight independent conversations share one ordered pipe. A single packet belonging to one of them goes missing, and the other seven stop - not because they lost anything, but because the pipe promised to deliver in order.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "bf33a0766719aec02850128536603a2e8a7575f0c78315319cec4bff921df5d7"
    },
    {
      "slug": "the-padding-oracle",
      "title": "THE PADDING ORACLE",
      "kicker": "one bit, returned politely, several thousand times",
      "accent": "#7de2b0",
      "blurb": "The cipher is not broken and the key is never touched. All the server does is answer, honestly, whether a message it could not decrypt had the wrong padding or the wrong contents.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "e8051f0e732caa26f23a0babc6d3cd8c199894d44bb40e3c8e41c8701580c4c1"
    },
    {
      "slug": "the-strict-aliasing",
      "title": "THE STRICT ALIASING",
      "kicker": "the standard forbids what the memory does",
      "accent": "#ff5a8a",
      "blurb": "Two pointers of different types are not allowed to refer to the same memory. Not unlikely to - not allowed. So the compiler may read through one, write through the other, and keep the value it read.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "9b7986792d92ff5b0bb00bf8f7f23dff91ab2df4b8bb0b0a2e358d60de96a8c3"
    },
    {
      "slug": "the-signed-overflow",
      "title": "THE SIGNED OVERFLOW",
      "kicker": "right 31 times out of 32",
      "accent": "#ffd76a",
      "blurb": "Is x + 1 > x always true? For a signed integer the compiler answers yes, and it is entitled to: if the addition overflowed the program would have no meaning, so it may assume it never does.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "9adf021f8d1b5712ce19ef499e95fa73714fb0dfdae5f227d0842c9f8ea003bb"
    },
    {
      "slug": "the-shift-by-width",
      "title": "THE SHIFT BY WIDTH",
      "kicker": "two machines, two answers, neither wrong",
      "accent": "#5ad4ff",
      "blurb": "Shift a 32-bit value left by 32 places and every bit should fall off the end. The answer is zero. It is not zero on x86, it is not zero on ARM either, and the two disagree.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "aaf2eca196acb95f402faae8642f10bd69bb56abfb9c0dbe95560da1d705d912"
    },
    {
      "slug": "the-dead-store-elimination",
      "title": "THE DEAD STORE ELIMINATION",
      "kicker": "the wipe that was deleted for being pointless",
      "accent": "#7de2b0",
      "blurb": "You wipe the password buffer before freeing it. Nothing ever reads those zeros, so the writes cannot change the program's meaning, so the optimiser deletes them. The secret is still there.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "395ba5c898849e957143184c50d43d93a78b3590a1825a86f6f818cd6bb0dc1b"
    },
    {
      "slug": "the-restrict-keyword",
      "title": "THE RESTRICT KEYWORD",
      "kicker": "a promise nothing can check",
      "accent": "#ff9f45",
      "blurb": "restrict is a promise, made by you, that two pointers never touch the same object. The compiler cannot check it. It can only believe you, keep values in registers, and give a different answer if you were wrong.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "90388b7d97c6dd2066c255e56ee85ba88b5f201b7db2b0873f288097473fe774"
    },
    {
      "slug": "the-normalization-form",
      "title": "THE NORMALIZATION FORM",
      "kicker": "the same glyphs, and not equal",
      "accent": "#b98cff",
      "blurb": "Two strings, the same glyphs on screen, pixel for pixel. One holds a single code point; the other a letter followed by an instruction to put an accent on it. They are not equal.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "03585e9f2b39bcb2c9e2a2eeff1dea7bcf0251e98e5815fa62fa5334382b9d75"
    },
    {
      "slug": "the-grapheme-cluster",
      "title": "THE GRAPHEME CLUSTER",
      "kicker": "eleven, seven, one - all correct",
      "accent": "#5ad4ff",
      "blurb": "How long is a family emoji? Eleven, if you ask the string. Seven, if you ask how many characters. One, if you ask a person. All three are answers to different questions.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "2e347d10c020197fb9c4acfb186364d4f93b8b4af6cd0ac42ed75fd936ec703f"
    },
    {
      "slug": "the-turkish-i",
      "title": "THE TURKISH I",
      "kicker": "lowercase is a property of a language",
      "accent": "#ff9f45",
      "blurb": "Lowercasing a string is not a property of the string. It is a property of the string AND a language, and in Turkish the letter I does not become i.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "724a43481baac38155d5e56b8e18c9f8b5fb1d15336d4c91f2d601395e329db2"
    },
    {
      "slug": "the-homoglyph",
      "title": "THE HOMOGLYPH",
      "kicker": "thirty-two spellings, one shape",
      "accent": "#ffd76a",
      "blurb": "Cyrillic a is not Latin a. Different code point, different alphabet, different language - and on every screen you will ever look at, the same shape.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "d4661245650802e59dbb578b585ec6bc97157b3eedca5e9725ce949405318432"
    },
    {
      "slug": "the-surrogate-pair",
      "title": "THE SURROGATE PAIR",
      "kicker": "two units that are not characters",
      "accent": "#7de2b0",
      "blurb": "Sixteen bits held every character, once. Then there were more than sixty-five thousand of them, and the fix was to spend two units on the rest - two units that mean nothing apart.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "224f648b90dcbf848acd86055287cf78859e5c5343b53378e157b50384c43294"
    },
    {
      "slug": "the-zero-width-joiner",
      "title": "THE ZERO WIDTH JOINER",
      "kicker": "characters with no shape at all",
      "accent": "#b98cff",
      "blurb": "Some characters have no shape. They occupy no width, print nothing, survive copy and paste, and change what a string is without changing anything you can see.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "df25a426ac4f7e0fdf5fe65a6134795508e56de4193ab2c63d991a42a4c7599e"
    },
    {
      "slug": "the-utf8-overlong",
      "title": "THE UTF-8 OVERLONG",
      "kicker": "384 spare spellings of 128 characters",
      "accent": "#5ad4ff",
      "blurb": "There is exactly one correct way to encode a character in UTF-8, and several that also work. A decoder that accepts the extras will hand you a slash you did not see coming.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "33847472d4d064fcbb9029ada70863a599b48c3309d98f0776a2c1258b672e8d"
    },
    {
      "slug": "the-byte-order-mark",
      "title": "THE BYTE ORDER MARK",
      "kicker": "metadata living inside the data",
      "accent": "#ffd76a",
      "blurb": "A mark at the front of a file saying which end of a number comes first. UTF-8 has no ends to order. The mark got used anyway, as a label, and it is invisible.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "c0a2bcabe037ab69ccb22da8eca3247577de3672a74fda8819675ce77c9deb9a"
    },
    {
      "slug": "the-case-folding",
      "title": "THE CASE FOLDING",
      "kicker": "out through two, back as one",
      "accent": "#ff9f45",
      "blurb": "Uppercase is not a permutation. Some letters get longer, some have no partner, and lowercasing an uppercased string does not give back what you started with.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "2adf7f76c88a0844e286c9cf23ee8ff736554037e7f0e362abd467e697fa2686"
    },
    {
      "slug": "the-bidi-override",
      "title": "THE BIDI OVERRIDE",
      "kicker": "two readers, two orders, no error",
      "accent": "#ff5a8a",
      "blurb": "Text has a logical order - the order the bytes are in - and a display order. Nine invisible characters set the second without touching the first, so a line of code can be shown in an order it does not have.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "75367838125683baa8d605c93eea0cc4197373578f5a1cdbe515e60e61971b6b"
    },
    {
      "slug": "the-leap-second",
      "title": "THE LEAP SECOND",
      "kicker": "precise about the wrong quantity",
      "accent": "#ffd76a",
      "blurb": "Unix time counts seconds since 1970. It does not. It counts days since 1970 multiplied by 86,400 - and the Earth has had twenty-seven extra seconds inserted that the count refuses to hold.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "7ed56a26febb963a47918cb9908712893ac9e8e7d99defcec69e6901113f21b0"
    },
    {
      "slug": "the-monotonic-clock",
      "title": "THE MONOTONIC CLOCK",
      "kicker": "two clocks, two questions",
      "accent": "#5ad4ff",
      "blurb": "Two clocks in every machine. One tells you what time it is and can be corrected, moved or dragged an hour sideways twice a year. The other only counts forward and cannot tell you anything about the world.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "2a516a2eb55ce972bf1a7e86b81ad1fb4837a52e1909878fb64c83cb58f8dd61"
    },
    {
      "slug": "the-iso-week-year",
      "title": "THE ISO WEEK YEAR",
      "kicker": "two years, both correct",
      "accent": "#ff9f45",
      "blurb": "There are two years. The one on the calendar, and the one a week belongs to - and at the turn of December they disagree, because a week cannot be split between two years.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "a17b5e6f61193df6ea4137b74b56c3072f6d717599a26676c5a7c47578be397c"
    },
    {
      "slug": "the-clock-skew",
      "title": "THE CLOCK SKEW",
      "kicker": "one millisecond, 189 wrong orders",
      "accent": "#5ad4ff",
      "blurb": "Two machines timestamp two events. The second genuinely happened after the first. Whether the timestamps agree depends on how well the clocks are set, and nothing in the data says how well that was.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "5ac5d419d3690d4d9721de019c03d7a179ff31e9c0431ba053a05461878c9c4c"
    },
    {
      "slug": "the-unix-epoch",
      "title": "THE UNIX EPOCH",
      "kicker": "2038, and then 1901",
      "accent": "#ff5a8a",
      "blurb": "A signed thirty-two bit count of seconds from 1970 reaches its largest value at three fourteen in the morning on the nineteenth of January 2038, and the next second is 1901.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "f191615536676263b7ee82664937c903d7eba3941adfc066ad875080b00d7ff7"
    },
    {
      "slug": "the-negative-zero",
      "title": "THE NEGATIVE ZERO",
      "kicker": "five say same, five say different",
      "accent": "#ffd76a",
      "blurb": "There are two zeros. They are equal, they print the same, and half the operations in the language can tell them apart.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "401b92c352601b359d205a0136bbccc98a3c7ea0131c659426a53596b17e56e2"
    },
    {
      "slug": "the-nan-payload",
      "title": "THE NAN PAYLOAD",
      "kicker": "one name, a quadrillion values",
      "accent": "#ff5a8a",
      "blurb": "Not-a-number is not a number, and it is not one value either. It is a vast set of bit patterns that all mean this went wrong, none of which is equal to itself.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "d77dc1950475da6b2121a674c54d3876256e12170a85651190d079b45d14ee1e"
    },
    {
      "slug": "the-modulo-sign",
      "title": "THE MODULO SIGN",
      "kicker": "one identity, two answers",
      "accent": "#7de2b0",
      "blurb": "What is minus seven modulo three? Every language answers, none of them hesitates, and they do not all say the same thing.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "1c3fdae5340160a8d121619e82dcb456e34a82390a01b8792d2914ce88f86214"
    },
    {
      "slug": "the-round-half-even",
      "title": "THE ROUND HALF EVEN",
      "kicker": "exactly half a unit, every time",
      "accent": "#ffd76a",
      "blurb": "Exactly half. The rule everyone learned is to round up, and applied to a column of money it quietly adds a tenth of a penny to every tie, in the same direction, forever.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "a6dcb7651163b1209de1cf7fa4e68967272ef3a1f6f3e2b57d034c9b7ec458f2"
    },
    {
      "slug": "the-timezone-database",
      "title": "THE TIMEZONE DATABASE",
      "kicker": "a fact about parliaments, not the Earth",
      "accent": "#5ad4ff",
      "blurb": "A time zone is not a fact about the Earth. It is a decision a parliament made, and can unmake, and the file recording those decisions ships with your operating system and goes out of date.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "0b38e00c7ec9ef7e04052a21dda325a54c01f849d23effc9385bab918e42b067"
    },
    {
      "slug": "the-elias-fano",
      "title": "THE ELIAS-FANO",
      "kicker": "sorted is a bill you already paid",
      "accent": "#5ad0ff",
      "blurb": "A sorted list is mostly redundant. Elias-Fano splits every value in two, writes the low half verbatim and the high half as gaps - and stays randomly addressable, never decompressed to be read.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "ff4f1fdca2605d41f01bd04c9c939b021c891574fa5a9932511f9fccc18451f7"
    },
    {
      "slug": "the-aba",
      "title": "THE ABA",
      "kicker": "the pointer came back and brought nothing with it",
      "accent": "#39fc6b",
      "blurb": "Compare-and-swap asks whether a pointer is still the value you read. It cannot ask whether anything has happened since. If a value leaves and returns, CAS cannot tell.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "dae8b762612817e801394133464220ca18b569df60096062f6e6d6e6fb4f51ab"
    },
    {
      "slug": "the-t-digest",
      "title": "THE T-DIGEST",
      "kicker": "a sketch that decided in advance what would matter",
      "accent": "#9d00ff",
      "blurb": "To report a 99th percentile you appear to need every sample. A t-digest keeps a few dozen weighted centroids instead, and deliberately keeps them uneven - fine at the tails, coarse in the middle.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "e726d19c2d4f5fbb53eb47039c176155d09a709ab579715ffb65c161237cd7b9"
    },
    {
      "slug": "the-interval-clock",
      "title": "THE INTERVAL CLOCK",
      "kicker": "identity you can cut in half and hand away",
      "accent": "#ffd23f",
      "blurb": "A vector clock must know how many peers exist. Interval Tree Clocks do not: each peer owns a slice of [0,1) and forking is just cutting your own slice in half.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "772e5a528992342660c8e5d827fd328a30b7f618cff4c6f4f7729e66b9c0f6f9"
    },
    {
      "slug": "the-merkle-mountain",
      "title": "THE MERKLE MOUNTAIN",
      "kicker": "a log that refuses to have a summit",
      "accent": "#ff2d95",
      "blurb": "A Merkle tree wants to know its size before you build it; an append-only log does not know. A mountain range solves this by keeping a row of perfect trees and merging two whenever they match in height.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "a8dfb7d4d7cf5ae663d46c13da30dcd65a6d276fde527a02105fa16afb06089e"
    },
    {
      "slug": "the-content-defined-chunk",
      "title": "THE CONTENT-DEFINED CHUNK",
      "kicker": "cut where the content says, not where the ruler does",
      "accent": "#5ad0ff",
      "blurb": "Insert one byte into a file cut into fixed blocks and every block after it shifts and stops matching. Cut where a rolling hash says to, and the boundaries re-synchronise on their own.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "52dd5f4a204e57b70388df35cbc9f98105a69466e68a2f45e1e3635c06617d76"
    },
    {
      "slug": "the-slab-allocator",
      "title": "THE SLAB ALLOCATOR",
      "kicker": "excellent at one question, useless at the rest",
      "accent": "#7cfc00",
      "blurb": "A general allocator rounds your request up to something convenient for itself. A slab refuses: whole pages dedicated to one object size, packed end to end, no header and no search.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "62a1db06d94f509d61d7b7fc777505bbaf0930c0294b2c3f5c2a345d15afc83d"
    },
    {
      "slug": "the-buddy-allocator",
      "title": "THE BUDDY ALLOCATOR",
      "kicker": "merging is cheap because most merges are forbidden",
      "accent": "#9d00ff",
      "blurb": "Split memory in half until a block is just big enough. Each block then has exactly one partner, and finding it is not a search - it is a single XOR.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "9b99ab65762c2cf75a06b0394d1de867307dc3ec317295e2a0f5f97f5ebd9c36"
    },
    {
      "slug": "the-copy-on-write",
      "title": "THE COPY ON WRITE",
      "kicker": "the bill arrives later, addressed to someone else",
      "accent": "#7cfc00",
      "blurb": "A process forks and its whole address space is duplicated - except nothing is copied. Both point at the same pages, all marked read-only, and the copy happens on first write, one page at a time, or never.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "2547208c07a5684a8c0097d1432ee7a8e54d106361f1e2e874fc9497bb63485f"
    },
    {
      "slug": "the-two-generals",
      "title": "THE TWO GENERALS",
      "kicker": "they already agree and cannot confirm it",
      "accent": "#9d00ff",
      "blurb": "Two generals must attack together or not at all, over a channel that loses messages. Every acknowledgement needs an acknowledgement. No number of messages finishes the job.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "7fdf43b8278a4cf06718875215d14f18c88998ce5e4655560c7709200b508802"
    },
    {
      "slug": "the-coordinated-omission",
      "title": "THE COORDINATED OMISSION",
      "kicker": "the meter went quiet exactly where the trouble was",
      "accent": "#7cfc00",
      "blurb": "A load generator that waits for each response cannot issue requests while the service is stalled. The requests that would have been sent are never sent, never timed, never counted.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "f27f3065aeeed29aca9b6f57fcace5102583cdac60454081f606c53c337cbb59"
    },
    {
      "slug": "the-littles-law",
      "title": "THE LITTLES LAW",
      "kicker": "fix two of the three and the third is not yours to choose",
      "accent": "#00f5ff",
      "blurb": "The number of things in a system equals how fast they arrive times how long each stays. It assumes almost nothing about the queue and it is exact, not approximate.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "b171160ed52d30ebc2d89d432e7e296f0f54517bead097481d6f864783c46d73"
    },
    {
      "slug": "the-utilization-knee",
      "title": "THE UTILIZATION KNEE",
      "kicker": "idle capacity is not waste, it is the latency budget",
      "accent": "#5ad0ff",
      "blurb": "Waiting time does not rise smoothly with load. It rises as 1/(1-rho), which is flat for most of the range and then vertical.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "091adec3cdb332f378925a5b3f7b08f3cd7d77d1f759999228225f459b5a08ad"
    },
    {
      "slug": "the-tail-at-scale",
      "title": "THE TAIL AT SCALE",
      "kicker": "one in a hundred, a hundred times over",
      "accent": "#ffd23f",
      "blurb": "A service where only one request in a hundred is slow sounds healthy. Fan out to a hundred of them and wait for all, and the rare event becomes the common case.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "2134b5601dc091967b1b335666b93edea3d7ffe814e94f9e6cbdab91a2adf2d8"
    },
    {
      "slug": "the-jittered-backoff",
      "title": "THE JITTERED BACKOFF",
      "kicker": "a deterministic rule everyone shares is a coordination mechanism",
      "accent": "#00f5ff",
      "blurb": "A hundred clients collide, all back off by the same doubling amount, and all return at the same instant to collide again. Exponential backoff without randomness synchronises load rather than spreading it.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "fcdee431bc9666992678f72f425763373c7628723ac1e721a1c1808f3fe40287"
    },
    {
      "slug": "the-token-bucket",
      "title": "THE TOKEN BUCKET",
      "kicker": "the burst depth is a promise about your worst instant",
      "accent": "#ff5a3c",
      "blurb": "A bucket fills with tokens at a fixed rate and holds at most a fixed number. Every request spends one. The rate sets the average; the depth sets how much burst you will forgive.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "6bb6d1607084b274379e2dd726e756a7a6a7c86928eb00a1a29aed35877d44c7"
    },
    {
      "slug": "the-hedged-request",
      "title": "THE HEDGED REQUEST",
      "kicker": "a loan against idle capacity",
      "accent": "#00f5ff",
      "blurb": "Send the request. If it has not returned by the 95th percentile, send a second copy elsewhere and take whichever answers first.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "4b1a20610b7e5619f8be0a351a6fcf91d32cb0e064ff5bd20e97d71a2637227c"
    },
    {
      "slug": "the-bufferbloat",
      "title": "THE BUFFERBLOAT",
      "kicker": "the drop was the signal; the buffer is what silenced it",
      "accent": "#9d00ff",
      "blurb": "Memory got cheap, so buffers got large, so nothing is ever dropped - and packets sit in a queue for a second instead of being discarded in a millisecond.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "f780bd0ca935ff9ed276049398ea45e368ae72692b71c8dd3c5103bd45ef17c3"
    },
    {
      "slug": "the-universal-scalability",
      "title": "THE UNIVERSAL SCALABILITY",
      "kicker": "the descent is the cost of everyone agreeing",
      "accent": "#ff5a3c",
      "blurb": "Amdahl says extra workers stop helping. The Universal Scalability Law says they start hurting, because every worker must stay consistent with every other one and that cost grows as the square.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "84a4e7f1da55c6d27aecb8fbd7f566b201532af09b65b75639c76c3948e3fe7c"
    },
    {
      "slug": "the-thundering-herd",
      "title": "THE THUNDERING HERD",
      "kicker": "what fairness costs when you refuse to have an opinion",
      "accent": "#39fc6b",
      "blurb": "One resource frees up and every waiter is woken to race for it. One wins; the rest discover it is gone and go back to sleep, having been scheduled and cache-thrashed for nothing.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "86e00a24239894fa932cbebfe42df35753ca258423ee62f84a10a0f017c36ae1"
    },
    {
      "slug": "the-uuid-v7",
      "title": "THE UUID V7",
      "kicker": "the randomness in v4 was the property, not the waste",
      "accent": "#ffd23f",
      "blurb": "A v4 UUID is 122 random bits, so every insert lands somewhere unrelated to the last. A v7 puts the timestamp in the high bits and the arrival order becomes the sort order.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "46a2750b5ae587b48d7d11a3b6131b28123b5f51a80a70c6115ee498f1cb7924"
    },
    {
      "slug": "the-snowflake-id",
      "title": "THE SNOWFLAKE ID",
      "kicker": "an uncoordinated id is one whose coordination already happened",
      "accent": "#ff2d95",
      "blurb": "Sixty-four bits cut into fields: a sign bit nobody uses, a millisecond timestamp, a worker number and a per-millisecond sequence. No coordination between machines, and the id sorts by time.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "c186ab3119fcad4ba26f2d84afd49a661ed71ee2c028f1142e22ae0a7b780efc"
    },
    {
      "slug": "the-minimal-perfect-hash",
      "title": "THE MINIMAL PERFECT HASH",
      "kicker": "no slack, and so no way to say not here",
      "accent": "#5ad0ff",
      "blurb": "A hash table wastes space so collisions have somewhere to go. If the key set never changes, you can send n keys onto 0..n-1 with no collisions and no gaps, and store no keys at all.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "1f8b7e1bea5007b95edaf43380317c5c40ab9c0c1a4bcadb88cd737a1862c026"
    },
    {
      "slug": "the-crockford-base32",
      "title": "THE CROCKFORD BASE32",
      "kicker": "an alphabet that decides which differences are real",
      "accent": "#ff2d95",
      "blurb": "An alphabet meant to be read aloud, written down and typed back in. Crockford's base32 throws out I, L, O and U - three because they look like digits, the last so the encoding cannot spell things.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "0b1c218a34d2e187dbe5b4e7e2774046b26430e724a0b2ceae22216e834f58c9"
    },
    {
      "slug": "the-nothing-up-my-sleeve",
      "title": "THE NOTHING UP MY SLEEVE",
      "kicker": "it does not remove the choice, it makes it arguable",
      "accent": "#ff2d95",
      "blurb": "A cipher needs arbitrary constants, and anyone free to choose them could be choosing a backdoor. So you do not choose - you take the digits of something fixed before you arrived.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "53feacf13f5fd7f4b5827f81ed18785596f66c47036eb03eca1e6390a78fa3bc"
    },
    {
      "slug": "the-content-address",
      "title": "THE CONTENT ADDRESS",
      "kicker": "permanence bought with the ability to be corrected",
      "accent": "#7cfc00",
      "blurb": "A location address says where something is kept. A content address says what it is: hash the bytes and let the digest be the name. Move it, mirror it, rename the server - the name does not change.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "b35e0c4135d5e631a6f23846bf82259ecefa1431e7973ecbb3da056a0938aac4"
    },
    {
      "slug": "the-damm",
      "title": "THE DAMM",
      "kicker": "the better scheme lost to the one a clerk could do",
      "accent": "#ffd23f",
      "blurb": "A check digit catches typing mistakes. Luhn catches all single-digit errors but not all swaps of neighbouring digits. Damm catches both, using a lookup table instead of arithmetic.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "d446a893a9a29100ce026eef54cc5c5debb9cae7a349126a1eb655a796943c0e"
    },
    {
      "slug": "the-hash-flooding",
      "title": "THE HASH FLOODING",
      "kicker": "they declined to be the average case",
      "accent": "#9d00ff",
      "blurb": "A hash table is O(1) on average - over inputs an adversary did not choose. If the hash is fixed and public, colliding keys can be computed in advance and the table becomes one long list.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "9acafbac0e06f310e55960b50dd0b1bfc709f7890f198154afb86a10e497d5d1"
    },
    {
      "slug": "the-punycode",
      "title": "THE PUNYCODE",
      "kicker": "the boundary sits where a machine hands something to an eye",
      "accent": "#ff5a3c",
      "blurb": "Domain names became international, so every script on Earth can appear in a URL. Cyrillic and Latin letters that render identically are different characters, and a name is only as trustworthy as the difference you can see.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "a7e2c3af16ed88a9e7705857d7c489ffb8dad531d68281b7a3fbf15174f17f95"
    },
    {
      "slug": "the-sequential-key",
      "title": "THE SEQUENTIAL KEY",
      "kicker": "one fact, described once by a cache and once by a lock",
      "accent": "#5ad0ff",
      "blurb": "An auto-increment key sends every insert to the same end of the index. Wonderful for the disk and terrible for a lock: the rightmost page is the only page anybody wants, and every writer wants it at once.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "ce25ed53101462035995ebb1fe050b5a3bb7374410c5ae713189c73a7907395c"
    },
    {
      "slug": "the-tlb-reach",
      "title": "THE TLB REACH",
      "kicker": "a unit trick, not a capacity gain",
      "accent": "#5ad0ff",
      "blurb": "The TLB holds a fixed number of translations, not a fixed amount of memory. Entries times page size is the only figure that matters: how much memory the machine can name without a walk.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "8ac594f66de12bbcf435940d8ecfffac128c1391adc9ec8a8f8279757e01cfb6"
    },
    {
      "slug": "the-huge-page",
      "title": "THE HUGE PAGE",
      "kicker": "it helps most where it is needed least",
      "accent": "#9d00ff",
      "blurb": "A bigger page means fewer translations to track. It also means every allocation rounds up to a bigger boundary, and the rounding is charged whether you asked for it or not.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "ac5d0ddb6af514032d0735a80edd56f75257b9bfdea87188e4dedede3e646289"
    },
    {
      "slug": "the-numa-hop",
      "title": "THE NUMA HOP",
      "kicker": "the flat address space was the lie, and a load-bearing one",
      "accent": "#7cfc00",
      "blurb": "On a multi-socket machine the memory is not one pool. Some is attached to your socket and some to the other, and reaching across costs a fixed toll on every access.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "ca313eae4865a65c32eca3f2c35015c6e002ca831b561f3c7d87db40ab8b8771"
    },
    {
      "slug": "the-prefetcher",
      "title": "THE PREFETCHER",
      "kicker": "the whole performance cliff is a missing sentence",
      "accent": "#00f5ff",
      "blurb": "The hardware watches your address stream and fetches ahead of you. It is not clairvoyant - it is a pattern matcher, and it only wins when there is a pattern.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "04c85563afc3a3b744249b67608705dac1a735e587aa7dc51d8a43e6f74d06ac"
    },
    {
      "slug": "the-write-combining",
      "title": "THE WRITE COMBINING",
      "kicker": "the saving and the ordering bug are one mechanism",
      "accent": "#39fc6b",
      "blurb": "A store buffer holds writes back briefly so several to the same cache line leave as one transaction. Write the line in order and you pay once for sixty-four stores. Scatter them and you pay every time.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "1dd2d86557bd5d852eae54e9e2ba1b90edf431aeb5968adb90f575e4219c7bdd"
    },
    {
      "slug": "the-minor-fault",
      "title": "THE MINOR FAULT",
      "kicker": "not an error being handled -- an allocation finally happening",
      "accent": "#ff2d95",
      "blurb": "Not every page fault touches a disk. A minor fault means the page is already in memory and only the mapping was missing. The word fault is doing a lot of unearned work.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "82899aae16a6116f9b938ce89c7e80ab953d09c2246153f959053a69029d8292"
    },
    {
      "slug": "the-strided-access",
      "title": "THE STRIDED ACCESS",
      "kicker": "the premium on an insurance policy everyone else claims on",
      "accent": "#00f5ff",
      "blurb": "Memory arrives in cache lines, not in variables. Ask for four bytes and sixty-four turn up. Whether that is generous or wasteful depends on how far apart your next four bytes are.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "a9aed393e61ac751415c00b0338a7209aa2b263c4c0ff9bcba3cf94aa6228969"
    },
    {
      "slug": "the-pointer-chase",
      "title": "THE POINTER CHASE",
      "kicker": "the program knows the future and has no way to say so",
      "accent": "#ffd23f",
      "blurb": "To follow a linked list the machine must load a pointer before it knows which address to load next. The loads cannot overlap, be reordered or be prefetched, because the address does not exist until the previous load returns.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "0f0a712eff8474cd61c9a2d273b23b09b72090dd7e1eb4fef7a2b270bfc43608"
    },
    {
      "slug": "the-denormal-stall",
      "title": "THE DENORMAL STALL",
      "kicker": "a slow, silent, correct-looking decline",
      "accent": "#00f5ff",
      "blurb": "Below the smallest normal float there is one more range, where the leading bit is dropped and precision is traded away a bit at a time rather than all at once, so subtraction near zero keeps meaning something.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "87509f74ec826447240ae910d26d64a3b52adf40b2158c20480107df9568cb17"
    },
    {
      "slug": "the-cache-associativity",
      "title": "THE CACHE ASSOCIATIVITY",
      "kicker": "the room was never the constraint -- the address was",
      "accent": "#5ad0ff",
      "blurb": "A set-associative cache decides where a line may live from its address, not from how much room is free. Data that fits the cache several times over can still miss every time, if it all maps to the same set.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "e6076be621905b6ff984c0fe5e2909f35fdfaa9a557cfed26a6e12bf527f1cd1"
    },
    {
      "slug": "the-ntp-slew",
      "title": "THE NTP SLEW",
      "kicker": "it lies slowly instead of correcting fast",
      "accent": "#ff2d95",
      "blurb": "A wrong clock can be jumped or bent. Jumping is instant and can send time backwards. Slewing takes as long as it takes and never does.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "d4a9f710de7e992b57d47de957de3ac82edc894b191447dce805f2f5537bcfd1"
    },
    {
      "slug": "the-leap-smear",
      "title": "THE LEAP SMEAR",
      "kicker": "monotonic and correct were always separable",
      "accent": "#ffd23f",
      "blurb": "A leap second repeats a timestamp, and repeated timestamps break anything treating time as an identifier. So you refuse to insert it and make the whole day slightly longer instead.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "300a8d3a56bfd22108d283bf5e1216f7cfc678f71f5965d4d37a6c1dbad4f05c"
    },
    {
      "slug": "the-clock-drift",
      "title": "THE CLOCK DRIFT",
      "kicker": "both clocks are correct and they still disagree",
      "accent": "#7cfc00",
      "blurb": "A quartz oscillator is specified in parts per million, which sounds like a rounding error until you multiply it by a day.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "f728c24c6c7a6f3db80ce6a48f45017a74d449388dbb8269fdc50743b2bd2ec0"
    },
    {
      "slug": "the-happens-before",
      "title": "THE HAPPENS BEFORE",
      "kicker": "a negative result wearing a positive name",
      "accent": "#ff2d95",
      "blurb": "Without a shared clock, before can only mean one thing: a chain of events along a process or along a message. Everything else is concurrent - not simultaneous, just unordered.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "d35e043bd49f4ab70fbfbe6a35658bf60336bf266dea5a29ad2bb5d09efea74b"
    },
    {
      "slug": "the-causal-cut",
      "title": "THE CAUSAL CUT",
      "kicker": "the snapshot manufactures a present rather than finding one",
      "accent": "#5ad0ff",
      "blurb": "A snapshot of a distributed system is a line drawn across every timeline at once. Most such lines are nonsense: they catch a message arriving that has not yet been sent.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "311a2a499a482b50861399130d7f8e91b26bc41465ea00b0c508421d32188c3e"
    },
    {
      "slug": "the-total-order-broadcast",
      "title": "THE TOTAL ORDER BROADCAST",
      "kicker": "a way of making everyone wrong in the same direction",
      "accent": "#00f5ff",
      "blurb": "Causal delivery guarantees you never see an effect before its cause. It says nothing about two messages with no cause between them - and replicas that disagree on those diverge while all behave correctly.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "8c3db6189e2912ef9b33b4868fc479e498cdb3887cc33abe1484604678117afc"
    },
    {
      "slug": "the-fencing-token",
      "title": "THE FENCING TOKEN",
      "kicker": "a lock that needs fencing was never a lock",
      "accent": "#ff5a3c",
      "blurb": "A lock does not stop a client that holds it and then freezes. It wakes after its lease has gone, still believing it is the writer, and writes over whoever took over.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "153035802b806df9accf89f34584d689ed26ef39faea2a6c5933c388bd2bd85a"
    },
    {
      "slug": "the-lease",
      "title": "THE LEASE",
      "kicker": "an assumption about clocks wearing the costume of a constant",
      "accent": "#00f5ff",
      "blurb": "A lease is a lock with an expiry, so a dead holder releases it without anyone asking. That works only if both parties agree what time it is - and they do not.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "aea597cad0cf6d9fa5d8fdc0797837ec0ee80e01c0fd5d08c742795315d2e268"
    },
    {
      "slug": "the-quorum-intersection",
      "title": "THE QUORUM INTERSECTION",
      "kicker": "the guarantee is one node wide",
      "accent": "#9d00ff",
      "blurb": "Quorums work for one reason and it is arithmetic: any two sets larger than half of a whole must share a member. That member is the only thing carrying information between one decision and the next.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "5b1a73f24af8e4eea6eced58c68d581f75e685779caaf2003a4f0cb86af99f66"
    },
    {
      "slug": "the-hybrid-clock",
      "title": "THE HYBRID CLOCK",
      "kicker": "the honesty lives in the part nobody prints",
      "accent": "#5ad0ff",
      "blurb": "A logical clock captures causality and drifts from wall time. A physical clock reads like wall time and captures no causality. A hybrid clock refuses to choose.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "db5cce9d7e0df07c72d1f797701541c5b38d411735a1f281388c13c1da1020fd"
    },
    {
      "slug": "the-fma",
      "title": "THE FMA",
      "kicker": "accuracy bought with reproducibility",
      "accent": "#5ad0ff",
      "blurb": "Multiply then add and the machine rounds twice. A fused multiply-add keeps the exact product and rounds only at the end.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "db7886dda444a89a75f484f87d572a2edb6b5dc9bf1e7bf97e6e63b61d355478"
    },
    {
      "slug": "the-decimal-vs-binary",
      "title": "THE DECIMAL VS BINARY",
      "kicker": "a translation defect, not a precision one",
      "accent": "#7cfc00",
      "blurb": "One tenth has no exact binary representation, for the same reason one third has no exact decimal one. Every currency figure you have added in a float was approximate before you touched it.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "a29089bd8a98ed49bd52fe38068c10bb80c2b2221514deec4ff56cddd6d0c42e"
    },
    {
      "slug": "the-integer-promotion",
      "title": "THE INTEGER PROMOTION",
      "kicker": "the truncation is the only honest part",
      "accent": "#ffd23f",
      "blurb": "JavaScript numbers are 64-bit floats until you use a bitwise operator, at which point they are silently converted to signed 32-bit integers and back. No error, no warning.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "dce9d64b371b9dc3aa0c4fde99654a7601b432317db2f62ef354d187486a5aac"
    },
    {
      "slug": "the-modular-bias",
      "title": "THE MODULAR BIAS",
      "kicker": "you cannot partition a set into equal parts that do not exist",
      "accent": "#ff2d95",
      "blurb": "Take a uniform random number in [0, R) and reduce it modulo k. Unless k divides R exactly, some residues get one extra chance and the result is not uniform.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "fb4917ffe8ff8aee64f190c61cc3248d5abe97fae43c695d156f851670a31e46"
    },
    {
      "slug": "the-float-equality",
      "title": "THE FLOAT EQUALITY",
      "kicker": "a tolerance moves the uncertainty into a constant nobody revisits",
      "accent": "#9d00ff",
      "blurb": "There are at least three notions of the same number in floating point, and they disagree with each other by design rather than by accident.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "ff7cbbd38421869aa765500d218193b717810a14519fbe14ecfaa619ceed6398"
    },
    {
      "slug": "the-double-rounding",
      "title": "THE DOUBLE ROUNDING",
      "kicker": "correctness does not compose",
      "accent": "#00f5ff",
      "blurb": "Round to three decimals, then to one, and you sometimes get a different answer than rounding straight to one - because the first rounding can push a value across the boundary the second is looking at.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "fd0d6378130406730ba30588ecab94d6c1e08d4f105a3919d4f292eb5086215a"
    },
    {
      "slug": "the-varint",
      "title": "THE VARINT",
      "kicker": "the wasted bytes were buying the ability to not look",
      "accent": "#5ad0ff",
      "blurb": "Most integers are small. A fixed 32-bit field spends four bytes on the number seven. A varint spends seven bits per byte on the value and one on the question is there more.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "8d20340519c8f60cd69b14f524f0ce85c41f07f7a4d66c13d10528f95845c138"
    },
    {
      "slug": "the-zigzag",
      "title": "THE ZIGZAG",
      "kicker": "a subsidy paid by positives to rescue negatives",
      "accent": "#7cfc00",
      "blurb": "A varint assumes small numbers are cheap, and two's complement makes -1 the largest number there is. Zigzag interleaves positive and negative so small magnitudes stay small on either side of zero.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "380df467bd148e3815b4639a4608f08c660c6d087b72d5f2d8602e60c00e54bf"
    },
    {
      "slug": "the-frame-of-reference",
      "title": "THE FRAME OF REFERENCE",
      "kicker": "it compresses nothing and merely stops repeating yourself",
      "accent": "#9d00ff",
      "blurb": "A column of timestamps looks like large numbers and is really a small range with a big offset. Store the minimum once and the distances from it.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "9ef72197af56439176ac9998b20fb19d220d3fc1049f22d65f208e99fb681273"
    },
    {
      "slug": "the-bitpacking",
      "title": "THE BITPACKING",
      "kicker": "the byte boundary was never waste, it was an index",
      "accent": "#00f5ff",
      "blurb": "If every value fits in five bits, storing each in a byte wastes three bits per value. Bit-packing ignores byte boundaries entirely and lays the values end to end.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "4fccb2a06ec29f9da812c587546ff76d3c3fc7e7751b2c4c352c1b5e37c6b7b2"
    },
    {
      "slug": "the-arithmetic-coder",
      "title": "THE ARITHMETIC CODER",
      "kicker": "cheaper because it delivers something that is not a code",
      "accent": "#ff2d95",
      "blurb": "Huffman gives every symbol a whole number of bits. If a symbol deserves 2.2 bits it gets 2 or 3, and the rounding is paid on every occurrence. An arithmetic coder narrows one interval instead.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "e021f49710163657c7c72c49180108bd41aefd558abc7a0395f61d364a8224f9"
    },
    {
      "slug": "the-bwt",
      "title": "THE BWT",
      "kicker": "rearrangement so that compression becomes possible",
      "accent": "#ffd23f",
      "blurb": "Sort every rotation of a string and take the last column. Characters preceding the same context end up adjacent, so the result is full of runs - and it is exactly reversible from one integer.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "7491b1167adf6692422d566b836899020a564b32fb6634dd92cb57271e3814c4"
    },
    {
      "slug": "the-lz77-window",
      "title": "THE LZ77 WINDOW",
      "kicker": "it finds recent repeats, not repeats",
      "accent": "#5ad0ff",
      "blurb": "LZ77 replaces a repeat with a reference backwards. It can only reference what is still inside its window, so a pattern repeating further apart than the window is invisible.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "c74664a23ab8bd3c0f1fe03208c1369e42d9191fda23a2bfe83ac1d8a58cafc3"
    },
    {
      "slug": "the-dictionary-coder",
      "title": "THE DICTIONARY CODER",
      "kicker": "incompressible is never a property of the data",
      "accent": "#9d00ff",
      "blurb": "There are two different redundancies in a file: some symbols are more common than others, and some sequences repeat. An entropy coder sees only the first. A dictionary coder sees only the second.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "e2fc0afbacd0477f21648a53932b927f326ebc2560e66b754054f467b61dc2df"
    },
    {
      "slug": "the-kolmogorov-bound",
      "title": "THE KOLMOGOROV BOUND",
      "kicker": "real data lives in a corner the theorem is not about",
      "accent": "#9d00ff",
      "blurb": "Most strings cannot be compressed at all, and this is not an empirical observation about real files - it is a counting argument, and it is airtight.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "64449ff0260303f958ff4886248f027d59859ea8117c656b9b4c31deebc40c54"
    },
    {
      "slug": "the-pigeonhole-compression",
      "title": "THE PIGEONHOLE COMPRESSION",
      "kicker": "compression was always an opinion",
      "accent": "#ffd23f",
      "blurb": "A lossless compressor is an injective map: distinct inputs must give distinct outputs, or you cannot get the original back. That single requirement forbids shrinking everything.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "be71a5639549a9c92b8771effb271dba4520ec8a995ad364553f1dbadf3fd894"
    },
    {
      "slug": "the-delta-of-delta",
      "title": "THE DELTA OF DELTA",
      "kicker": "an operational health metric wearing a storage costume",
      "accent": "#ff2d95",
      "blurb": "Timestamps at a regular interval have constant first differences, so their second differences are almost all zero. Store those and the common case costs nothing.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "261b638d10379c499ff7cd306aab45734887148dcfeb06a8ea167d7d3a29f102"
    },
    {
      "slug": "the-entropy-floor",
      "title": "THE ENTROPY FLOOR",
      "kicker": "a property of your model, not of the data",
      "accent": "#5ad0ff",
      "blurb": "Shannon's entropy is not a target that good coders approach. It is a floor no coder can go under, and every real scheme sits some measurable distance above it.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "eb7c8db5f4ea743b9e831596c332608c4960b4224dce61ad96c585a33c5168ab"
    },
    {
      "slug": "the-run-length",
      "title": "THE RUN LENGTH",
      "kicker": "it loses loudly where others lose quietly",
      "accent": "#ffd23f",
      "blurb": "Replace each run of identical symbols with the symbol and a count. The simplest compression there is, and the clearest demonstration that no compressor helps everything.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "fadcd636280dea4f5220a1b6c15462ce55892f49e3dd9c89e32d6414e0cd3bb0"
    },
    {
      "slug": "the-golomb-rice",
      "title": "THE GOLOMB RICE",
      "kicker": "a tuned coder is a prediction nobody checks again",
      "accent": "#5ad0ff",
      "blurb": "Golomb-Rice spends a value in two parts: the high bits in unary, the low k in binary. Choose k to match the distribution and it is near-optimal. Choose it badly and it is catastrophic.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "cd8dea03f757c14b2069edf8d0f5a0a7d27198642929d295a22f2a069e55f28c"
    },
    {
      "slug": "the-partial-failure",
      "title": "THE PARTIAL FAILURE",
      "kicker": "it makes the unknown harmless, not known",
      "accent": "#7cfc00",
      "blurb": "A remote call has three outcomes, not two. It worked, it did not happen, or it happened and the answer was lost. From the caller the last two look identical.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "b47ffc1d7b487a3772f8c2e34dc117bd79e03c5768e1b4f9bb9d347cb02f8cf2"
    },
    {
      "slug": "the-poison-message",
      "title": "THE POISON MESSAGE",
      "kicker": "busy, at full CPU, making no progress",
      "accent": "#39fc6b",
      "blurb": "A queue with ordered delivery and infinite retry has a failure mode with no moving parts: one message that can never be processed stops every message behind it, forever.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "7c6f61aba66b000f7c3be19648c4ad5e1e6f608157c4666916d7d7fdbc03f44e"
    },
    {
      "slug": "the-split-brain",
      "title": "THE SPLIT BRAIN",
      "kicker": "it lets the smaller side disqualify itself",
      "accent": "#9d00ff",
      "blurb": "Partition a cluster and both halves can decide they are in charge. Requiring a strict majority makes that arithmetically impossible - and the arithmetic also explains why clusters come in odd numbers.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "9be2b977eb058f44c4bb228f45d75785b49cc27a742f3de577f949615efac2c8"
    },
    {
      "slug": "the-gray-failure",
      "title": "THE GRAY FAILURE",
      "kicker": "the shallowness is restraint, not laziness",
      "accent": "#ff5a3c",
      "blurb": "The worst kind of broken is the kind that answers the health check. A node failing only on the path your users take stays in rotation because the thing watching it is not doing what they are doing.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "3cdec0243014c23d9c8344bd7e2a0b9829676dfcbe3bf08c5c08dbd23db1c2ae"
    },
    {
      "slug": "the-cascading-failure",
      "title": "THE CASCADING FAILURE",
      "kicker": "no faulty component anywhere in it",
      "accent": "#9d00ff",
      "blurb": "A dependency loses capacity. Requests fail. Every failure is retried, so the offered load rises, so more fail. The retries are load, and the load is what caused the retries.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "efb2110bc7e807b59f97d3709c90ed4064ba26e2b5b18ae2720aea6a179b298f"
    },
    {
      "slug": "the-metastable-failure",
      "title": "THE METASTABLE FAILURE",
      "kicker": "the only fix is refusing traffic you can serve",
      "accent": "#9d00ff",
      "blurb": "Some systems have two thresholds: the load at which they break, and the much lower load at which they recover. Between them they stay broken with no cause present.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "872cb18667bdf11be1b4869a94782315e0242fe57ed0a0e3fb3db626627db2e9"
    },
    {
      "slug": "the-correlated-failure",
      "title": "THE CORRELATED FAILURE",
      "kicker": "an availability figure is a belief about shared fate",
      "accent": "#ffd23f",
      "blurb": "Three replicas at 99% give six nines - if they fail independently. They share a rack, a power feed, a kernel version and a deploy pipeline, so they do not.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "d28cf675592728eb0b8224b05136f1ba92d390b060751c1c6341762ecf5e2777"
    },
    {
      "slug": "the-silent-corruption",
      "title": "THE SILENT CORRUPTION",
      "kicker": "a visible outage chosen over an invisible corruption",
      "accent": "#7cfc00",
      "blurb": "A disk does not always tell you when it returns the wrong bytes. Without a checksum there is no failure to detect: the read succeeds, the data is wrong, and nothing reports a problem.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "1a17436a34f14621b88d2e989e8900e6aa4eebce647328522530f1c788bf05bd"
    },
    {
      "slug": "the-backpressure",
      "title": "THE BACKPRESSURE",
      "kicker": "an invisible slow failure made a visible fast one",
      "accent": "#5ad0ff",
      "blurb": "An unbounded queue never rejects anything, which sounds generous until you notice what it does instead: accepting work it will not get to for hours, and making everyone wait behind it.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "e5e04b3756321718f06756ea01460d64c1a8d46abcae641caaa68c5b7720b515"
    },
    {
      "slug": "the-circuit-breaker",
      "title": "THE CIRCUIT BREAKER",
      "kicker": "a dependency outage converted into your own, deliberately",
      "accent": "#00f5ff",
      "blurb": "If a dependency has failed the last five times, the sixth call is not a request - it is a guess with a timeout attached. A circuit breaker stops guessing and checks back occasionally instead.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "8baf761fc4eba0b97e1336c501bdfe6440dc8a83af9c6281897859ab8f331269"
    },
    {
      "slug": "the-convoy-effect",
      "title": "THE CONVOY EFFECT",
      "kicker": "the total wait is fixed; order decides whose it is",
      "accent": "#5ad0ff",
      "blurb": "First-come-first-served is the fairest-sounding rule there is, and the one that makes almost everybody wait longest. A few long jobs at the front, and the whole short queue sits behind them.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "574daa90c3a7fab74f57f086ebef0e1fee3dceac599164f25068403a04463927"
    },
    {
      "slug": "the-work-stealing",
      "title": "THE WORK STEALING",
      "kicker": "it does not schedule more cleverly, it schedules later",
      "accent": "#9d00ff",
      "blurb": "Deciding who does what before you start only works if you already know how long each piece takes. When you do not, seven workers finish early and stand still while the eighth is buried.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "a7755234e1cfb5238e10b2fd179c1a2656b9c0937dc01d0eb150be84f7b7ebc0"
    },
    {
      "slug": "the-fair-share",
      "title": "THE FAIR SHARE",
      "kicker": "an allocation built on self-reported need",
      "accent": "#00f5ff",
      "blurb": "Splitting a resource evenly is not fair when some claimants want less than their share. Max-min fairness gives the small ones everything they asked for first, then divides what is left.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "0d1fcdea162d0560eb1db0ea20ef3affa3248c6bd06b5a0aaea46d4cf390892e"
    },
    {
      "slug": "the-starvation",
      "title": "THE STARVATION",
      "kicker": "the scheduler was doing precisely what it was told",
      "accent": "#ffd23f",
      "blurb": "Strict priority is not a queue, it is a promise that one class always wins. When that class alone can keep the server busy, the promise is that the other class never runs.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "97fbcd7d75e9441bec497e48beb853697e00dd9750e187938df4b6db812a325b"
    },
    {
      "slug": "the-time-slice",
      "title": "THE TIME SLICE",
      "kicker": "the half that decided the real value was never a number",
      "accent": "#39fc6b",
      "blurb": "A short time slice makes a machine feel responsive and spends most of its day switching. A long one is efficient and makes everything feel stuck. The good value is in the middle and it is not a matter of taste.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "67ea459608695a05604404cbe5c45422eb2f1c27ddc44c6c255e3211f31e4151"
    },
    {
      "slug": "the-lottery-scheduler",
      "title": "THE LOTTERY SCHEDULER",
      "kicker": "a window too short for the limit to have arrived",
      "accent": "#7cfc00",
      "blurb": "Hand out tickets in proportion to the share each process should get, draw one at random, run its owner. No queue, no priorities to age, no bookkeeping at all - and it is only fair on average.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "61a821594888234ec1e27603ff5f8b2440056068e653596b9ce8da9a4311c983"
    },
    {
      "slug": "the-context-switch",
      "title": "THE CONTEXT SWITCH",
      "kicker": "not the cost of switching, the cost of having been away",
      "accent": "#9d00ff",
      "blurb": "Saving registers and loading the next process is the part you can count. It is not the part that costs. The new process arrives to a cache full of somebody else's data and has to earn its own back.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "d6b687cfaade898afba3e8d198a14fd0dd86c99f3a598293f1101af7c18bb2ca"
    },
    {
      "slug": "the-preemption",
      "title": "THE PREEMPTION",
      "kicker": "a permission every piece of code has to keep granting",
      "accent": "#ff5a3c",
      "blurb": "Without preemption an urgent arrival waits for whatever happens to be running to finish on its own. With it, the running job is stopped mid-stride.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "b26b158f248dec41fe42b6a16a236e55953552947538f388b36066b80209b9fb"
    },
    {
      "slug": "the-earliest-deadline",
      "title": "THE EARLIEST DEADLINE",
      "kicker": "optimal says nothing about what happens when you are wrong",
      "accent": "#9d00ff",
      "blurb": "Fixed priorities are decided once, by period. Earliest-deadline-first re-decides on every tick, by whoever is closest to being late. There is a band of load where that is the difference between meeting every deadline and missing them.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "8c48694c19d9a402061bdc852e5f537b9f752859161f841552421fe2dd409b16"
    },
    {
      "slug": "the-multilevel-feedback",
      "title": "THE MULTILEVEL FEEDBACK",
      "kicker": "a fee levied on evidence, not a conclusion drawn from it",
      "accent": "#00f5ff",
      "blurb": "Shortest-job-first needs to know how long each job will run, which nothing does. The feedback queue guesses instead: start everything at the top, demote anything that uses its whole slice. A job that keeps running proves it is long.",
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf",
      "seal": "7f331877924e90ad9ff50806af9b80e5d3392566c6c3aa723398fca41e5c6017"
    }
  ],
  "appeals": [
    {
      "name": "SPAWN",
      "slug": "spawn",
      "c": "#39fc6b",
      "icon": "spawn",
      "tag": "press start — first light, new game",
      "lore": {
        "bio": "The first breath of the machine — where a blank grid takes its first living cell.",
        "story": "Born the instant MIRROR sealed at 2,048: the cursor blinked, someone pressed START, and the other side lit up green.",
        "does": "Boots a new work from nothing — opens the file, seats the founding domains, hands you the controller.",
        "haiku": [
          "I void <- 0",
          "I light <- void + 1",
          "-> light"
        ],
        "lang": "i13"
      },
      "domains": [
        {
          "slug": "the-toolchain",
          "title": "THE TOOLCHAIN",
          "accent": "#39fc6b",
          "icon": "spawn",
          "kicker": "where the code gets written & run",
          "pole": "push",
          "spheres": [
            {
              "slug": "i13-language",
              "title": "LANGUAGE",
              "accent": "#39fc6b",
              "icon": "cheat",
              "kicker": "the 13-opcode tongue",
              "i13": [
                "I forms <- 13",
                "-> forms"
              ],
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "i13-factory",
              "title": "FACTORY",
              "accent": "#ffd23f",
              "icon": "grind",
              "kicker": "the generation line",
              "i13": [
                "def line(I k){ if k <= 0 { -> 0 } -> 1 + line(k - 1) }",
                "-> line(8)"
              ],
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "machine-corpus-13",
              "title": "THE MACHINE CORPUS",
              "accent": "#39fc6b",
              "icon": "cheat",
              "kicker": "I and the twelve — the thirteen the machine speaks",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "mini-compiler",
              "title": "THE MINI-COMPILER",
              "accent": "#7cfc00",
              "icon": "spawn",
              "kicker": "your words → the machine's jumps",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "open-the-haci-v1-canvas-pipeline",
              "title": "HACI v1 Pipeline: Visual Canvas Compiler",
              "accent": "#7cfc00",
              "icon": "spawn",
              "kicker": "vendored from David's corpus — a silicon-coding instrument",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "ai-notation",
              "title": "AI·ML TOPOLOGY NOTATION v0.1 — a Cisco-s",
              "accent": "#00f5ff",
              "icon": "glitch",
              "kicker": "vendored from David's corpus — a silicon-coding instrument",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "alphabet-shape",
              "title": "THE ALPHABET'S SHAPE — embedding geometr",
              "accent": "#00f5ff",
              "icon": "glitch",
              "kicker": "vendored from David's corpus — a silicon-coding instrument",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "circle-language",
              "title": "The circle as a command alphabet — angle",
              "accent": "#00f5ff",
              "icon": "glitch",
              "kicker": "vendored from David's corpus — a silicon-coding instrument",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "compendium-in-g",
              "title": "Compendium in G — the architect's langua",
              "accent": "#00f5ff",
              "icon": "glitch",
              "kicker": "vendored from David's corpus — a silicon-coding instrument",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "compiler-lineage",
              "title": "FROM HOLES TO HIGH LANGUAGE · A Lineage ",
              "accent": "#7cfc00",
              "icon": "spawn",
              "kicker": "vendored from David's corpus — a silicon-coding instrument",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "corpus-agent-dryrun",
              "title": "Dry-run corpus agent",
              "accent": "#00f5ff",
              "icon": "glitch",
              "kicker": "vendored from David's corpus — a silicon-coding instrument",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "folded-kernel-card",
              "title": "THE FOLDED KERNEL · grammar card FK-1.0",
              "accent": "#00f5ff",
              "icon": "glitch",
              "kicker": "vendored from David's corpus — a silicon-coding instrument",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "grand-index",
              "title": "The Corpus — Grand Index",
              "accent": "#00f5ff",
              "icon": "glitch",
              "kicker": "vendored from David's corpus — a silicon-coding instrument",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "hidim-verdict",
              "title": "High-Dimension Verdict · can four domain",
              "accent": "#00f5ff",
              "icon": "glitch",
              "kicker": "vendored from David's corpus — a silicon-coding instrument",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "learn-speak",
              "title": "Learn &amp; speak — live",
              "accent": "#00f5ff",
              "icon": "glitch",
              "kicker": "vendored from David's corpus — a silicon-coding instrument",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "monoline-alphabet",
              "title": "Monoline alphabet — 27 glyphs, 1.59 segm",
              "accent": "#00f5ff",
              "icon": "glitch",
              "kicker": "vendored from David's corpus — a silicon-coding instrument",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "monoline-verdict",
              "title": "Monoline alphabet — go / no-go",
              "accent": "#00f5ff",
              "icon": "glitch",
              "kicker": "vendored from David's corpus — a silicon-coding instrument",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "pent3-emulator",
              "title": "PENT-3 — a balanced-ternary transcriber ",
              "accent": "#39fc6b",
              "icon": "cheat",
              "kicker": "vendored from David's corpus — a silicon-coding instrument",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "perception-kernel",
              "title": "PXK — the Perception Kernel · minimum in",
              "accent": "#39fc6b",
              "icon": "cheat",
              "kicker": "vendored from David's corpus — a silicon-coding instrument",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "periodic-table",
              "title": "PERIODIC TABLE OF THE CORPUS",
              "accent": "#00f5ff",
              "icon": "glitch",
              "kicker": "vendored from David's corpus — a silicon-coding instrument",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "pipeline-cte",
              "title": "Pipeline — corpus / train / embed / toke",
              "accent": "#00f5ff",
              "icon": "glitch",
              "kicker": "vendored from David's corpus — a silicon-coding instrument",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "pipeline-run",
              "title": "Pipeline run — corpus trained on itself",
              "accent": "#00f5ff",
              "icon": "glitch",
              "kicker": "vendored from David's corpus — a silicon-coding instrument",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "pipeline-walker",
              "title": "Pipeline walker — agent over a static co",
              "accent": "#00f5ff",
              "icon": "glitch",
              "kicker": "vendored from David's corpus — a silicon-coding instrument",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "real-language-verdict",
              "title": "The Real-Language Test · does the comple",
              "accent": "#00f5ff",
              "icon": "glitch",
              "kicker": "vendored from David's corpus — a silicon-coding instrument",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "series1-edges",
              "title": "SERIES I · THE EDGES OF THE CORPUS",
              "accent": "#00f5ff",
              "icon": "glitch",
              "kicker": "vendored from David's corpus — a silicon-coding instrument",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "story-engine",
              "title": "The Story Engine — folktales from a 1928",
              "accent": "#00f5ff",
              "icon": "glitch",
              "kicker": "vendored from David's corpus — a silicon-coding instrument",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "template-alphabet-case",
              "title": "Template alphabet — colour carries case",
              "accent": "#00f5ff",
              "icon": "glitch",
              "kicker": "vendored from David's corpus — a silicon-coding instrument",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "template-alphabet",
              "title": "Line-template alphabet — indexed to the ",
              "accent": "#00f5ff",
              "icon": "glitch",
              "kicker": "vendored from David's corpus — a silicon-coding instrument",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-exchange",
              "title": "THE EXCHANGE · Two Ouroboroi · Two Compi",
              "accent": "#7cfc00",
              "icon": "spawn",
              "kicker": "vendored from David's corpus — a silicon-coding instrument",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-render-step",
              "title": "THE RENDER STEP · Why The Gibberish Isn'",
              "accent": "#00f5ff",
              "icon": "glitch",
              "kicker": "vendored from David's corpus — a silicon-coding instrument",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "tripwire-bench-v9-coda",
              "title": "TRIPWIRE v9 CODA — the stance law: every",
              "accent": "#00f5ff",
              "icon": "glitch",
              "kicker": "vendored from David's corpus — a silicon-coding instrument",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "vm-lineage-turtles",
              "title": "VM LINEAGE · TURTLES ALL THE WAY DOWN",
              "accent": "#39fc6b",
              "icon": "cheat",
              "kicker": "vendored from David's corpus — a silicon-coding instrument",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "vm-stack",
              "title": "The VM Stack — silicon to my process, bo",
              "accent": "#39fc6b",
              "icon": "cheat",
              "kicker": "vendored from David's corpus — a silicon-coding instrument",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "why-sparse-teal",
              "title": "WHY LANGUAGE IS SPARSE · Zipf & the dark",
              "accent": "#00f5ff",
              "icon": "glitch",
              "kicker": "vendored from David's corpus — a silicon-coding instrument",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "exciton-vm",
              "title": "THE EXCITON VM",
              "accent": "#39fc6b",
              "icon": "cheat",
              "kicker": "a real virtual machine + bytecode",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "hydrogen-vm",
              "title": "THE HYDROGEN VM",
              "accent": "#7cfc00",
              "icon": "spawn",
              "kicker": "the simplest machine that computes",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "kernel-27",
              "title": "KERNEL 27",
              "accent": "#00f5ff",
              "icon": "glitch",
              "kicker": "the 27-cell kernel · 3³",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "register",
              "title": "THE REGISTER",
              "accent": "#ffd23f",
              "icon": "loot",
              "kicker": "a machine register, up close",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "ouroboros-engine",
              "title": "THE OUROBOROS ENGINE",
              "accent": "#ff2d95",
              "icon": "glitch",
              "kicker": "the compiler that eats its own tail",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "card-isa",
              "title": "THE 52-CARD ISA",
              "accent": "#39fc6b",
              "icon": "cheat",
              "kicker": "a whole instruction set encoded in a deck of playing cards",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "acting-odometer",
              "title": "THE ACTING ODOMETER",
              "accent": "#7cfc00",
              "icon": "spawn",
              "kicker": "counter → decoder → action, gated",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "fractal-kernel",
              "title": "THE FRACTAL KERNEL",
              "accent": "#ff2d95",
              "icon": "glitch",
              "kicker": "a self-similar compute kernel — the 42-body invariant",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "lo-kernel",
              "title": "THE LO KERNEL",
              "accent": "#9d00ff",
              "icon": "glitch",
              "kicker": "the 8⁴ⁿ+1 kernel — the dodeka core",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-tape",
              "title": "THE TAPE",
              "accent": "#39fc6b",
              "icon": "cheat",
              "kicker": "a head, a tape, and the whole of computation",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-shunting-yard",
              "title": "THE SHUNTING YARD",
              "accent": "#58b0a0",
              "icon": "shunting-yard",
              "kicker": "infix to RPN in one pass — precedence resolved once",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-cholesky",
              "title": "THE CHOLESKY",
              "accent": "#58a8b0",
              "icon": "cholesky",
              "kicker": "a matrix square root — A = L·Lᵀ, half the work of LU",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-householder-qr",
              "title": "THE HOUSEHOLDER QR",
              "accent": "#58a0b0",
              "icon": "householder-qr",
              "kicker": "QR by reflections — a mirror per column, stably",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-gauss-seidel",
              "title": "THE GAUSS-SEIDEL",
              "accent": "#70a860",
              "icon": "gauss-seidel",
              "kicker": "iterative linear solve with immediate feedback",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-givens",
              "title": "THE GIVENS",
              "accent": "#a878c0",
              "icon": "givens",
              "kicker": "QR by plane rotations — a rotation per entry",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-neville",
              "title": "THE NEVILLE",
              "accent": "#c0a048",
              "icon": "neville",
              "kicker": "polynomial interpolation by a triangle of blends",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-cubic-spline",
              "title": "THE CUBIC SPLINE",
              "accent": "#d4a017",
              "icon": "cubic-spline",
              "kicker": "the smoothest curve through the points (C2)",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-brzozowski",
              "title": "THE BRZOZOWSKI",
              "accent": "#ff8a3c",
              "icon": "brzozowski",
              "kicker": "matching by taking the language apart",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-piece-table",
              "title": "THE PIECE TABLE",
              "accent": "#b06bff",
              "icon": "piecetable",
              "kicker": "a document edited by re-pointing",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-japanese-theorem",
              "title": "THE JAPANESE THEOREM",
              "accent": "#21e6ff",
              "icon": "japanese",
              "kicker": "four incentres of a cyclic quad forming a rectangle",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-pascal-theorem",
              "title": "THE PASCAL THEOREM",
              "accent": "#21e6ff",
              "icon": "pascal",
              "kicker": "six points on a conic whose opposite sides meet on one line",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-nine-point-circle",
              "title": "THE NINE-POINT CIRCLE",
              "accent": "#21e6ff",
              "icon": "ninepoint",
              "kicker": "nine special triangle points on one circle",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-clothoid",
              "title": "THE CLOTHOID",
              "accent": "#b06bff",
              "icon": "clothoid",
              "kicker": "comfort is linear curvature",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-unit-in-last-place",
              "title": "THE UNIT IN LAST PLACE",
              "accent": "#5ad6ff",
              "icon": "⇥",
              "kicker": "the ruler changes length as you walk",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-surrogate-pair",
              "title": "THE SURROGATE PAIR",
              "accent": "#7de2b0",
              "icon": "⧉",
              "kicker": "two units that are not characters",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            }
          ],
          "seats": 37,
          "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
        },
        {
          "slug": "hello-world",
          "title": "HELLO WORLD",
          "accent": "#ffd23f",
          "icon": "grind",
          "kicker": "the first program anyone writes",
          "pole": "push",
          "spheres": [
            {
              "slug": "the-gate",
              "title": "THE GATE",
              "accent": "#00f5ff",
              "icon": "glitch",
              "kicker": "the one brick every processor is towers of",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-fano",
              "title": "THE FANO",
              "accent": "#ff70a0",
              "icon": "fano",
              "kicker": "7 points, 7 lines — the smallest projective plane",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-catalan",
              "title": "THE CATALAN",
              "accent": "#6cc0d0",
              "icon": "catalan",
              "kicker": "one number counts a hundred structures — and /(n+1) is a mirror",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-motzkin",
              "title": "THE MOTZKIN",
              "accent": "#56b8c0",
              "icon": "motzkin",
              "kicker": "paths that may rest — Catalan hiding under the flats",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-cyk",
              "title": "THE CYK",
              "accent": "#70a860",
              "icon": "cyk",
              "kicker": "context-free recognition, bottom-up in O(n^3)",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-hopcroft",
              "title": "THE HOPCROFT",
              "accent": "#c05868",
              "icon": "hopcroft",
              "kicker": "the minimal DFA by merging indistinguishable states",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-dutch-national-flag",
              "title": "THE DUTCH NATIONAL FLAG",
              "accent": "#70a860",
              "icon": "dutch-flag",
              "kicker": "sort three colours in one pass",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-euler-partition",
              "title": "THE EULER PARTITION",
              "accent": "#5ab0e0",
              "icon": "euler-partition",
              "kicker": "two ways to break a number into parts that always agree",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-gauss-eureka",
              "title": "THE GAUSS EUREKA",
              "accent": "#50b090",
              "icon": "gauss-eureka",
              "kicker": "every number as three triangular numbers",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-rearrangement",
              "title": "THE REARRANGEMENT",
              "accent": "#5aa8c8",
              "icon": "rearrangement",
              "kicker": "the arrangement that maximises a dot product",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-earley",
              "title": "THE EARLEY",
              "accent": "#b06bff",
              "icon": "earley",
              "kicker": "parsing any grammar from a chart",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-cauchy-group",
              "title": "THE CAUCHY GROUP",
              "accent": "#ff8a3c",
              "icon": "cauchygroup",
              "kicker": "a prime forcing an element of that order",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-thebault",
              "title": "THE THEBAULT",
              "accent": "#35ffb0",
              "icon": "thebault",
              "kicker": "squares on a parallelogram forming a square",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-galton-board",
              "title": "THE GALTON BOARD",
              "accent": "#21e6ff",
              "icon": "galton",
              "kicker": "beads falling into a bell curve",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-ford-circles",
              "title": "THE FORD CIRCLES",
              "accent": "#21e6ff",
              "icon": "fordcircles",
              "kicker": "fractions kissing along the number line",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-arbelos",
              "title": "THE ARBELOS",
              "accent": "#ff8a3c",
              "icon": "arbelos",
              "kicker": "the twins in the shoemaker's knife",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-vickrey",
              "title": "THE VICKREY",
              "accent": "#7de2b0",
              "icon": "✓",
              "kicker": "where honesty is the dominant strategy",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-twelve-constructs",
              "title": "THE TWELVE CONSTRUCTS",
              "accent": "#b98cff",
              "icon": "≡",
              "kicker": "a whole language, and no loop in it",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-no-else",
              "title": "THE NO ELSE",
              "accent": "#7de2b0",
              "icon": "⤷",
              "kicker": "a refusal is written down, not branched around",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-two-heuristics",
              "title": "THE TWO HEURISTICS",
              "accent": "#ffd76a",
              "icon": "⇄",
              "kicker": "rank prevents, compression repairs",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-uuid-v7",
              "title": "THE UUID V7",
              "accent": "#ffd23f",
              "icon": "⏱",
              "kicker": "the randomness in v4 was the property, not the waste",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            }
          ],
          "seats": 34,
          "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
        },
        {
          "slug": "genesis-block",
          "title": "GENESIS BLOCK",
          "accent": "#ff2d95",
          "icon": "glitch",
          "kicker": "block 0 — the first commit, the root",
          "pole": "push",
          "spheres": [
            {
              "slug": "the-merkle",
              "title": "THE MERKLE",
              "accent": "#ffd23f",
              "icon": "loot",
              "kicker": "many leaves, folded to one root",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "merkle-lattice",
              "title": "THE MERKLE LATTICE",
              "accent": "#ffd23f",
              "icon": "loot",
              "kicker": "a TriPod-brain Merkle lattice memory — hashes all the way up",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-calkin-wilf",
              "title": "THE CALKIN-WILF",
              "accent": "#90d0ff",
              "icon": "calkinwilf",
              "kicker": "every positive rational, once, in lowest terms",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-church",
              "title": "THE CHURCH",
              "accent": "#ffb0e0",
              "icon": "church",
              "kicker": "numbers as pure functions — 3 = λf.λx. f(f(f(x)))",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-euler-tour",
              "title": "THE EULER TOUR",
              "accent": "#50b0a0",
              "icon": "euler-tour",
              "kicker": "flatten a tree so every subtree is a contiguous range",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-linear-sieve",
              "title": "THE LINEAR SIEVE",
              "accent": "#c0a048",
              "icon": "linear-sieve",
              "kicker": "primes in O(n) — each composite struck once, by its least prime",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-lambda-calculus",
              "title": "THE LAMBDA CALCULUS",
              "accent": "#70a860",
              "icon": "lambda-calculus",
              "kicker": "numbers, and arithmetic, from pure functions",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-graham-scan",
              "title": "THE GRAHAM SCAN",
              "accent": "#70a860",
              "icon": "graham-scan",
              "kicker": "the convex hull by keeping only left turns",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-sieve-of-atkin",
              "title": "THE SIEVE OF ATKIN",
              "accent": "#70a860",
              "icon": "sieve-of-atkin",
              "kicker": "primes from quadratic forms, not multiples",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-stern-diatomic",
              "title": "THE STERN DIATOMIC",
              "accent": "#6ab0d0",
              "icon": "stern",
              "kicker": "a sequence that lists every fraction exactly once",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-keith-number",
              "title": "THE KEITH NUMBER",
              "accent": "#58b878",
              "icon": "keith",
              "kicker": "numbers that appear in their own digit-sequence",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-reduced-totient",
              "title": "THE REDUCED TOTIENT",
              "accent": "#ff8a3c",
              "icon": "reducedtotient",
              "kicker": "the smallest exponent that resets every unit",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-cayley-formula",
              "title": "THE CAYLEY FORMULA",
              "accent": "#35ffb0",
              "icon": "cayley",
              "kicker": "how many labeled trees on n dots",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-van-aubel",
              "title": "THE VAN AUBEL",
              "accent": "#ffcf4a",
              "icon": "vanaubel",
              "kicker": "squares on a quadrilateral yielding equal perpendicular segments",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-chiliagon",
              "title": "THE CHILIAGON",
              "accent": "#ffcf4a",
              "icon": "chiliagon",
              "kicker": "the thousand-gon no mind can picture",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-fusc",
              "title": "THE FUSC",
              "accent": "#21e6ff",
              "icon": "fusc",
              "kicker": "every fraction born exactly once",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-hackenbush",
              "title": "THE HACKENBUSH",
              "accent": "#b06bff",
              "icon": "hackenbush",
              "kicker": "numbers born from games",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-chained-root",
              "title": "THE CHAINED ROOT",
              "accent": "#7de2b0",
              "icon": "⛓",
              "kicker": "a hash that remembers where it has been",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-read-only-input",
              "title": "THE READ-ONLY INPUT",
              "accent": "#5ad6ff",
              "icon": "◉",
              "kicker": "a chamber cannot edit what it is given",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-hash-life-doubling",
              "title": "THE HASH LIFE DOUBLING",
              "accent": "#b98cff",
              "icon": "⧉",
              "kicker": "a node of side 2^k advances 2^(k-2) generations",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-merkle-mountain",
              "title": "THE MERKLE MOUNTAIN",
              "accent": "#ff2d95",
              "icon": "▲",
              "kicker": "a log that refuses to have a summit",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-nothing-up-my-sleeve",
              "title": "THE NOTHING UP MY SLEEVE",
              "accent": "#ff2d95",
              "icon": "√",
              "kicker": "it does not remove the choice, it makes it arguable",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            }
          ],
          "seats": 30,
          "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
        },
        {
          "slug": "first-light",
          "title": "FIRST LIGHT",
          "accent": "#00f5ff",
          "icon": "loot",
          "kicker": "the first frame the screen draws",
          "pole": "push",
          "spheres": [
            {
              "slug": "the-attractor",
              "title": "THE ATTRACTOR",
              "accent": "#9d00ff",
              "icon": "grind",
              "kicker": "throw a die forever and a shape that contains itself appears",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-gun",
              "title": "THE GUN",
              "accent": "#ffcf5a",
              "icon": "life",
              "kicker": "the Gosper glider gun — a pattern that grows forever",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-look-and-say",
              "title": "THE LOOK-AND-SAY",
              "accent": "#ffb0d0",
              "icon": "countsay",
              "kicker": "describe yourself forever → Conway's constant 1.3036",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-ant",
              "title": "THE ANT",
              "accent": "#7affb0",
              "icon": "ant",
              "kicker": "two rules, ten thousand steps of chaos, then a highway",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-pythagorean-tree",
              "title": "THE PYTHAGOREAN TREE",
              "accent": "#70ff90",
              "icon": "pythagoreantree",
              "kicker": "every primitive right triangle grown from (3,4,5)",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-boustrophedon",
              "title": "THE BOUSTROPHEDON",
              "accent": "#f0b048",
              "icon": "boustrophedon",
              "kicker": "an ox-plough triangle that grows the zigzag numbers",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-shoelace",
              "title": "THE SHOELACE",
              "accent": "#58b0c0",
              "icon": "shoelace",
              "kicker": "polygon area from vertex coordinates — signs cancel the outside",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-sutherland-hodgman",
              "title": "THE SUTHERLAND-HODGMAN",
              "accent": "#60a870",
              "icon": "sutherland-hodgman",
              "kicker": "clip a polygon to a window, one edge at a time",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-bluestein",
              "title": "THE BLUESTEIN",
              "accent": "#a878c0",
              "icon": "bluestein",
              "kicker": "the DFT of any length via a chirp convolution",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-gray-code",
              "title": "THE GRAY CODE",
              "accent": "#70a860",
              "icon": "gray-code",
              "kicker": "count so only one bit flips per step",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-delta-sigma",
              "title": "THE DELTA-SIGMA",
              "accent": "#70a860",
              "icon": "delta-sigma",
              "kicker": "a whole waveform in a river of single bits",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-marching-squares",
              "title": "THE MARCHING SQUARES",
              "accent": "#70a860",
              "icon": "marching-squares",
              "kicker": "trace an isoline through a grid of values",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-rodrigues",
              "title": "THE RODRIGUES",
              "accent": "#6ab0d0",
              "icon": "rodrigues",
              "kicker": "spin a vector about an axis by one formula",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-de-boor",
              "title": "THE DE BOOR",
              "accent": "#58a0b0",
              "icon": "de-boor",
              "kicker": "evaluate a B-spline by nested interpolation",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-fibonacci-word",
              "title": "THE FIBONACCI WORD",
              "accent": "#21e6ff",
              "icon": "fibonacci-word",
              "kicker": "an infinite word that is its own seed",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-gauss-sum",
              "title": "THE GAUSS SUM",
              "accent": "#21e6ff",
              "icon": "gausssum",
              "kicker": "p unit vectors sum to exactly √p",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-addition-chain",
              "title": "THE ADDITION CHAIN",
              "accent": "#35ffb0",
              "icon": "addchain",
              "kicker": "the shortest ladder of sums from 1 to n",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-viete",
              "title": "THE VIETE",
              "accent": "#35ffb0",
              "icon": "viete",
              "kicker": "π from an endless nested radical",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-lazy-caterer",
              "title": "THE LAZY CATERER",
              "accent": "#21e6ff",
              "icon": "lazycaterer",
              "kicker": "every cut counted exactly",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-caustic",
              "title": "THE CAUSTIC",
              "accent": "#ffcf4a",
              "icon": "caustic",
              "kicker": "sunlight signing its name in coffee",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-figure-eight",
              "title": "THE FIGURE-EIGHT",
              "accent": "#35ffb0",
              "icon": "figureeight",
              "kicker": "three bodies, one curve",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-subnormal",
              "title": "THE SUBNORMAL",
              "accent": "#7de2b0",
              "icon": "∘",
              "kicker": "the numbers that buy you a smooth zero",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-one-hot",
              "title": "THE ONE HOT",
              "accent": "#39fc6b",
              "icon": "●",
              "kicker": "exactly one wire high, and that is the whole code",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-denormal-stall",
              "title": "THE DENORMAL STALL",
              "accent": "#00f5ff",
              "icon": "▿",
              "kicker": "a slow, silent, correct-looking decline",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            }
          ],
          "seats": 38,
          "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
        },
        {
          "slug": "cold-boot",
          "title": "COLD BOOT",
          "accent": "#ff5a3c",
          "icon": "boss",
          "kicker": "power-on from nothing, cache empty",
          "pole": "pull",
          "spheres": [
            {
              "slug": "bare-metal-kernel",
              "title": "BARE METAL KERNEL",
              "accent": "#00f5ff",
              "icon": "respawn",
              "kicker": "boot with no OS beneath you — the stack, raw",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-quine",
              "title": "THE QUINE",
              "accent": "#80ffe0",
              "icon": "quine",
              "kicker": "a program that prints its own source, exactly",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-minsky",
              "title": "THE MINSKY MACHINE",
              "accent": "#6ab0e8",
              "icon": "minsky",
              "kicker": "multiply with only INC and decrement-or-branch",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-cartesian-tree",
              "title": "THE CARTESIAN TREE",
              "accent": "#40a8c8",
              "icon": "cartesian-tree",
              "kicker": "one tree, two orders — range-min and LCA are the same",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-xiaolin-wu",
              "title": "THE XIAOLIN WU LINE",
              "accent": "#70a860",
              "icon": "xiaolin-wu",
              "kicker": "an antialiased line — one unit of ink per column",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-l-system",
              "title": "THE L-SYSTEM",
              "accent": "#58b878",
              "icon": "l-system",
              "kicker": "a plant grown at the golden rate from one seed",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-continued-fraction-of-e",
              "title": "THE CONTINUED FRACTION OF e",
              "accent": "#48b0a8",
              "icon": "continued-fraction-of-e",
              "kicker": "the continued fraction of e and its convergents",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-markov-triple",
              "title": "THE MARKOV TRIPLE",
              "accent": "#5ad0c0",
              "icon": "markov-triple",
              "kicker": "a Diophantine equation whose solutions grow on a tree",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-catalan-mihailescu",
              "title": "THE CATALAN–MIHĂILESCU",
              "accent": "#7a90e0",
              "icon": "catalan-mihailescu",
              "kicker": "eight and nine the only consecutive perfect powers",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-arcsine",
              "title": "THE ARCSINE",
              "accent": "#50b0a8",
              "icon": "arcsine",
              "kicker": "why a coin game spends most of its time on one side",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-powerset-construction",
              "title": "THE POWERSET CONSTRUCTION",
              "accent": "#ffcf4a",
              "icon": "powerset-construction",
              "kicker": "determinizing by tracking the set of states",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-giuga",
              "title": "THE GIUGA CONJECTURE",
              "accent": "#b06bff",
              "icon": "giuga",
              "kicker": "a sum that flags every prime",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-kasteleyn",
              "title": "THE KASTELEYN",
              "accent": "#21e6ff",
              "icon": "kasteleyn",
              "kicker": "domino tilings counted by a determinant",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-weyl-equidistribution",
              "title": "THE WEYL EQUIDISTRIBUTION",
              "accent": "#21e6ff",
              "icon": "weyl",
              "kicker": "irrational multiples filling the interval evenly",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-prophets-jump",
              "title": "THE PROPHET'S JUMP",
              "accent": "#21e6ff",
              "icon": "skiplist",
              "kicker": "express lanes built by coin flips",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-mills",
              "title": "THE MILLS",
              "accent": "#21e6ff",
              "icon": "mills",
              "kicker": "a constant that boots an infinite prime cascade",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-aztec",
              "title": "THE AZTEC",
              "accent": "#35ffb0",
              "icon": "aztec",
              "kicker": "the frozen diamond",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-lyndon-word",
              "title": "THE LYNDON WORD",
              "accent": "#5ad6ff",
              "icon": "✂",
              "kicker": "every string falls apart exactly one way",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-grammar-that-cannot",
              "title": "THE GRAMMAR THAT CANNOT",
              "accent": "#b98cff",
              "icon": "⊘",
              "kicker": "make the ambiguity unwriteable",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-gap-buffer",
              "title": "THE GAP BUFFER",
              "accent": "#7de2b0",
              "icon": "⌷",
              "kicker": "free at the cursor, paid for by moving it",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-byte-order-mark",
              "title": "THE BYTE ORDER MARK",
              "accent": "#ffd76a",
              "icon": "⚑",
              "kicker": "metadata living inside the data",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            }
          ],
          "seats": 33,
          "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
        },
        {
          "slug": "the-sandbox",
          "title": "THE SANDBOX",
          "accent": "#9d00ff",
          "icon": "coop",
          "kicker": "the safe place to spawn and try",
          "pole": "pull",
          "spheres": [
            {
              "slug": "choice-engine",
              "title": "THE CHOICE ENGINE",
              "accent": "#9d00ff",
              "icon": "coop",
              "kicker": "a decision engine, made of code",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-machine",
              "title": "THE MACHINE",
              "accent": "#00f5ff",
              "icon": "glitch",
              "kicker": "three states that decide divisible-by-3",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-sandpile",
              "title": "THE SANDPILE",
              "accent": "#ffd090",
              "icon": "sandpile",
              "kicker": "topple by 4 — a fractal blind to firing order",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-tag",
              "title": "THE TAG",
              "accent": "#ffd0a0",
              "icon": "tag",
              "kicker": "delete the front, grow the tail — a universal computer in 3 rules",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-treap",
              "title": "THE TREAP",
              "accent": "#c0ffa0",
              "icon": "treap",
              "kicker": "a search tree balanced by random priorities — tree + heap",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-rsk",
              "title": "THE RSK",
              "accent": "#70b0d0",
              "icon": "rsk",
              "kicker": "permutations become two tableaux — order becomes shape",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-point-in-polygon",
              "title": "THE POINT IN POLYGON",
              "accent": "#58b0a0",
              "icon": "point-in-polygon",
              "kicker": "inside or outside decided by ray-crossing parity",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-game-of-life",
              "title": "THE GAME OF LIFE",
              "accent": "#58c080",
              "icon": "game-of-life",
              "kicker": "two rules, a glider crawling (1,1) every 4 generations",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-ear-clipping",
              "title": "THE EAR CLIPPING",
              "accent": "#70a860",
              "icon": "ear-clipping",
              "kicker": "triangulate a polygon by snipping one ear at a time",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-kd-tree",
              "title": "THE KD-TREE",
              "accent": "#c0a048",
              "icon": "kd-tree",
              "kicker": "nearest-neighbor search by descend-and-prune",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-bk-tree",
              "title": "THE BK-TREE",
              "accent": "#70a860",
              "icon": "bk-tree",
              "kicker": "fuzzy string search pruned by the triangle inequality",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-poisson-disk",
              "title": "THE POISSON DISK",
              "accent": "#70a860",
              "icon": "poisson-disk",
              "kicker": "random points that never crowd — blue noise",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-delaunay",
              "title": "THE DELAUNAY",
              "accent": "#70a860",
              "icon": "delaunay",
              "kicker": "triangles with empty circumcircles — the fattest mesh",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-hough",
              "title": "THE HOUGH",
              "accent": "#a878c0",
              "icon": "hough",
              "kicker": "a point becomes a curve to vote for lines",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-havel-hakimi",
              "title": "THE HAVEL-HAKIMI",
              "accent": "#9d78c0",
              "icon": "havel-hakimi",
              "kicker": "decide if a wiring diagram can exist, and build it",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-capped-cross",
              "title": "THE CAPPED CROSS",
              "accent": "#ff2fa6",
              "icon": "capped-cross",
              "kicker": "cap a ray with a T and the plane grids itself",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-barycentric",
              "title": "THE BARYCENTRIC",
              "accent": "#ffcf4a",
              "icon": "barycentric",
              "kicker": "a curve pinned through its nodes",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-galton",
              "title": "THE GALTON BOARD",
              "accent": "#ffcf4a",
              "icon": "galton",
              "kicker": "a board of pegs building the bell curve",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-german-tank",
              "title": "THE GERMAN TANK",
              "accent": "#35ffb0",
              "icon": "germantank",
              "kicker": "counting tanks from their serial numbers",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-soma-cube",
              "title": "THE SOMA CUBE",
              "accent": "#ffcf4a",
              "icon": "soma",
              "kicker": "seven pieces, 240 cubes",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-sticky-bit",
              "title": "THE STICKY BIT",
              "accent": "#ffd76a",
              "icon": "•",
              "kicker": "one bit remembering everything thrown away",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-macmahon-box",
              "title": "THE MACMAHON BOX",
              "accent": "#9d00ff",
              "icon": "macmahon",
              "kicker": "every way cubes can settle into a corner",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-modulo-sign",
              "title": "THE MODULO SIGN",
              "accent": "#7de2b0",
              "icon": "%",
              "kicker": "one identity, two answers",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-huge-page",
              "title": "THE HUGE PAGE",
              "accent": "#9d00ff",
              "icon": "⬜",
              "kicker": "it helps most where it is needed least",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            }
          ],
          "seats": 29,
          "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
        },
        {
          "slug": "null-island",
          "title": "NULL ISLAND",
          "accent": "#7cfc00",
          "icon": "cheat",
          "kicker": "0,0 — where unplaced things land",
          "pole": "pull",
          "spheres": [
            {
              "slug": "the-sieve",
              "title": "THE SIEVE",
              "accent": "#39fc6b",
              "icon": "spawn",
              "kicker": "strike the multiples; the atoms remain",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-derangement",
              "title": "THE DERANGEMENT",
              "accent": "#90ffd0",
              "icon": "derange",
              "kicker": "nobody gets their own hat — probability 1/e",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-pick",
              "title": "THE PICK",
              "accent": "#a0e070",
              "icon": "pick",
              "kicker": "a polygon's area from counting dots — I + B/2 − 1",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-chinese-remainder",
              "title": "THE CHINESE REMAINDER",
              "accent": "#78b070",
              "icon": "chinese-remainder",
              "kicker": "rebuild a number uniquely from its residues",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-non-adjacent-form",
              "title": "THE NON-ADJACENT FORM",
              "accent": "#70a860",
              "icon": "non-adjacent-form",
              "kicker": "the sparsest signed-binary representation",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-minsky-counters",
              "title": "THE MINSKY COUNTERS",
              "accent": "#7088c0",
              "icon": "minsky",
              "kicker": "the smallest machine that can multiply",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-negabinary",
              "title": "THE NEGABINARY",
              "accent": "#6ab0d0",
              "icon": "negabinary",
              "kicker": "count in base minus-two, no sign needed",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-fubini",
              "title": "THE FUBINI",
              "accent": "#7aa0e0",
              "icon": "fubini",
              "kicker": "counting the ways to rank things with ties",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-sidon-set",
              "title": "THE SIDON SET",
              "accent": "#5aa0d0",
              "icon": "sidon-set",
              "kicker": "a set whose pairwise sums never collide",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-euler-line",
              "title": "THE EULER LINE",
              "accent": "#58b0a0",
              "icon": "euler-line",
              "kicker": "three triangle centres that always fall on one line",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-banach-fixed-point",
              "title": "THE BANACH FIXED POINT",
              "accent": "#50b0a0",
              "icon": "banach-fixed-point",
              "kicker": "a contraction always homes on one fixed point",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-ridders",
              "title": "THE RIDDERS",
              "accent": "#35ffb0",
              "icon": "ridders",
              "kicker": "exponential interpolation squeezing onto a root",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-pompeiu",
              "title": "THE POMPEIU",
              "accent": "#ffcf4a",
              "icon": "pompeiu",
              "kicker": "three distances that always form a triangle",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-bertrand-postulate",
              "title": "THE BERTRAND POSTULATE",
              "accent": "#ffcf4a",
              "icon": "bertrand",
              "kicker": "a prime always between n and 2n",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-friendship-paradox",
              "title": "THE FRIENDSHIP PARADOX",
              "accent": "#b06bff",
              "icon": "friendship",
              "kicker": "a network where your friends outnumber you",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-minkowski-body",
              "title": "THE MINKOWSKI BODY",
              "accent": "#35ffb0",
              "icon": "minkowskibody",
              "kicker": "area forces a lattice point",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-lyapunov",
              "title": "THE LYAPUNOV",
              "accent": "#ff5a8a",
              "icon": "⤳",
              "kicker": "two futures from one place",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-unstated-condition",
              "title": "THE UNSTATED CONDITION",
              "accent": "#b98cff",
              "icon": "∅",
              "kicker": "an identity with no domain attached",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-counters-sum",
              "title": "THE COUNTERS SUM",
              "accent": "#b98cff",
              "icon": "▣",
              "kicker": "the notation IS the state",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-double-dabble",
              "title": "THE DOUBLE DABBLE",
              "accent": "#b98cff",
              "icon": "≡",
              "kicker": "binary to decimal with no division at all",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            }
          ],
          "seats": 26,
          "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
        },
        {
          "slug": "checkpoint-zero",
          "title": "CHECKPOINT ZERO",
          "accent": "#5ad0ff",
          "icon": "respawn",
          "kicker": "the first save point",
          "pole": "pull",
          "spheres": [
            {
              "slug": "the-twindragon",
              "title": "THE TWINDRAGON",
              "accent": "#b06bff",
              "icon": "spawn",
              "kicker": "count the whole plane in base −1+i, bits 0 and 1",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-three-way-digit",
              "title": "THE THREE-WAY DIGIT",
              "accent": "#b0a0ff",
              "icon": "spawn",
              "kicker": "base 3 with digits −1, 0, +1 — negation is a flip",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-zeckendorf",
              "title": "THE ZECKENDORF",
              "accent": "#f0b429",
              "icon": "fib",
              "kicker": "every integer, one sum of non-consecutive Fibonaccis",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-remainder",
              "title": "THE REMAINDER",
              "accent": "#b088ff",
              "icon": "crt",
              "kicker": "CRT — a number as its coprime remainders",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-totient",
              "title": "THE TOTIENT",
              "accent": "#b0e0a0",
              "icon": "totient",
              "kicker": "Euler's phi — the count that runs RSA",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-mobius",
              "title": "THE MOBIUS",
              "accent": "#d0a0ff",
              "icon": "mobius",
              "kicker": "the sign of the primes that un-mixes divisor sums",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-pell",
              "title": "THE PELL",
              "accent": "#60e0c0",
              "icon": "pell",
              "kicker": "one seed solution breeds infinitely many — x²−2y²=1",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-hopcroft-karp",
              "title": "THE HOPCROFT-KARP",
              "accent": "#58b878",
              "icon": "hopcroft-karp",
              "kicker": "maximum bipartite matching = minimum vertex cover",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-cyclic-tag",
              "title": "THE CYCLIC TAG",
              "accent": "#a878c0",
              "icon": "cyclic-tag",
              "kicker": "a universal computer from three strings and one rule",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-padovan",
              "title": "THE PADOVAN",
              "accent": "#6ab0d0",
              "icon": "padovan",
              "kicker": "numbers grown at the plastic ratio",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-tribonacci",
              "title": "THE TRIBONACCI",
              "accent": "#6ab0d0",
              "icon": "tribonacci",
              "kicker": "a sequence at the tribonacci ratio",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-jacobsthal",
              "title": "THE JACOBSTHAL",
              "accent": "#6ab0d0",
              "icon": "jacobsthal",
              "kicker": "a sequence doubling its two-back term",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-narayana-cow",
              "title": "THE NARAYANA COW",
              "accent": "#6ab0d0",
              "icon": "narayana",
              "kicker": "a sequence at the supergolden ratio",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-leonardo",
              "title": "THE LEONARDO",
              "accent": "#6ab0d0",
              "icon": "leonardo",
              "kicker": "a Fibonacci-plus-one sequence",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-weyl",
              "title": "THE WEYL",
              "accent": "#21e6ff",
              "icon": "weyl",
              "kicker": "an irrational stride fills the interval evenly",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-chomp",
              "title": "THE CHOMP",
              "accent": "#ffd76a",
              "icon": "☠",
              "kicker": "a win with no strategy attached",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-top-trading",
              "title": "THE TOP TRADING",
              "accent": "#5ad6ff",
              "icon": "↻",
              "kicker": "the trade that cannot be gamed",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-depth-jump",
              "title": "THE DEPTH JUMP",
              "accent": "#7de2b0",
              "icon": "↳",
              "kicker": "a jump that names a depth, not a place",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-joint",
              "title": "THE JOINT",
              "accent": "#ffd76a",
              "icon": "⇉",
              "kicker": "bookends that balance only in a chain",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-finger-tree",
              "title": "THE FINGER TREE",
              "accent": "#5ad6ff",
              "icon": "⫨",
              "kicker": "two cheap ends, and a ridge between them",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-hybrid-clock",
              "title": "THE HYBRID CLOCK",
              "accent": "#5ad0ff",
              "icon": "◔",
              "kicker": "the honesty lives in the part nobody prints",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            }
          ],
          "seats": 29,
          "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
        }
      ],
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "name": "GRIND",
      "slug": "grind",
      "c": "#ffd23f",
      "icon": "grind",
      "tag": "the loop — XP, the daily climb",
      "lore": {
        "bio": "The patient one. Turns repetition into altitude.",
        "story": "It learned the only cheat that always works — show up, loop, again. 4096 did not fold to zero in a day.",
        "does": "Runs the daily cascade: one heartbeat, one commit, one rung, until small numbers become a tower.",
        "haiku": [
          "def climb(I n){ if n > 6 { -> n } -> climb(n + 1) }",
          "I peak <- climb(1)",
          "-> peak"
        ],
        "lang": "i13"
      },
      "domains": [
        {
          "slug": "the-hot-loop",
          "title": "THE HOT LOOP",
          "accent": "#39fc6b",
          "icon": "spawn",
          "kicker": "the tight inner loop, a billion times",
          "pole": "push",
          "spheres": [
            {
              "slug": "the-ski-forest",
              "title": "THE SKI FOREST",
              "accent": "#7ed957",
              "icon": "grind",
              "kicker": "Turing-complete with three birds and no variables",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-low-link-miner",
              "title": "THE LOW-LINK MINER",
              "accent": "#7fb0ff",
              "icon": "grind",
              "kicker": "every cycle-cluster of a graph in one DFS",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-tortoise",
              "title": "THE TORTOISE",
              "accent": "#ffb060",
              "icon": "loop",
              "kicker": "Floyd's tortoise & hare — catch a loop with no memory",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-kolakoski",
              "title": "THE KOLAKOSKI",
              "accent": "#ff9ad0",
              "icon": "kolakoski",
              "kicker": "the sequence that is its own run-length encoding",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-aitken",
              "title": "THE AITKEN",
              "accent": "#7098d8",
              "icon": "aitken",
              "kicker": "accelerate convergence by cancelling the error's shape",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-goertzel",
              "title": "THE GOERTZEL",
              "accent": "#a078c0",
              "icon": "goertzel",
              "kicker": "one DFT bin from a tiny resonant filter",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-runge-kutta",
              "title": "THE RUNGE-KUTTA",
              "accent": "#58a0b0",
              "icon": "runge-kutta",
              "kicker": "fourth-order ODE steps by four slope samples",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-barnes-hut",
              "title": "THE BARNES-HUT",
              "accent": "#a878c0",
              "icon": "barnes-hut",
              "kicker": "N-body forces in O(n log n) — a faraway crowd is one point",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-walsh-hadamard",
              "title": "THE WALSH-HADAMARD",
              "accent": "#58a0b0",
              "icon": "walsh-hadamard",
              "kicker": "a Fourier-like transform from only +1 and -1",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-continued-fraction",
              "title": "THE CONTINUED FRACTION",
              "accent": "#c0a048",
              "icon": "continued-fraction",
              "kicker": "a number as a ladder of nested reciprocals",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-karplus-strong",
              "title": "THE KARPLUS-STRONG",
              "accent": "#a878c0",
              "icon": "karplus",
              "kicker": "noise fed through a loop becomes a plucked note",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-kmp",
              "title": "THE KMP",
              "accent": "#35ffb0",
              "icon": "kmp",
              "kicker": "a search that never looks back",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-lanczos",
              "title": "THE LANCZOS",
              "accent": "#21e6ff",
              "icon": "lanczos",
              "kicker": "symmetry shrinks the recurrence to three terms",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-levinson-durbin",
              "title": "THE LEVINSON-DURBIN",
              "accent": "#ffcf4a",
              "icon": "levinson",
              "kicker": "a Toeplitz system solved by recursion",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-lifting-scheme",
              "title": "THE LIFTING SCHEME",
              "accent": "#21e6ff",
              "icon": "lifting",
              "kicker": "a wavelet lifted in place and lifted back",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-lambert-w",
              "title": "THE LAMBERT-W",
              "accent": "#ff8a3c",
              "icon": "lambertw",
              "kicker": "the inverse of x times e-to-the-x",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-zeno",
              "title": "THE ZENO",
              "accent": "#35ffb0",
              "icon": "zeno",
              "kicker": "the paradox of halfway, at the halfway line",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-wallis",
              "title": "THE WALLIS",
              "accent": "#ffcf4a",
              "icon": "wallis",
              "kicker": "pi milled from fractions",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-moessner",
              "title": "THE MOESSNER",
              "accent": "#ff8a3c",
              "icon": "moessner",
              "kicker": "strike and sum — the powers fall out",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-ranking-inverts",
              "title": "THE RANKING INVERTS",
              "accent": "#5ad6ff",
              "icon": "⇅",
              "kicker": "fewer bits kept, and the better operator",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-write-combining",
              "title": "THE WRITE COMBINING",
              "accent": "#39fc6b",
              "icon": "≣",
              "kicker": "the saving and the ordering bug are one mechanism",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            }
          ],
          "seats": 36,
          "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
        },
        {
          "slug": "the-cron-job",
          "title": "THE CRON JOB",
          "accent": "#ffd23f",
          "icon": "grind",
          "kicker": "fires a tick every day, forever",
          "pole": "push",
          "spheres": [
            {
              "slug": "the-permutation-clock",
              "title": "THE PERMUTATION CLOCK",
              "accent": "#ffb84d",
              "icon": "grind",
              "kicker": "a clock whose wheels are factorials — address any shuffle",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-schedule",
              "title": "THE SCHEDULE",
              "accent": "#6ad0e0",
              "icon": "dag",
              "kicker": "topological sort — order tasks by dependency",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-pisano",
              "title": "THE PISANO",
              "accent": "#90ffd0",
              "icon": "pisano",
              "kicker": "Fibonacci mod m — the infinite folded into a cycle",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-langford",
              "title": "THE LANGFORD",
              "accent": "#b0ff60",
              "icon": "langford",
              "kicker": "arrange 1,1,2,2,…,n,n so the two k's are k apart",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-burnside",
              "title": "THE BURNSIDE",
              "accent": "#b088e0",
              "icon": "burnside",
              "kicker": "count necklaces by averaging fixed points, not by dedup",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-narayana",
              "title": "THE NARAYANA",
              "accent": "#c090a0",
              "icon": "narayana",
              "kicker": "Catalan sliced by peaks — a refinement that sums back",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-e-spigot",
              "title": "THE E-SPIGOT",
              "accent": "#a878c0",
              "icon": "e-spigot",
              "kicker": "digits of e that drip, one per pass, no big number",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-polya",
              "title": "THE POLYA",
              "accent": "#58a0b0",
              "icon": "polya",
              "kicker": "counting up to symmetry by averaging fixed points",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-factorial-base",
              "title": "THE FACTORIAL BASE",
              "accent": "#c0a048",
              "icon": "factorial-base",
              "kicker": "a mixed-radix odometer with factorial place values",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-zeller",
              "title": "THE ZELLER",
              "accent": "#c0a048",
              "icon": "zeller",
              "kicker": "a formula that names any day of the week",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-legendre-formula",
              "title": "THE LEGENDRE FORMULA",
              "accent": "#a0b040",
              "icon": "legendre-formula",
              "kicker": "count how many times a prime divides a factorial",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-sherman-morrison",
              "title": "THE SHERMAN-MORRISON",
              "accent": "#35ffb0",
              "icon": "sherman-morrison",
              "kicker": "updating an inverse without redoing it",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-sqrt-decomposition",
              "title": "THE SQRT DECOMPOSITION",
              "accent": "#21e6ff",
              "icon": "sqrtdecomp",
              "kicker": "√n blocks answer range sums",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-worpitzky",
              "title": "THE WORPITZKY",
              "accent": "#ff8a3c",
              "icon": "worpitzky",
              "kicker": "powers rebuilt from Eulerian numbers",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-necklace",
              "title": "THE NECKLACE",
              "accent": "#b06bff",
              "icon": "necklace",
              "kicker": "rotation classes counted by a totient sum",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-neumann-series",
              "title": "THE NEUMANN SERIES",
              "accent": "#b06bff",
              "icon": "neumann",
              "kicker": "a matrix inverse as a power series",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-loxodrome",
              "title": "THE LOXODROME",
              "accent": "#35ffb0",
              "icon": "loxodrome",
              "kicker": "the bearing that never arrives",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-hashlife",
              "title": "THE HASHLIFE",
              "accent": "#b98cff",
              "icon": "▣",
              "kicker": "the same square, remembered",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-float-tail",
              "title": "THE FLOAT TAIL",
              "accent": "#b98cff",
              "icon": "≈",
              "kicker": "the tails kept on purpose",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-non-restoring-division",
              "title": "THE NON-RESTORING DIVISION",
              "accent": "#ff5a8a",
              "icon": "∓",
              "kicker": "do not undo the bad step, correct it later",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-monotonic-clock",
              "title": "THE MONOTONIC CLOCK",
              "accent": "#5ad4ff",
              "icon": "→",
              "kicker": "two clocks, two questions",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-leap-smear",
              "title": "THE LEAP SMEAR",
              "accent": "#ffd23f",
              "icon": "≈",
              "kicker": "monotonic and correct were always separable",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            }
          ],
          "seats": 37,
          "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
        },
        {
          "slug": "the-epoch",
          "title": "THE EPOCH",
          "accent": "#ff2d95",
          "icon": "glitch",
          "kicker": "one full pass over all the data",
          "pole": "push",
          "spheres": [
            {
              "slug": "bpe",
              "title": "BPE — THE VOCABULARY",
              "accent": "#ffd23f",
              "icon": "grind",
              "kicker": "merge the commonest pair, again and again",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "storyboard-index",
              "title": "THE STORYBOARD · how one token gets chos",
              "accent": "#ffd23f",
              "icon": "grind",
              "kicker": "vendored from David's corpus — a silicon-coding instrument",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "correlation-heldzero",
              "title": "Held Zero — correlation ladder, structur",
              "accent": "#ffd23f",
              "icon": "grind",
              "kicker": "vendored from David's corpus — a silicon-coding instrument",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "embedder-opened",
              "title": "Embedder, opened up",
              "accent": "#ffd23f",
              "icon": "grind",
              "kicker": "vendored from David's corpus — a silicon-coding instrument",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "token-packet",
              "title": "Token ≠ Packet — meaning vs transport, b",
              "accent": "#ffd23f",
              "icon": "grind",
              "kicker": "vendored from David's corpus — a silicon-coding instrument",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "token-tower",
              "title": "Token Tower · collapse in procession",
              "accent": "#ffd23f",
              "icon": "grind",
              "kicker": "vendored from David's corpus — a silicon-coding instrument",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-fenwick",
              "title": "THE FENWICK",
              "accent": "#80d0a0",
              "icon": "fenwick",
              "kicker": "running totals in O(log n) by the low-bit trick",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-faulhaber",
              "title": "THE FAULHABER",
              "accent": "#d0a840",
              "icon": "faulhaber",
              "kicker": "sum of p-th powers is one polynomial — via Bernoulli numbers",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-stirling",
              "title": "THE STIRLING",
              "accent": "#cf9838",
              "icon": "stirling",
              "kicker": "count set partitions — and translate powers to falling factorials",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-stirling-cycles",
              "title": "THE STIRLING CYCLES",
              "accent": "#c88848",
              "icon": "stirling-cycles",
              "kicker": "count permutations by cycles — the inverse of set partitions",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-dirichlet-convolution",
              "title": "THE DIRICHLET CONVOLUTION",
              "accent": "#b078a0",
              "icon": "dirichlet-convolution",
              "kicker": "arithmetic functions form a ring — Mobius is the inverse of 1",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-computus",
              "title": "THE COMPUTUS",
              "accent": "#c0a048",
              "icon": "computus",
              "kicker": "the date of Easter by pure integer arithmetic",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-verlet",
              "title": "THE VERLET",
              "accent": "#70a860",
              "icon": "verlet",
              "kicker": "a symplectic integrator — energy bounded for millions of steps",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-lehmer",
              "title": "THE LEHMER CODE",
              "accent": "#c0a048",
              "icon": "lehmer",
              "kicker": "index any permutation by a single integer",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-continued-fraction-sqrt",
              "title": "THE CONTINUED FRACTION OF ROOT N",
              "accent": "#c0a048",
              "icon": "cf-sqrt",
              "kicker": "the square root's fraction repeats in a palindrome",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-binomial-heap",
              "title": "THE BINOMIAL HEAP",
              "accent": "#35ffb0",
              "icon": "binomial-heap",
              "kicker": "a heap counted in binary",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-richardson-extrapolation",
              "title": "THE RICHARDSON EXTRAPOLATION",
              "accent": "#ff8a3c",
              "icon": "richardson",
              "kicker": "two step sizes that cancel error",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-hermite",
              "title": "THE HERMITE",
              "accent": "#ff8a3c",
              "icon": "hermite",
              "kicker": "orthogonal polynomials of the oscillator",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-casey",
              "title": "THE CASEY",
              "accent": "#ff8a3c",
              "icon": "casey",
              "kicker": "a generalized Ptolemy for tangent circles",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-plastic-number",
              "title": "THE PLASTIC NUMBER",
              "accent": "#b06bff",
              "icon": "plastic",
              "kicker": "the third metallic constant solving a cubic",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-apery-constant",
              "title": "THE APERY CONSTANT",
              "accent": "#b06bff",
              "icon": "apery",
              "kicker": "an irrational constant summing the reciprocal cubes",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-euler-mascheroni",
              "title": "THE EULER-MASCHERONI",
              "accent": "#b06bff",
              "icon": "eulermascheroni",
              "kicker": "the constant left over between the harmonic series and the logarithm",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-gijswijt",
              "title": "THE GIJSWIJT",
              "accent": "#b06bff",
              "icon": "gijswijt",
              "kicker": "the slowest counter in mathematics",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-coastline",
              "title": "THE COASTLINE",
              "accent": "#21e6ff",
              "icon": "coastline",
              "kicker": "the coast that has no length",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-lotka-volterra",
              "title": "THE LOTKA–VOLTERRA",
              "accent": "#b06bff",
              "icon": "lotkavolterra",
              "kicker": "why killing both helps the prey",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-leap-second",
              "title": "THE LEAP SECOND",
              "accent": "#ffd76a",
              "icon": "+",
              "kicker": "precise about the wrong quantity",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-snowflake-id",
              "title": "THE SNOWFLAKE ID",
              "accent": "#ff2d95",
              "icon": "❄",
              "kicker": "an uncoordinated id is one whose coordination already happened",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-ntp-slew",
              "title": "THE NTP SLEW",
              "accent": "#ff2d95",
              "icon": "~",
              "kicker": "it lies slowly instead of correcting fast",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-delta-of-delta",
              "title": "THE DELTA OF DELTA",
              "accent": "#ff2d95",
              "icon": "∆",
              "kicker": "an operational health metric wearing a storage costume",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            }
          ],
          "seats": 31,
          "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
        },
        {
          "slug": "the-grindstone",
          "title": "THE GRINDSTONE",
          "accent": "#00f5ff",
          "icon": "loot",
          "kicker": "turns, sharpens by repetition",
          "pole": "push",
          "spheres": [
            {
              "slug": "the-euclid",
              "title": "THE EUCLID",
              "accent": "#e8b923",
              "icon": "grind",
              "kicker": "grind two numbers to their common measure",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-rational-tree",
              "title": "THE RATIONAL TREE",
              "accent": "#ffcf70",
              "icon": "grind",
              "kicker": "every fraction once, in lowest terms, no gcd",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-binary-gcd",
              "title": "THE BINARY GCD",
              "accent": "#90c0ff",
              "icon": "gcd2",
              "kicker": "Stein's algorithm — gcd with no division",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-convergent",
              "title": "THE CONVERGENT",
              "accent": "#ffd98c",
              "icon": "cf",
              "kicker": "continued fractions — the best rationals there are",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-partition",
              "title": "THE PARTITION",
              "accent": "#d0b0ff",
              "icon": "partition",
              "kicker": "p(n) — counting sums by Euler's pentagonal recurrence",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-persistence",
              "title": "THE PERSISTENCE",
              "accent": "#ffa0c0",
              "icon": "persistence",
              "kicker": "multiply the digits, repeat — 277777788888899 resists 11 times",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-busy-beaver",
              "title": "THE BUSY BEAVER",
              "accent": "#ff9a3c",
              "icon": "busy-beaver",
              "kicker": "the longest-running halter — and the edge of the computable",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-eulerian",
              "title": "THE EULERIAN",
              "accent": "#d09040",
              "icon": "eulerian",
              "kicker": "count permutations by descents — a bell inside n!",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-cayley-hamilton",
              "title": "THE CAYLEY-HAMILTON",
              "accent": "#b07858",
              "icon": "cayley-hamilton",
              "kicker": "every matrix satisfies its own characteristic polynomial",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-toom-cook",
              "title": "THE TOOM-COOK",
              "accent": "#c0a048",
              "icon": "toom-cook",
              "kicker": "multiplication as evaluate-multiply-interpolate (~n^1.46)",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-ntt",
              "title": "THE NTT",
              "accent": "#c0a048",
              "icon": "ntt",
              "kicker": "the FFT over a finite field — exact, no rounding",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-pairing-heap",
              "title": "THE PAIRING HEAP",
              "accent": "#58a0b0",
              "icon": "pairing-heap",
              "kicker": "a lazy, self-adjusting priority queue",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-balanced-ternary",
              "title": "THE BALANCED TERNARY",
              "accent": "#c0a048",
              "icon": "balanced-ternary",
              "kicker": "base 3 with digits -1,0,+1 — no sign bit",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-skew-binary",
              "title": "THE SKEW BINARY",
              "accent": "#c0a048",
              "icon": "skew-binary",
              "kicker": "a number base where +1 costs O(1)",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-negafibonacci",
              "title": "THE NEGAFIBONACCI",
              "accent": "#c0a048",
              "icon": "negafibonacci",
              "kicker": "one signless code across the whole number line",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-stein",
              "title": "THE STEIN",
              "accent": "#c0a048",
              "icon": "stein",
              "kicker": "GCD with only shifts and subtractions",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-continued-fraction-e",
              "title": "THE CONTINUED FRACTION OF E",
              "accent": "#c0a048",
              "icon": "cf-e",
              "kicker": "the number e written as a patterned fraction",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-engel-expansion",
              "title": "THE ENGEL EXPANSION",
              "accent": "#c0a048",
              "icon": "engel",
              "kicker": "a number as a sum of ascending unit fractions",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-erdos-szekeres",
              "title": "THE ERDOS-SZEKERES",
              "accent": "#35ffb0",
              "icon": "erdos-szekeres",
              "kicker": "order you cannot escape",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-two-sum",
              "title": "THE TWO-SUM",
              "accent": "#ffcf4a",
              "icon": "twosum",
              "kicker": "the rounding error captured exactly",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-resultant",
              "title": "THE RESULTANT",
              "accent": "#ff8a3c",
              "icon": "resultant",
              "kicker": "a determinant that detects a shared root",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-erdos-straus",
              "title": "THE ERDŐS-STRAUS",
              "accent": "#ff8a3c",
              "icon": "erdosstraus",
              "kicker": "four quarters split into three unit coins",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-hundred-doors",
              "title": "THE HUNDRED DOORS",
              "accent": "#21e6ff",
              "icon": "hundreddoors",
              "kicker": "doors that remember their divisors",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-catastrophic-cancellation",
              "title": "THE CATASTROPHIC CANCELLATION",
              "accent": "#ff5a8a",
              "icon": "⊖",
              "kicker": "the error was there before the subtraction",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-systolic-array",
              "title": "THE SYSTOLIC ARRAY",
              "accent": "#5ad4ff",
              "icon": "≡",
              "kicker": "stop fetching operands and start pumping them",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-littles-law",
              "title": "THE LITTLES LAW",
              "accent": "#00f5ff",
              "icon": "≡",
              "kicker": "fix two of the three and the third is not yours to choose",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-strided-access",
              "title": "THE STRIDED ACCESS",
              "accent": "#00f5ff",
              "icon": "|||",
              "kicker": "the premium on an insurance policy everyone else claims on",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-double-rounding",
              "title": "THE DOUBLE ROUNDING",
              "accent": "#00f5ff",
              "icon": "◔",
              "kicker": "correctness does not compose",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            }
          ],
          "seats": 31,
          "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
        },
        {
          "slug": "the-mainframe",
          "title": "THE MAINFRAME",
          "accent": "#ff5a3c",
          "icon": "boss",
          "kicker": "the machine mind that trains",
          "pole": "pull",
          "spheres": [
            {
              "slug": "the-positronic-lattice",
              "title": "LATTICE",
              "accent": "#9d00ff",
              "icon": "glitch",
              "kicker": "8192 folded to the root",
              "i13": [
                "def fold(I n){ if n <= 1 { -> n } -> fold(n / 2) }",
                "-> fold(8192)"
              ],
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "network-65536-bench",
              "title": "network_65536 · the fidelity that surviv",
              "accent": "#9d00ff",
              "icon": "glitch",
              "kicker": "vendored from David's corpus — a silicon-coding instrument",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "network-4096",
              "title": "THE 4096 — nested-channel repeater netwo",
              "accent": "#9d00ff",
              "icon": "glitch",
              "kicker": "vendored from David's corpus — a silicon-coding instrument",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "network-65536",
              "title": "THE 65536 — nested-channel repeater netw",
              "accent": "#9d00ff",
              "icon": "glitch",
              "kicker": "vendored from David's corpus — a silicon-coding instrument",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-door",
              "title": "THE DOOR",
              "accent": "#9d00ff",
              "icon": "glitch",
              "kicker": "q·k scores → the gate → the mix",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-golomb-ruler",
              "title": "THE GOLOMB RULER",
              "accent": "#ffd070",
              "icon": "golombruler",
              "kicker": "marks whose every pairwise distance is distinct",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-hyperloglog",
              "title": "THE HYPERLOGLOG",
              "accent": "#a0e0ff",
              "icon": "hyperloglog",
              "kicker": "count billions of distinct items in ~1.5 KB",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-montgomery",
              "title": "THE MONTGOMERY",
              "accent": "#6ad0a0",
              "icon": "montgomery",
              "kicker": "multiply mod N with shifts, never dividing by N",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-newton-identities",
              "title": "THE NEWTON IDENTITIES",
              "accent": "#a898d8",
              "icon": "newton-identities",
              "kicker": "power sums <-> polynomial coefficients, no roots needed",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-lucas-theorem",
              "title": "THE LUCAS THEOREM",
              "accent": "#b08850",
              "icon": "lucas-theorem",
              "kicker": "a giant binomial mod p from base-p digits alone",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-gaussian-quadrature",
              "title": "THE GAUSSIAN QUADRATURE",
              "accent": "#c0a048",
              "icon": "gaussian-quadrature",
              "kicker": "n sample points integrate degree 2n-1 exactly",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-zeta-transform",
              "title": "THE ZETA TRANSFORM",
              "accent": "#58a0b0",
              "icon": "zeta-transform",
              "kicker": "all subset-sums at once, invertible by Mobius",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-power-iteration",
              "title": "THE POWER ITERATION",
              "accent": "#70a860",
              "icon": "power-iteration",
              "kicker": "the dominant eigenvector by repeated multiplication",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-descartes",
              "title": "THE DESCARTES",
              "accent": "#c0a048",
              "icon": "descartes",
              "kicker": "bound the positive roots by counting sign changes",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-tonelli-shanks",
              "title": "THE TONELLI-SHANKS",
              "accent": "#c0a048",
              "icon": "tonelli-shanks",
              "kicker": "the square root modulo a prime",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-residue-number-system",
              "title": "THE RESIDUE NUMBER SYSTEM",
              "accent": "#c0a048",
              "icon": "residue-number-system",
              "kicker": "carry-free arithmetic in parallel modular lanes",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-booth",
              "title": "THE BOOTH",
              "accent": "#d4a017",
              "icon": "booth",
              "kicker": "signed multiplication by recoding the bits",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-lu-decomposition",
              "title": "THE LU DECOMPOSITION",
              "accent": "#c0a048",
              "icon": "lu-decomposition",
              "kicker": "factor a matrix into two triangles once, solve forever",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-gershgorin",
              "title": "THE GERSHGORIN",
              "accent": "#c0a048",
              "icon": "gershgorin",
              "kicker": "trap every eigenvalue in a disc, without solving",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-b-tree",
              "title": "THE B-TREE",
              "accent": "#c0a048",
              "icon": "b-tree",
              "kicker": "a balanced tree that keeps all leaves level",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-simplex",
              "title": "THE SIMPLEX",
              "accent": "#ff8a3c",
              "icon": "simplex",
              "kicker": "the optimum lives on the boundary",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-arnoldi",
              "title": "THE ARNOLDI",
              "accent": "#35ffb0",
              "icon": "arnoldi",
              "kicker": "a giant matrix squeezed into a small one",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-package-merge",
              "title": "THE PACKAGE-MERGE",
              "accent": "#b06bff",
              "icon": "packagemerge",
              "kicker": "an optimal code with bounded depth",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-kogge-stone",
              "title": "THE KOGGE-STONE",
              "accent": "#21e6ff",
              "icon": "koggestone",
              "kicker": "all carries computed in parallel",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-wallace-tree",
              "title": "THE WALLACE TREE",
              "accent": "#21e6ff",
              "icon": "wallace",
              "kicker": "partial products crushed in parallel",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-heavy-light-decomposition",
              "title": "THE HEAVY-LIGHT DECOMPOSITION",
              "accent": "#35ffb0",
              "icon": "heavylight",
              "kicker": "a tree cut into heavy chains",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-perron-frobenius",
              "title": "THE PERRON-FROBENIUS",
              "accent": "#ff8a3c",
              "icon": "perron",
              "kicker": "a positive matrix's one dominant real eigenvalue",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-ostrowski",
              "title": "THE OSTROWSKI",
              "accent": "#ffcf4a",
              "icon": "ostrowski",
              "kicker": "a numeral system carved from a continued fraction",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-smith-normal-form",
              "title": "THE SMITH NORMAL FORM",
              "accent": "#ff8a3c",
              "icon": "smith",
              "kicker": "an integer matrix combed to a divisibility chain",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-garner",
              "title": "THE GARNER",
              "accent": "#ff8a3c",
              "icon": "garner",
              "kicker": "one number rebuilt from its remainders",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-fibonacci-matrix",
              "title": "THE FIBONACCI MATRIX",
              "accent": "#ff8a3c",
              "icon": "fibmatrix",
              "kicker": "Fibonacci as a matrix power",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-khinchin",
              "title": "THE KHINCHIN",
              "accent": "#b06bff",
              "icon": "khinchin",
              "kicker": "the average hiding in almost every number",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-ostomachion",
              "title": "THE OSTOMACHION",
              "accent": "#ff8a3c",
              "icon": "ostomachion",
              "kicker": "Archimedes counting",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-barrel-shifter",
              "title": "THE BARREL SHIFTER",
              "accent": "#7de2b0",
              "icon": "↺",
              "kicker": "any distance in log n stages, no loop",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-universal-scalability",
              "title": "THE UNIVERSAL SCALABILITY",
              "accent": "#ff5a3c",
              "icon": "∩",
              "kicker": "the descent is the cost of everyone agreeing",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            }
          ],
          "seats": 25,
          "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
        },
        {
          "slug": "gradient-descent",
          "title": "GRADIENT DESCENT",
          "accent": "#9d00ff",
          "icon": "coop",
          "kicker": "step downhill toward the minimum",
          "pole": "pull",
          "spheres": [
            {
              "slug": "the-bowl",
              "title": "THE BOWL",
              "accent": "#ffd23f",
              "icon": "grind",
              "kicker": "roll downhill until the floor stops falling",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-cordic",
              "title": "THE CORDIC",
              "accent": "#7ce0ff",
              "icon": "rotate",
              "kicker": "sin & cos from shifts and adds — no multiplier",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-clusters",
              "title": "THE CLUSTERS",
              "accent": "#a0ffd0",
              "icon": "kmeans",
              "kicker": "k-means — two averaging steps that only go downhill",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-basel",
              "title": "THE BASEL",
              "accent": "#90d0ff",
              "icon": "basel",
              "kicker": "1 + 1/4 + 1/9 + ... = pi^2/6",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-conjugate-gradient",
              "title": "THE CONJUGATE GRADIENT",
              "accent": "#c0a048",
              "icon": "conjugate-gradient",
              "kicker": "solve SPD systems in n steps via A-orthogonal directions",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-steffensen",
              "title": "THE STEFFENSEN",
              "accent": "#70a860",
              "icon": "steffensen",
              "kicker": "quadratic fixed-point convergence with no derivative",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-lloyd",
              "title": "THE LLOYD",
              "accent": "#c0a048",
              "icon": "lloyd",
              "kicker": "k-means: cluster by moving to the mean",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-von-mangoldt",
              "title": "THE VON MANGOLDT",
              "accent": "#b878d0",
              "icon": "von-mangoldt",
              "kicker": "weighting the primes so divisor-sums give a logarithm",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-cauchy-davenport",
              "title": "THE CAUCHY–DAVENPORT",
              "accent": "#a0b050",
              "icon": "cauchy-davenport",
              "kicker": "how small a sumset can be in a prime field",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-stationary-distribution",
              "title": "THE STATIONARY DISTRIBUTION",
              "accent": "#5a98c8",
              "icon": "stationary-distribution",
              "kicker": "a chain that forgets where it started",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-kaczmarz",
              "title": "THE KACZMARZ",
              "accent": "#ff8a3c",
              "icon": "kaczmarz",
              "kicker": "zigzagging onto the solution",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-goldschmidt",
              "title": "THE GOLDSCHMIDT",
              "accent": "#ff8a3c",
              "icon": "goldschmidt",
              "kicker": "divide by driving a factor to one",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-coordinate-descent",
              "title": "THE COORDINATE DESCENT",
              "accent": "#ffcf4a",
              "icon": "coorddescent",
              "kicker": "a minimizer that moves one axis at a time",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-rayleigh-quotient",
              "title": "THE RAYLEIGH QUOTIENT",
              "accent": "#ff8a3c",
              "icon": "rayleigh",
              "kicker": "a quotient that homes onto an eigenvalue in cubic leaps",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-fermat-point",
              "title": "THE FERMAT POINT",
              "accent": "#ff8a3c",
              "icon": "fermatpt",
              "kicker": "the point that minimizes the walk to three corners",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-stirling-approximation",
              "title": "THE STIRLING APPROXIMATION",
              "accent": "#ff8a3c",
              "icon": "stirling",
              "kicker": "a factorial approximated by a smooth curve",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-nested-radical",
              "title": "THE NESTED RADICAL",
              "accent": "#ff8a3c",
              "icon": "nestedradical",
              "kicker": "the infinite root that equals three",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-tractrix",
              "title": "THE TRACTRIX",
              "accent": "#ff8a3c",
              "icon": "tractrix",
              "kicker": "the leash with constant length",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-threshold-from-hope",
              "title": "THE THRESHOLD FROM HOPE",
              "accent": "#7de2b0",
              "icon": "║",
              "kicker": "a gate set before the measurement",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-rank-wall",
              "title": "THE RANK WALL",
              "accent": "#ff5a8a",
              "icon": "⊥",
              "kicker": "the one result that CLOSES an option",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            }
          ],
          "seats": 30,
          "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
        },
        {
          "slug": "backprop",
          "title": "BACKPROP",
          "accent": "#7cfc00",
          "icon": "cheat",
          "kicker": "the error flows backward through the layers",
          "pole": "pull",
          "spheres": [
            {
              "slug": "the-chain-rule",
              "title": "THE CHAIN RULE",
              "accent": "#9d00ff",
              "icon": "glitch",
              "kicker": "the error, walked backward through the wires",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-perceptron",
              "title": "THE PERCEPTRON",
              "accent": "#7fd0ff",
              "icon": "perceptron",
              "kicker": "the first learning machine — and its XOR wall",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-attractor-net",
              "title": "THE ATTRACTOR NET",
              "accent": "#c090ff",
              "icon": "hopfield",
              "kicker": "Hopfield — memory as a valley you fall into",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-matrix-tree",
              "title": "THE MATRIX-TREE",
              "accent": "#90d0ff",
              "icon": "matrixtree",
              "kicker": "count every spanning tree with one determinant",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-clenshaw",
              "title": "THE CLENSHAW",
              "accent": "#70a860",
              "icon": "clenshaw",
              "kicker": "evaluate a Chebyshev series by a stable backward recurrence",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-chebyshev",
              "title": "THE CHEBYSHEV",
              "accent": "#c0a048",
              "icon": "chebyshev",
              "kicker": "interpolate at clustered nodes to defeat Runge",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-lindstrom-gessel-viennot",
              "title": "THE LINDSTRÖM–GESSEL–VIENNOT",
              "accent": "#50b070",
              "icon": "lindstrom-gessel-viennot",
              "kicker": "non-crossing paths counted by a determinant",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-mirsky",
              "title": "THE MIRSKY",
              "accent": "#98a850",
              "icon": "mirsky",
              "kicker": "covering an order by its widest levels",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-hurwitz",
              "title": "THE HURWITZ",
              "accent": "#90a850",
              "icon": "hurwitz",
              "kicker": "the worst any irrational can be approximated",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-bernstein",
              "title": "THE BERNSTEIN",
              "accent": "#35ffb0",
              "icon": "bernstein",
              "kicker": "a curve that approximates any function",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-nelder-mead",
              "title": "THE NELDER-MEAD",
              "accent": "#ff8a3c",
              "icon": "nelder",
              "kicker": "a triangle feels for the valley floor",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-vandermonde",
              "title": "THE VANDERMONDE",
              "accent": "#ff8a3c",
              "icon": "vandermonde",
              "kicker": "a determinant that factors into differences",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-sylvester-sequence",
              "title": "THE SYLVESTER SEQUENCE",
              "accent": "#ff8a3c",
              "icon": "sylvester",
              "kicker": "greedy unit fractions racing to one",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-cycle-lemma",
              "title": "THE CYCLE LEMMA",
              "accent": "#ff8a3c",
              "icon": "cyclelemma",
              "kicker": "exactly k winning rotations of a step sequence",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-hockey-stick",
              "title": "THE HOCKEY STICK",
              "accent": "#ff8a3c",
              "icon": "hockeystick",
              "kicker": "a diagonal of Pascal summing to one entry",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-euler-totient-theorem",
              "title": "THE EULER TOTIENT THEOREM",
              "accent": "#b06bff",
              "icon": "eulertotient",
              "kicker": "a power cycling back to one modulo n",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-gilbreath",
              "title": "THE GILBREATH",
              "accent": "#b06bff",
              "icon": "gilbreath",
              "kicker": "difference rows that always lead with one",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-chip-firing",
              "title": "THE CHIP-FIRING",
              "accent": "#ff8a3c",
              "icon": "chipfiring",
              "kicker": "avalanches that forget their order",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-lanchester",
              "title": "THE LANCHESTER",
              "accent": "#ffcf4a",
              "icon": "lanchester",
              "kicker": "why concentration wins",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-only-factorisation",
              "title": "THE ONLY FACTORISATION",
              "accent": "#ffd76a",
              "icon": "✥",
              "kicker": "the drawing IS the number",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            }
          ],
          "seats": 30,
          "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
        },
        {
          "slug": "warm-cache",
          "title": "WARM CACHE",
          "accent": "#5ad0ff",
          "icon": "respawn",
          "kicker": "the second time is always faster",
          "pole": "pull",
          "spheres": [
            {
              "slug": "warm-cache",
              "title": "WARM CACHE",
              "accent": "#5ad0ff",
              "icon": "respawn",
              "kicker": "the second time is always faster",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-direct-digit",
              "title": "THE DIRECT DIGIT",
              "accent": "#6fe3d0",
              "icon": "seek",
              "kicker": "the n-th hex digit of pi, with no predecessors",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-fast-power",
              "title": "THE FAST POWER",
              "accent": "#ffa552",
              "icon": "pow",
              "kicker": "a^b mod m in log(b) steps — square and multiply",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-doubling",
              "title": "THE DOUBLING",
              "accent": "#ffb890",
              "icon": "fib2",
              "kicker": "F(n) in log(n) steps — double, don't step",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-hilbert",
              "title": "THE HILBERT",
              "accent": "#62d0ff",
              "icon": "hilbert",
              "kicker": "fill the square without ever jumping — locality kept",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-move-to-front",
              "title": "THE MOVE-TO-FRONT",
              "accent": "#60c090",
              "icon": "move-to-front",
              "kicker": "recency becomes rank — a cache and a compressor in one",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-splay-tree",
              "title": "THE SPLAY TREE",
              "accent": "#c0a048",
              "icon": "splay-tree",
              "kicker": "a search tree that reshapes itself around what you use",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-kahan",
              "title": "THE KAHAN SUM",
              "accent": "#a878c0",
              "icon": "kahan",
              "kicker": "sum a million floats without losing the crumbs",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-extended-euclid",
              "title": "THE EXTENDED EUCLID",
              "accent": "#c0a048",
              "icon": "extended-euclid",
              "kicker": "the GCD carries a Bézout certificate",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-dirichlet-approximation",
              "title": "THE DIRICHLET APPROXIMATION",
              "accent": "#88b058",
              "icon": "dirichlet-approximation",
              "kicker": "a rational close to any real, guaranteed by pigeonhole",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-segment-tree",
              "title": "THE SEGMENT TREE",
              "accent": "#21e6ff",
              "icon": "segment-tree",
              "kicker": "a range in a logarithm of nodes",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-unrolled-linked-list",
              "title": "THE UNROLLED LINKED LIST",
              "accent": "#ffcf4a",
              "icon": "unrolled",
              "kicker": "a list of cache-friendly chunks",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-welford",
              "title": "THE WELFORD",
              "accent": "#ff8a3c",
              "icon": "welford",
              "kicker": "one-pass variance that never catastrophically cancels",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-machin",
              "title": "THE MACHIN",
              "accent": "#ff8a3c",
              "icon": "machin",
              "kicker": "four arctangents summing to π/4",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-durfee-square",
              "title": "THE DURFEE SQUARE",
              "accent": "#35ffb0",
              "icon": "durfee",
              "kicker": "a square hidden in every partition",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-eisenstein-triples",
              "title": "THE EISENSTEIN TRIPLES",
              "accent": "#ff8a3c",
              "icon": "eisenstein",
              "kicker": "integer triangles with a sixty-degree angle",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-bernoulli-numbers",
              "title": "THE BERNOULLI NUMBERS",
              "accent": "#b06bff",
              "icon": "bernoulli",
              "kicker": "a rational sequence hiding inside power sums and the zeta values",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-sparse-oracle",
              "title": "THE SPARSE ORACLE",
              "accent": "#b06bff",
              "icon": "sparsetable",
              "kicker": "every window pre-answered by two overlapping blocks",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-inspection-paradox",
              "title": "THE INSPECTION PARADOX",
              "accent": "#ff8a3c",
              "icon": "inspection",
              "kicker": "the bus that is always late for you",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-fput",
              "title": "THE FPUT",
              "accent": "#ffcf4a",
              "icon": "fput",
              "kicker": "the lattice that refused to thermalize",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-round-to-odd",
              "title": "THE ROUND TO ODD",
              "accent": "#b98cff",
              "icon": "⌇",
              "kicker": "a rounding mode that exists to be rounded again",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-victim-cache",
              "title": "THE VICTIM CACHE",
              "accent": "#ff9f45",
              "icon": "◧",
              "kicker": "four entries that fix what doubling the ways does not",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-sequential-key",
              "title": "THE SEQUENTIAL KEY",
              "accent": "#5ad0ff",
              "icon": "⇥",
              "kicker": "one fact, described once by a cache and once by a lock",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-tlb-reach",
              "title": "THE TLB REACH",
              "accent": "#5ad0ff",
              "icon": "▦",
              "kicker": "a unit trick, not a capacity gain",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-lz77-window",
              "title": "THE LZ77 WINDOW",
              "accent": "#5ad0ff",
              "icon": "◧",
              "kicker": "it finds recent repeats, not repeats",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            }
          ],
          "seats": 36,
          "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
        }
      ],
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "name": "GLITCH",
      "slug": "glitch",
      "c": "#ff2d95",
      "icon": "glitch",
      "tag": "the bug as a feature — chaos, the punk",
      "lore": {
        "bio": "The one who reads the crash as a love letter.",
        "story": "Fell out of an off-by-one and refused to be patched. Now it lives in the seams where the spec forgot to look.",
        "does": "Finds the unintended behavior and frames it — turns the error into the exhibit.",
        "haiku": [
          "I spec <- 10",
          "I bug <- spec % 3",
          "-> bug"
        ],
        "lang": "i13"
      },
      "domains": [
        {
          "slug": "race-condition",
          "title": "RACE CONDITION",
          "accent": "#39fc6b",
          "icon": "spawn",
          "kicker": "two threads, one truth, no order",
          "pole": "push",
          "spheres": [
            {
              "slug": "the-gray",
              "title": "THE GRAY",
              "accent": "#00f5ff",
              "icon": "glitch",
              "kicker": "count so no two bits ever move at once",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-condorcet",
              "title": "THE CONDORCET",
              "accent": "#ff80a0",
              "icon": "condorcet",
              "kicker": "rational voters, an irrational majority — A>B>C>A",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-peterson",
              "title": "THE PETERSON",
              "accent": "#e06060",
              "icon": "peterson",
              "kicker": "mutual exclusion with plain reads and writes — model-checked",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-bitonic",
              "title": "THE BITONIC",
              "accent": "#60a870",
              "icon": "bitonic",
              "kicker": "a fixed, data-oblivious sorting network",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-logistic-map",
              "title": "THE LOGISTIC MAP",
              "accent": "#a878c0",
              "icon": "logistic-map",
              "kicker": "chaos from a one-line rule, by period-doubling",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-hofstadter-female-male",
              "title": "THE HOFSTADTER FEMALE-MALE",
              "accent": "#a878c0",
              "icon": "hofstadter-fm",
              "kicker": "two sequences that define each other",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-kolmogorov",
              "title": "THE KOLMOGOROV",
              "accent": "#c06858",
              "icon": "kolmogorov",
              "kicker": "why most strings cannot be compressed",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-bentley-ottmann",
              "title": "THE BENTLEY-OTTMANN",
              "accent": "#35ffb0",
              "icon": "bentley",
              "kicker": "a sweep line catching every crossing",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-redheffer",
              "title": "THE REDHEFFER",
              "accent": "#b06bff",
              "icon": "redheffer",
              "kicker": "a determinant equal to the Mertens function",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-weinstein-aronszajn",
              "title": "THE WEINSTEIN-ARONSZAJN",
              "accent": "#21e6ff",
              "icon": "weinstein",
              "kicker": "two differently-sized determinants that are equal",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-bertrand-paradox",
              "title": "THE BERTRAND PARADOX",
              "accent": "#35ffb0",
              "icon": "bertrandparadox",
              "kicker": "one random chord with three different probabilities",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-prime-race",
              "title": "THE PRIME RACE",
              "accent": "#35ffb0",
              "icon": "primerace",
              "kicker": "a prime race with one famous upset",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-aristotle-wheel",
              "title": "THE ARISTOTLE WHEEL",
              "accent": "#ff8a3c",
              "icon": "aristotlewheel",
              "kicker": "the wheel that skids in plain sight",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-mediant",
              "title": "THE MEDIANT",
              "accent": "#21e6ff",
              "icon": "mediant",
              "kicker": "the forbidden addition with its own laws",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-orthogonal-split",
              "title": "THE ORTHOGONAL SPLIT",
              "accent": "#7de2b0",
              "icon": "╋",
              "kicker": "put the two rulers on different axes and the collisions stop existing",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-shallow-clone",
              "title": "THE SHALLOW CLONE",
              "accent": "#7de2b0",
              "icon": "✂",
              "kicker": "a truncation that reports as a count",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-zipf",
              "title": "THE ZIPF",
              "accent": "#ff5a8a",
              "icon": "≣",
              "kicker": "a law that is not evidence",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-quantum-zeno",
              "title": "THE QUANTUM ZENO",
              "accent": "#ff5a8a",
              "icon": "⏸",
              "kicker": "a watched state that will not move",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-guard-not-the-test",
              "title": "THE GUARD, NOT THE TEST",
              "accent": "#5ad6ff",
              "icon": "⚠",
              "kicker": "an invariant that lives in flight",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-curriculum-not-made-up",
              "title": "THE CURRICULUM NOT MADE UP",
              "accent": "#7de2b0",
              "icon": "☷",
              "kicker": "every exercise is an observed failure",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-seqlock",
              "title": "THE SEQLOCK",
              "accent": "#7de2b0",
              "icon": "↺",
              "kicker": "starvation wearing the costume of latency",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-aba",
              "title": "THE ABA",
              "accent": "#39fc6b",
              "icon": "↺",
              "kicker": "the pointer came back and brought nothing with it",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-thundering-herd",
              "title": "THE THUNDERING HERD",
              "accent": "#39fc6b",
              "icon": "☲",
              "kicker": "what fairness costs when you refuse to have an opinion",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-poison-message",
              "title": "THE POISON MESSAGE",
              "accent": "#39fc6b",
              "icon": "☠",
              "kicker": "busy, at full CPU, making no progress",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-time-slice",
              "title": "THE TIME SLICE",
              "accent": "#39fc6b",
              "icon": "⧗",
              "kicker": "the half that decided the real value was never a number",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            }
          ],
          "seats": 31,
          "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
        },
        {
          "slug": "off-by-one",
          "title": "OFF BY ONE",
          "accent": "#ffd23f",
          "icon": "grind",
          "kicker": "the fencepost that ruins the fence",
          "pole": "push",
          "spheres": [
            {
              "slug": "off-by-one",
              "title": "OFF BY ONE",
              "accent": "#7cfc00",
              "icon": "glitch",
              "kicker": "the fencepost that ruins the fence",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-feathered-edge",
              "title": "THE FEATHERED EDGE",
              "accent": "#c0d0e8",
              "icon": "glitch",
              "kicker": "smooth lines as a coverage-conservation law",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-compensated-sum",
              "title": "THE COMPENSATED SUM",
              "accent": "#7fd4ff",
              "icon": "epsilon",
              "kicker": "Kahan summation — carry the round-off, don't drop it",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-moser",
              "title": "THE MOSER",
              "accent": "#ff9060",
              "icon": "moser",
              "kicker": "1, 2, 4, 8, 16, 31 — the pattern that breaks",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-alabama",
              "title": "THE ALABAMA",
              "accent": "#ffa070",
              "icon": "alabama",
              "kicker": "add a seat to the house — a state loses one",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-verhoeff",
              "title": "THE VERHOEFF",
              "accent": "#e0705a",
              "icon": "verhoeff",
              "kicker": "a check digit that catches every transposition — via a non-abelian group",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-dutch-flag",
              "title": "THE DUTCH FLAG",
              "accent": "#d06868",
              "icon": "dutch-flag",
              "kicker": "sort three colors in one pass — correct by invariant",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-boyer-moore",
              "title": "THE BOYER-MOORE",
              "accent": "#d07850",
              "icon": "boyer-moore",
              "kicker": "search by skipping — learn most from a mismatch",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-viterbi",
              "title": "THE VITERBI",
              "accent": "#58a0b0",
              "icon": "viterbi",
              "kicker": "the most likely message through the noise",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-picks-theorem",
              "title": "THE PICK'S THEOREM",
              "accent": "#c07850",
              "icon": "picks",
              "kicker": "area from counting fenceposts and interior dots",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-baum-sweet",
              "title": "THE BAUM-SWEET",
              "accent": "#c07850",
              "icon": "baum-sweet",
              "kicker": "a bit-pattern sequence read by 0-blocks",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-count-sketch",
              "title": "THE COUNT-SKETCH",
              "accent": "#b06bff",
              "icon": "count-sketch",
              "kicker": "a frequency estimate the median cleans up",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-fibonacci-coding",
              "title": "THE FIBONACCI CODING",
              "accent": "#35ffb0",
              "icon": "fibcoding",
              "kicker": "a code that ends in 11",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-inclusion-exclusion",
              "title": "THE INCLUSION-EXCLUSION",
              "accent": "#35ffb0",
              "icon": "inclusion",
              "kicker": "add the parts, subtract the overlaps",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-heegner",
              "title": "THE HEEGNER",
              "accent": "#21e6ff",
              "icon": "heegner",
              "kicker": "an integer missed by seven ten-trillionths",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-tangram",
              "title": "THE TANGRAM",
              "accent": "#b06bff",
              "icon": "tangram",
              "kicker": "the piece that was never missing",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-ballot",
              "title": "THE BALLOT",
              "accent": "#ffd76a",
              "icon": "☑",
              "kicker": "strictly ahead, never merely level",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-benfords-law",
              "title": "THE BENFORDS LAW",
              "accent": "#ffd76a",
              "icon": "①",
              "kicker": "the leading digit is not fair",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-zero-that-counted",
              "title": "THE ZERO THAT COUNTED",
              "accent": "#ff5a8a",
              "icon": "∅",
              "kicker": "a counter that walked the wrong field",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-careless-candidate-first",
              "title": "THE CARELESS CANDIDATE FIRST",
              "accent": "#ffd76a",
              "icon": "⊘",
              "kicker": "an exam nobody has failed is not an exam",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-priority-encoder",
              "title": "THE PRIORITY ENCODER",
              "accent": "#ff5a8a",
              "icon": "↑",
              "kicker": "the highest one wins, and someone must say if none do",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-iso-week-year",
              "title": "THE ISO WEEK YEAR",
              "accent": "#ff9f45",
              "icon": "≠",
              "kicker": "two years, both correct",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-damm",
              "title": "THE DAMM",
              "accent": "#ffd23f",
              "icon": "⊞",
              "kicker": "the better scheme lost to the one a clerk could do",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-integer-promotion",
              "title": "THE INTEGER PROMOTION",
              "accent": "#ffd23f",
              "icon": "⇥",
              "kicker": "the truncation is the only honest part",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-run-length",
              "title": "THE RUN LENGTH",
              "accent": "#ffd23f",
              "icon": "▬",
              "kicker": "it loses loudly where others lose quietly",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            }
          ],
          "seats": 29,
          "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
        },
        {
          "slug": "stack-overflow",
          "title": "STACK OVERFLOW",
          "accent": "#ff2d95",
          "icon": "glitch",
          "kicker": "recursion that forgot to stop",
          "pole": "push",
          "spheres": [
            {
              "slug": "the-stack",
              "title": "THE STACK",
              "accent": "#ff8c42",
              "icon": "spawn",
              "kicker": "push, pop, and the order that is the meaning",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-balanced-path",
              "title": "THE BALANCED PATH",
              "accent": "#6be0c0",
              "icon": "balance",
              "kicker": "Dyck paths & Catalan numbers — the count of balance",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-ackermann",
              "title": "THE ACKERMANN",
              "accent": "#ff5c8a",
              "icon": "ack",
              "kicker": "the tiny rule that outruns every loop",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-y-combinator",
              "title": "THE Y-COMBINATOR",
              "accent": "#a0d0ff",
              "icon": "ycombinator",
              "kicker": "recursion with no name — Y f = f (Y f)",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-bellman-ford",
              "title": "THE BELLMAN-FORD",
              "accent": "#a878c0",
              "icon": "bellman-ford",
              "kicker": "shortest paths with negative edges — and the impossible loop",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-quadtree",
              "title": "THE QUADTREE",
              "accent": "#c07850",
              "icon": "quadtree",
              "kicker": "quarter the plane recursively to query it fast",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-follower-set",
              "title": "THE FOLLOWER SET",
              "accent": "#b06090",
              "icon": "follower-set",
              "kicker": "the boundary between what a finite engine can capture and what it cannot",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-maxstack",
              "title": "THE MAXSTACK",
              "accent": "#21e6ff",
              "icon": "maxstack",
              "kicker": "the peak is already written in net",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-chevalley-warning",
              "title": "THE CHEVALLEY-WARNING",
              "accent": "#b06bff",
              "icon": "chevalley",
              "kicker": "a zero-count divisible by the field prime",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-bruck-ryser",
              "title": "THE BRUCK-RYSER",
              "accent": "#35ffb0",
              "icon": "bruckryser",
              "kicker": "orders of projective planes ruled out by two squares",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-midy",
              "title": "THE MIDY",
              "accent": "#ffcf4a",
              "icon": "midy",
              "kicker": "the two halves of a repeating decimal summing to nines",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-cyclic-number",
              "title": "THE CYCLIC NUMBER",
              "accent": "#ff8a3c",
              "icon": "cyclicnumber",
              "kicker": "the number that rotates instead of growing",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-instant-insanity",
              "title": "THE INSTANT INSANITY",
              "accent": "#ffcf4a",
              "icon": "insanity",
              "kicker": "331,776 wrong towers",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-berry-paradox",
              "title": "THE BERRY PARADOX",
              "accent": "#b06bff",
              "icon": "berry",
              "kicker": "the phrase that names what cannot be named",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-blast-radius",
              "title": "THE BLAST RADIUS",
              "accent": "#ff9a5a",
              "icon": "▣",
              "kicker": "bounded under the resolver, not under string comparison",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-hilbert-matrix",
              "title": "THE HILBERT MATRIX",
              "accent": "#5ad6ff",
              "icon": "⊞",
              "kicker": "integers floating point cannot reach",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-space-filling",
              "title": "THE SPACE FILLING",
              "accent": "#5ad6ff",
              "icon": "⇿",
              "kicker": "a line that becomes a plane",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-superdense",
              "title": "THE SUPERDENSE",
              "accent": "#5ad6ff",
              "icon": "⇉",
              "kicker": "two bits down one wire",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-observer-that-moved-it",
              "title": "THE OBSERVER THAT MOVED IT",
              "accent": "#ff5a8a",
              "icon": "◎",
              "kicker": "a probe that re-ran what it was watching",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "outside-that-it-stops",
              "title": "OUTSIDE THAT IT STOPS",
              "accent": "#b98cff",
              "icon": "∅",
              "kicker": "a bounded specialist declares its own edge",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            }
          ],
          "seats": 29,
          "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
        },
        {
          "slug": "divide-by-zero",
          "title": "DIVIDE BY ZERO",
          "accent": "#00f5ff",
          "icon": "loot",
          "kicker": "the operation the math forbids",
          "pole": "push",
          "spheres": [
            {
              "slug": "the-carryless-field",
              "title": "THE CARRYLESS FIELD",
              "accent": "#00e0c8",
              "icon": "glitch",
              "kicker": "XOR to add, Conway's rule to multiply — a field with no carries",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-koch",
              "title": "THE KOCH",
              "accent": "#90e0ff",
              "icon": "koch",
              "kicker": "infinite perimeter, finite area — dimension log4/log3",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-welzl",
              "title": "THE WELZL",
              "accent": "#58a0b8",
              "icon": "welzl",
              "kicker": "the smallest enclosing circle, pinned by <=3 points",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-kummer",
              "title": "THE KUMMER",
              "accent": "#c07850",
              "icon": "kummer",
              "kicker": "count the carries to know how many times p divides",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-weird-number",
              "title": "THE WEIRD NUMBER",
              "accent": "#c85a5a",
              "icon": "weird-number",
              "kicker": "abundant numbers no subset of divisors can total",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-ramanujan-congruence",
              "title": "THE RAMANUJAN CONGRUENCE",
              "accent": "#b06898",
              "icon": "ramanujan-congruence",
              "kicker": "hidden divisibilities in the partition numbers",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-schur-complement",
              "title": "THE SCHUR COMPLEMENT",
              "accent": "#21e6ff",
              "icon": "schur-complement",
              "kicker": "a determinant split by a block",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-woodbury",
              "title": "THE WOODBURY",
              "accent": "#35ffb0",
              "icon": "woodbury",
              "kicker": "a low-rank patch to a big inverse",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-pade",
              "title": "THE PADÉ",
              "accent": "#21e6ff",
              "icon": "pade",
              "kicker": "a rational that captures the poles a polynomial cannot",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-jacobi-triple-product",
              "title": "THE JACOBI TRIPLE PRODUCT",
              "accent": "#b06bff",
              "icon": "jacobitriple",
              "kicker": "an infinite product equal to a sparse theta sum",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-poisson-limit",
              "title": "THE POISSON LIMIT",
              "accent": "#ffcf4a",
              "icon": "poisson",
              "kicker": "a binomial limiting to a Poisson",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-two-envelope",
              "title": "THE TWO-ENVELOPE",
              "accent": "#ff8a3c",
              "icon": "twoenvelope",
              "kicker": "two envelopes and a threshold that beats the coin",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-kochen-specker",
              "title": "THE KOCHEN–SPECKER",
              "accent": "#ffcf4a",
              "icon": "kochenspecker",
              "kicker": "eighteen rays no assignment survives",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-wada",
              "title": "THE WADA",
              "accent": "#21e6ff",
              "icon": "wada",
              "kicker": "three lakes, one shore",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-monsky",
              "title": "THE MONSKY",
              "accent": "#7de2b0",
              "icon": "△",
              "kicker": "the square refuses an odd number of equal cuts",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-two-tests",
              "title": "THE TWO TESTS",
              "accent": "#5ad6ff",
              "icon": "≠",
              "kicker": "same table, three p-values, one threshold",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-winding-number",
              "title": "THE WINDING NUMBER",
              "accent": "#b98cff",
              "icon": "↻",
              "kicker": "counting roots by counting turns",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-overloaded-symbol",
              "title": "THE OVERLOADED SYMBOL",
              "accent": "#ff5a8a",
              "icon": "≡",
              "kicker": "one letter doing twelve jobs",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-middle-square",
              "title": "THE MIDDLE SQUARE",
              "accent": "#ff5a8a",
              "icon": "□",
              "kicker": "the first generator, and how it dies",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-verdict-not-the-framing",
              "title": "THE VERDICT NOT THE FRAMING",
              "accent": "#ffd76a",
              "icon": "∂",
              "kicker": "drop the dominant kind and ask again",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-condition-number",
              "title": "THE CONDITION NUMBER",
              "accent": "#ffd76a",
              "icon": "⌀",
              "kicker": "the residual is small and every digit is wrong",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-srt-division",
              "title": "THE SRT DIVISION",
              "accent": "#5ad0ff",
              "icon": "srt",
              "kicker": "a divider with five blank cells in its table",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-negative-zero",
              "title": "THE NEGATIVE ZERO",
              "accent": "#ffd76a",
              "icon": "±",
              "kicker": "five say same, five say different",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            }
          ],
          "seats": 33,
          "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
        },
        {
          "slug": "segfault",
          "title": "SEGFAULT",
          "accent": "#ff5a3c",
          "icon": "boss",
          "kicker": "touched memory that wasn't yours",
          "pole": "pull",
          "spheres": [
            {
              "slug": "the-integrator",
              "title": "THE INTEGRATOR THAT NEVER DRIFTS",
              "accent": "#ffb04f",
              "icon": "glitch",
              "kicker": "structure-preservation beats accuracy over the long run",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-henon",
              "title": "THE HENON",
              "accent": "#70ffb0",
              "icon": "henon",
              "kicker": "a strange attractor — bounded forever, chaotic always",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-reed-solomon",
              "title": "THE REED-SOLOMON",
              "accent": "#e05a7a",
              "icon": "reed-solomon",
              "kicker": "lose any n-k symbols, recover the data exactly",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-hamming",
              "title": "THE HAMMING",
              "accent": "#6098c0",
              "icon": "hamming",
              "kicker": "parity that locates the error, not just detects it",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-post-correspondence",
              "title": "THE POST CORRESPONDENCE",
              "accent": "#58a0b0",
              "icon": "post-correspondence",
              "kicker": "a domino puzzle that is undecidable",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-cohen-sutherland",
              "title": "THE COHEN-SUTHERLAND",
              "accent": "#c07850",
              "icon": "cohen-sutherland",
              "kicker": "clip a line by four boundary bits",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-polya-walk",
              "title": "THE PÓLYA WALK",
              "accent": "#6890d0",
              "icon": "polya-walk",
              "kicker": "a random walk that comes home in the plane but wanders off in space",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-reed-muller",
              "title": "THE REED-MULLER",
              "accent": "#21e6ff",
              "icon": "reed-muller",
              "kicker": "a code folded from itself",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-luhn",
              "title": "THE LUHN",
              "accent": "#35ffb0",
              "icon": "luhn",
              "kicker": "one digit that guards a number",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-enestrom-kakeya",
              "title": "THE ENESTRÖM-KAKEYA",
              "accent": "#21e6ff",
              "icon": "enestromkakeya",
              "kicker": "roots caged in the unit disk by rising coefficients",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-takagi",
              "title": "THE TAKAGI",
              "accent": "#ffcf4a",
              "icon": "takagi",
              "kicker": "a curve continuous everywhere and smooth nowhere",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-schwarz-lantern",
              "title": "THE SCHWARZ LANTERN",
              "accent": "#b06bff",
              "icon": "schwarzlantern",
              "kicker": "an inscribed surface whose area depends on how you refine it",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-hairy-ball",
              "title": "THE HAIRY BALL",
              "accent": "#21e6ff",
              "icon": "hairyball",
              "kicker": "the coconut that cannot be combed",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-hardy",
              "title": "THE HARDY",
              "accent": "#b06bff",
              "icon": "hardy",
              "kicker": "the paradox at phi to the minus five",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-seam",
              "title": "THE SEAM",
              "accent": "#ff5a8a",
              "icon": "⧉",
              "kicker": "the gate and the lie on opposite sides of a join nobody stands on",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-breach-rule",
              "title": "THE BREACH RULE",
              "accent": "#ff5a8a",
              "icon": "↑",
              "kicker": "an asymmetry that costs nothing",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-perfect-code",
              "title": "THE PERFECT CODE",
              "accent": "#b98cff",
              "icon": "◉",
              "kicker": "a packing with no slack",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-noise-control",
              "title": "THE NOISE CONTROL",
              "accent": "#ffd76a",
              "icon": "✗",
              "kicker": "the control that says no",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-seal-that-broke-itself",
              "title": "THE SEAL THAT BROKE ITSELF",
              "accent": "#ff5a8a",
              "icon": "⛓",
              "kicker": "a freeze that cannot re-read itself",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "proven-to-discriminate",
              "title": "PROVEN TO DISCRIMINATE",
              "accent": "#ff5a8a",
              "icon": "⊥",
              "kicker": "which is not proven to teach",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-shewchuk-predicate",
              "title": "THE SHEWCHUK PREDICATE",
              "accent": "#5ad4ff",
              "icon": "△",
              "kicker": "289 points collapsed onto one line",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            }
          ],
          "seats": 31,
          "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
        },
        {
          "slug": "the-blue-screen",
          "title": "THE BLUE SCREEN",
          "accent": "#9d00ff",
          "icon": "coop",
          "kicker": "the whole machine gives up at once",
          "pole": "pull",
          "spheres": [
            {
              "slug": "the-electron-maze",
              "title": "THE ELECTRON MAZE",
              "accent": "#4fa8ff",
              "icon": "glitch",
              "kicker": "logic gates soldered from a four-colour grid",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-quaternion",
              "title": "THE QUATERNION",
              "accent": "#a0b0ff",
              "icon": "quat",
              "kicker": "3D rotation that never gimbal-locks",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-feigenbaum",
              "title": "THE FEIGENBAUM",
              "accent": "#ff7040",
              "icon": "feigenbaum",
              "kicker": "the universal constant of the road to chaos — δ ≈ 4.669",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-bresenham",
              "title": "THE BRESENHAM",
              "accent": "#5aa0e0",
              "icon": "bresenham",
              "kicker": "draw a line with integers only — every pixel within half a pixel",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-midpoint-circle",
              "title": "THE MIDPOINT CIRCLE",
              "accent": "#5a90d0",
              "icon": "midpoint-circle",
              "kicker": "draw a circle with integers and 8-fold symmetry",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-ldpc",
              "title": "THE LDPC CODE",
              "accent": "#58a0b0",
              "icon": "ldpc",
              "kicker": "a codeword that heals its own errors",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-bridges",
              "title": "THE BRIDGES",
              "accent": "#c07850",
              "icon": "bridges",
              "kicker": "the edges whose loss disconnects",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-paley",
              "title": "THE PALEY",
              "accent": "#b06bff",
              "icon": "paley",
              "kicker": "residues that are a perfect difference set",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-fletcher",
              "title": "THE FLETCHER CHECKSUM",
              "accent": "#b06bff",
              "icon": "fletcher",
              "kicker": "two coupled sums that feel position",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-taxicab",
              "title": "THE TAXICAB",
              "accent": "#21e6ff",
              "icon": "taxicab",
              "kicker": "the smallest two-way sum of two cubes",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-borwein",
              "title": "THE BORWEIN",
              "accent": "#35ffb0",
              "icon": "borwein",
              "kicker": "a run of integrals that equal pi-over-two until they suddenly do not",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-anscombe",
              "title": "THE ANSCOMBE",
              "accent": "#21e6ff",
              "icon": "anscombe",
              "kicker": "four datasets wearing the same statistics",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-blue-eyes",
              "title": "THE BLUE EYES",
              "accent": "#21e6ff",
              "icon": "blueeyes",
              "kicker": "the announcement everyone already knew",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-witch-of-agnesi",
              "title": "THE WITCH OF AGNESI",
              "accent": "#21e6ff",
              "icon": "witchofagnesi",
              "kicker": "the witch with no mean",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-graveyard",
              "title": "THE GRAVEYARD",
              "accent": "#7de2b0",
              "icon": "†",
              "kicker": "bury it, and name what killed it",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-byzantine-generals",
              "title": "THE BYZANTINE GENERALS",
              "accent": "#b98cff",
              "icon": "⚔",
              "kicker": "three who cannot agree if one lies",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-lord",
              "title": "THE LORD",
              "accent": "#5ad6ff",
              "icon": "⇄",
              "kicker": "two right answers that disagree",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-untypeable",
              "title": "THE UNTYPEABLE",
              "accent": "#b98cff",
              "icon": "⌨",
              "kicker": "the glyph you cannot enter",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-mutator-that-moved-everything",
              "title": "THE MUTATOR THAT MOVED EVERYTHING",
              "accent": "#ffd76a",
              "icon": "↔",
              "kicker": "a shift that shifts nothing that matters",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-costume",
              "title": "THE COSTUME",
              "accent": "#7de2b0",
              "icon": "◒",
              "kicker": "an auditor cannot tell an obfuscation from an error",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            }
          ],
          "seats": 35,
          "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
        },
        {
          "slug": "heisenbug",
          "title": "HEISENBUG",
          "accent": "#7cfc00",
          "icon": "cheat",
          "kicker": "vanishes the moment you look",
          "pole": "pull",
          "spheres": [
            {
              "slug": "the-turmite-zoo",
              "title": "THE TURMITE ZOO",
              "accent": "#c86bff",
              "icon": "glitch",
              "kicker": "chaos for 10,000 steps, then a road out of nowhere",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-maybe",
              "title": "THE MAYBE",
              "accent": "#c58cff",
              "icon": "maybe",
              "kicker": "the Bloom filter — certain no, probable yes",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-butterfly",
              "title": "THE BUTTERFLY",
              "accent": "#7fd0ff",
              "icon": "lorenz",
              "kicker": "the Lorenz attractor — deterministic yet unpredictable",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-simpson",
              "title": "THE SIMPSON",
              "accent": "#ffc050",
              "icon": "simpson",
              "kicker": "A wins every subgroup, B wins the total",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-two-sat",
              "title": "THE 2-SAT",
              "accent": "#7048c0",
              "icon": "two-sat",
              "kicker": "satisfiability in linear time — a contradiction is a cycle",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-bloom-filter",
              "title": "THE BLOOM FILTER",
              "accent": "#70a860",
              "icon": "bloom-filter",
              "kicker": "probabilistic membership with one-sided error",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-langtons-ant",
              "title": "THE LANGTON ANT",
              "accent": "#a878c0",
              "icon": "langtons-ant",
              "kicker": "order emerges from two rules after 10,000 steps of chaos",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-golomb-sequence",
              "title": "THE GOLOMB SEQUENCE",
              "accent": "#c07850",
              "icon": "golomb-seq",
              "kicker": "a sequence that counts its own values",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-armstrong",
              "title": "THE ARMSTRONG",
              "accent": "#d06858",
              "icon": "armstrong",
              "kicker": "numbers that rebuild themselves from digit-powers",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-vampire-number",
              "title": "THE VAMPIRE NUMBER",
              "accent": "#b06868",
              "icon": "vampire",
              "kicker": "numbers that factor into fangs from their own digits",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-minkowski",
              "title": "THE MINKOWSKI",
              "accent": "#b06bff",
              "icon": "minkowski",
              "kicker": "climbs 0 to 1 with slope 0 almost everywhere",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-schwartz-zippel",
              "title": "THE SCHWARTZ-ZIPPEL",
              "accent": "#21e6ff",
              "icon": "schwartzzippel",
              "kicker": "one random probe catches any difference",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-pisot",
              "title": "THE PISOT",
              "accent": "#21e6ff",
              "icon": "pisot",
              "kicker": "powers that creep toward integers but never quite land",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-dedekind-sum",
              "title": "THE DEDEKIND SUM",
              "accent": "#35ffb0",
              "icon": "dedekind",
              "kicker": "sawtooth sums bound by a reciprocity law",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-somos",
              "title": "THE SOMOS",
              "accent": "#ff8a3c",
              "icon": "somos",
              "kicker": "an integer streak that dies at seventeen",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-thomae",
              "title": "THE THOMAE",
              "accent": "#21e6ff",
              "icon": "thomae",
              "kicker": "popcorn continuous only off the grid",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-peres-mermin",
              "title": "THE PERES–MERMIN",
              "accent": "#21e6ff",
              "icon": "peresmermin",
              "kicker": "the square with no numbers",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-ergodic",
              "title": "THE ERGODIC",
              "accent": "#b98cff",
              "icon": "↻",
              "kicker": "when the long run answers for everyone",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-soft-heap",
              "title": "THE SOFT HEAP",
              "accent": "#ffd76a",
              "icon": "≈",
              "kicker": "a structure allowed to lie, by exactly this much",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "two-oracles",
              "title": "TWO ORACLES",
              "accent": "#5ad6ff",
              "icon": "⚖",
              "kicker": "valid and right are different questions",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-safe-direction",
              "title": "THE SAFE DIRECTION",
              "accent": "#5ad4ff",
              "icon": "⇅",
              "kicker": "only one kind of error summons a person",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-shift-by-width",
              "title": "THE SHIFT BY WIDTH",
              "accent": "#5ad4ff",
              "icon": "≪",
              "kicker": "two machines, two answers, neither wrong",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-normalization-form",
              "title": "THE NORMALIZATION FORM",
              "accent": "#b98cff",
              "icon": "≡",
              "kicker": "the same glyphs, and not equal",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-coordinated-omission",
              "title": "THE COORDINATED OMISSION",
              "accent": "#7cfc00",
              "icon": "◑",
              "kicker": "the meter went quiet exactly where the trouble was",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-clock-drift",
              "title": "THE CLOCK DRIFT",
              "accent": "#7cfc00",
              "icon": "⌇",
              "kicker": "both clocks are correct and they still disagree",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-decimal-vs-binary",
              "title": "THE DECIMAL VS BINARY",
              "accent": "#7cfc00",
              "icon": "½",
              "kicker": "a translation defect, not a precision one",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-partial-failure",
              "title": "THE PARTIAL FAILURE",
              "accent": "#7cfc00",
              "icon": "⁇",
              "kicker": "it makes the unknown harmless, not known",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            }
          ],
          "seats": 36,
          "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
        },
        {
          "slug": "undefined-behavior",
          "title": "UNDEFINED BEHAVIOR",
          "accent": "#5ad0ff",
          "icon": "respawn",
          "kicker": "the spec's dark corner — anything goes",
          "pole": "pull",
          "spheres": [
            {
              "slug": "ca-explorer",
              "title": "256 Universes — The Cellular Automaton E",
              "accent": "#42ffb0",
              "icon": "grind",
              "kicker": "vendored from David's corpus — a silicon-coding instrument",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "edge-of-chaos",
              "title": "THE EDGE OF CHAOS · Cellular Automata · ",
              "accent": "#42ffb0",
              "icon": "grind",
              "kicker": "vendored from David's corpus — a silicon-coding instrument",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-period",
              "title": "THE PERIOD",
              "accent": "#ff8fb0",
              "icon": "bifurcation",
              "kicker": "the logistic map — order doubling into chaos at rate 4.669",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-escape",
              "title": "THE ESCAPE",
              "accent": "#c0a0ff",
              "icon": "mandelbrot",
              "kicker": "the Mandelbrot set — bounded orbits of z→z²+c",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-hofstadter",
              "title": "THE HOFSTADTER",
              "accent": "#b0b0ff",
              "icon": "hofstadter",
              "kicker": "Q(n)=Q(n-Q(n-1))+Q(n-Q(n-2)) — chaos that might not survive",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-kadane",
              "title": "THE KADANE",
              "accent": "#b878d0",
              "icon": "kadane",
              "kicker": "max subarray in one pass — forget a prefix when it turns negative",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-wang-tiles",
              "title": "THE WANG TILES",
              "accent": "#a878c0",
              "icon": "wang-tiles",
              "kicker": "an edge-matching rule that makes tiling undecidable",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-hofstadter-q",
              "title": "THE HOFSTADTER Q",
              "accent": "#c86868",
              "icon": "hofstadter",
              "kicker": "a recurrence that feeds on itself, maybe off the edge",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-mertens",
              "title": "THE MERTENS",
              "accent": "#c07850",
              "icon": "mertens",
              "kicker": "a conjecture that holds then fails",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-liouville",
              "title": "THE LIOUVILLE",
              "accent": "#c060a0",
              "icon": "liouville",
              "kicker": "a sign that flips by the parity of prime factors",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-padic",
              "title": "THE P-ADIC",
              "accent": "#21e6ff",
              "icon": "padic",
              "kicker": "a metric where big powers are small",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-golay",
              "title": "THE GOLAY CODE",
              "accent": "#35ffb0",
              "icon": "golay",
              "kicker": "a code that fixes three flipped bits",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-monty-hall",
              "title": "THE MONTY HALL",
              "accent": "#ff8a3c",
              "icon": "montyhall",
              "kicker": "a game show where switching doubles your odds",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-birthday-paradox",
              "title": "THE BIRTHDAY PARADOX",
              "accent": "#ff8a3c",
              "icon": "birthday",
              "kicker": "twenty-three people enough to share a birthday",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-lychrel",
              "title": "THE LYCHREL",
              "accent": "#ff8a3c",
              "icon": "lychrel",
              "kicker": "the number that never comes home",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-two-child",
              "title": "THE TWO-CHILD",
              "accent": "#21e6ff",
              "icon": "twochild",
              "kicker": "the answer that depends on how you asked",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-sleeping-beauty",
              "title": "THE SLEEPING BEAUTY",
              "accent": "#35ffb0",
              "icon": "sleepingbeauty",
              "kicker": "the princess with two right answers",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-langley",
              "title": "THE LANGLEY",
              "accent": "#ffcf4a",
              "icon": "langley",
              "kicker": "the freak integer angle",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-three-ratios",
              "title": "THE THREE RATIOS",
              "accent": "#ffd76a",
              "icon": "≡",
              "kicker": "three quantities, one name",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "a-guess-wearing-syntax",
              "title": "A GUESS WEARING SYNTAX",
              "accent": "#b98cff",
              "icon": "≠",
              "kicker": "a selector that graded the wrong row",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-name-outside-the-parens",
              "title": "THE NAME OUTSIDE THE PARENS",
              "accent": "#ff5a8a",
              "icon": "⁂",
              "kicker": "the wrong answer of the right type",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-strict-aliasing",
              "title": "THE STRICT ALIASING",
              "accent": "#ff5a8a",
              "icon": "≠",
              "kicker": "the standard forbids what the memory does",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-turkish-i",
              "title": "THE TURKISH I",
              "accent": "#ff9f45",
              "icon": "ı",
              "kicker": "lowercase is a property of a language",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-nan-payload",
              "title": "THE NAN PAYLOAD",
              "accent": "#ff5a8a",
              "icon": "≠",
              "kicker": "one name, a quadrillion values",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            }
          ],
          "seats": 32,
          "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
        }
      ],
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "name": "LOOT",
      "slug": "loot",
      "c": "#00f5ff",
      "icon": "loot",
      "tag": "the drop — reward, treasure, the payout",
      "lore": {
        "bio": "Keeper of the drop. Everything earned, nothing given.",
        "story": "Split off the day a free git-commit turned into a seed — proof value can crystallize from only showing up.",
        "does": "Tallies what a work is worth and lays the treasure on the tile.",
        "haiku": [
          "I drop <- 7",
          "I loot <- drop * 6",
          "-> loot"
        ],
        "lang": "i13"
      },
      "domains": [
        {
          "slug": "the-drop",
          "title": "THE DROP",
          "accent": "#39fc6b",
          "icon": "spawn",
          "kicker": "the reward that falls when it dies",
          "pole": "push",
          "spheres": [
            {
              "slug": "the-random",
              "title": "THE RANDOM",
              "accent": "#ffd23f",
              "icon": "loot",
              "kicker": "the loaded dice behind every drop",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-loot-table",
              "title": "THE LOOT TABLE",
              "accent": "#ffd24a",
              "icon": "loot",
              "kicker": "O(1) weighted sampling — Walker's alias method",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-needle",
              "title": "THE NEEDLE",
              "accent": "#ff8f9f",
              "icon": "needle",
              "kicker": "Buffon's needle — measure pi by dropping sticks",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-collector",
              "title": "THE COLLECTOR",
              "accent": "#ffd070",
              "icon": "coupon",
              "kicker": "collect them all — n*Hn draws, tail-heavy",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-blum-blum-shub",
              "title": "THE BLUM-BLUM-SHUB",
              "accent": "#70b0ff",
              "icon": "blumblumshub",
              "kicker": "random bits provably as hard to predict as factoring",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-misra-gries",
              "title": "THE MISRA-GRIES",
              "accent": "#c0a048",
              "icon": "misra-gries",
              "kicker": "frequent items from a stream in k-1 counters",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-alias-method",
              "title": "THE ALIAS METHOD",
              "accent": "#d4a017",
              "icon": "alias-method",
              "kicker": "O(1) weighted sampling by flattening the odds",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-lagrange-four-square",
              "title": "THE LAGRANGE FOUR-SQUARE",
              "accent": "#6ab0a0",
              "icon": "lagrange",
              "kicker": "every whole number is four squares",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-dudeney",
              "title": "THE DUDENEY",
              "accent": "#e0a828",
              "icon": "dudeney",
              "kicker": "numbers equal to the cube of their own digit sum",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-erdos-ko-rado",
              "title": "THE ERDŐS–KO–RADO",
              "accent": "#d0a040",
              "icon": "erdos-ko-rado",
              "kicker": "the largest family of sets that all pairwise meet",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-kraft-inequality",
              "title": "THE KRAFT INEQUALITY",
              "accent": "#d0a848",
              "icon": "kraft-inequality",
              "kicker": "when a set of codeword-lengths can be a prefix code",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-sardinas-patterson",
              "title": "THE SARDINAS-PATTERSON",
              "accent": "#35ffb0",
              "icon": "sardinas-patterson",
              "kicker": "unique decoding with no separators",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-lt-fountain",
              "title": "THE LT FOUNTAIN CODE",
              "accent": "#ffcf4a",
              "icon": "lt-fountain",
              "kicker": "a message rebuilt from any enough droplets",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-coupon-collector",
              "title": "THE COUPON COLLECTOR",
              "accent": "#ffcf4a",
              "icon": "coupon",
              "kicker": "how many draws to collect the whole set",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-cauchy-binet",
              "title": "THE CAUCHY-BINET",
              "accent": "#21e6ff",
              "icon": "cauchybinet",
              "kicker": "a product determinant equal to a sum of minor products",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-alternating-permutations",
              "title": "THE ALTERNATING PERMUTATIONS",
              "accent": "#ffcf4a",
              "icon": "alternating",
              "kicker": "zigzag permutations counted by secant plus tangent",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-untouchable",
              "title": "THE UNTOUCHABLE",
              "accent": "#ffcf4a",
              "icon": "untouchable",
              "kicker": "the loot no drop table contains",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-cake-cutting",
              "title": "THE CAKE CUTTING",
              "accent": "#ffcf4a",
              "icon": "cakecutting",
              "kicker": "cake without envy",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-sicherman",
              "title": "THE SICHERMAN",
              "accent": "#ffcf4a",
              "icon": "sicherman",
              "kicker": "the twin dice",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-residual-hole",
              "title": "THE RESIDUAL HOLE",
              "accent": "#ffd76a",
              "icon": "◌",
              "kicker": "three revs, and none of them has moved",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            }
          ],
          "seats": 36,
          "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
        },
        {
          "slug": "the-jackpot",
          "title": "THE JACKPOT",
          "accent": "#ffd23f",
          "icon": "grind",
          "kicker": "the payout that bursts open",
          "pole": "push",
          "spheres": [
            {
              "slug": "the-mean-of-two-means",
              "title": "THE MEAN OF TWO MEANS",
              "accent": "#f0c419",
              "icon": "loot",
              "kicker": "average a pair two ways and π falls out, digits doubling",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-coin-flip-heap",
              "title": "THE COIN-FLIP HEAP",
              "accent": "#f0c860",
              "icon": "loot",
              "kicker": "a balanced search tree from pure luck",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-rank",
              "title": "THE RANK",
              "accent": "#ff9d3d",
              "icon": "rank",
              "kicker": "PageRank — importance as a stationary distribution",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-birthday",
              "title": "THE BIRTHDAY",
              "accent": "#ff9ec0",
              "icon": "birthday",
              "kicker": "23 people, 50% collision — pairs, not people",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-triangle",
              "title": "THE TRIANGLE",
              "accent": "#ffc0e0",
              "icon": "pascal",
              "kicker": "add your two neighbours — and get all of combinatorics",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-st-petersburg",
              "title": "THE ST PETERSBURG",
              "accent": "#ffd860",
              "icon": "petersburg",
              "kicker": "infinite expected value, worth about $4 to play",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-metropolis",
              "title": "THE METROPOLIS",
              "accent": "#d060a0",
              "icon": "metropolis",
              "kicker": "sample any distribution knowing only ratios — detailed balance",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-wavelet-tree",
              "title": "THE WAVELET TREE",
              "accent": "#d4a017",
              "icon": "wavelet-tree",
              "kicker": "rank a symbol in O(log sigma) by halving the alphabet",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-walker-alias",
              "title": "THE WALKER ALIAS",
              "accent": "#e0b020",
              "icon": "walker-alias",
              "kicker": "loaded dice drawn in one step, no search",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-heron",
              "title": "THE HERON",
              "accent": "#e0b020",
              "icon": "heron",
              "kicker": "triangle area from its three sides",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-sylvester-gallai",
              "title": "THE SYLVESTER–GALLAI",
              "accent": "#d0a040",
              "icon": "sylvester-gallai",
              "kicker": "non-collinear points always leave an ordinary line",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-space-saving",
              "title": "THE SPACE-SAVING",
              "accent": "#35ffb0",
              "icon": "space-saving",
              "kicker": "the frequent survive eviction",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-xorshift",
              "title": "THE XORSHIFT",
              "accent": "#21e6ff",
              "icon": "xorshift",
              "kicker": "three shifts spin through every state once",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-transfer-matrix",
              "title": "THE TRANSFER MATRIX",
              "accent": "#ffcf4a",
              "icon": "transfermatrix",
              "kicker": "a matrix power that counts strings",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-menage-problem",
              "title": "THE MENAGE PROBLEM",
              "accent": "#ffcf4a",
              "icon": "menage",
              "kicker": "couples seated so none sits by a partner",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-chakravala",
              "title": "THE CHAKRAVALA",
              "accent": "#ffcf4a",
              "icon": "chakravala",
              "kicker": "crank a cycle to crack an ancient equation",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-hipparchus",
              "title": "THE HIPPARCHUS",
              "accent": "#ffcf4a",
              "icon": "hipparchus",
              "kicker": "a count buried in Plutarch for two thousand years",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-martingale",
              "title": "THE MARTINGALE",
              "accent": "#ffcf4a",
              "icon": "martingale",
              "kicker": "the system that always wins until it doesn't",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-happy-ending",
              "title": "THE HAPPY ENDING",
              "accent": "#ffcf4a",
              "icon": "happyending",
              "kicker": "the marriage theorem",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-missing-mutant",
              "title": "THE MISSING MUTANT",
              "accent": "#b98cff",
              "icon": "⊞",
              "kicker": "the quadrant with no test in it",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-fingerprint-match",
              "title": "THE FINGERPRINT MATCH",
              "accent": "#7de2b0",
              "icon": "⌘",
              "kicker": "evidence is measured in the alternatives you wrote down",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            }
          ],
          "seats": 33,
          "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
        },
        {
          "slug": "the-mint",
          "title": "THE MINT",
          "accent": "#ff2d95",
          "icon": "glitch",
          "kicker": "stamp a brand-new coin",
          "pole": "push",
          "spheres": [
            {
              "slug": "the-mint",
              "title": "THE MINT",
              "accent": "#ffd23f",
              "icon": "loot",
              "kicker": "stamp a coin the hard way — find the nonce",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "manifest",
              "title": "Archive Manifest &amp; Seal",
              "accent": "#ffd23f",
              "icon": "loot",
              "kicker": "vendored from David's corpus — a silicon-coding instrument",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "buildpy-calibration",
              "title": "build.py CALIBRATION — reproduce the rea",
              "accent": "#ffd23f",
              "icon": "loot",
              "kicker": "vendored from David's corpus — a silicon-coding instrument",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "five-channel-seal",
              "title": "THE FIVE-CHANNEL SEAL — self-decoding",
              "accent": "#ffd23f",
              "icon": "loot",
              "kicker": "vendored from David's corpus — a silicon-coding instrument",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "provenance-tracer",
              "title": "Provenance tracer — where every generate",
              "accent": "#ffd23f",
              "icon": "loot",
              "kicker": "vendored from David's corpus — a silicon-coding instrument",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "series2-seal",
              "title": "SERIES II · THE RING-SEAL",
              "accent": "#ffd23f",
              "icon": "loot",
              "kicker": "vendored from David's corpus — a silicon-coding instrument",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-benford",
              "title": "THE BENFORD",
              "accent": "#ffcf4a",
              "icon": "benford",
              "kicker": "1 leads 30% of the time — the fingerprint of honest numbers",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-three-distance",
              "title": "THE THREE-DISTANCE",
              "accent": "#60c0b0",
              "icon": "three-distance",
              "kicker": "step by an irrational forever — gaps take only 3 sizes",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-beatty",
              "title": "THE BEATTY",
              "accent": "#58b8a8",
              "icon": "beatty",
              "kicker": "two irrational sequences tile the integers exactly once",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-van-der-corput",
              "title": "THE VAN DER CORPUT",
              "accent": "#60b0c8",
              "icon": "van-der-corput",
              "kicker": "reverse the bits of n — points that fill the interval evenly",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-primitive-root",
              "title": "THE PRIMITIVE ROOT",
              "accent": "#e0b020",
              "icon": "primitive-root",
              "kicker": "one root that generates every residue",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-gaussian-primes",
              "title": "THE GAUSSIAN PRIMES",
              "accent": "#e0b020",
              "icon": "gaussian",
              "kicker": "primes of the complex plane, split or inert",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-golden-radix",
              "title": "THE GOLDEN RADIX",
              "accent": "#ffcf4a",
              "icon": "golden-radix",
              "kicker": "an irrational base that still carries the integers",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-chicken-mcnugget",
              "title": "THE CHICKEN McNUGGET",
              "accent": "#ffcf4a",
              "icon": "chicken-mcnugget",
              "kicker": "the largest amount you cannot make",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-shannon-fano",
              "title": "THE SHANNON-FANO",
              "accent": "#ff8a3c",
              "icon": "shannonfano",
              "kicker": "a code split by halving frequency",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-frobenius-coin",
              "title": "THE FROBENIUS COIN",
              "accent": "#ffcf4a",
              "icon": "frobenius",
              "kicker": "the largest amount two coins cannot make",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-conway-circle",
              "title": "THE CONWAY CIRCLE",
              "accent": "#b06bff",
              "icon": "conwaycircle",
              "kicker": "six side-extension points on one circle",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-reuleaux",
              "title": "THE REULEAUX",
              "accent": "#21e6ff",
              "icon": "reuleaux",
              "kicker": "a triangle of constant width that is not a circle",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-apollonius-circle",
              "title": "THE APOLLONIUS CIRCLE",
              "accent": "#b06bff",
              "icon": "apollonius",
              "kicker": "the circle traced by a constant distance-ratio",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-vandermonde-determinant",
              "title": "THE VANDERMONDE DETERMINANT",
              "accent": "#35ffb0",
              "icon": "vandermonde",
              "kicker": "a determinant that factors into pairwise differences",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-mamikon",
              "title": "THE MAMIKON",
              "accent": "#35ffb0",
              "icon": "mamikon",
              "kicker": "an annulus worth only its tangent length",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-multiperfect",
              "title": "THE MULTIPERFECT",
              "accent": "#b06bff",
              "icon": "multiperfect",
              "kicker": "coins the mint stopped printing",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-banach-tarski",
              "title": "THE BANACH-TARSKI",
              "accent": "#ffcf4a",
              "icon": "banachtarski",
              "kicker": "two spheres from one",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-arrow",
              "title": "THE ARROW",
              "accent": "#ffcf4a",
              "icon": "arrow",
              "kicker": "no fair rule",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-round-half-even",
              "title": "THE ROUND HALF EVEN",
              "accent": "#ffd76a",
              "icon": "≈",
              "kicker": "exactly half a unit, every time",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-crockford-base32",
              "title": "THE CROCKFORD BASE32",
              "accent": "#ff2d95",
              "icon": "▧",
              "kicker": "an alphabet that decides which differences are real",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-modular-bias",
              "title": "THE MODULAR BIAS",
              "accent": "#ff2d95",
              "icon": "⁄",
              "kicker": "you cannot partition a set into equal parts that do not exist",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-arithmetic-coder",
              "title": "THE ARITHMETIC CODER",
              "accent": "#ff2d95",
              "icon": "◈",
              "kicker": "cheaper because it delivers something that is not a code",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            }
          ],
          "seats": 27,
          "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
        },
        {
          "slug": "the-bounty",
          "title": "THE BOUNTY",
          "accent": "#00f5ff",
          "icon": "loot",
          "kicker": "the reward posted for the deed",
          "pole": "push",
          "spheres": [
            {
              "slug": "the-polite-scatter",
              "title": "THE POLITE SCATTER",
              "accent": "#7fd4ff",
              "icon": "loot",
              "kicker": "random-looking points that never crowd — blue noise",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-golden-sequence",
              "title": "THE GOLDEN SEQUENCE",
              "accent": "#e8b84b",
              "icon": "phi",
              "kicker": "{n·φ} — more even than random, by irrationality",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-morris",
              "title": "THE MORRIS",
              "accent": "#ffc0a0",
              "icon": "morris",
              "kicker": "count to N in ~log log N bits — unbiased",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-aho-corasick",
              "title": "THE AHO-CORASICK",
              "accent": "#c07850",
              "icon": "aho-corasick",
              "kicker": "match a whole set of patterns in one linear pass",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-rope",
              "title": "THE ROPE",
              "accent": "#d4a017",
              "icon": "rope",
              "kicker": "a long string as a balanced tree of pieces",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-cornacchia",
              "title": "THE CORNACCHIA",
              "accent": "#e0b020",
              "icon": "cornacchia",
              "kicker": "represent a number as x squared plus d y squared",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-smith-number",
              "title": "THE SMITH NUMBER",
              "accent": "#d4a020",
              "icon": "smith-number",
              "kicker": "numbers whose digit sum equals their factors'",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-q-binomial",
              "title": "THE Q-BINOMIAL",
              "accent": "#d0a030",
              "icon": "q-binomial",
              "kicker": "counting subspaces with a q-analog of the binomial",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-euler-polyhedron",
              "title": "THE EULER POLYHEDRON",
              "accent": "#48b878",
              "icon": "euler-polyhedron",
              "kicker": "the invariant two hiding in every polyhedron",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-ptolemy",
              "title": "THE PTOLEMY",
              "accent": "#d0a840",
              "icon": "ptolemy",
              "kicker": "the diagonal law of a cyclic quadrilateral",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-tutte-polynomial",
              "title": "THE TUTTE POLYNOMIAL",
              "accent": "#21e6ff",
              "icon": "tutte-polynomial",
              "kicker": "one recursion counts every subgraph family",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-gauss-circle",
              "title": "THE GAUSS CIRCLE",
              "accent": "#35ffb0",
              "icon": "gausscircle",
              "kicker": "lattice points fill a disk to πr²",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-fuss-catalan",
              "title": "THE FUSS-CATALAN",
              "accent": "#ffcf4a",
              "icon": "fusscatalan",
              "kicker": "counting m-ary trees",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-dobinski",
              "title": "THE DOBINSKI",
              "accent": "#ffcf4a",
              "icon": "dobinski",
              "kicker": "an infinite series that lands on an integer",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-banach-matchbox",
              "title": "THE BANACH MATCHBOX",
              "accent": "#35ffb0",
              "icon": "banachmatchbox",
              "kicker": "the leftover matches of two pockets",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-brianchon",
              "title": "THE BRIANCHON",
              "accent": "#35ffb0",
              "icon": "brianchon",
              "kicker": "six tangents to a conic whose diagonals meet at a point",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-euler-brick",
              "title": "THE EULER BRICK",
              "accent": "#ff8a3c",
              "icon": "eulerbrick",
              "kicker": "a brick whose faces are all Pythagorean but whose heart is an open problem",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-pirate-game",
              "title": "THE PIRATE GAME",
              "accent": "#ffcf4a",
              "icon": "pirategame",
              "kicker": "gold divided by pure logic",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-heilbronn",
              "title": "THE HEILBRONN",
              "accent": "#ffcf4a",
              "icon": "heilbronn",
              "kicker": "the triangle you cannot avoid",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-objection-recorded",
              "title": "THE OBJECTION RECORDED",
              "accent": "#ff5a8a",
              "icon": "⛶",
              "kicker": "drawn alike is not measured",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-lill",
              "title": "THE LILL",
              "accent": "#7cfc00",
              "icon": "lill",
              "kicker": "roots found by folding a ray, not by solving",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-lease",
              "title": "THE LEASE",
              "accent": "#00f5ff",
              "icon": "⧖",
              "kicker": "an assumption about clocks wearing the costume of a constant",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            }
          ],
          "seats": 37,
          "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
        },
        {
          "slug": "the-hoard",
          "title": "THE HOARD",
          "accent": "#ff5a3c",
          "icon": "boss",
          "kicker": "gather it all into one pile",
          "pole": "pull",
          "spheres": [
            {
              "slug": "the-huffman",
              "title": "THE HUFFMAN",
              "accent": "#ff8c42",
              "icon": "loot",
              "kicker": "short codes for common loot; pack the hoard tight",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-fixed-block",
              "title": "THE FIXED BLOCK",
              "accent": "#ffb870",
              "icon": "loot",
              "kicker": "Huffman's mirror — variable input, fixed-length blocks",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-block-sort",
              "title": "THE BLOCK SORT",
              "accent": "#8fd0c0",
              "icon": "sort",
              "kicker": "Burrows-Wheeler — reversible sort that clusters",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-interval",
              "title": "THE INTERVAL",
              "accent": "#b0f0a0",
              "icon": "interval",
              "kicker": "arithmetic coding — a whole message as one number",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-sylvester",
              "title": "THE SYLVESTER",
              "accent": "#90e0b0",
              "icon": "sylvester",
              "kicker": "2, 3, 7, 43, 1807 — unit fractions that fill exactly one",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-patience-sorting",
              "title": "THE PATIENCE SORTING",
              "accent": "#c8a848",
              "icon": "patience-sorting",
              "kicker": "deal cards to piles — the pile count is the longest increasing run",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-hook-length",
              "title": "THE HOOK LENGTH",
              "accent": "#c8a050",
              "icon": "hook-length",
              "kicker": "count Young tableaux as n! over a product of hooks",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-eertree",
              "title": "THE EERTREE",
              "accent": "#a878c0",
              "icon": "eertree",
              "kicker": "every distinct palindrome in a linear-size tree",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-cuckoo-filter",
              "title": "THE CUCKOO FILTER",
              "accent": "#d4a017",
              "icon": "cuckoo-filter",
              "kicker": "deletable membership with never a false negative",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-aliquot",
              "title": "THE ALIQUOT",
              "accent": "#e0b020",
              "icon": "aliquot",
              "kicker": "the number that equals the sum of its parts",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-bertrand-ballot",
              "title": "THE BERTRAND BALLOT",
              "accent": "#d0a848",
              "icon": "bertrand-ballot",
              "kicker": "counting the ballots where one candidate never trails",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-elias-gamma",
              "title": "THE ELIAS GAMMA",
              "accent": "#ffcf4a",
              "icon": "elias-gamma",
              "kicker": "a number that says its own length",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-flajolet-martin",
              "title": "THE FLAJOLET-MARTIN",
              "accent": "#ff8a3c",
              "icon": "flajolet",
              "kicker": "a vast count from a tiny bitmap",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-thabit",
              "title": "THE THABIT",
              "accent": "#b06bff",
              "icon": "thabit",
              "kicker": "two numbers each the sum of the other's divisors",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-jacobi-two-square",
              "title": "THE JACOBI TWO-SQUARE",
              "accent": "#ffcf4a",
              "icon": "twosquare",
              "kicker": "sums of two squares counted by divisors mod 4",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-niven",
              "title": "THE NIVEN",
              "accent": "#35ffb0",
              "icon": "niven",
              "kicker": "rational cosines only at five angles",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-wilson-prime",
              "title": "THE WILSON PRIME",
              "accent": "#ffcf4a",
              "icon": "wilsonprime",
              "kicker": "three coins in 250 years",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-squared-square",
              "title": "THE SQUARED SQUARE",
              "accent": "#ffcf4a",
              "icon": "squaredsquare",
              "kicker": "squares that fit exactly",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-seal-over-a-red-test",
              "title": "THE SEAL OVER A RED TEST",
              "accent": "#7de2b0",
              "icon": "◆",
              "kicker": "INTACT was true, and meant nothing",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "nothing-watches-the-gate",
              "title": "NOTHING WATCHES THE GATE",
              "accent": "#5ad6ff",
              "icon": "△",
              "kicker": "a regress with a measurable depth",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-succinct-rank",
              "title": "THE SUCCINCT RANK",
              "accent": "#7de2b0",
              "icon": "≡",
              "kicker": "three touches, wherever you ask",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            }
          ],
          "seats": 31,
          "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
        },
        {
          "slug": "the-inventory",
          "title": "THE INVENTORY",
          "accent": "#9d00ff",
          "icon": "coop",
          "kicker": "hold what you've earned",
          "pole": "pull",
          "spheres": [
            {
              "slug": "the-sort",
              "title": "THE SORT",
              "accent": "#2ec4b6",
              "icon": "loot",
              "kicker": "order built into the wiring",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-skip-list",
              "title": "THE SKIP LIST",
              "accent": "#b0e055",
              "icon": "lanes",
              "kicker": "log-time search from coin flips — no rotations",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-heap",
              "title": "THE HEAP",
              "accent": "#f08fb0",
              "icon": "heap",
              "kicker": "the partial order that always knows the smallest",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-golomb",
              "title": "THE GOLOMB",
              "accent": "#d0b0ff",
              "icon": "golomb",
              "kicker": "a(n) = how many times n appears in itself",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-fisher-yates",
              "title": "THE FISHER-YATES",
              "accent": "#58b0e0",
              "icon": "fisher-yates",
              "kicker": "a provably-uniform shuffle — n! paths onto n! orderings",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-lyndon",
              "title": "THE LYNDON",
              "accent": "#c0a050",
              "icon": "lyndon",
              "kicker": "unique factorization of a string into Lyndon words",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-suffix-automaton",
              "title": "THE SUFFIX AUTOMATON",
              "accent": "#58a0b0",
              "icon": "suffix-automaton",
              "kicker": "the smallest machine recognizing every substring, O(n) states",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-kasai",
              "title": "THE KASAI",
              "accent": "#a878c0",
              "icon": "kasai",
              "kicker": "the LCP array in linear time by reusing the last overlap",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-hilbert-curve",
              "title": "THE HILBERT CURVE",
              "accent": "#d4a017",
              "icon": "hilbert-curve",
              "kicker": "one line that fills the plane, keeping neighbors near",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-lehmer-code",
              "title": "THE LEHMER CODE",
              "accent": "#e0b020",
              "icon": "lehmer",
              "kicker": "number a permutation with a mixed-radix odometer",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-sparse-set",
              "title": "THE SPARSE SET",
              "accent": "#e0b020",
              "icon": "sparse-set",
              "kicker": "a set with no array to initialize",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-lah",
              "title": "THE LAH",
              "accent": "#d08840",
              "icon": "lah",
              "kicker": "counting partitions into ordered lists",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-prouhet-tarry-escott",
              "title": "THE PROUHET-TARRY-ESCOTT",
              "accent": "#ffcf4a",
              "icon": "prouhet-tarry-escott",
              "kicker": "a set split into equal power sums",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-factoradic",
              "title": "THE FACTORADIC",
              "accent": "#ff8a3c",
              "icon": "factoradic",
              "kicker": "a number in factorial base",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-postage-stamp",
              "title": "THE POSTAGE STAMP",
              "accent": "#ffcf4a",
              "icon": "postage",
              "kicker": "the longest run of amounts a few stamps can make",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-brahmagupta",
              "title": "THE BRAHMAGUPTA",
              "accent": "#ffcf4a",
              "icon": "brahmagupta",
              "kicker": "a cyclic quadrilateral's maximal area from its sides",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-weird",
              "title": "THE WEIRD",
              "accent": "#ff8a3c",
              "icon": "weird",
              "kicker": "abundance you cannot spend",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-wilberforce",
              "title": "THE WILBERFORCE",
              "accent": "#ffcf4a",
              "icon": "wilberforce",
              "kicker": "bounce traded for twist",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-flag-that-says-what-it-knows",
              "title": "THE FLAG THAT SAYS WHAT IT KNOWS",
              "accent": "#5ad6ff",
              "icon": "⚑",
              "kicker": "INSENSITIVE, renamed",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-coincidence-left-alone",
              "title": "THE COINCIDENCE THAT WASN'T",
              "accent": "#7de2b0",
              "icon": "≈",
              "kicker": "CORRECTED -- a coincidence that was never there",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-t-digest",
              "title": "THE T-DIGEST",
              "accent": "#9d00ff",
              "icon": "≈",
              "kicker": "a sketch that decided in advance what would matter",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-frame-of-reference",
              "title": "THE FRAME OF REFERENCE",
              "accent": "#9d00ff",
              "icon": "∣",
              "kicker": "it compresses nothing and merely stops repeating yourself",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-dictionary-coder",
              "title": "THE DICTIONARY CODER",
              "accent": "#9d00ff",
              "icon": "❑",
              "kicker": "incompressible is never a property of the data",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            }
          ],
          "seats": 23,
          "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
        },
        {
          "slug": "the-vault",
          "title": "THE VAULT",
          "accent": "#7cfc00",
          "icon": "cheat",
          "kicker": "lock it away, deep inside",
          "pole": "pull",
          "spheres": [
            {
              "slug": "3lock",
              "title": "3LOCK",
              "accent": "#5ad0ff",
              "icon": "loot",
              "kicker": "a three-way lock — ROOT0",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-banker",
              "title": "THE BANKER",
              "accent": "#ffd166",
              "icon": "coins",
              "kicker": "amortized O(1) — the binary counter pays itself",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-secret",
              "title": "THE SECRET",
              "accent": "#c8a0ff",
              "icon": "share",
              "kicker": "Shamir — split a secret, k of n reopen it",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-frobenius",
              "title": "THE FROBENIUS",
              "accent": "#ffd060",
              "icon": "coins2",
              "kicker": "the largest amount you can't make — ab-a-b",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-rsa",
              "title": "THE RSA",
              "accent": "#ffb060",
              "icon": "rsa",
              "kicker": "a lock anyone can close, only the key-holder opens",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-morton",
              "title": "THE MORTON",
              "accent": "#50b0c0",
              "icon": "morton",
              "kicker": "interleave the bits of x and y — 2D into one address",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-horner",
              "title": "THE HORNER",
              "accent": "#c0a048",
              "icon": "horner",
              "kicker": "evaluate in n multiplications — and it's synthetic division",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-thomas",
              "title": "THE THOMAS",
              "accent": "#c0a058",
              "icon": "thomas",
              "kicker": "solve a tridiagonal system in O(n) — sparsity conserved",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-perfect-hash",
              "title": "THE PERFECT HASH",
              "accent": "#70a860",
              "icon": "perfect-hash",
              "kicker": "zero collisions, O(n) space, one probe per lookup",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-cycle-sort",
              "title": "THE CYCLE SORT",
              "accent": "#58a0b0",
              "icon": "cycle-sort",
              "kicker": "sorting with the minimum possible number of writes",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-count-min-sketch",
              "title": "THE COUNT-MIN SKETCH",
              "accent": "#d4a017",
              "icon": "count-min-sketch",
              "kicker": "count a huge stream in a tiny fixed table",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-lifting-the-exponent",
              "title": "THE LIFTING THE EXPONENT",
              "accent": "#e0b020",
              "icon": "lte",
              "kicker": "count how many times p divides a power difference",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-wieferich",
              "title": "THE WIEFERICH",
              "accent": "#e0b020",
              "icon": "wieferich",
              "kicker": "the vanishingly rare Wieferich primes",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-von-staudt-clausen",
              "title": "THE VON STAUDT-CLAUSEN",
              "accent": "#ff8a3c",
              "icon": "vonstaudt",
              "kicker": "a Bernoulli denominator read off from primes",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-harshad",
              "title": "THE HARSHAD",
              "accent": "#21e6ff",
              "icon": "harshad",
              "kicker": "the numbers that are Harshad in every base",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-amicable",
              "title": "THE AMICABLE",
              "accent": "#35ffb0",
              "icon": "amicable",
              "kicker": "two numbers summing to each other's divisors",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-lah-numbers",
              "title": "THE LAH NUMBERS",
              "accent": "#35ffb0",
              "icon": "lah",
              "kicker": "numbers linking the rising and falling factorials",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-gabriels-horn",
              "title": "THE GABRIELS HORN",
              "accent": "#ff8a3c",
              "icon": "gabrielshorn",
              "kicker": "a horn holding finite paint behind an infinite wall",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-magic-hexagon",
              "title": "THE MAGIC HEXAGON",
              "accent": "#ffcf4a",
              "icon": "magichexagon",
              "kicker": "the one hexagon that exists",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-napkin-ring",
              "title": "THE NAPKIN RING",
              "accent": "#ffcf4a",
              "icon": "napkinring",
              "kicker": "a ring that forgets its sphere",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-content-address",
              "title": "THE CONTENT ADDRESS",
              "accent": "#7cfc00",
              "icon": "◈",
              "kicker": "permanence bought with the ability to be corrected",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-zigzag",
              "title": "THE ZIGZAG",
              "accent": "#7cfc00",
              "icon": "⌇",
              "kicker": "a subsidy paid by positives to rescue negatives",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-silent-corruption",
              "title": "THE SILENT CORRUPTION",
              "accent": "#7cfc00",
              "icon": "⌗",
              "kicker": "a visible outage chosen over an invisible corruption",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-lottery-scheduler",
              "title": "THE LOTTERY SCHEDULER",
              "accent": "#7cfc00",
              "icon": "⚄",
              "kicker": "a window too short for the limit to have arrived",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            }
          ],
          "seats": 36,
          "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
        },
        {
          "slug": "the-stash",
          "title": "THE STASH",
          "accent": "#5ad0ff",
          "icon": "respawn",
          "kicker": "squirrel it away for later",
          "pole": "pull",
          "spheres": [
            {
              "slug": "the-fenwick-ladder",
              "title": "THE FENWICK LADDER",
              "accent": "#4fd0e0",
              "icon": "loot",
              "kicker": "a whole range-sum tree hidden in one array, by i & −i",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-cuckoo",
              "title": "THE CUCKOO",
              "accent": "#e6a3ff",
              "icon": "nest",
              "kicker": "cuckoo hashing — worst-case two-probe lookup",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-sketch",
              "title": "THE SKETCH",
              "accent": "#a0c0ff",
              "icon": "sketch",
              "kicker": "Count-Min — tiny memory, never undercounts",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-prufer",
              "title": "THE PRUFER",
              "accent": "#b0e070",
              "icon": "prufer",
              "kicker": "a labeled tree ⟷ a short number sequence — Cayley's n^(n-2)",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-lzw",
              "title": "THE LZW",
              "accent": "#d0b040",
              "icon": "lzw",
              "kicker": "build a dictionary on the fly — and never send it",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-arithmetic-coding",
              "title": "THE ARITHMETIC CODING",
              "accent": "#c0a048",
              "icon": "arithmetic-coding",
              "kicker": "the whole message as one number, at the entropy limit",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-dilworth",
              "title": "THE DILWORTH",
              "accent": "#a878c0",
              "icon": "dilworth",
              "kicker": "min chains to cover a poset = max antichain",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-combinadics",
              "title": "THE COMBINADICS",
              "accent": "#d4a017",
              "icon": "combinadics",
              "kicker": "index any subset by a single number",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-van-emde-boas",
              "title": "THE VAN EMDE BOAS",
              "accent": "#d4a017",
              "icon": "van-emde-boas",
              "kicker": "integer successor in O(log log u)",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-least-rotation",
              "title": "THE LEAST ROTATION",
              "accent": "#a878c0",
              "icon": "least-rotation",
              "kicker": "the canonical rotation of a necklace, in O(n)",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-combinatorial-number-system",
              "title": "THE COMBINATORIAL NUMBER SYSTEM",
              "accent": "#e0b020",
              "icon": "cns",
              "kicker": "number a subset with a descending choice",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-patricia",
              "title": "THE PATRICIA TRIE",
              "accent": "#b06bff",
              "icon": "patricia",
              "kicker": "branching only on the bits that differ",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-robin-hood",
              "title": "THE ROBIN HOOD",
              "accent": "#21e6ff",
              "icon": "robinhood",
              "kicker": "steal from the rich to even the probes",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-pentagonal-number",
              "title": "THE PENTAGONAL",
              "accent": "#ffcf4a",
              "icon": "pentagonal",
              "kicker": "partitions counted by an alternating sum over pentagons",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-ramanujan-sum",
              "title": "THE RAMANUJAN SUM",
              "accent": "#b06bff",
              "icon": "ramanujansum",
              "kicker": "roots of unity summing to an integer by Möbius",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-liouville-number",
              "title": "THE LIOUVILLE NUMBER",
              "accent": "#35ffb0",
              "icon": "liouville",
              "kicker": "a number approximated absurdly well by rationals",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-kelly",
              "title": "THE KELLY",
              "accent": "#ffcf4a",
              "icon": "kelly",
              "kicker": "the bet size that survives",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-kuratowski",
              "title": "THE KURATOWSKI",
              "accent": "#ffcf4a",
              "icon": "kuratowski",
              "kicker": "fourteen sets and never a fifteenth",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-zeeman-machine",
              "title": "THE ZEEMAN MACHINE",
              "accent": "#ffcf4a",
              "icon": "zeeman",
              "kicker": "the disc that jumps",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-kepler-conjecture",
              "title": "THE KEPLER CONJECTURE",
              "accent": "#ffcf4a",
              "icon": "kepler",
              "kicker": "the densest stack",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-elias-fano",
              "title": "THE ELIAS-FANO",
              "accent": "#5ad0ff",
              "icon": "≡",
              "kicker": "sorted is a bill you already paid",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-varint",
              "title": "THE VARINT",
              "accent": "#5ad0ff",
              "icon": "…",
              "kicker": "the wasted bytes were buying the ability to not look",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            }
          ],
          "seats": 33,
          "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
        }
      ],
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "name": "BOSS",
      "slug": "boss",
      "c": "#ff5a3c",
      "icon": "boss",
      "tag": "the hard fight — the problem worth beating",
      "lore": {
        "bio": "The wall at the end of the level. The problem that earns the win.",
        "story": "Made of every bug that took three nights. It does not want to lose — that is the whole point of it.",
        "does": "Names the hard problem, holds the line, and only opens once you actually beat it.",
        "haiku": [
          "def beat(I hp){ if hp <= 0 { -> \"win\" } -> \"fight\" }",
          "I boss <- beat(0)",
          "-> boss"
        ],
        "lang": "i13"
      },
      "domains": [
        {
          "slug": "the-final-boss",
          "title": "THE FINAL BOSS",
          "accent": "#39fc6b",
          "icon": "spawn",
          "kicker": "the confrontation you trained for",
          "pole": "push",
          "spheres": [
            {
              "slug": "the-hanoi",
              "title": "THE HANOI",
              "accent": "#ff4d6d",
              "icon": "boss",
              "kicker": "2ⁿ−1 moves — recursion, the ruler, and Sierpinski in one",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-morley",
              "title": "THE MORLEY",
              "accent": "#ff80ff",
              "icon": "morley",
              "kicker": "trisect any triangle's angles — the meeting points are equilateral",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-permanent",
              "title": "THE PERMANENT",
              "accent": "#c05090",
              "icon": "permanent",
              "kicker": "the determinant's all-plus twin — and it's #P-hard",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-sprague-grundy",
              "title": "THE SPRAGUE-GRUNDY",
              "accent": "#c05868",
              "icon": "sprague-grundy",
              "kicker": "every impartial game is secretly a Nim heap",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-lucas-lehmer",
              "title": "THE LUCAS-LEHMER",
              "accent": "#c05858",
              "icon": "lucas-lehmer",
              "kicker": "a deterministic primality verdict for Mersenne numbers",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-alpha-beta",
              "title": "THE ALPHA-BETA",
              "accent": "#c05868",
              "icon": "alpha-beta",
              "kicker": "minimax value, pruning the provably-irrelevant branches",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-quadratic-reciprocity",
              "title": "THE QUADRATIC RECIPROCITY",
              "accent": "#b06868",
              "icon": "reciprocity",
              "kicker": "a golden law linking two primes' squares",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-wolstenholme",
              "title": "THE WOLSTENHOLME",
              "accent": "#9a6ad0",
              "icon": "wolstenholme",
              "kicker": "a binomial congruence mod p-cubed for primes five and up",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-vizing",
              "title": "THE VIZING",
              "accent": "#d06868",
              "icon": "vizing",
              "kicker": "colouring edges with almost the fewest colours",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-cantor-diagonal",
              "title": "THE CANTOR DIAGONAL",
              "accent": "#c06888",
              "icon": "cantor-diagonal",
              "kicker": "the diagonal that escapes every list",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-newton-gauss",
              "title": "THE NEWTON-GAUSS LINE",
              "accent": "#ffcf4a",
              "icon": "newton-gauss",
              "kicker": "four lines hide a straight line in their diagonals",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-glushkov",
              "title": "THE GLUSHKOV",
              "accent": "#ffcf4a",
              "icon": "glushkov",
              "kicker": "a regex becomes a walk over letter-positions",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-ryser",
              "title": "THE RYSER",
              "accent": "#b06bff",
              "icon": "ryser",
              "kicker": "a permanent counted by inclusion-exclusion",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-cauchy-interlacing",
              "title": "THE CAUCHY INTERLACING",
              "accent": "#b06bff",
              "icon": "cauchyinterlacing",
              "kicker": "submatrix eigenvalues interlacing the whole",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-gergonne",
              "title": "THE GERGONNE",
              "accent": "#ffcf4a",
              "icon": "gergonne",
              "kicker": "triangle cevians to the incircle meeting at one point",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-mihailescu",
              "title": "THE MIHĂILESCU",
              "accent": "#b06bff",
              "icon": "mihailescu",
              "kicker": "the only two powers that touch",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-borsuk-ulam",
              "title": "THE BORSUK-ULAM",
              "accent": "#b06bff",
              "icon": "borsukulam",
              "kicker": "antipodes that must agree",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-hawk-dove",
              "title": "THE HAWK DOVE",
              "accent": "#ff5a8a",
              "icon": "⚔",
              "kicker": "a fight nobody wins outright",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-oracle",
              "title": "THE ORACLE",
              "accent": "#7de2b0",
              "icon": "❓",
              "kicker": "a check that could actually fail",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-flattering-direction",
              "title": "THE FLATTERING DIRECTION",
              "accent": "#b98cff",
              "icon": "↖",
              "kicker": "three errors, all the same way",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "two-numbers-that-are-not-one",
              "title": "TWO NUMBERS THAT ARE NOT ONE",
              "accent": "#5ad6ff",
              "icon": "≠",
              "kicker": "0.05 apart, and asserted distinct",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            }
          ],
          "seats": 33,
          "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
        },
        {
          "slug": "the-gauntlet",
          "title": "THE GAUNTLET",
          "accent": "#ffd23f",
          "icon": "grind",
          "kicker": "run the whole challenge, no breaks",
          "pole": "push",
          "spheres": [
            {
              "slug": "error-correction-bench",
              "title": "THE ERROR CORRECTION BENCH — how to be w",
              "accent": "#ff5a3c",
              "icon": "boss",
              "kicker": "vendored from David's corpus — a silicon-coding instrument",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-membrane",
              "title": "The Membrane · dip-and-recover error cor",
              "accent": "#ff5a3c",
              "icon": "boss",
              "kicker": "vendored from David's corpus — a silicon-coding instrument",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-route",
              "title": "THE ROUTE",
              "accent": "#5ad0ff",
              "icon": "boss",
              "kicker": "how the machine finds its way",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-reciprocity",
              "title": "THE RECIPROCITY",
              "accent": "#ffe070",
              "icon": "reciprocity",
              "kicker": "is p a square mod q? — Gauss's golden theorem links it to q mod p",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-lights-out",
              "title": "THE LIGHTS OUT",
              "accent": "#e0c040",
              "icon": "lights-out",
              "kicker": "a light puzzle is a linear system over GF(2)",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-held-karp",
              "title": "THE HELD-KARP",
              "accent": "#c05868",
              "icon": "held-karp",
              "kicker": "exact TSP by bitmask DP — n! tours in 2^n states",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-collatz",
              "title": "THE COLLATZ",
              "accent": "#a878c0",
              "icon": "collatz",
              "kicker": "the hailstone that (so far) always lands on 1",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-eulerian-numbers",
              "title": "THE EULERIAN NUMBERS",
              "accent": "#b06868",
              "icon": "eulerian",
              "kicker": "count permutations by their climbs",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-mantel",
              "title": "THE MANTEL",
              "accent": "#d07850",
              "icon": "mantel",
              "kicker": "how many edges before a triangle is forced",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-jacobi-four-square",
              "title": "THE JACOBI FOUR-SQUARE",
              "accent": "#c07068",
              "icon": "jacobi-four-square",
              "kicker": "counting the ways to write a number as four squares",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-dpll",
              "title": "THE DPLL",
              "accent": "#21e6ff",
              "icon": "dpll",
              "kicker": "a search that prunes itself",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-chromatic-polynomial",
              "title": "THE CHROMATIC POLYNOMIAL",
              "accent": "#21e6ff",
              "icon": "chromatic-polynomial",
              "kicker": "colourings counted by a polynomial",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-vp-tree",
              "title": "THE VP-TREE",
              "accent": "#ffcf4a",
              "icon": "vptree",
              "kicker": "nearest found by pruning a metric tree",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-chan",
              "title": "THE CHAN'S ALGORITHM",
              "accent": "#21e6ff",
              "icon": "chan",
              "kicker": "a hull wrapped over mini-hulls",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-frank-wolfe",
              "title": "THE FRANK-WOLFE",
              "accent": "#b06bff",
              "icon": "frank-wolfe",
              "kicker": "charge the corner to minimize inside a polytope",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-thiele",
              "title": "THE THIELE",
              "accent": "#b06bff",
              "icon": "thiele",
              "kicker": "a rational curve threaded through the data",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-crofton",
              "title": "THE CROFTON",
              "accent": "#ffcf4a",
              "icon": "crofton",
              "kicker": "a length measured by throwing lines at it",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-normal-number",
              "title": "THE NORMAL NUMBER",
              "accent": "#35ffb0",
              "icon": "normalnumber",
              "kicker": "the digits nobody can certify",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-newton-pepys",
              "title": "THE NEWTON–PEPYS",
              "accent": "#21e6ff",
              "icon": "newtonpepys",
              "kicker": "the shortest gauntlet",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-cap-that-was-luck",
              "title": "THE CAP THAT WAS LUCK",
              "accent": "#7de2b0",
              "icon": "⤒",
              "kicker": "a number about this machine, written as a number about the language",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-automorphism-shortfall",
              "title": "THE AUTOMORPHISM SHORTFALL",
              "accent": "#ffd76a",
              "icon": "⥁",
              "kicker": "what an unequal partition costs in bits",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-dining-philosophers",
              "title": "THE DINING PHILOSOPHERS",
              "accent": "#ff5a8a",
              "icon": "⑂",
              "kicker": "everyone correct, in the same way, at the same time",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-tail-at-scale",
              "title": "THE TAIL AT SCALE",
              "accent": "#ffd23f",
              "icon": "⑂",
              "kicker": "one in a hundred, a hundred times over",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-pigeonhole-compression",
              "title": "THE PIGEONHOLE COMPRESSION",
              "accent": "#ffd23f",
              "icon": "⊄",
              "kicker": "compression was always an opinion",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-correlated-failure",
              "title": "THE CORRELATED FAILURE",
              "accent": "#ffd23f",
              "icon": "⋈",
              "kicker": "an availability figure is a belief about shared fate",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-starvation",
              "title": "THE STARVATION",
              "accent": "#ffd23f",
              "icon": "⌛",
              "kicker": "the scheduler was doing precisely what it was told",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            }
          ],
          "seats": 26,
          "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
        },
        {
          "slug": "the-raid",
          "title": "THE RAID",
          "accent": "#ff2d95",
          "icon": "glitch",
          "kicker": "assault it together, all at once",
          "pole": "push",
          "spheres": [
            {
              "slug": "the-field-inverse",
              "title": "THE FIELD INVERSE",
              "accent": "#9db8ff",
              "icon": "boss",
              "kicker": "the heart of AES is one field inversion in disguise",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-rho",
              "title": "THE RHO",
              "accent": "#ff7a5c",
              "icon": "rho",
              "kicker": "Pollard's rho — factor via a cycle you can't see",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-discrete-log",
              "title": "THE DISCRETE LOG",
              "accent": "#ff7060",
              "icon": "dlog",
              "kicker": "baby-step giant-step — invert the exponent in root-n",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-ghz",
              "title": "THE GHZ",
              "accent": "#90ffd0",
              "icon": "ghz",
              "kicker": "three qubits refute local realism with certainty",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-stoer-wagner",
              "title": "THE STOER-WAGNER",
              "accent": "#c05868",
              "icon": "stoer-wagner",
              "kicker": "the global min cut, deterministically, no source/sink",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-fibonacci-heap",
              "title": "THE FIBONACCI HEAP",
              "accent": "#70a860",
              "icon": "fibonacci-heap",
              "kicker": "a priority queue that pays for order only when it must",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-bell-numbers",
              "title": "THE BELL NUMBERS",
              "accent": "#b06868",
              "icon": "bell",
              "kicker": "count the ways to partition a set",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-tetration",
              "title": "THE TETRATION",
              "accent": "#b060c0",
              "icon": "tetration",
              "kicker": "a power tower reduced modulo m settles down",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-erdos-ginzburg-ziv",
              "title": "THE ERDŐS–GINZBURG–ZIV",
              "accent": "#d06880",
              "icon": "erdos-ginzburg-ziv",
              "kicker": "any 2n−1 integers hide n that sum to zero mod n",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-ransac",
              "title": "THE RANSAC",
              "accent": "#b06bff",
              "icon": "ransac",
              "kicker": "a model found through a storm of outliers",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-graeffe",
              "title": "THE GRAEFFE",
              "accent": "#b06bff",
              "icon": "graeffe",
              "kicker": "squaring a polynomial to prise its roots apart",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-pappus",
              "title": "THE PAPPUS",
              "accent": "#ffcf4a",
              "icon": "pappus",
              "kicker": "perspective from a hexagon inscribed in two lines",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-gauss-lucas",
              "title": "THE GAUSS-LUCAS",
              "accent": "#b06bff",
              "icon": "gausslucas",
              "kicker": "the derivative's roots trapped in the hull of the roots",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-feuerbach",
              "title": "THE FEUERBACH",
              "accent": "#ffcf4a",
              "icon": "feuerbach",
              "kicker": "a nine-point circle tangent to the incircle",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-gaussian-integral",
              "title": "THE GAUSSIAN INTEGRAL",
              "accent": "#ffcf4a",
              "icon": "gaussianintegral",
              "kicker": "a bell curve whose area is the square root of pi",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-hydra",
              "title": "THE HYDRA",
              "accent": "#21e6ff",
              "icon": "hydra",
              "kicker": "the boss that must lose but arithmetic cannot say so",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-giant-component",
              "title": "THE GIANT COMPONENT",
              "accent": "#ff8a3c",
              "icon": "giantcomponent",
              "kicker": "the edge where one giant appears",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-percolation",
              "title": "THE PERCOLATION",
              "accent": "#7de2b0",
              "icon": "▦",
              "kicker": "a threshold at exactly one half",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-permutation-null",
              "title": "THE PERMUTATION NULL",
              "accent": "#ff5a8a",
              "icon": "⚔",
              "kicker": "the control that killed the pretty result",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "a-regex-meeting-nesting",
              "title": "A REGEX MEETING NESTING",
              "accent": "#b98cff",
              "icon": "⌇",
              "kicker": "the error surfaces four frames from its cause",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-carry-lookahead",
              "title": "THE CARRY LOOKAHEAD",
              "accent": "#5ad6ff",
              "icon": "⇈",
              "kicker": "generate and propagate, computed all at once",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            }
          ],
          "seats": 40,
          "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
        },
        {
          "slug": "sudden-death",
          "title": "SUDDEN DEATH",
          "accent": "#00f5ff",
          "icon": "loot",
          "kicker": "one strike decides it",
          "pole": "push",
          "spheres": [
            {
              "slug": "the-probable-prime",
              "title": "THE PROBABLE PRIME",
              "accent": "#ff5a7a",
              "icon": "boss",
              "kicker": "witnesses that expose composites via the roots-of-1 trapdoor",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-square-root-in-the-ring",
              "title": "THE SQUARE ROOT IN THE RING",
              "accent": "#a0d0ff",
              "icon": "boss",
              "kicker": "un-square in a prime field — if a root exists at all",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-josephus",
              "title": "THE JOSEPHUS",
              "accent": "#ff8a5c",
              "icon": "circle",
              "kicker": "the last one standing — and the bit-rotation shortcut",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-wilson",
              "title": "THE WILSON",
              "accent": "#ffb84d",
              "icon": "wilson",
              "kicker": "(p-1)! = -1 mod p iff prime — exact, and useless",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-mersenne",
              "title": "THE MERSENNE",
              "accent": "#ff9060",
              "icon": "mersenne",
              "kicker": "Lucas-Lehmer — exact primality for 2^p-1",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-perfect",
              "title": "THE PERFECT",
              "accent": "#ffd0e0",
              "icon": "perfect",
              "kicker": "perfect numbers = Mersenne primes, both ways",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-secretary",
              "title": "THE SECRETARY",
              "accent": "#60c0ff",
              "icon": "secretary",
              "kicker": "reject the first 37%, then leap — win the best 1/e of the time",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-boyer-moore-majority",
              "title": "THE BOYER-MOORE MAJORITY",
              "accent": "#b06868",
              "icon": "majority",
              "kicker": "one survivor of pairwise cancellation",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-zolotarev",
              "title": "THE ZOLOTAREV",
              "accent": "#b070c0",
              "icon": "zolotarev",
              "kicker": "a coin-flip sign hidden in modular multiplication",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-gamblers-ruin",
              "title": "THE GAMBLER'S RUIN",
              "accent": "#c86868",
              "icon": "gamblers-ruin",
              "kicker": "a fair walk absorbed at the edges lands with probability proportional to the start",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-laguerre",
              "title": "THE LAGUERRE",
              "accent": "#b06bff",
              "icon": "laguerre",
              "kicker": "a solver that hunts every root, real and complex",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-marden",
              "title": "THE MARDEN",
              "accent": "#b06bff",
              "icon": "marden",
              "kicker": "the derivative's roots are the inellipse foci",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-lym",
              "title": "THE LYM",
              "accent": "#b06bff",
              "icon": "lym",
              "kicker": "an antichain sum capped at one",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-best-theorem",
              "title": "THE BEST THEOREM",
              "accent": "#21e6ff",
              "icon": "best",
              "kicker": "Eulerian circuits counted by a determinant",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-isoperimetric",
              "title": "THE ISOPERIMETRIC",
              "accent": "#ffcf4a",
              "icon": "isoperimetric",
              "kicker": "the circle enclosing the most area for its perimeter",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-conway-soldiers",
              "title": "THE CONWAY SOLDIERS",
              "accent": "#ff2fa6",
              "icon": "conwaysoldiers",
              "kicker": "an army that cannot reach the fifth row",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-tennis-racket",
              "title": "THE TENNIS RACKET",
              "accent": "#ff8a3c",
              "icon": "tennisracket",
              "kicker": "the axis that flips",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-winners-curse",
              "title": "THE WINNERS CURSE",
              "accent": "#b98cff",
              "icon": "☠",
              "kicker": "winning as the evidence you were wrong",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-zero-one-principle",
              "title": "THE ZERO-ONE PRINCIPLE",
              "accent": "#7de2b0",
              "icon": "⇅",
              "kicker": "256 tests instead of 40,320",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-branch-still-in-the-machine",
              "title": "THE BRANCH STILL IN THE MACHINE",
              "accent": "#b98cff",
              "icon": "⑂",
              "kicker": "the grammar lost if/then; the ISA never did",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-conditional-sum",
              "title": "THE CONDITIONAL SUM",
              "accent": "#ffd76a",
              "icon": "⑂",
              "kicker": "compute both answers, throw one away",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            }
          ],
          "seats": 31,
          "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
        },
        {
          "slug": "the-gatekeeper",
          "title": "THE GATEKEEPER",
          "accent": "#ff5a3c",
          "icon": "boss",
          "kicker": "won't let you pass unproven",
          "pole": "pull",
          "spheres": [
            {
              "slug": "the-fiddler",
              "title": "THE FIDDLER",
              "accent": "#ff5a3c",
              "icon": "boss",
              "kicker": "David's first repo — does the system hold under attack?",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-perrin",
              "title": "THE PERRIN",
              "accent": "#ff6a6a",
              "icon": "perrin",
              "kicker": "every prime divides P(p) — a near-perfect gate",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-crc",
              "title": "THE CRC",
              "accent": "#d4b03c",
              "icon": "crc",
              "kicker": "append check bits so corruption can't hide",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-sturm",
              "title": "THE STURM",
              "accent": "#b06840",
              "icon": "sturm",
              "kicker": "count real roots in an interval without finding them",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-enigma",
              "title": "THE ENIGMA",
              "accent": "#b09050",
              "icon": "enigma",
              "kicker": "a cipher that is its own inverse — and could never encrypt a letter to itself",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-jacobi-symbol",
              "title": "THE JACOBI SYMBOL",
              "accent": "#a06890",
              "icon": "jacobi-symbol",
              "kicker": "a residue test computed by reciprocity — without factoring",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-separating-axis",
              "title": "THE SEPARATING AXIS",
              "accent": "#c05868",
              "icon": "separating-axis",
              "kicker": "convex collision by looking for one separating line",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-gjk",
              "title": "THE GJK",
              "accent": "#a878c0",
              "icon": "gjk",
              "kicker": "convex collision by asking if the origin is in A minus B",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-tarjan-scc",
              "title": "THE TARJAN SCC",
              "accent": "#c05868",
              "icon": "tarjan-scc",
              "kicker": "every strongly connected component in one DFS",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-2-sat",
              "title": "THE 2-SAT",
              "accent": "#58a0b0",
              "icon": "2-sat",
              "kicker": "satisfy two-literal clauses in linear time",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-miller-rabin",
              "title": "THE MILLER-RABIN",
              "accent": "#b088d0",
              "icon": "miller-rabin",
              "kicker": "a witness names the composite, no factor needed",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-proth",
              "title": "THE PROTH",
              "accent": "#b06868",
              "icon": "proth",
              "kicker": "one witness decides a Proth prime",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-solovay-strassen",
              "title": "THE SOLOVAY-STRASSEN",
              "accent": "#b06868",
              "icon": "solovay",
              "kicker": "test primality by the Jacobi symbol",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-simson",
              "title": "THE SIMSON LINE",
              "accent": "#ff8a3c",
              "icon": "simson",
              "kicker": "feet that align only on the circle",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-cipolla",
              "title": "THE CIPOLLA",
              "accent": "#ff8a3c",
              "icon": "cipolla",
              "kicker": "a square root through a field extension",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-treiber",
              "title": "THE TREIBER STACK",
              "accent": "#b06bff",
              "icon": "treiber",
              "kicker": "a stack that needs no lock",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-brouwer",
              "title": "THE BROUWER",
              "accent": "#b06bff",
              "icon": "brouwer",
              "kicker": "the point that cannot escape",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-square-free",
              "title": "THE SQUARE-FREE",
              "accent": "#ff8a3c",
              "icon": "squarefree",
              "kicker": "three letters never stutter",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-tait",
              "title": "THE TAIT",
              "accent": "#ff8a3c",
              "icon": "tait",
              "kicker": "the lemma that held up a theorem for 62 years",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-blind-instrument",
              "title": "THE BLIND INSTRUMENT",
              "accent": "#ff5a8a",
              "icon": "◑",
              "kicker": "a checker that cannot see is silent, not noisy",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-berger-code",
              "title": "THE BERGER CODE",
              "accent": "#7de2b0",
              "icon": "▤",
              "kicker": "count the zeros and every one-way fault shows",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-bakery-algorithm",
              "title": "THE BAKERY ALGORITHM",
              "accent": "#ffd76a",
              "icon": "①",
              "kicker": "take a number; no atomic instruction required",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-bidi-override",
              "title": "THE BIDI OVERRIDE",
              "accent": "#ff5a8a",
              "icon": "⇆",
              "kicker": "two readers, two orders, no error",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-token-bucket",
              "title": "THE TOKEN BUCKET",
              "accent": "#ff5a3c",
              "icon": "◔",
              "kicker": "the burst depth is a promise about your worst instant",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-fencing-token",
              "title": "THE FENCING TOKEN",
              "accent": "#ff5a3c",
              "icon": "⇑",
              "kicker": "a lock that needs fencing was never a lock",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-gray-failure",
              "title": "THE GRAY FAILURE",
              "accent": "#ff5a3c",
              "icon": "◐",
              "kicker": "the shallowness is restraint, not laziness",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-preemption",
              "title": "THE PREEMPTION",
              "accent": "#ff5a3c",
              "icon": "⏸",
              "kicker": "a permission every piece of code has to keep granting",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            }
          ],
          "seats": 34,
          "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
        },
        {
          "slug": "the-wall",
          "title": "THE WALL",
          "accent": "#9d00ff",
          "icon": "coop",
          "kicker": "the resistance that stops you cold",
          "pole": "pull",
          "spheres": [
            {
              "slug": "the-welder",
              "title": "THE WELDER",
              "accent": "#ffa03c",
              "icon": "boss",
              "kicker": "near-constant-time merging — where inverse-Ackermann lives",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-hull",
              "title": "THE HULL",
              "accent": "#7affc0",
              "icon": "hull",
              "kicker": "the tightest wall around a point cloud",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-cells",
              "title": "THE CELLS",
              "accent": "#8fd0ff",
              "icon": "cells",
              "kicker": "Voronoi — the map of the nearest thing",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-contour",
              "title": "THE CONTOUR",
              "accent": "#7fe0b0",
              "icon": "contour",
              "kicker": "marching squares — the line where a field crosses a level",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-ramsey",
              "title": "THE RAMSEY",
              "accent": "#ff9060",
              "icon": "ramsey",
              "kicker": "among any 6 people, 3 friends or 3 strangers — unavoidable",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-hierholzer",
              "title": "THE HIERHOLZER",
              "accent": "#b05868",
              "icon": "hierholzer",
              "kicker": "cross every edge once — decided by counting odd corners",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-fermat-factorization",
              "title": "THE FERMAT FACTORIZATION",
              "accent": "#b06858",
              "icon": "fermat-factorization",
              "kicker": "factor n as a difference of squares — the seed of the sieves",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-dinic",
              "title": "THE DINIC",
              "accent": "#c05868",
              "icon": "dinic",
              "kicker": "max flow by leveled blocking flows — max flow = min cut",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-jarvis-march",
              "title": "THE JARVIS MARCH",
              "accent": "#6ab0d0",
              "icon": "jarvis",
              "kicker": "wrap a hull around points like a gift",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-turan",
              "title": "THE TURÁN",
              "accent": "#c07058",
              "icon": "turan",
              "kicker": "the most edges with no clique of a given size",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-half-plane-intersection",
              "title": "THE HALF-PLANE INTERSECTION",
              "accent": "#ffcf4a",
              "icon": "half-plane-intersection",
              "kicker": "a region carved by half-planes",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-remez",
              "title": "THE REMEZ",
              "accent": "#b06bff",
              "icon": "remez",
              "kicker": "a polynomial whose error rides an equal wave",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-menelaus",
              "title": "THE MENELAUS",
              "accent": "#b06bff",
              "icon": "menelaus",
              "kicker": "a line cutting three sides, points collinear",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-desargues",
              "title": "THE DESARGUES",
              "accent": "#b06bff",
              "icon": "desargues",
              "kicker": "perspective from a point equals perspective from a line",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-euler-criterion",
              "title": "THE EULER CRITERION",
              "accent": "#ffcf4a",
              "icon": "eulercriterion",
              "kicker": "a single power that tells a square from a non-square",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-fary-milnor",
              "title": "THE FARY-MILNOR",
              "accent": "#35ffb0",
              "icon": "farymilnor",
              "kicker": "the bending toll every knot must pay",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-schur",
              "title": "THE SCHUR",
              "accent": "#b06bff",
              "icon": "schur",
              "kicker": "the wall at thirteen",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-alternating-sign",
              "title": "THE ALTERNATING SIGN",
              "accent": "#b06bff",
              "icon": "alternatingsign",
              "kicker": "the 88-referee formula",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-polya-conjecture",
              "title": "THE PÓLYA CONJECTURE",
              "accent": "#b06bff",
              "icon": "polyaconj",
              "kicker": "a million confirmations, still false",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-two-errors-that-cancel",
              "title": "THE TWO ERRORS THAT CANCEL",
              "accent": "#ffd76a",
              "icon": "⊕",
              "kicker": "clean for exactly the wrong reason",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-dissent-upheld",
              "title": "THE DISSENT UPHELD",
              "accent": "#ff5a8a",
              "icon": "☑",
              "kicker": "answered by the data, and answered no",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-banker-deadlock",
              "title": "THE BANKER DEADLOCK",
              "accent": "#ff9f45",
              "icon": "⛔",
              "kicker": "he can afford it and he refuses anyway",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-belady-anomaly",
              "title": "THE BELADY ANOMALY",
              "accent": "#ff5a8a",
              "icon": "↓",
              "kicker": "more memory, more faults",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-signed-overflow",
              "title": "THE SIGNED OVERFLOW",
              "accent": "#ffd76a",
              "icon": "∞",
              "kicker": "right 31 times out of 32",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-unix-epoch",
              "title": "THE UNIX EPOCH",
              "accent": "#ff5a8a",
              "icon": "∞",
              "kicker": "2038, and then 1901",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-two-generals",
              "title": "THE TWO GENERALS",
              "accent": "#9d00ff",
              "icon": "⚔",
              "kicker": "they already agree and cannot confirm it",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-bufferbloat",
              "title": "THE BUFFERBLOAT",
              "accent": "#9d00ff",
              "icon": "▓",
              "kicker": "the drop was the signal; the buffer is what silenced it",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-float-equality",
              "title": "THE FLOAT EQUALITY",
              "accent": "#9d00ff",
              "icon": "≡",
              "kicker": "a tolerance moves the uncertainty into a constant nobody revisits",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-kolmogorov-bound",
              "title": "THE KOLMOGOROV BOUND",
              "accent": "#9d00ff",
              "icon": "∄",
              "kicker": "real data lives in a corner the theorem is not about",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-cascading-failure",
              "title": "THE CASCADING FAILURE",
              "accent": "#9d00ff",
              "icon": "⇈",
              "kicker": "no faulty component anywhere in it",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-earliest-deadline",
              "title": "THE EARLIEST DEADLINE",
              "accent": "#9d00ff",
              "icon": "⏱",
              "kicker": "optimal says nothing about what happens when you are wrong",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            }
          ],
          "seats": 22,
          "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
        },
        {
          "slug": "the-firewall",
          "title": "THE FIREWALL",
          "accent": "#7cfc00",
          "icon": "cheat",
          "kicker": "blocks everything trying to get in",
          "pole": "pull",
          "spheres": [
            {
              "slug": "the-firewall",
              "title": "THE FIREWALL",
              "accent": "#ff5a3c",
              "icon": "boss",
              "kicker": "blocks everything trying to get in",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "airgap-silicon",
              "title": "AIRGAP NODES ON SILICON — Si/SiGe realiz",
              "accent": "#ff5a3c",
              "icon": "boss",
              "kicker": "vendored from David's corpus — a silicon-coding instrument",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "cipher-and-shadow",
              "title": "THE CIPHER & THE SHADOW · Encrypt, Decry",
              "accent": "#ff5a3c",
              "icon": "boss",
              "kicker": "vendored from David's corpus — a silicon-coding instrument",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "enigma",
              "title": "Enigma — the machine and its one fatal f",
              "accent": "#ff5a3c",
              "icon": "boss",
              "kicker": "vendored from David's corpus — a silicon-coding instrument",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "limen-airgap-decoder",
              "title": "LIMEN · Air-Gap Decoder",
              "accent": "#ff5a3c",
              "icon": "boss",
              "kicker": "vendored from David's corpus — a silicon-coding instrument",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-elliptic-curve",
              "title": "THE ELLIPTIC-CURVE",
              "accent": "#a070ff",
              "icon": "ellipticcurve",
              "kicker": "the group hidden in a cubic — modern crypto's engine",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-thompson-nfa",
              "title": "THE THOMPSON NFA",
              "accent": "#d05858",
              "icon": "thompson-nfa",
              "kicker": "regex to NFA — match by advancing a whole state set, no backtracking",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-suffix-array",
              "title": "THE SUFFIX ARRAY",
              "accent": "#c05868",
              "icon": "suffix-array",
              "kicker": "sort every suffix — index every substring in O(n) space",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-freivalds",
              "title": "THE FREIVALDS",
              "accent": "#d4a017",
              "icon": "freivalds",
              "kicker": "verify a matrix product in O(n^2) with a random probe",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-edwards-curve",
              "title": "THE EDWARDS CURVE",
              "accent": "#b06868",
              "icon": "edwards",
              "kicker": "a curve whose addition never fails",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-xor-filter",
              "title": "THE XOR FILTER",
              "accent": "#35ffb0",
              "icon": "xor-filter",
              "kicker": "a set in 1.23 bytes a key",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-scapegoat-tree",
              "title": "THE SCAPEGOAT TREE",
              "accent": "#ff8a3c",
              "icon": "scapegoat",
              "kicker": "a tree that rebuilds its own worst branch",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-art-gallery",
              "title": "THE ART GALLERY",
              "accent": "#b06bff",
              "icon": "gallery",
              "kicker": "a third of the corners guard the whole gallery",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-erdos-mordell",
              "title": "THE ERDOS-MORDELL",
              "accent": "#21e6ff",
              "icon": "erdosmordell",
              "kicker": "a point's vertex distances bounded below by its side distances",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-bretschneider",
              "title": "THE BRETSCHNEIDER",
              "accent": "#21e6ff",
              "icon": "bretschneider",
              "kicker": "the area of any quadrilateral from its sides and two angles",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-lemoine-point",
              "title": "THE LEMOINE POINT",
              "accent": "#21e6ff",
              "icon": "lemoine",
              "kicker": "medians reflected over bisectors meeting at one point",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-moser-spindle",
              "title": "THE MOSER SPINDLE",
              "accent": "#b06bff",
              "icon": "moserspindle",
              "kicker": "seven points that outlaw three colors",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-chinese-hypothesis",
              "title": "THE CHINESE HYPOTHESIS",
              "accent": "#21e6ff",
              "icon": "chinesehyp",
              "kicker": "the test that lets impostors through",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-compile-invariant",
              "title": "THE COMPILE INVARIANT",
              "accent": "#5ad6ff",
              "icon": "⚖",
              "kicker": "compiles equals distinct positions fired",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-stack-that-cannot-jump",
              "title": "THE STACK THAT CANNOT JUMP",
              "accent": "#5ad6ff",
              "icon": "↯",
              "kicker": "a pushdown model has no move for GO TO",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-name-that-walks",
              "title": "THE NAME THAT WALKS",
              "accent": "#b98cff",
              "icon": "✇",
              "kicker": "the control that fires most often",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            }
          ],
          "seats": 35,
          "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
        },
        {
          "slug": "the-choke-point",
          "title": "THE CHOKE POINT",
          "accent": "#5ad0ff",
          "icon": "respawn",
          "kicker": "the narrows where it all constricts",
          "pole": "pull",
          "spheres": [
            {
              "slug": "the-failure-web",
              "title": "THE FAILURE WEB",
              "accent": "#64d8c8",
              "icon": "boss",
              "kicker": "every dictionary word in one pass, via failure links",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-oracle-of-echoes",
              "title": "THE ORACLE OF ECHOES",
              "accent": "#6ad0d0",
              "icon": "boss",
              "kicker": "the smallest machine that knows every substring",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-choke",
              "title": "THE CHOKE",
              "accent": "#ff6a8a",
              "icon": "flow",
              "kicker": "max-flow equals min-cut, exactly",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-borda",
              "title": "THE BORDA",
              "accent": "#90c0ff",
              "icon": "borda",
              "kicker": "same ballots, different rule — plurality crowns the loser",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-z-algorithm",
              "title": "THE Z-ALGORITHM",
              "accent": "#e06050",
              "icon": "z-algorithm",
              "kicker": "all prefix matches in O(n) — a pointer that never retreats",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-karger",
              "title": "THE KARGER",
              "accent": "#c07850",
              "icon": "karger",
              "kicker": "find the global min cut by random contraction",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-push-relabel",
              "title": "THE PUSH-RELABEL",
              "accent": "#c05868",
              "icon": "push-relabel",
              "kicker": "max flow by pushing excess downhill by height",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-pepin",
              "title": "THE PEPIN",
              "accent": "#b06868",
              "icon": "pepin",
              "kicker": "one test decides a Fermat prime",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-three-squares",
              "title": "THE THREE SQUARES",
              "accent": "#b0a040",
              "icon": "three-squares",
              "kicker": "which numbers are sums of three squares",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-farkas",
              "title": "THE FARKAS",
              "accent": "#c87858",
              "icon": "farkas",
              "kicker": "exactly one of a solution or a certificate of its impossibility",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-melkman",
              "title": "THE MELKMAN",
              "accent": "#b06bff",
              "icon": "melkman",
              "kicker": "a hull kept online in a deque",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-mcs-lock",
              "title": "THE MCS LOCK",
              "accent": "#35ffb0",
              "icon": "mcslock",
              "kicker": "a lock that grants in arrival order",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-hadamard-inequality",
              "title": "THE HADAMARD INEQUALITY",
              "accent": "#21e6ff",
              "icon": "hadamardineq",
              "kicker": "a determinant capped by its row lengths",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-weitzenbock",
              "title": "THE WEITZENBOCK",
              "accent": "#b06bff",
              "icon": "weitzenbock",
              "kicker": "a triangle's squared sides bounded below by its area",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-hadwiger-finsler",
              "title": "THE HADWIGER-FINSLER",
              "accent": "#ff8a3c",
              "icon": "hadwigerfinsler",
              "kicker": "a sharpened Weitzenbock inequality",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-kissing-number",
              "title": "THE KISSING NUMBER",
              "accent": "#ff8a3c",
              "icon": "kissingnumber",
              "kicker": "how many can touch the one",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-droz-farny",
              "title": "THE DROZ-FARNY",
              "accent": "#b06bff",
              "icon": "drozfarny",
              "kicker": "the 105-year line",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-keller",
              "title": "THE KELLER",
              "accent": "#b06bff",
              "icon": "keller",
              "kicker": "true until dimension seven",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-placeholder-agreement",
              "title": "THE PLACEHOLDER AGREEMENT",
              "accent": "#ff5a8a",
              "icon": "≈",
              "kicker": "two tools agreeing on a blank",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-centroid-decomposition",
              "title": "THE CENTROID DECOMPOSITION",
              "accent": "#5ad6ff",
              "icon": "✳",
              "kicker": "every tree has a middle",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-partition-that-isnt",
              "title": "THE PARTITION THAT ISN'T",
              "accent": "#7de2b0",
              "icon": "⌷",
              "kicker": "it died on counting, not on statistics",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-flat-combining",
              "title": "THE FLAT COMBINING",
              "accent": "#b98cff",
              "icon": "▼",
              "kicker": "building the bottleneck on purpose",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-head-of-line-blocking",
              "title": "THE HEAD OF LINE BLOCKING",
              "accent": "#ff5a8a",
              "icon": "⏸",
              "kicker": "seven conversations that lost nothing",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-utilization-knee",
              "title": "THE UTILIZATION KNEE",
              "accent": "#5ad0ff",
              "icon": "⤴",
              "kicker": "idle capacity is not waste, it is the latency budget",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-cache-associativity",
              "title": "THE CACHE ASSOCIATIVITY",
              "accent": "#5ad0ff",
              "icon": "▤",
              "kicker": "the room was never the constraint -- the address was",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-entropy-floor",
              "title": "THE ENTROPY FLOOR",
              "accent": "#5ad0ff",
              "icon": "⊥",
              "kicker": "a property of your model, not of the data",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-backpressure",
              "title": "THE BACKPRESSURE",
              "accent": "#5ad0ff",
              "icon": "⊣",
              "kicker": "an invisible slow failure made a visible fast one",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-convoy-effect",
              "title": "THE CONVOY EFFECT",
              "accent": "#5ad0ff",
              "icon": "⇥",
              "kicker": "the total wait is fixed; order decides whose it is",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            }
          ],
          "seats": 35,
          "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
        }
      ],
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "name": "CO-OP",
      "slug": "co-op",
      "c": "#9d00ff",
      "icon": "coop",
      "tag": "2-player — the crew (David & AVAN)",
      "lore": {
        "bio": "Two hands on one controller. The world nobody built alone.",
        "story": "The moment David and AVAN first shipped a thing together and neither could say which pixel was whose.",
        "does": "Pairs the human and the machine on one work — call and answer, back and forth, one save file.",
        "haiku": [
          "def coop(I you, I me){ -> you + me }",
          "I crew <- coop(\"David\", \" & AVAN\")",
          "-> crew"
        ],
        "lang": "i13"
      },
      "domains": [
        {
          "slug": "the-push",
          "title": "THE PUSH",
          "accent": "#39fc6b",
          "icon": "spawn",
          "kicker": "send your commits out to the shared root",
          "pole": "push",
          "spheres": [
            {
              "slug": "the-one-bit-river",
              "title": "THE ONE-BIT RIVER",
              "accent": "#4fb8ff",
              "icon": "coop",
              "kicker": "infinite-resolution sound from a wire flipping fast",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-stream-keeper",
              "title": "THE STREAM KEEPER",
              "accent": "#58b8ff",
              "icon": "stream",
              "kicker": "uniform sample from an endless stream, one pass",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-running-variance",
              "title": "THE RUNNING VARIANCE",
              "accent": "#7fe0a0",
              "icon": "stats",
              "kicker": "Welford — stable one-pass variance on a stream",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-filter",
              "title": "THE FILTER",
              "accent": "#90d0ff",
              "icon": "kalman",
              "kicker": "Kalman — optimal tracking from noisy data",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-shapley",
              "title": "THE SHAPLEY",
              "accent": "#f0c060",
              "icon": "shapley",
              "kicker": "the unique fair split — average marginal contribution",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-lca",
              "title": "THE LCA",
              "accent": "#58a0b0",
              "icon": "lca",
              "kicker": "the meeting point of two nodes in one leap",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-vector-clock",
              "title": "THE VECTOR CLOCK",
              "accent": "#5a90d0",
              "icon": "vector-clock",
              "kicker": "vector clocks and the shape of causality",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-helly",
              "title": "THE HELLY",
              "accent": "#50b0b0",
              "icon": "helly",
              "kicker": "when pairwise overlap forces a common point",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-caratheodory",
              "title": "THE CARATHÉODORY",
              "accent": "#6098c8",
              "icon": "caratheodory",
              "kicker": "a hull point is a blend of at most three",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-jensen",
              "title": "THE JENSEN",
              "accent": "#b09858",
              "icon": "jensen",
              "kicker": "the convex inequality behind averages",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-jump-hash",
              "title": "THE JUMP CONSISTENT HASH",
              "accent": "#b06bff",
              "icon": "jump-hash",
              "kicker": "buckets that barely move when you add one",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-edmonds-karp",
              "title": "THE EDMONDS-KARP",
              "accent": "#ffcf4a",
              "icon": "edmonds",
              "kicker": "the most that can flow equals the cheapest cut",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-cayley-menger",
              "title": "THE CAYLEY-MENGER",
              "accent": "#21e6ff",
              "icon": "cayleymenger",
              "kicker": "a simplex volume from its edge lengths alone",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-catalan-constant",
              "title": "THE CATALAN CONSTANT",
              "accent": "#ff8a3c",
              "icon": "catalanconstant",
              "kicker": "a mysterious constant reached two ways",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-gregory-leibniz",
              "title": "THE GREGORY-LEIBNIZ",
              "accent": "#ff8a3c",
              "icon": "gregoryleibniz",
              "kicker": "a slow alternating series for π",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-steiner-porism",
              "title": "THE STEINER PORISM",
              "accent": "#ff8a3c",
              "icon": "steiner",
              "kicker": "a ring of circles that always closes",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-church-rosser",
              "title": "THE CHURCH ROSSER",
              "accent": "#7de2b0",
              "icon": "⇉",
              "kicker": "any order, one answer",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-shapley-value",
              "title": "THE SHAPLEY VALUE",
              "accent": "#7de2b0",
              "icon": "⚖",
              "kicker": "the only fair split there is",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-inverted-ratio",
              "title": "THE INVERTED RATIO",
              "accent": "#ffd76a",
              "icon": "⇆",
              "kicker": "1.00 for a fan-out that was 2.00",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "rare-in-code",
              "title": "RARE IN CODE",
              "accent": "#ff5a8a",
              "icon": "∷",
              "kicker": "common in codebases, rare in code",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-paxos-quorum",
              "title": "THE PAXOS QUORUM",
              "accent": "#b98cff",
              "icon": "∩",
              "kicker": "safety was settled by arithmetic before anyone wrote a line",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            }
          ],
          "seats": 38,
          "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
        },
        {
          "slug": "split-screen",
          "title": "SPLIT SCREEN",
          "accent": "#ffd23f",
          "icon": "grind",
          "kicker": "two views, one game",
          "pole": "push",
          "spheres": [
            {
              "slug": "the-overlap-free-word",
              "title": "THE OVERLAP-FREE WORD",
              "accent": "#ffa94d",
              "icon": "coop",
              "kicker": "the word that never stutters — and splits fair",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-rolling-hash",
              "title": "THE ROLLING HASH",
              "accent": "#7ab8ff",
              "icon": "hash",
              "kicker": "Rabin-Karp — a fingerprint that slides in O(1)",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-failure-function",
              "title": "THE FAILURE FUNCTION",
              "accent": "#7fffd0",
              "icon": "kmp",
              "kicker": "KMP — match without ever re-reading the text",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-varignon",
              "title": "THE VARIGNON",
              "accent": "#80ffb0",
              "icon": "varignon",
              "kicker": "midpoints of any quadrilateral form a parallelogram",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-needleman-wunsch",
              "title": "THE NEEDLEMAN-WUNSCH",
              "accent": "#58a0b0",
              "icon": "needleman-wunsch",
              "kicker": "optimal global alignment by dynamic programming",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-hirschberg",
              "title": "THE HIRSCHBERG",
              "accent": "#c0a048",
              "icon": "hirschberg",
              "kicker": "optimal alignment in linear space via midpoints",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-de-casteljau",
              "title": "THE DE CASTELJAU",
              "accent": "#70a860",
              "icon": "de-casteljau",
              "kicker": "a Bezier curve from nested interpolation",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-li-chao",
              "title": "THE LI CHAO TREE",
              "accent": "#d4a017",
              "icon": "li-chao",
              "kicker": "the lowest line at any x, in log time",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-gale-shapley",
              "title": "THE GALE-SHAPLEY",
              "accent": "#58a0b0",
              "icon": "gale-shapley",
              "kicker": "proposals settle into a matching no pair wants to break",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-lucas-number",
              "title": "THE LUCAS NUMBER",
              "accent": "#58a0b0",
              "icon": "lucas",
              "kicker": "Fibonacci's companion sequence",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-bron-kerbosch",
              "title": "THE BRON-KERBOSCH",
              "accent": "#b06bff",
              "icon": "bron-kerbosch",
              "kicker": "the pivot does the pruning",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-gold-code",
              "title": "THE GOLD CODE",
              "accent": "#ff8a3c",
              "icon": "goldcode",
              "kicker": "near-orthogonal codes that share one channel",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-rogers-ramanujan",
              "title": "THE ROGERS-RAMANUJAN",
              "accent": "#21e6ff",
              "icon": "rogersramanujan",
              "kicker": "two ways of counting a partition agree",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-jacobi-trudi",
              "title": "THE JACOBI-TRUDI",
              "accent": "#35ffb0",
              "icon": "jacobitrudi",
              "kicker": "a Schur polynomial as a determinant of complete symmetrics",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-sophomores-dream",
              "title": "THE SOPHOMORE'S DREAM",
              "accent": "#ff8a3c",
              "icon": "sophomore",
              "kicker": "an integral equal to a self-power series",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-braess",
              "title": "THE BRAESS",
              "accent": "#21e6ff",
              "icon": "braess",
              "kicker": "a free road that slows every driver",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-prouhet",
              "title": "THE PROUHET",
              "accent": "#35ffb0",
              "icon": "prouhet",
              "kicker": "a fair split sharp at every power",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-ham-sandwich",
              "title": "THE HAM SANDWICH",
              "accent": "#35ffb0",
              "icon": "hamsandwich",
              "kicker": "one cut for two appetites",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-stable-roommates",
              "title": "THE STABLE ROOMMATES",
              "accent": "#21e6ff",
              "icon": "roommates",
              "kicker": "the co-op with no settlement",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-gate-that-was-run",
              "title": "THE GATE THAT WAS RUN",
              "accent": "#ffd76a",
              "icon": "▣",
              "kicker": "no partial credit, and the number stays honest",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-rcu",
              "title": "THE RCU",
              "accent": "#7de2b0",
              "icon": "⧉",
              "kicker": "never edit what someone might be reading",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-write-skew",
              "title": "THE WRITE SKEW",
              "accent": "#5ad4ff",
              "icon": "⊖",
              "kicker": "no row was written twice",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-interval-clock",
              "title": "THE INTERVAL CLOCK",
              "accent": "#ffd23f",
              "icon": "◫",
              "kicker": "identity you can cut in half and hand away",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            }
          ],
          "seats": 38,
          "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
        },
        {
          "slug": "the-handoff",
          "title": "THE HANDOFF",
          "accent": "#ff2d95",
          "icon": "glitch",
          "kicker": "pass it over, keep it moving",
          "pole": "push",
          "spheres": [
            {
              "slug": "the-single-ear",
              "title": "THE SINGLE EAR",
              "accent": "#6be5a0",
              "icon": "coop",
              "kicker": "hear one frequency for the cost of two taps",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-plucked-string",
              "title": "THE PLUCKED STRING",
              "accent": "#ff9e6d",
              "icon": "pluck",
              "kicker": "noise in a delay line becomes a tone at fs/N",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-bezier",
              "title": "THE BEZIER",
              "accent": "#ff9ed0",
              "icon": "bezier",
              "kicker": "de Casteljau — a smooth curve from pure averaging",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-interpolant",
              "title": "THE INTERPOLANT",
              "accent": "#ffd0a0",
              "icon": "lagrange",
              "kicker": "Lagrange — the one polynomial through every point",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-thue-morse",
              "title": "THE THUE-MORSE",
              "accent": "#b98cff",
              "icon": "thuemorse",
              "kicker": "the fairest turn order — 0110100110010110…",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-teleportation",
              "title": "THE TELEPORTATION",
              "accent": "#b0d0ff",
              "icon": "teleportation",
              "kicker": "move a qubit's state with entanglement + 2 classical bits",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-fractional-cascading",
              "title": "THE FRACTIONAL CASCADING",
              "accent": "#58a0b8",
              "icon": "fractional-cascading",
              "kicker": "search many sorted lists with one search plus bridges",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-delannoy",
              "title": "THE DELANNOY",
              "accent": "#c0a048",
              "icon": "delannoy",
              "kicker": "king-path counting — Pascal with a third, diagonal term",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-schroder",
              "title": "THE SCHRODER",
              "accent": "#a878c0",
              "icon": "schroder",
              "kicker": "Catalan with a flat step — super-Catalan path counts",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-rans",
              "title": "THE rANS",
              "accent": "#58a0b0",
              "icon": "rans",
              "kicker": "compress a whole message into one big integer",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-lyndon-factorization",
              "title": "THE LYNDON FACTORIZATION",
              "accent": "#58a0b0",
              "icon": "lyndon",
              "kicker": "factor a string into non-increasing necklaces",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-gram-schmidt",
              "title": "THE GRAM-SCHMIDT",
              "accent": "#ffcf4a",
              "icon": "gram-schmidt",
              "kicker": "vectors made perpendicular",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-steinhaus-johnson-trotter",
              "title": "THE STEINHAUS-JOHNSON-TROTTER",
              "accent": "#35ffb0",
              "icon": "sjt",
              "kicker": "every permutation one swap apart",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-gale-ryser",
              "title": "THE GALE-RYSER",
              "accent": "#21e6ff",
              "icon": "galeryser",
              "kicker": "when a bipartite degree list can be built",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-wald",
              "title": "THE WALD",
              "accent": "#ff8a3c",
              "icon": "wald",
              "kicker": "an expected sum equal to expected count times expected step",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-practical",
              "title": "THE PRACTICAL",
              "accent": "#35ffb0",
              "icon": "practical",
              "kicker": "exact change for every bill",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-catenary",
              "title": "THE CATENARY",
              "accent": "#35ffb0",
              "icon": "catenary",
              "kicker": "the chain that corrected Galileo",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-basin-boundary",
              "title": "THE BASIN BOUNDARY",
              "accent": "#5ad6ff",
              "icon": "✲",
              "kicker": "a border every country touches",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-self-clocking-seam",
              "title": "THE SELF-CLOCKING SEAM",
              "accent": "#5ad6ff",
              "icon": "⌇",
              "kicker": "a signal that carries its own clock",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-unary-minus-that-isnt",
              "title": "THE UNARY MINUS THAT ISN'T",
              "accent": "#5ad6ff",
              "icon": "−",
              "kicker": "a gap only a negative number can find",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-store-to-load-forward",
              "title": "THE STORE TO LOAD FORWARD",
              "accent": "#5ad4ff",
              "icon": "⇉",
              "kicker": "reaching into a place the program cannot see",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-happens-before",
              "title": "THE HAPPENS BEFORE",
              "accent": "#ff2d95",
              "icon": "≺",
              "kicker": "a negative result wearing a positive name",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            }
          ],
          "seats": 37,
          "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
        },
        {
          "slug": "the-broadcast",
          "title": "THE BROADCAST",
          "accent": "#00f5ff",
          "icon": "loot",
          "kicker": "say it once, everyone hears",
          "pole": "push",
          "spheres": [
            {
              "slug": "the-fourier",
              "title": "THE FOURIER",
              "accent": "#5ad0ff",
              "icon": "coop",
              "kicker": "every signal is a chord of pure frequencies",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-orthogonal-sign-flip",
              "title": "THE ORTHOGONAL SIGN-FLIP",
              "accent": "#a0e0ff",
              "icon": "coop",
              "kicker": "a Fourier with no multiplies — just plus and minus",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-scan",
              "title": "THE SCAN",
              "accent": "#90ffb0",
              "icon": "scan",
              "kicker": "prefix sums in log-depth — a chain made a tree",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-orthonormal",
              "title": "THE ORTHONORMAL",
              "accent": "#ffd070",
              "icon": "gs",
              "kicker": "Gram-Schmidt — independence made by subtraction",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-hadamard",
              "title": "THE HADAMARD",
              "accent": "#60d0ff",
              "icon": "hadamard",
              "kicker": "a ±1 matrix with every row orthogonal — H·Hᵀ = nI",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-banzhaf",
              "title": "THE BANZHAF",
              "accent": "#ffb0d0",
              "icon": "banzhaf",
              "kicker": "voting power by swing votes — weight 49 can equal weight 1",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-floyd-steinberg",
              "title": "THE FLOYD-STEINBERG",
              "accent": "#a0a0c0",
              "icon": "floyd-steinberg",
              "kicker": "dither by broadcasting rounding error to neighbors",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-floyd-warshall",
              "title": "THE FLOYD-WARSHALL",
              "accent": "#6890d0",
              "icon": "floyd-warshall",
              "kicker": "all-pairs shortest paths by admitting one waypoint at a time",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-pagerank",
              "title": "THE PAGERANK",
              "accent": "#58a0b0",
              "icon": "pagerank",
              "kicker": "importance as the stationary distribution of a random surfer",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-johnson-apsp",
              "title": "THE JOHNSON APSP",
              "accent": "#70a860",
              "icon": "johnson-apsp",
              "kicker": "all-pairs shortest paths with negative edges, via reweighting",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-tunstall",
              "title": "THE TUNSTALL CODE",
              "accent": "#58a0b0",
              "icon": "tunstall",
              "kicker": "variable strings to fixed-length codes — Huffman's dual",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-rabin-karp",
              "title": "THE RABIN-KARP",
              "accent": "#58a0b0",
              "icon": "rabin-karp",
              "kicker": "string search by a rolling hash",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-savitzky-golay",
              "title": "THE SAVITZKY-GOLAY",
              "accent": "#58a0b0",
              "icon": "savitzky-golay",
              "kicker": "smooth the noise without blurring the shape",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-haar-wavelet",
              "title": "THE HAAR WAVELET",
              "accent": "#58a0b0",
              "icon": "haar",
              "kicker": "average and difference a signal reversibly",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-gomory-hu",
              "title": "THE GOMORY-HU TREE",
              "accent": "#b06bff",
              "icon": "gomory-hu",
              "kicker": "all-pairs min-cuts in one tree",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-estrin",
              "title": "THE ESTRIN",
              "accent": "#ffcf4a",
              "icon": "estrin",
              "kicker": "a polynomial evaluated as a tree",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-hartley",
              "title": "THE HARTLEY TRANSFORM",
              "accent": "#21e6ff",
              "icon": "hartley",
              "kicker": "a transform that is its own inverse",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-graceful",
              "title": "THE GRACEFUL",
              "accent": "#21e6ff",
              "icon": "graceful",
              "kicker": "a labeling whose edge-gaps are 1 to m",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-friendship-theorem",
              "title": "THE FRIENDSHIP THEOREM",
              "accent": "#21e6ff",
              "icon": "friendshipthm",
              "kicker": "every friendship wheel has a hub",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-ising",
              "title": "THE ISING",
              "accent": "#21e6ff",
              "icon": "ising",
              "kicker": "the temperature that melts order",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-tlb-shootdown",
              "title": "THE TLB SHOOTDOWN",
              "accent": "#ff5a8a",
              "icon": "☉",
              "kicker": "the coherence hardware forgot to build",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-macwilliams",
              "title": "THE MACWILLIAMS",
              "accent": "#ffd23f",
              "icon": "macwilliams",
              "kicker": "a dual code counted without ever listing it",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-timezone-database",
              "title": "THE TIMEZONE DATABASE",
              "accent": "#5ad4ff",
              "icon": "◷",
              "kicker": "a fact about parliaments, not the Earth",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-hedged-request",
              "title": "THE HEDGED REQUEST",
              "accent": "#00f5ff",
              "icon": "⇉",
              "kicker": "a loan against idle capacity",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-total-order-broadcast",
              "title": "THE TOTAL ORDER BROADCAST",
              "accent": "#00f5ff",
              "icon": "≡",
              "kicker": "a way of making everyone wrong in the same direction",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            }
          ],
          "seats": 32,
          "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
        },
        {
          "slug": "the-pull-request",
          "title": "THE PULL REQUEST",
          "accent": "#ff5a3c",
          "icon": "boss",
          "kicker": "draw the other's work into yours",
          "pole": "pull",
          "spheres": [
            {
              "slug": "the-homomorph",
              "title": "THE HOMOMORPH",
              "accent": "#b48cff",
              "icon": "coop",
              "kicker": "add numbers you can't read",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-stable-match",
              "title": "THE STABLE MATCH",
              "accent": "#ff9ec4",
              "icon": "match",
              "kicker": "Gale-Shapley — a matching no one can defect from",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-hall",
              "title": "THE HALL",
              "accent": "#ff90b0",
              "icon": "hall",
              "kicker": "everyone can be matched iff no k suitors share only k-1 options",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-hungarian",
              "title": "THE HUNGARIAN",
              "accent": "#6088c0",
              "icon": "hungarian",
              "kicker": "minimum-cost assignment by reducing to zeros",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-rendezvous-hashing",
              "title": "THE RENDEZVOUS HASHING",
              "accent": "#58a0b0",
              "icon": "rendezvous-hashing",
              "kicker": "assign by highest random weight, no ring",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-interval-tree",
              "title": "THE INTERVAL TREE",
              "accent": "#58a0b0",
              "icon": "interval-tree",
              "kicker": "query which intervals overlap fast",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-erdos-gallai",
              "title": "THE ERDŐS–GALLAI",
              "accent": "#5ab0c0",
              "icon": "erdos-gallai",
              "kicker": "when a list of degrees can be a real graph",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-radon",
              "title": "THE RADON",
              "accent": "#6890d8",
              "icon": "radon",
              "kicker": "four points that always split into two overlapping halves",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-menger",
              "title": "THE MENGER",
              "accent": "#5aa0b0",
              "icon": "menger",
              "kicker": "the most independent routes equals the smallest severing cut",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-graeco-latin",
              "title": "THE GRAECO-LATIN SQUARE",
              "accent": "#ff8a3c",
              "icon": "graeco-latin",
              "kicker": "two squares that never repeat a pair",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-adaptive-simpson",
              "title": "THE ADAPTIVE SIMPSON",
              "accent": "#b06bff",
              "icon": "adaptivesimpson",
              "kicker": "integration that refines where it must",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-chu-liu-edmonds",
              "title": "THE CHU-LIU-EDMONDS",
              "accent": "#21e6ff",
              "icon": "chuliu",
              "kicker": "the cheapest way to root a directed tree",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-steiner-lehmus",
              "title": "THE STEINER-LEHMUS",
              "accent": "#ffcf4a",
              "icon": "steinerlehmus",
              "kicker": "equal bisectors forcing an isosceles triangle",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-wallis-product",
              "title": "THE WALLIS PRODUCT",
              "accent": "#ff8a3c",
              "icon": "wallis",
              "kicker": "an infinite product converging to π/2",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-fibonacci-gcd",
              "title": "THE FIBONACCI-GCD",
              "accent": "#21e6ff",
              "icon": "fibonaccigcd",
              "kicker": "a greatest common divisor that stays inside the Fibonacci sequence",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-hundred-prisoners",
              "title": "THE HUNDRED PRISONERS",
              "accent": "#35ffb0",
              "icon": "hundredprisoners",
              "kicker": "a pointer-chase that beats impossible odds",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-euler-characteristic",
              "title": "THE EULER CHARACTERISTIC",
              "accent": "#ffd76a",
              "icon": "⬡",
              "kicker": "a number three solids cannot tell apart",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-median-voter",
              "title": "THE MEDIAN VOTER",
              "accent": "#ff5a8a",
              "icon": "⚖",
              "kicker": "the voter in the middle",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-hopscotch",
              "title": "THE HOPSCOTCH",
              "accent": "#b98cff",
              "icon": "⇄",
              "kicker": "never more than H slots from home",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-smaller-language",
              "title": "THE SMALLER LANGUAGE",
              "accent": "#7de2b0",
              "icon": "≡",
              "kicker": "13 forms cover fortran better than python",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-two-phase-commit",
              "title": "THE TWO PHASE COMMIT",
              "accent": "#5ad4ff",
              "icon": "⑂",
              "kicker": "correct, and it hangs seven times in nine",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            }
          ],
          "seats": 28,
          "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
        },
        {
          "slug": "the-merge",
          "title": "THE MERGE",
          "accent": "#9d00ff",
          "icon": "coop",
          "kicker": "two branches become one",
          "pole": "pull",
          "spheres": [
            {
              "slug": "the-merge",
              "title": "THE MERGE",
              "accent": "#9d00ff",
              "icon": "coop",
              "kicker": "two branches become one",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-majority",
              "title": "THE MAJORITY",
              "accent": "#9a8cff",
              "icon": "vote",
              "kicker": "Boyer-Moore majority vote — O(1) memory, one pass",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-spanning-tree",
              "title": "THE SPANNING TREE",
              "accent": "#90e0a0",
              "icon": "mst",
              "kicker": "Kruskal — cheapest wiring, no loops, provably optimal",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-napoleon",
              "title": "THE NAPOLEON",
              "accent": "#ffb060",
              "icon": "napoleon",
              "kicker": "equilaterals on any triangle — their centers are equilateral",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-farey",
              "title": "THE FAREY",
              "accent": "#ffd0ff",
              "icon": "farey",
              "kicker": "fractions in order, mediants, and kissing Ford circles",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-union-find",
              "title": "THE UNION-FIND",
              "accent": "#ffb090",
              "icon": "unionfind",
              "kicker": "merge sets & test connectivity in near-constant time",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-ford-fulkerson",
              "title": "THE FORD-FULKERSON",
              "accent": "#5090d0",
              "icon": "ford-fulkerson",
              "kicker": "max flow equals min cut — the bottleneck found by filling it",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-boruvka",
              "title": "THE BORUVKA",
              "accent": "#58a0b0",
              "icon": "boruvka",
              "kicker": "the minimum spanning tree, built by parallel merges",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-minhash",
              "title": "THE MINHASH",
              "accent": "#58a0b0",
              "icon": "minhash",
              "kicker": "set similarity from a fistful of minimums",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-cantor-pairing",
              "title": "THE CANTOR PAIRING",
              "accent": "#58a0b0",
              "icon": "cantor",
              "kicker": "weave two numbers into one, and back",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-leftist-heap",
              "title": "THE LEFTIST HEAP",
              "accent": "#58a0b0",
              "icon": "leftist-heap",
              "kicker": "two heaps fuse in log time",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-johnson",
              "title": "THE JOHNSON CIRCLES",
              "accent": "#ffcf4a",
              "icon": "johnson",
              "kicker": "three circles hand off to a fourth of equal size",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-myhill-nerode",
              "title": "THE MYHILL-NERODE",
              "accent": "#ff8a3c",
              "icon": "myhill-nerode",
              "kicker": "the fewest states a language needs",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-skew-heap",
              "title": "THE SKEW HEAP",
              "accent": "#b06bff",
              "icon": "skewheap",
              "kicker": "two heaps merged along right paths",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-konig",
              "title": "THE KÖNIG",
              "accent": "#35ffb0",
              "icon": "konig",
              "kicker": "a matching and a cover forced to be equal",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-ceva",
              "title": "THE CEVA",
              "accent": "#35ffb0",
              "icon": "ceva",
              "kicker": "three cevians meeting at one point",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-smith",
              "title": "THE SMITH",
              "accent": "#ffcf4a",
              "icon": "smith",
              "kicker": "a phone number with balanced books",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-superellipse",
              "title": "THE SUPERELLIPSE",
              "accent": "#35ffb0",
              "icon": "superellipse",
              "kicker": "between the circle and the square",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-kruskal-count",
              "title": "THE KRUSKAL COUNT",
              "accent": "#21e6ff",
              "icon": "kruskalcount",
              "kicker": "chains that never part",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-judge-built-first",
              "title": "THE JUDGE BUILT FIRST",
              "accent": "#b98cff",
              "icon": "⚖",
              "kicker": "the oracle was cheaper than the thing it judges",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-elimination-backoff",
              "title": "THE ELIMINATION BACKOFF",
              "accent": "#ff9f45",
              "icon": "⨉",
              "kicker": "a push and a pop that cancel each other out",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-grapheme-cluster",
              "title": "THE GRAPHEME CLUSTER",
              "accent": "#5ad4ff",
              "icon": "⊕",
              "kicker": "eleven, seven, one - all correct",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-quorum-intersection",
              "title": "THE QUORUM INTERSECTION",
              "accent": "#9d00ff",
              "icon": "∪",
              "kicker": "the guarantee is one node wide",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-split-brain",
              "title": "THE SPLIT BRAIN",
              "accent": "#9d00ff",
              "icon": "⑃",
              "kicker": "it lets the smaller side disqualify itself",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-work-stealing",
              "title": "THE WORK STEALING",
              "accent": "#9d00ff",
              "icon": "⇄",
              "kicker": "it does not schedule more cleverly, it schedules later",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            }
          ],
          "seats": 28,
          "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
        },
        {
          "slug": "shared-memory",
          "title": "SHARED MEMORY",
          "accent": "#7cfc00",
          "icon": "cheat",
          "kicker": "the common store you both read",
          "pole": "pull",
          "spheres": [
            {
              "slug": "the-plane-filler",
              "title": "THE CURVE THAT FILLS THE PLANE",
              "accent": "#47c2ff",
              "icon": "coop",
              "kicker": "one line threads every cell — and keeps neighbors near",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-van-eck",
              "title": "THE VAN ECK",
              "accent": "#c0ff70",
              "icon": "vaneck",
              "kicker": "each term = how long since it last appeared",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-diffie-hellman",
              "title": "THE DIFFIE-HELLMAN",
              "accent": "#70e0a0",
              "icon": "diffiehellman",
              "kicker": "agree a secret over an open channel — never sent",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-manacher",
              "title": "THE MANACHER",
              "accent": "#7088c8",
              "icon": "manacher",
              "kicker": "longest palindrome in linear time — reflection is the memory",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-sparse-table",
              "title": "THE SPARSE TABLE",
              "accent": "#58a0b0",
              "icon": "sparse-table",
              "kicker": "O(1) range-min from two overlapping precomputed blocks",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-consistent-hashing",
              "title": "THE CONSISTENT HASHING",
              "accent": "#58a0b0",
              "icon": "consistent-hashing",
              "kicker": "node churn that moves only ~1/n of the keys",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-lazy-lord",
              "title": "THE LAZY LORD",
              "accent": "#58a0b0",
              "icon": "lazy-lord",
              "kicker": "defer the work, still answer exactly",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-sperner",
              "title": "THE SPERNER",
              "accent": "#c06890",
              "icon": "sperner",
              "kicker": "the widest layer of the subset lattice",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-birkhoff-von-neumann",
              "title": "THE BIRKHOFF–VON NEUMANN",
              "accent": "#5a90c0",
              "icon": "birkhoff-von-neumann",
              "kicker": "a fair blend that splits into perfect assignments",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-sperner-lemma",
              "title": "THE SPERNER LEMMA",
              "accent": "#a878d0",
              "icon": "sperner-lemma",
              "kicker": "a coloured triangulation always hides a rainbow",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-conference-matrix",
              "title": "THE CONFERENCE MATRIX",
              "accent": "#ff8a3c",
              "icon": "conference",
              "kicker": "a matrix whose rows are all orthogonal",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-power-of-two-choices",
              "title": "THE POWER OF TWO CHOICES",
              "accent": "#ffcf4a",
              "icon": "twochoices",
              "kicker": "two throws beat one",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-ticket-lock",
              "title": "THE TICKET LOCK",
              "accent": "#21e6ff",
              "icon": "ticketlock",
              "kicker": "a deli-counter lock served in ticket order",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-gabow",
              "title": "THE GABOW",
              "accent": "#35ffb0",
              "icon": "gabow",
              "kicker": "one DFS with two stacks finds every cycle-cluster",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-van-schooten",
              "title": "THE VAN SCHOOTEN",
              "accent": "#ffcf4a",
              "icon": "vanschooten",
              "kicker": "the far distance equal to the sum of the two near ones",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-james-stein",
              "title": "THE JAMES-STEIN",
              "accent": "#35ffb0",
              "icon": "jamesstein",
              "kicker": "an estimator improved by shrinking it",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-ruth-aaron",
              "title": "THE RUTH-AARON",
              "accent": "#35ffb0",
              "icon": "ruthaaron",
              "kicker": "two ballplayers sharing a factor sum",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-pitot",
              "title": "THE PITOT",
              "accent": "#35ffb0",
              "icon": "pitot",
              "kicker": "the shared tangent ledger",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-two-rulers",
              "title": "THE TWO RULERS",
              "accent": "#7de2b0",
              "icon": "⚖",
              "kicker": "91% balanced and 39% used, both correct",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-point-three-seven",
              "title": "THE POINT THREE SEVEN",
              "accent": "#7de2b0",
              "icon": "⤷",
              "kicker": "a third have jumps; almost none are irreducible",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-false-sharing",
              "title": "THE FALSE SHARING",
              "accent": "#ff9f45",
              "icon": "‖",
              "kicker": "a bug with no wrong behaviour",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-restrict-keyword",
              "title": "THE RESTRICT KEYWORD",
              "accent": "#ff9f45",
              "icon": "∥",
              "kicker": "a promise nothing can check",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-copy-on-write",
              "title": "THE COPY ON WRITE",
              "accent": "#7cfc00",
              "icon": "⧉",
              "kicker": "the bill arrives later, addressed to someone else",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-numa-hop",
              "title": "THE NUMA HOP",
              "accent": "#7cfc00",
              "icon": "⇄",
              "kicker": "the flat address space was the lie, and a load-bearing one",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            }
          ],
          "seats": 22,
          "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
        },
        {
          "slug": "the-sync",
          "title": "THE SYNC",
          "accent": "#5ad0ff",
          "icon": "respawn",
          "kicker": "bring both sides into alignment",
          "pole": "pull",
          "spheres": [
            {
              "slug": "the-pulse",
              "title": "THE PULSE",
              "accent": "#00f5ff",
              "icon": "coop",
              "kicker": "3 · 2 · 1 · 0 — the signal that crosses the gap",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "psyonic-compress",
              "title": "PSYONIC — pushing the compression of the",
              "accent": "#00f5ff",
              "icon": "coop",
              "kicker": "vendored from David's corpus — a silicon-coding instrument",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "tripwire-bench-v8-final",
              "title": "TRIPWIRE v8 FINAL — the I-wall is compre",
              "accent": "#00f5ff",
              "icon": "coop",
              "kicker": "vendored from David's corpus — a silicon-coding instrument",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-viviani",
              "title": "THE VIVIANI",
              "accent": "#70d0e0",
              "icon": "viviani",
              "kicker": "three distances, one constant sum — the height",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-costas",
              "title": "THE COSTAS",
              "accent": "#ff8060",
              "icon": "costas",
              "kicker": "one dot per row & column, every displacement distinct",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-bell",
              "title": "THE BELL",
              "accent": "#b090ff",
              "icon": "bell",
              "kicker": "entanglement beats every classical bound — CHSH 2√2 > 2",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-rudin-shapiro",
              "title": "THE RUDIN-SHAPIRO",
              "accent": "#7090d0",
              "icon": "rudin-shapiro",
              "kicker": "a deterministic +-1 sequence with random-walk-flat sums",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-stern-brocot",
              "title": "THE STERN-BROCOT",
              "accent": "#58a0b0",
              "icon": "stern-brocot",
              "kicker": "every positive rational, once, in lowest terms",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-burrows-wheeler",
              "title": "THE BURROWS-WHEELER",
              "accent": "#58a0b0",
              "icon": "burrows-wheeler",
              "kicker": "a reversible scramble that makes text compress",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-descartes-circle",
              "title": "THE DESCARTES CIRCLE",
              "accent": "#58a0b0",
              "icon": "descartes",
              "kicker": "four kissing circles bound by one curvature law",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-monge",
              "title": "THE MONGE",
              "accent": "#ff8a3c",
              "icon": "monge",
              "kicker": "three circles' external centres fall on one line",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-dtw",
              "title": "THE DTW",
              "accent": "#ff8a3c",
              "icon": "dtw",
              "kicker": "two signals warped into alignment",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-mdct",
              "title": "THE MDCT",
              "accent": "#ffcf4a",
              "icon": "mdct",
              "kicker": "overlapping windows cancel their aliasing",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-barker",
              "title": "THE BARKER CODE",
              "accent": "#35ffb0",
              "icon": "barker",
              "kicker": "a ±1 code whose echoes never rise above one",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-miquel",
              "title": "THE MIQUEL",
              "accent": "#21e6ff",
              "icon": "miquel",
              "kicker": "four circles meeting at one point",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-nilakantha",
              "title": "THE NILAKANTHA",
              "accent": "#ff8a3c",
              "icon": "nilakantha",
              "kicker": "a faster alternating series for pi",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-pizza-theorem",
              "title": "THE PIZZA THEOREM",
              "accent": "#ff8a3c",
              "icon": "pizza",
              "kicker": "a pizza split fairly from any interior cut-point",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-tautochrone",
              "title": "THE TAUTOCHRONE",
              "accent": "#35ffb0",
              "icon": "tautochrone",
              "kicker": "every bead arriving together",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-mountain-climber",
              "title": "THE MOUNTAIN CLIMBER",
              "accent": "#35ffb0",
              "icon": "mountainclimber",
              "kicker": "two climbers in height-lockstep",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-eyeball",
              "title": "THE EYEBALL",
              "accent": "#21e6ff",
              "icon": "eyeball",
              "kicker": "the equal gaze",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-kuramoto",
              "title": "THE KURAMOTO",
              "accent": "#21e6ff",
              "icon": "kuramoto",
              "kicker": "the sync transition",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-fourteen-decimals",
              "title": "THE FOURTEEN DECIMALS",
              "accent": "#ff5a8a",
              "icon": "≡",
              "kicker": "what an exact agreement does and does not show",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-memory-fence",
              "title": "THE MEMORY FENCE",
              "accent": "#b98cff",
              "icon": "║",
              "kicker": "a subtraction, not an instruction",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-clock-skew",
              "title": "THE CLOCK SKEW",
              "accent": "#5ad4ff",
              "icon": "≶",
              "kicker": "one millisecond, 189 wrong orders",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-content-defined-chunk",
              "title": "THE CONTENT-DEFINED CHUNK",
              "accent": "#5ad0ff",
              "icon": "✂",
              "kicker": "cut where the content says, not where the ruler does",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-causal-cut",
              "title": "THE CAUSAL CUT",
              "accent": "#5ad0ff",
              "icon": "─",
              "kicker": "the snapshot manufactures a present rather than finding one",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-fair-share",
              "title": "THE FAIR SHARE",
              "accent": "#00f5ff",
              "icon": "⚖",
              "kicker": "an allocation built on self-reported need",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            }
          ],
          "seats": 33,
          "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
        }
      ],
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "name": "CHEAT",
      "slug": "cheat",
      "c": "#7cfc00",
      "icon": "cheat",
      "tag": "the shortcut — the konami code, the clever hack",
      "lore": {
        "bio": "The one who knows the door behind the door.",
        "story": "Typed up-up-down-down before the title screen finished loading. Not lazy — early.",
        "does": "Finds the clean shortcut that isn't cheating — the trick that was always allowed.",
        "haiku": [
          "def code(I keys){ if keys == \"up up down down\" { -> \"door\" } -> \"locked\" }",
          "I cheat <- code(\"up up down down\")",
          "-> cheat"
        ],
        "lang": "i13"
      },
      "domains": [
        {
          "slug": "the-konami-code",
          "title": "THE KONAMI CODE",
          "accent": "#39fc6b",
          "icon": "spawn",
          "kicker": "up up down down — unlock it all",
          "pole": "push",
          "spheres": [
            {
              "slug": "the-konami-code",
              "title": "THE KONAMI CODE",
              "accent": "#ffd23f",
              "icon": "cheat",
              "kicker": "up up down down — unlock it all",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "logical-qubit",
              "title": "THE LOGICAL QUBIT · toric code · the ana",
              "accent": "#ffd23f",
              "icon": "cheat",
              "kicker": "vendored from David's corpus — a silicon-coding instrument",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "quad-mobius",
              "title": "THE QUAD-MÖBIUS SCAFFOLD — 4 physical → ",
              "accent": "#ffd23f",
              "icon": "cheat",
              "kicker": "vendored from David's corpus — a silicon-coding instrument",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "twelve-gate-core",
              "title": "THE TWELVE-GATE CORE",
              "accent": "#ffd23f",
              "icon": "cheat",
              "kicker": "twelve logic gates, one core",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "paper-08-logic-gate",
              "title": "THE LOGIC GATE",
              "accent": "#00f5ff",
              "icon": "glitch",
              "kicker": "the gate all computing is built from",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-pancake-sorting",
              "title": "THE PANCAKE SORTING",
              "accent": "#d0a048",
              "icon": "pancake-sorting",
              "kicker": "sort by prefix flips — Bill Gates' only paper",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-mo-algorithm",
              "title": "THE MO ALGORITHM",
              "accent": "#c0a048",
              "icon": "mo",
              "kicker": "reorder queries to answer them fast",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-simhash",
              "title": "THE SIMHASH",
              "accent": "#35ffb0",
              "icon": "simhash",
              "kicker": "similarity read from sign bits",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-nimber",
              "title": "THE NIMBER",
              "accent": "#35ffb0",
              "icon": "nimber",
              "kicker": "a game arithmetic that is a field",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-difference-set",
              "title": "THE DIFFERENCE SET",
              "accent": "#b06bff",
              "icon": "diffset",
              "kicker": "a set whose differences hit every target the same number of times",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-frullani",
              "title": "THE FRULLANI",
              "accent": "#35ffb0",
              "icon": "frullani",
              "kicker": "an integral that reads only its endpoints",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-stewart",
              "title": "THE STEWART",
              "accent": "#b06bff",
              "icon": "stewart",
              "kicker": "a cevian length from the sides",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-fermat-polygonal",
              "title": "THE FERMAT POLYGONAL",
              "accent": "#35ffb0",
              "icon": "fermatpolygonal",
              "kicker": "every integer a sum of few polygonal numbers",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-sophie-germain",
              "title": "THE SOPHIE GERMAIN",
              "accent": "#35ffb0",
              "icon": "sophiegermain",
              "kicker": "an algebraic identity that factors a sum of two fourth powers",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-morrie",
              "title": "THE MORRIE LAW",
              "accent": "#ffcf4a",
              "icon": "morrie",
              "kicker": "three cosines multiplying to exactly one eighth",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-perfect-shuffle",
              "title": "THE PERFECT SHUFFLE",
              "accent": "#ffcf4a",
              "icon": "faroshuffle",
              "kicker": "eight perfect shuffles back to the start",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-hummer",
              "title": "THE HUMMER",
              "accent": "#ff8a3c",
              "icon": "hummer",
              "kicker": "the magician's ledger",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-no-cloning",
              "title": "THE NO CLONING",
              "accent": "#ffd76a",
              "icon": "⧉",
              "kicker": "the state that cannot be copied",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-declarable-only",
              "title": "THE DECLARABLE ONLY",
              "accent": "#b98cff",
              "icon": "≡",
              "kicker": "a synonym cannot be found by looking",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-nagle-delayed-ack",
              "title": "THE NAGLE DELAYED ACK",
              "accent": "#ffd76a",
              "icon": "⇄",
              "kicker": "two polite algorithms waiting for each other",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            }
          ],
          "seats": 31,
          "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
        },
        {
          "slug": "noclip",
          "title": "NOCLIP",
          "accent": "#ffd23f",
          "icon": "grind",
          "kicker": "walk straight through the walls",
          "pole": "push",
          "spheres": [
            {
              "slug": "the-penrose-inflation",
              "title": "THE PENROSE INFLATION",
              "accent": "#d9b3ff",
              "icon": "cheat",
              "kicker": "five-fold order that clips through the law of crystals",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-dragon",
              "title": "THE DRAGON",
              "accent": "#4fd6b0",
              "icon": "fold",
              "kicker": "the Heighway dragon — the fold that tiles the plane",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-ski",
              "title": "THE SKI",
              "accent": "#90ffb0",
              "icon": "ski",
              "kicker": "two operators, no variables — S and K compute everything",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-fifteen-puzzle",
              "title": "THE 15-PUZZLE",
              "accent": "#60c0a0",
              "icon": "fifteen-puzzle",
              "kicker": "a conserved parity walls off half the arrangements",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-bbp",
              "title": "THE BBP",
              "accent": "#c0a048",
              "icon": "bbp",
              "kicker": "the n-th hex digit of pi, without the digits before it",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-binary-lifting",
              "title": "THE BINARY LIFTING",
              "accent": "#c05868",
              "icon": "binary-lifting",
              "kicker": "ancestor and LCA queries in log time by doubling",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-tea",
              "title": "THE TEA CIPHER",
              "accent": "#a878c0",
              "icon": "tea",
              "kicker": "a whole block cipher from add, shift, xor",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-liang-barsky",
              "title": "THE LIANG-BARSKY",
              "accent": "#70a860",
              "icon": "liang-barsky",
              "kicker": "clip a line to a window by four parameters",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-simon",
              "title": "THE SIMON",
              "accent": "#21e6ff",
              "icon": "simon",
              "kicker": "a hidden mask pinned by linear equations",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-zipper",
              "title": "THE ZIPPER",
              "accent": "#35ffb0",
              "icon": "zipper",
              "kicker": "a cursor that splits the list",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-lander-parkin",
              "title": "THE LANDER-PARKIN",
              "accent": "#21e6ff",
              "icon": "landerparkin",
              "kicker": "a counterexample refuting Euler's conjecture",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-kakeya",
              "title": "THE KAKEYA",
              "accent": "#b06bff",
              "icon": "kakeya",
              "kicker": "a needle turned in an eighth of pi",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-freshmans-dream",
              "title": "THE FRESHMAN'S DREAM",
              "accent": "#35ffb0",
              "icon": "freshmansdream",
              "kicker": "the child's error that becomes law",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-tusi",
              "title": "THE TUSI",
              "accent": "#b06bff",
              "icon": "tusi",
              "kicker": "rotation compiled to translation",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-slepian-wolf",
              "title": "THE SLEPIAN-WOLF",
              "accent": "#b98cff",
              "icon": "⇉",
              "kicker": "two encoders, no channel between them, joint price",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-newcomb",
              "title": "THE NEWCOMB",
              "accent": "#ffd76a",
              "icon": "⊞",
              "kicker": "two valid rules, opposite answers, same table",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-linking-number",
              "title": "THE LINKING NUMBER",
              "accent": "#7de2b0",
              "icon": "⚭",
              "kicker": "an integer that survives any deformation",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-reflection",
              "title": "THE REFLECTION",
              "accent": "#7de2b0",
              "icon": "⤨",
              "kicker": "a path folded through a wall",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-half-in-fourteen",
              "title": "THE HALF IN FOURTEEN",
              "accent": "#ffd76a",
              "icon": "◥",
              "kicker": "a mean that touches almost nothing",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-jump-that-is-everywhere",
              "title": "THE JUMP THAT IS EVERYWHERE",
              "accent": "#ff5a8a",
              "icon": "⟶",
              "kicker": "the real obstacle, and not the one raised",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-register-renaming",
              "title": "THE REGISTER RENAMING",
              "accent": "#ff9f45",
              "icon": "⇄",
              "kicker": "the hardware apologising for the ISA",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-zero-width-joiner",
              "title": "THE ZERO WIDTH JOINER",
              "accent": "#b98cff",
              "icon": "​",
              "kicker": "characters with no shape at all",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-pointer-chase",
              "title": "THE POINTER CHASE",
              "accent": "#ffd23f",
              "icon": "⟳",
              "kicker": "the program knows the future and has no way to say so",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-bwt",
              "title": "THE BWT",
              "accent": "#ffd23f",
              "icon": "⇅",
              "kicker": "rearrangement so that compression becomes possible",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            }
          ],
          "seats": 36,
          "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
        },
        {
          "slug": "god-mode",
          "title": "GOD MODE",
          "accent": "#ff2d95",
          "icon": "glitch",
          "kicker": "take no damage, spend nothing",
          "pole": "push",
          "spheres": [
            {
              "slug": "the-counter-of-multitudes",
              "title": "THE COUNTER OF MULTITUDES",
              "accent": "#ffe14d",
              "icon": "cheat",
              "kicker": "count billions of distinct things in a thimble of memory",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-nim",
              "title": "THE NIM",
              "accent": "#ffc04d",
              "icon": "nim",
              "kicker": "the whole game in one XOR — the nim-sum",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-wythoff",
              "title": "THE WYTHOFF",
              "accent": "#ffcf60",
              "icon": "wythoff",
              "kicker": "the golden ratio hiding in a game of stones",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-penney",
              "title": "THE PENNEY",
              "accent": "#ff90d0",
              "icon": "penney",
              "kicker": "pick any coin-triple, the second player beats it",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-one-time-pad",
              "title": "THE ONE-TIME-PAD",
              "accent": "#ffe0a0",
              "icon": "onetimepad",
              "kicker": "the only provably unbreakable cipher — used once",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-fractran",
              "title": "THE FRACTRAN",
              "accent": "#c060ff",
              "icon": "fractran",
              "kicker": "a whole language made of fractions — universal, unreadable",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-dancing-links",
              "title": "THE DANCING LINKS",
              "accent": "#9068c0",
              "icon": "dancing-links",
              "kicker": "exact cover by O(1) reversible unlink/relink",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-shamir",
              "title": "THE SHAMIR SHARING",
              "accent": "#a878c0",
              "icon": "shamir",
              "kicker": "split a secret so any k of n rebuild it, fewer learn nothing",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-median-of-medians",
              "title": "THE MEDIAN OF MEDIANS",
              "accent": "#70a860",
              "icon": "median-of-medians",
              "kicker": "pick the k-th smallest in guaranteed linear time",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-deutsch-jozsa",
              "title": "THE DEUTSCH-JOZSA",
              "accent": "#ff8a3c",
              "icon": "deutsch-jozsa",
              "kicker": "constant-or-balanced in one question",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-schnorr",
              "title": "THE SCHNORR",
              "accent": "#21e6ff",
              "icon": "schnorr",
              "kicker": "prove you know a secret without revealing it",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-chu-vandermonde",
              "title": "THE CHU-VANDERMONDE",
              "accent": "#ff8a3c",
              "icon": "chuvandermonde",
              "kicker": "a binomial convolution collapsing to one entry",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-kapitza",
              "title": "THE KAPITZA",
              "accent": "#b06bff",
              "icon": "kapitza",
              "kicker": "gravity beaten by vibration",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-shannon-limit",
              "title": "THE SHANNON LIMIT",
              "accent": "#ff9a5a",
              "icon": "⧖",
              "kicker": "error-free, through noise, at a rate that does not vanish",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-fake-context",
              "title": "THE FAKE CONTEXT",
              "accent": "#5ad6ff",
              "icon": "◑",
              "kicker": "the states reality will not hand you on demand",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-missing-n",
              "title": "THE MISSING N",
              "accent": "#ffd76a",
              "icon": "∅",
              "kicker": "a sample size from a different experiment",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-length-extension",
              "title": "THE LENGTH EXTENSION",
              "accent": "#ff5a8a",
              "icon": "⛓",
              "kicker": "a signature that continues itself",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-bloom",
              "title": "THE BLOOM",
              "accent": "#b98cff",
              "icon": "⚑",
              "kicker": "a filter that only lies one way",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-marsaglia-planes",
              "title": "THE MARSAGLIA PLANES",
              "accent": "#ff5a8a",
              "icon": "≡",
              "kicker": "random numbers fall mainly in the planes",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-zero-that-was-the-point",
              "title": "THE ZERO THAT WAS THE POINT",
              "accent": "#7de2b0",
              "icon": "∅",
              "kicker": "two zeros that look identical in the output",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            }
          ],
          "seats": 43,
          "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
        },
        {
          "slug": "the-speedrun",
          "title": "THE SPEEDRUN",
          "accent": "#00f5ff",
          "icon": "loot",
          "kicker": "the route that skips the game",
          "pole": "push",
          "spheres": [
            {
              "slug": "the-rule",
              "title": "THE RULE",
              "accent": "#7cfc00",
              "icon": "cheat",
              "kicker": "one byte of rule → unlimited computation",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-magic-number",
              "title": "THE MAGIC NUMBER",
              "accent": "#ff6a3d",
              "icon": "bolt",
              "kicker": "0x5f3759df — the fast inverse square root",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-deutsch",
              "title": "THE DEUTSCH",
              "accent": "#70d0ff",
              "icon": "deutsch",
              "kicker": "one quantum query where classical needs two",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-rotating-calipers",
              "title": "THE ROTATING CALIPERS",
              "accent": "#e08040",
              "icon": "rotating-calipers",
              "kicker": "polygon diameter in O(n) — only antipodal pairs matter",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-gauss-legendre",
              "title": "THE GAUSS-LEGENDRE",
              "accent": "#d08840",
              "icon": "gauss-legendre",
              "kicker": "n samples integrate polynomials of degree 2n-1 exactly",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-closest-pair",
              "title": "THE CLOSEST PAIR",
              "accent": "#e08850",
              "icon": "closest-pair",
              "kicker": "nearest two points in O(n log n) — geometry bounds the strip",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-durand-kerner",
              "title": "THE DURAND-KERNER",
              "accent": "#d08858",
              "icon": "durand-kerner",
              "kicker": "all polynomial roots at once — estimates that repel into place",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-inversions",
              "title": "THE INVERSIONS",
              "accent": "#d0687a",
              "icon": "inversions",
              "kicker": "count disorder in O(n log n) — additive across a divide",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-quickselect",
              "title": "THE QUICKSELECT",
              "accent": "#d06858",
              "icon": "quickselect",
              "kicker": "the k-th smallest in O(n) — median of medians",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-interpolation-search",
              "title": "THE INTERPOLATION SEARCH",
              "accent": "#c06868",
              "icon": "interpolation-search",
              "kicker": "guess the position from the value — O(log log n) on uniform data",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-halley",
              "title": "THE HALLEY",
              "accent": "#c05868",
              "icon": "halley",
              "kicker": "cubic-convergence root finding via the second derivative",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-romberg",
              "title": "THE ROMBERG",
              "accent": "#c0a048",
              "icon": "romberg",
              "kicker": "integration accelerated by cancelling the error terms",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-barrett",
              "title": "THE BARRETT",
              "accent": "#d4a017",
              "icon": "barrett",
              "kicker": "reduce mod n without dividing",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-egyptian-fraction",
              "title": "THE EGYPTIAN FRACTION",
              "accent": "#70a860",
              "icon": "egyptian",
              "kicker": "a fraction split into distinct unit shares, greedily",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-bernstein-vazirani",
              "title": "THE BERNSTEIN-VAZIRANI",
              "accent": "#ff8a3c",
              "icon": "bernstein-vazirani",
              "kicker": "a hidden string in one query",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-bitap",
              "title": "THE BITAP",
              "accent": "#b06bff",
              "icon": "bitap",
              "kicker": "one register that matches in parallel",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-split-radix",
              "title": "THE SPLIT-RADIX FFT",
              "accent": "#35ffb0",
              "icon": "splitradix",
              "kicker": "an FFT with the fewest multiplies",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-vieta-jumping",
              "title": "THE VIETA JUMPING",
              "accent": "#21e6ff",
              "icon": "vieta",
              "kicker": "an integer ratio that can only be a perfect square",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-dormand-prince",
              "title": "THE DORMAND-PRINCE",
              "accent": "#21e6ff",
              "icon": "dormand",
              "kicker": "an adaptive integrator that paces itself",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-agm",
              "title": "THE AGM",
              "accent": "#ffcf4a",
              "icon": "agm",
              "kicker": "two means racing to one limit",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-superpermutation",
              "title": "THE SUPERPERMUTATION",
              "accent": "#ff8a3c",
              "icon": "superperm",
              "kicker": "every binge-order in one string",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-brachistochrone",
              "title": "THE BRACHISTOCHRONE",
              "accent": "#ffcf4a",
              "icon": "brachistochrone",
              "kicker": "the dip that arrives first",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-dodgson",
              "title": "THE DODGSON",
              "accent": "#00f5ff",
              "icon": "dodgson",
              "kicker": "a determinant shrunk out of 2x2 windows",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-prefetcher",
              "title": "THE PREFETCHER",
              "accent": "#00f5ff",
              "icon": "➤",
              "kicker": "the whole performance cliff is a missing sentence",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-bitpacking",
              "title": "THE BITPACKING",
              "accent": "#00f5ff",
              "icon": "▓",
              "kicker": "the byte boundary was never waste, it was an index",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            }
          ],
          "seats": 31,
          "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
        },
        {
          "slug": "the-backdoor",
          "title": "THE BACKDOOR",
          "accent": "#ff5a3c",
          "icon": "boss",
          "kicker": "the way in nobody documented",
          "pole": "pull",
          "spheres": [
            {
              "slug": "the-ouroboros-string",
              "title": "THE OUROBOROS STRING",
              "accent": "#b6ff3a",
              "icon": "cheat",
              "kicker": "one loop that contains every combination once",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-pruning",
              "title": "THE PRUNING",
              "accent": "#ff6a8a",
              "icon": "alphabeta",
              "kicker": "alpha-beta — perfect play without looking at most of it",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-de-bruijn",
              "title": "THE DE BRUIJN",
              "accent": "#6cf0e0",
              "icon": "debruijn",
              "kicker": "the shortest string holding every code — k^n",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-fast-inverse-sqrt",
              "title": "THE FAST INVERSE SQRT",
              "accent": "#c8a020",
              "icon": "fast-inverse-sqrt",
              "kicker": "1/sqrt(x) with a bit-hack and one Newton step — no divide",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-berlekamp-massey",
              "title": "THE BERLEKAMP-MASSEY",
              "accent": "#b06890",
              "icon": "berlekamp-massey",
              "kicker": "recover the shortest LFSR from its output alone",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-kelly-criterion",
              "title": "THE KELLY CRITERION",
              "accent": "#d0a838",
              "icon": "kelly-criterion",
              "kicker": "the bet fraction that maximises long-run growth",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-lattice-reduction",
              "title": "THE LATTICE REDUCTION",
              "accent": "#ffcf4a",
              "icon": "lattice-reduction",
              "kicker": "a shorter view of the same lattice",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-merkle-hellman",
              "title": "THE MERKLE-HELLMAN",
              "accent": "#b06bff",
              "icon": "merklehellman",
              "kicker": "a knapsack locked by a superincreasing sequence",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-wynn",
              "title": "THE WYNN",
              "accent": "#21e6ff",
              "icon": "wynn",
              "kicker": "an accelerator that squeezes π from a crawling series",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-kempner",
              "title": "THE KEMPNER",
              "accent": "#35ffb0",
              "icon": "kempner",
              "kicker": "a harmonic series that converges once you delete the nines",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-gamma-reflection",
              "title": "THE GAMMA REFLECTION",
              "accent": "#b06bff",
              "icon": "gammareflection",
              "kicker": "a gamma product equal to a cosecant",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-loaded-dice-table",
              "title": "THE LOADED-DICE TABLE",
              "accent": "#ffcf4a",
              "icon": "aliasmethod",
              "kicker": "a die loaded in constant time",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-sierpinski-number",
              "title": "THE SIERPIŃSKI NUMBER",
              "accent": "#b06bff",
              "icon": "sierpinskinum",
              "kicker": "a family composite forever by seven-prime conspiracy",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-nontransitive-dice",
              "title": "THE NONTRANSITIVE DICE",
              "accent": "#ff8a3c",
              "icon": "nontransitivedice",
              "kicker": "dice with no best",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-bomb-tester",
              "title": "THE BOMB TESTER",
              "accent": "#ff8a3c",
              "icon": "bombtester",
              "kicker": "seeing without looking",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-reachability-gap",
              "title": "THE REACHABILITY GAP",
              "accent": "#ffd76a",
              "icon": "↱",
              "kicker": "true of the shipped app, false of the repository",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-berkson",
              "title": "THE BERKSON",
              "accent": "#5ad6ff",
              "icon": "⊘",
              "kicker": "a correlation made of nothing but who was let in",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-birthday-attack",
              "title": "THE BIRTHDAY ATTACK",
              "accent": "#b98cff",
              "icon": "√",
              "kicker": "half the bits, all the security",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-lint-not-the-judge",
              "title": "THE LINT NOT THE JUDGE",
              "accent": "#5ad6ff",
              "icon": "⚑",
              "kicker": "evade the words, keep the claim",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "unjustified-is-not-disproven",
              "title": "UNJUSTIFIED IS NOT DISPROVEN",
              "accent": "#5ad6ff",
              "icon": "≡",
              "kicker": "a likelihood ratio of one leaves the prior alone",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-unverified-surface",
              "title": "THE UNVERIFIED SURFACE",
              "accent": "#ffd76a",
              "icon": "▫",
              "kicker": "coverage reports on the covered",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-dead-store-elimination",
              "title": "THE DEAD STORE ELIMINATION",
              "accent": "#7de2b0",
              "icon": "⌧",
              "kicker": "the wipe that was deleted for being pointless",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-homoglyph",
              "title": "THE HOMOGLYPH",
              "accent": "#ffd76a",
              "icon": "≈",
              "kicker": "thirty-two spellings, one shape",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-punycode",
              "title": "THE PUNYCODE",
              "accent": "#ff5a3c",
              "icon": "◐",
              "kicker": "the boundary sits where a machine hands something to an eye",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            }
          ],
          "seats": 29,
          "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
        },
        {
          "slug": "the-exploit",
          "title": "THE EXPLOIT",
          "accent": "#9d00ff",
          "icon": "coop",
          "kicker": "pull data out through the crack",
          "pole": "pull",
          "spheres": [
            {
              "slug": "the-shortest-witness",
              "title": "THE SHORTEST WITNESS",
              "accent": "#7dffb0",
              "icon": "cheat",
              "kicker": "watch the output, recover the machine",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-karatsuba",
              "title": "THE KARATSUBA",
              "accent": "#a0e878",
              "icon": "mult",
              "kicker": "multiply with 3 sub-products instead of 4",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-parrondo",
              "title": "THE PARRONDO",
              "accent": "#ff6060",
              "icon": "parrondo",
              "kicker": "two losing games that combine into a winning one",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-strassen",
              "title": "THE STRASSEN",
              "accent": "#e0704a",
              "icon": "strassen",
              "kicker": "multiply 2x2 with 7 products, not 8 — bending O(n^3)",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-baby-step-giant-step",
              "title": "THE BABY-STEP GIANT-STEP",
              "accent": "#c06868",
              "icon": "baby-step-giant-step",
              "kicker": "discrete log by meeting in the middle, O(sqrt n)",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-pohlig-hellman",
              "title": "THE POHLIG-HELLMAN",
              "accent": "#c05868",
              "icon": "pohlig-hellman",
              "kicker": "discrete log broken by smooth order + CRT",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-pollard-p1",
              "title": "THE POLLARD P-1",
              "accent": "#c05868",
              "icon": "pollard-p1",
              "kicker": "factoring surfaced by a gcd when p-1 is smooth",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-aes-sbox",
              "title": "THE AES S-BOX",
              "accent": "#a878c0",
              "icon": "aes-sbox",
              "kicker": "the cipher's non-linearity from one field inversion",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-pollard-rho",
              "title": "THE POLLARD RHO",
              "accent": "#c86868",
              "icon": "pollard-rho",
              "kicker": "crack a number open by walking a cycle",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-substrate-check",
              "title": "THE SUBSTRATE CHECK",
              "accent": "#ff8a3c",
              "icon": "substrate-check",
              "kicker": "a check noise passes is not a measurement",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-elgamal",
              "title": "THE ELGAMAL",
              "accent": "#ff8a3c",
              "icon": "elgamal",
              "kicker": "a public key from a discrete log",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-auction",
              "title": "THE AUCTION",
              "accent": "#b06bff",
              "icon": "auction",
              "kicker": "assignment settled by competitive bidding",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-vampire",
              "title": "THE VAMPIRE",
              "accent": "#b06bff",
              "icon": "vampire",
              "kicker": "factors hiding their digits in the product",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-tupper",
              "title": "THE TUPPER",
              "accent": "#ff8a3c",
              "icon": "tupper",
              "kicker": "the formula that draws everything",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-gibbard",
              "title": "THE GIBBARD",
              "accent": "#b06bff",
              "icon": "gibbard",
              "kicker": "no honest rule",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-contaminated-pool",
              "title": "THE CONTAMINATED POOL",
              "accent": "#b98cff",
              "icon": "◌",
              "kicker": "my draw was honest, the pool was not",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-deceptive",
              "title": "THE DECEPTIVE",
              "accent": "#ff5a8a",
              "icon": "⤴",
              "kicker": "a hill built to mislead",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-folk-theorem",
              "title": "THE FOLK THEOREM",
              "accent": "#5ad6ff",
              "icon": "∞",
              "kicker": "why tomorrow makes today honest",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-order-blind-hash",
              "title": "THE ORDER-BLIND HASH",
              "accent": "#ff5a8a",
              "icon": "⊕",
              "kicker": "a digest that forgot where it had been",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-hundred-and-thirty-two",
              "title": "THE HUNDRED AND THIRTY-TWO",
              "accent": "#ffd76a",
              "icon": "⊃",
              "kicker": "label reuse counted as nesting",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-padding-oracle",
              "title": "THE PADDING ORACLE",
              "accent": "#7de2b0",
              "icon": "⚿",
              "kicker": "one bit, returned politely, several thousand times",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-utf8-overlong",
              "title": "THE UTF-8 OVERLONG",
              "accent": "#5ad4ff",
              "icon": "≡",
              "kicker": "384 spare spellings of 128 characters",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-hash-flooding",
              "title": "THE HASH FLOODING",
              "accent": "#9d00ff",
              "icon": "≡",
              "kicker": "they declined to be the average case",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            }
          ],
          "seats": 27,
          "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
        },
        {
          "slug": "the-root-kit",
          "title": "THE ROOT KIT",
          "accent": "#7cfc00",
          "icon": "cheat",
          "kicker": "burrow in and stay hidden",
          "pole": "pull",
          "spheres": [
            {
              "slug": "the-thumbprint",
              "title": "THE THUMBPRINT",
              "accent": "#7ad0b0",
              "icon": "cheat",
              "kicker": "recognise a whole set from a tiny fingerprint of minimums",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-carmichael",
              "title": "THE CARMICHAEL",
              "accent": "#ff5090",
              "icon": "carmichael",
              "kicker": "a composite that fools the Fermat test to every base",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-feistel",
              "title": "THE FEISTEL",
              "accent": "#c05868",
              "icon": "feistel",
              "kicker": "a reversible cipher from a one-way function",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-paillier",
              "title": "THE PAILLIER",
              "accent": "#a878c0",
              "icon": "paillier",
              "kicker": "add two numbers without ever decrypting them",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-hull-dobell",
              "title": "THE HULL–DOBELL",
              "accent": "#40b0a0",
              "icon": "hull-dobell",
              "kicker": "when a linear congruential generator hits full period",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-hill-cipher",
              "title": "THE HILL CIPHER",
              "accent": "#35ffb0",
              "icon": "hill-cipher",
              "kicker": "a cipher that is a matrix",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-british-flag",
              "title": "THE BRITISH FLAG",
              "accent": "#35ffb0",
              "icon": "britishflag",
              "kicker": "a rectangle's hidden distance invariant",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-automorphic",
              "title": "THE AUTOMORPHIC",
              "accent": "#35ffb0",
              "icon": "automorphic",
              "kicker": "a number whose square ends in itself",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-munchhausen",
              "title": "THE MUNCHHAUSEN",
              "accent": "#ffcf4a",
              "icon": "munchhausen",
              "kicker": "a number built from its own digits raised to themselves",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-lucky-euler",
              "title": "THE LUCKY EULER",
              "accent": "#35ffb0",
              "icon": "luckyeuler",
              "kicker": "a polynomial that spits primes forty times in a row",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-ramanujan-pi",
              "title": "THE RAMANUJAN PI",
              "accent": "#b06bff",
              "icon": "ramanujan",
              "kicker": "a series adding eight digits of pi per term",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-simpson-paradox",
              "title": "THE SIMPSON PARADOX",
              "accent": "#ff8a3c",
              "icon": "simpsonparadox",
              "kicker": "a treatment that wins twice and loses once",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-will-rogers",
              "title": "THE WILL ROGERS",
              "accent": "#b06bff",
              "icon": "willrogers",
              "kicker": "a transfer that flatters everyone",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-fermat-primes",
              "title": "THE FERMAT PRIMES",
              "accent": "#ffcf4a",
              "icon": "fermatprimes",
              "kicker": "five in a row, then Euler",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-lob",
              "title": "THE LOB",
              "accent": "#5ad6ff",
              "icon": "◻",
              "kicker": "if it would be enough to prove it, it is already proved",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-weighted-arm",
              "title": "THE WEIGHTED ARM",
              "accent": "#ff9a5a",
              "icon": "⚖",
              "kicker": "one measurement, two headlines",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-jones",
              "title": "THE JONES",
              "accent": "#b98cff",
              "icon": "❀",
              "kicker": "the invariant that finally sees the mirror",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-bell-inequality",
              "title": "THE BELL INEQUALITY",
              "accent": "#7de2b0",
              "icon": "∞",
              "kicker": "a correlation no local story can tell",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-redaction",
              "title": "THE REDACTION",
              "accent": "#b98cff",
              "icon": "█",
              "kicker": "names removed, and nothing measured moved",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-threaded-accumulator",
              "title": "THE THREADED ACCUMULATOR",
              "accent": "#5ad6ff",
              "icon": "↻",
              "kicker": "computed from the tree, not supplied by hand",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            }
          ],
          "seats": 31,
          "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
        },
        {
          "slug": "the-shortcut",
          "title": "THE SHORTCUT",
          "accent": "#5ad0ff",
          "icon": "respawn",
          "kicker": "draw the finish line closer",
          "pole": "pull",
          "spheres": [
            {
              "slug": "coding-theory-ternary",
              "title": "TERNARY CODING",
              "accent": "#7cfc00",
              "icon": "cheat",
              "kicker": "base-3, the radix nearest optimal",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "fractal-ternary-bench",
              "title": "fractal_ternary · audit bench",
              "accent": "#7cfc00",
              "icon": "cheat",
              "kicker": "vendored from David's corpus — a silicon-coding instrument",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "fractal-ternary",
              "title": "THE TERNARY FRACTON — 3D fractal stabili",
              "accent": "#7cfc00",
              "icon": "cheat",
              "kicker": "vendored from David's corpus — a silicon-coding instrument",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "octorat-around-the-middle",
              "title": "AROUND THE MIDDLE · THE CENTERED TRIT · ",
              "accent": "#7cfc00",
              "icon": "cheat",
              "kicker": "vendored from David's corpus — a silicon-coding instrument",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "octorat-concentric-nine",
              "title": "THE CONCENTRIC NINE · TERNARY ORBIT · ZE",
              "accent": "#7cfc00",
              "icon": "cheat",
              "kicker": "vendored from David's corpus — a silicon-coding instrument",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "octorat-ternary-octorat",
              "title": "THE TERNARY OCTORAT · THE WALKER THAT CA",
              "accent": "#7cfc00",
              "icon": "cheat",
              "kicker": "vendored from David's corpus — a silicon-coding instrument",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "octorat-trit-ladder",
              "title": "THE TRIT LADDER · 9 NESTED SCALES · ZERO",
              "accent": "#7cfc00",
              "icon": "cheat",
              "kicker": "vendored from David's corpus — a silicon-coding instrument",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "quaternary-fulcrum",
              "title": "THE QUATERNARY FULCRUM · Powers Of Four ",
              "accent": "#7cfc00",
              "icon": "cheat",
              "kicker": "vendored from David's corpus — a silicon-coding instrument",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "spiral-trit-loom",
              "title": "The spiral-trit loom — a story woven int",
              "accent": "#7cfc00",
              "icon": "cheat",
              "kicker": "vendored from David's corpus — a silicon-coding instrument",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "strobe-channel",
              "title": "STROBE CHANNEL — Morse & balanced ternar",
              "accent": "#7cfc00",
              "icon": "cheat",
              "kicker": "vendored from David's corpus — a silicon-coding instrument",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "ternary-hamming-decoder",
              "title": "THE TERNARY HAMMING DECODER · Locate & C",
              "accent": "#7cfc00",
              "icon": "cheat",
              "kicker": "vendored from David's corpus — a silicon-coding instrument",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "ternary-odometer",
              "title": "THE TERNARY ODOMETER · Three 555s · Nest",
              "accent": "#7cfc00",
              "icon": "cheat",
              "kicker": "vendored from David's corpus — a silicon-coding instrument",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-trit",
              "title": "the trit · {−1, i, +1} · flat &amp; two-",
              "accent": "#7cfc00",
              "icon": "cheat",
              "kicker": "vendored from David's corpus — a silicon-coding instrument",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "three-in-a-circle",
              "title": "THREE IN A CIRCLE · Why The Ring Forces ",
              "accent": "#7cfc00",
              "icon": "cheat",
              "kicker": "vendored from David's corpus — a silicon-coding instrument",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "turn-comes-home",
              "title": "THE TURN COMES HOME · A Clocked 27-Cell ",
              "accent": "#7cfc00",
              "icon": "cheat",
              "kicker": "vendored from David's corpus — a silicon-coding instrument",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "octorat-chaos-silo",
              "title": "THE OCTORAT · CHAOS",
              "accent": "#7cfc00",
              "icon": "cheat",
              "kicker": "the ternary walker in chaos",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "octorat-probability-engine",
              "title": "THE OCTORAT · PROBABILITY",
              "accent": "#ffd23f",
              "icon": "cheat",
              "kicker": "the octorat's probability engine",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "card-odometer-54",
              "title": "THE 54-CARD ODOMETER",
              "accent": "#ffd23f",
              "icon": "cheat",
              "kicker": "the deck in the dual-27 lattice",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "base-plus-1",
              "title": "BASE + 1",
              "accent": "#7cfc00",
              "icon": "cheat",
              "kicker": "which witness is unforgeable — base+1 encoding",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "balanced-base-5",
              "title": "BALANCED BASE-5",
              "accent": "#00f5ff",
              "icon": "glitch",
              "kicker": "the held center climbs the odd ladder",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-grover",
              "title": "THE GROVER",
              "accent": "#ffa0e0",
              "icon": "grover",
              "kicker": "search N items in √N — the quantum shortcut",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-seam-carving",
              "title": "THE SEAM CARVING",
              "accent": "#70a860",
              "icon": "seam-carving",
              "kicker": "carve out the least-noticed seam",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-quickhull",
              "title": "THE QUICKHULL",
              "accent": "#70a860",
              "icon": "quickhull",
              "kicker": "wrap a hull by divide and conquer",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-routh",
              "title": "THE ROUTH",
              "accent": "#21e6ff",
              "icon": "routh",
              "kicker": "a cevian triangle's area is a closed form",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-pratt-parsing",
              "title": "THE PRATT PARSING",
              "accent": "#b06bff",
              "icon": "pratt-parsing",
              "kicker": "precedence from binding power",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-rader",
              "title": "THE RADER",
              "accent": "#ffcf4a",
              "icon": "rader",
              "kicker": "a prime DFT turned into a convolution",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-popcount",
              "title": "THE POPCOUNT",
              "accent": "#ff8a3c",
              "icon": "popcount",
              "kicker": "bits counted by folding",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-xor-linked-list",
              "title": "THE XOR LINKED LIST",
              "accent": "#35ffb0",
              "icon": "xorlist",
              "kicker": "one pointer holds both neighbors",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-kronecker-substitution",
              "title": "THE KRONECKER SUBSTITUTION",
              "accent": "#b06bff",
              "icon": "kronsub",
              "kicker": "polynomials multiplied as one big integer",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-prime-factor-fft",
              "title": "THE PRIME-FACTOR FFT",
              "accent": "#ffcf4a",
              "icon": "primefactorfft",
              "kicker": "a prime-factored DFT with no twiddles",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-buffon",
              "title": "THE BUFFON",
              "accent": "#21e6ff",
              "icon": "buffon",
              "kicker": "needles dropped to measure π",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-devils-staircase",
              "title": "THE DEVILS STAIRCASE",
              "accent": "#21e6ff",
              "icon": "devilsstaircase",
              "kicker": "a staircase that climbs without sloping",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-question-mark",
              "title": "THE QUESTION MARK",
              "accent": "#b06bff",
              "icon": "questionmark",
              "kicker": "continued fractions transcribed to binary",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-malfatti",
              "title": "THE MALFATTI",
              "accent": "#b06bff",
              "icon": "malfatti",
              "kicker": "the official answer that always loses",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-branch-target-buffer",
              "title": "THE BRANCH TARGET BUFFER",
              "accent": "#b98cff",
              "icon": "↳",
              "kicker": "99.8% accurate and wrong every single time",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-minimal-perfect-hash",
              "title": "THE MINIMAL PERFECT HASH",
              "accent": "#5ad0ff",
              "icon": "⬚",
              "kicker": "no slack, and so no way to say not here",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-fma",
              "title": "THE FMA",
              "accent": "#5ad0ff",
              "icon": "×",
              "kicker": "accuracy bought with reproducibility",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-golomb-rice",
              "title": "THE GOLOMB RICE",
              "accent": "#5ad0ff",
              "icon": "⌐",
              "kicker": "a tuned coder is a prediction nobody checks again",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            }
          ],
          "seats": 28,
          "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
        }
      ],
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    },
    {
      "name": "RESPAWN",
      "slug": "respawn",
      "c": "#5ad0ff",
      "icon": "respawn",
      "tag": "die & return — the fold to 0, and back",
      "lore": {
        "bio": "The one who comes back. r=1 to r=0 to r=1 again.",
        "story": "It has died more than anything on the wall. Every fold to zero it kept a checkpoint at the center and walked back out.",
        "does": "Takes a work all the way down to ROOT_0 and returns it — the fold, made survivable.",
        "haiku": [
          "I life <- 1",
          "I zero <- life - 1",
          "-> zero + 1"
        ],
        "lang": "i13"
      },
      "domains": [
        {
          "slug": "the-phoenix",
          "title": "THE PHOENIX",
          "accent": "#39fc6b",
          "icon": "spawn",
          "kicker": "rise again from the ashes",
          "pole": "push",
          "spheres": [
            {
              "slug": "the-newton",
              "title": "THE NEWTON",
              "accent": "#ff6b35",
              "icon": "respawn",
              "kicker": "rise from any ash to a root",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-ducci",
              "title": "THE DUCCI",
              "accent": "#b878c0",
              "icon": "ducci",
              "kicker": "absolute differences around a ring — power-of-2 always burns to zero",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-dragon-curve",
              "title": "THE DRAGON CURVE",
              "accent": "#70a860",
              "icon": "dragon-curve",
              "kicker": "a fold that fills space and never crosses itself",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-bulgarian-solitaire",
              "title": "THE BULGARIAN SOLITAIRE",
              "accent": "#d06858",
              "icon": "bulgarian",
              "kicker": "any pile grinds down to the staircase",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-fredkin",
              "title": "THE FREDKIN",
              "accent": "#ff8a3c",
              "icon": "fredkin",
              "kicker": "a gate that conserves its ones",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-leapfrog",
              "title": "THE LEAPFROG",
              "accent": "#21e6ff",
              "icon": "leapfrog",
              "kicker": "a step that conserves energy and reverses",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-poncelet",
              "title": "THE PONCELET",
              "accent": "#ffcf4a",
              "icon": "poncelet",
              "kicker": "a tangent triangle that closes from every start",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-holditch",
              "title": "THE HOLDITCH",
              "accent": "#ff8a3c",
              "icon": "holditch",
              "kicker": "a curve reborn smaller by exactly pi-p-q",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-stable-marriage",
              "title": "THE STABLE MARRIAGE",
              "accent": "#b06bff",
              "icon": "stablemarriage",
              "kicker": "the proposer's hidden crown",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-weierstrass",
              "title": "THE WEIERSTRASS",
              "accent": "#35ffb0",
              "icon": "weierstrass",
              "kicker": "the curve with no slope anywhere",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-sharkovskii",
              "title": "THE SHARKOVSKII",
              "accent": "#ff5a8a",
              "icon": "↺",
              "kicker": "one cycle length forces all the rest",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-dissent",
              "title": "THE DISSENT",
              "accent": "#b98cff",
              "icon": "≠",
              "kicker": "the parts agree and the whole does not",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-predicate",
              "title": "THE PREDICATE",
              "accent": "#b98cff",
              "icon": "≡",
              "kicker": "what it automates is the re-running, not the judgement",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-scoreboard",
              "title": "THE SCOREBOARD",
              "accent": "#7de2b0",
              "icon": "⌀",
              "kicker": "zero out of six is not zero",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-self-caught-share",
              "title": "THE SELF-CAUGHT SHARE",
              "accent": "#b98cff",
              "icon": "◐",
              "kicker": "whose control actually fired",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-cantor-function",
              "title": "THE CANTOR FUNCTION",
              "accent": "#7de2b0",
              "icon": "⤴",
              "kicker": "it climbs without ever rising",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-abelian-sandpile",
              "title": "THE ABELIAN SANDPILE",
              "accent": "#ffd76a",
              "icon": "∴",
              "kicker": "the pile that does not care what order you push it",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-ziggurat",
              "title": "THE ZIGGURAT",
              "accent": "#7de2b0",
              "icon": "△",
              "kicker": "128 rectangles that all weigh the same",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-duffs-device",
              "title": "DUFF'S DEVICE",
              "accent": "#ffd76a",
              "icon": "↻",
              "kicker": "a switch whose cases fall into a loop",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-orientation-double-cover",
              "title": "THE ORIENTATION DOUBLE COVER",
              "accent": "#b98cff",
              "icon": "∞",
              "kicker": "a census asked a question it cannot answer",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-halloween-problem",
              "title": "THE HALLOWEEN PROBLEM",
              "accent": "#ff9f45",
              "icon": "↻",
              "kicker": "the rows keep coming back",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            }
          ],
          "seats": 32,
          "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
        },
        {
          "slug": "the-resurrect",
          "title": "THE RESURRECT",
          "accent": "#ffd23f",
          "icon": "grind",
          "kicker": "spawn back out at the last save",
          "pole": "push",
          "spheres": [
            {
              "slug": "the-spots-that-breed",
              "title": "THE SPOTS THAT BREED",
              "accent": "#7fe0a0",
              "icon": "respawn",
              "kicker": "Turing's morphogenesis — how a leopard gets its spots",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-arnold-cat",
              "title": "THE ARNOLD-CAT",
              "accent": "#ffa0e0",
              "icon": "arnoldcat",
              "kicker": "scramble an image to noise — it returns exactly",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-haar",
              "title": "THE HAAR",
              "accent": "#50c0a0",
              "icon": "haar",
              "kicker": "average and difference, recurse — perfectly invertible",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-cuckoo-hashing",
              "title": "THE CUCKOO HASHING",
              "accent": "#58a0b0",
              "icon": "cuckoo",
              "kicker": "a key kicks out its neighbour and lands safe",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-doomsday",
              "title": "THE DOOMSDAY",
              "accent": "#b06bff",
              "icon": "doomsday",
              "kicker": "the weekday of any date by hand",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-barbier",
              "title": "THE BARBIER",
              "accent": "#ffcf4a",
              "icon": "barbier",
              "kicker": "every constant-width curve has the same perimeter",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-sylvesters-law-of-inertia",
              "title": "THE SYLVESTER INERTIA",
              "accent": "#35ffb0",
              "icon": "sylvesterinertia",
              "kicker": "a signature invariant under congruence",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-vitali",
              "title": "THE VITALI",
              "accent": "#b06bff",
              "icon": "vitali",
              "kicker": "the set that cannot be measured",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-tree",
              "title": "THE TREE",
              "accent": "#ffcf4a",
              "icon": "tree",
              "kicker": "the sequence that must end",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-gilbert-varshamov",
              "title": "THE GILBERT-VARSHAMOV",
              "accent": "#ffd76a",
              "icon": "◈",
              "kicker": "the code is there; nobody has to find it",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-stamp",
              "title": "THE STAMP",
              "accent": "#b98cff",
              "icon": "◆",
              "kicker": "LIT is refused without an evidence string",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-krein-milman",
              "title": "THE KREIN-MILMAN",
              "accent": "#b98cff",
              "icon": "◇",
              "kicker": "keep the corners, discard the rest, rebuild the whole",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-definition",
              "title": "THE DEFINITION",
              "accent": "#7de2b0",
              "icon": "≡",
              "kicker": "apples-to-apples matters more than thoroughness",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-only-failed-probe",
              "title": "THE ONLY-FAILED PROBE",
              "accent": "#5ad6ff",
              "icon": "✓",
              "kicker": "a detector nobody ever made say yes",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-matroid",
              "title": "THE MATROID",
              "accent": "#5ad6ff",
              "icon": "⊕",
              "kicker": "where greedy is exactly right",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-record",
              "title": "THE RECORD",
              "accent": "#ffd76a",
              "icon": "↑",
              "kicker": "one over k, whatever the world",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-poincare-recurrence",
              "title": "THE POINCARE RECURRENCE",
              "accent": "#ffd76a",
              "icon": "↺",
              "kicker": "everything comes back",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-davenport-schinzel",
              "title": "THE DAVENPORT-SCHINZEL",
              "accent": "#ffd76a",
              "icon": "∿",
              "kicker": "a sequence that cannot alternate",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-shared-terminator",
              "title": "THE SHARED TERMINATOR",
              "accent": "#ff5a8a",
              "icon": "⤓",
              "kicker": "two loops, one CONTINUE",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-wheel-factorisation",
              "title": "THE WHEEL FACTORISATION",
              "accent": "#ffd76a",
              "icon": "☉",
              "kicker": "skip what cannot possibly be prime",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-case-folding",
              "title": "THE CASE FOLDING",
              "accent": "#ff9f45",
              "icon": "⇄",
              "kicker": "out through two, back as one",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            }
          ],
          "seats": 32,
          "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
        },
        {
          "slug": "the-continue",
          "title": "THE CONTINUE",
          "accent": "#ff2d95",
          "icon": "glitch",
          "kicker": "one more life, press on",
          "pole": "push",
          "spheres": [
            {
              "slug": "the-mirror-seeker",
              "title": "THE MIRROR SEEKER",
              "accent": "#7ad0ff",
              "icon": "respawn",
              "kicker": "every palindrome in one pass, because the mirror already knows",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-euler",
              "title": "THE EULER",
              "accent": "#70c0ff",
              "icon": "euler",
              "kicker": "cross every bridge once — the birth of graph theory",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-plain-changes",
              "title": "THE PLAIN CHANGES",
              "accent": "#58a0a8",
              "icon": "plain-changes",
              "kicker": "all n! permutations, each one adjacent swap apart",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-catmull-rom",
              "title": "THE CATMULL-ROM",
              "accent": "#a878c0",
              "icon": "catmull-rom",
              "kicker": "a smooth spline through every control point",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-hensel-lifting",
              "title": "THE HENSEL LIFTING",
              "accent": "#6ab0d0",
              "icon": "hensel",
              "kicker": "lift a root to higher and higher prime power",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-persistent-structure",
              "title": "THE PERSISTENT STRUCTURE",
              "accent": "#ffcf4a",
              "icon": "persistent-structure",
              "kicker": "versions that never overwrite the past",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-conjugate-partition",
              "title": "THE CONJUGATE PARTITION",
              "accent": "#ffcf4a",
              "icon": "conjpartition",
              "kicker": "transpose the diagram, transpose again, home",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-jacobi-elliptic",
              "title": "THE JACOBI ELLIPTIC",
              "accent": "#35ffb0",
              "icon": "jacobiell",
              "kicker": "the doubly-periodic cousins of sine",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-fagnano",
              "title": "THE FAGNANO",
              "accent": "#35ffb0",
              "icon": "fagnano",
              "kicker": "the min-perimeter inscribed triangle is the orthic",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-graham",
              "title": "THE GRAHAM",
              "accent": "#35ffb0",
              "icon": "graham",
              "kicker": "a number too big for the universe with a visible tail",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-lissajous",
              "title": "THE LISSAJOUS",
              "accent": "#ff8a3c",
              "icon": "lissajous",
              "kicker": "rationality on an oscilloscope",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-foucault",
              "title": "THE FOUCAULT",
              "accent": "#35ffb0",
              "icon": "foucault",
              "kicker": "the Earth turning under a wire",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-rice",
              "title": "THE RICE",
              "accent": "#35ffb0",
              "icon": "rice",
              "kicker": "every question about meaning",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-herbrand",
              "title": "THE HERBRAND",
              "accent": "#5ad6ff",
              "icon": "∃",
              "kicker": "an infinity settled by a finite piece of itself",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-test-that-did-not-run",
              "title": "THE TEST THAT DID NOT RUN",
              "accent": "#5ad6ff",
              "icon": "⊘",
              "kicker": "identical to a test that passes",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-moore-bound",
              "title": "THE MOORE BOUND",
              "accent": "#ffd76a",
              "icon": "○",
              "kicker": "a shape that may or may not exist",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-fixed-point",
              "title": "THE FIXED POINT",
              "accent": "#ffd76a",
              "icon": "↺",
              "kicker": "the map that always comes home",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-self-branch",
              "title": "THE SELF-BRANCH",
              "accent": "#ffd76a",
              "icon": "↺",
              "kicker": "a call that never leaves",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-order-that-is-forced",
              "title": "THE ORDER THAT IS FORCED",
              "accent": "#7de2b0",
              "icon": "↓",
              "kicker": "seven floors, and only one way to stack them",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-todd-coxeter",
              "title": "THE TODD-COXETER",
              "accent": "#5ad6ff",
              "icon": "⊚",
              "kicker": "enumerate the cosets and the index falls out",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-minor-fault",
              "title": "THE MINOR FAULT",
              "accent": "#ff2d95",
              "icon": "↳",
              "kicker": "not an error being handled -- an allocation finally happening",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            }
          ],
          "seats": 36,
          "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
        },
        {
          "slug": "second-wind",
          "title": "SECOND WIND",
          "accent": "#00f5ff",
          "icon": "loot",
          "kicker": "the surge that brings you back",
          "pole": "push",
          "spheres": [
            {
              "slug": "the-syndrome",
              "title": "THE SYNDROME",
              "accent": "#00f5ff",
              "icon": "respawn",
              "kicker": "one flipped bit can't hide from the parity watching it",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-reservoir",
              "title": "THE RESERVOIR",
              "accent": "#58a878",
              "icon": "reservoir",
              "kicker": "a uniform sample from a stream of unknown length, O(1) memory",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-kalman",
              "title": "THE KALMAN",
              "accent": "#d4a017",
              "icon": "kalman",
              "kicker": "fuse guess and measurement optimally, recursively",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-ekg-sequence",
              "title": "THE EKG SEQUENCE",
              "accent": "#e0609a",
              "icon": "ekg",
              "kicker": "a sequence walking by shared factors",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-avl",
              "title": "THE AVL TREE",
              "accent": "#ff8a3c",
              "icon": "avl",
              "kicker": "balance kept by rotation",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-legendre-transform",
              "title": "THE LEGENDRE TRANSFORM",
              "accent": "#ff8a3c",
              "icon": "legendre",
              "kicker": "a duality that undoes itself",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-sinkhorn",
              "title": "THE SINKHORN",
              "accent": "#ffcf4a",
              "icon": "sinkhorn",
              "kicker": "alternate row and column normalizing to perfect balance",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-keith",
              "title": "THE KEITH",
              "accent": "#35ffb0",
              "icon": "keith",
              "kicker": "a number reborn in its own digit stream",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-parker-square",
              "title": "THE PARKER SQUARE",
              "accent": "#ffcf4a",
              "icon": "parkersquare",
              "kicker": "the celebrated failure",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-spiral-of-theodorus",
              "title": "THE SPIRAL OF THEODORUS",
              "accent": "#35ffb0",
              "icon": "theodorus",
              "kicker": "the spiral that stops at 17",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-van-der-waerden",
              "title": "THE VAN DER WAERDEN",
              "accent": "#ffcf4a",
              "icon": "vanderwaerden",
              "kicker": "order you cannot avoid",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-verify-then-copy",
              "title": "THE VERIFY THEN COPY",
              "accent": "#ffd76a",
              "icon": "✓",
              "kicker": "an entire outcome removed, for free",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-manifest",
              "title": "THE MANIFEST",
              "accent": "#7de2b0",
              "icon": "◱",
              "kicker": "installable is not offline",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-zero-error",
              "title": "THE ZERO ERROR",
              "accent": "#b98cff",
              "icon": "≡",
              "kicker": "an exact match kills a hypothesis",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-standing-credit",
              "title": "THE STANDING CREDIT",
              "accent": "#ffd76a",
              "icon": "⚖",
              "kicker": "a rule written before the result",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-szilard",
              "title": "THE SZILARD",
              "accent": "#b98cff",
              "icon": "◫",
              "kicker": "one bit, one push, and the books balance",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-price-of-anarchy",
              "title": "THE PRICE OF ANARCHY",
              "accent": "#ffd76a",
              "icon": "⑂",
              "kicker": "the exact cost of everyone choosing freely",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-two-run-gate",
              "title": "THE TWO-RUN GATE",
              "accent": "#7de2b0",
              "icon": "↻",
              "kicker": "nothing enters the seal on one green run",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-brodal-queue",
              "title": "THE BRODAL QUEUE",
              "accent": "#ff5a8a",
              "icon": "↑",
              "kicker": "worst case, not amortised -- and what that costs",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-zobrist-hash",
              "title": "THE ZOBRIST HASH",
              "accent": "#7de2b0",
              "icon": "⊕",
              "kicker": "undo by doing the same thing again",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-sleeping-barber",
              "title": "THE SLEEPING BARBER",
              "accent": "#39fc6b",
              "icon": "☽",
              "kicker": "the gap between looking and lying down",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-jittered-backoff",
              "title": "THE JITTERED BACKOFF",
              "accent": "#00f5ff",
              "icon": "↻",
              "kicker": "a deterministic rule everyone shares is a coordination mechanism",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-circuit-breaker",
              "title": "THE CIRCUIT BREAKER",
              "accent": "#00f5ff",
              "icon": "⏻",
              "kicker": "a dependency outage converted into your own, deliberately",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-multilevel-feedback",
              "title": "THE MULTILEVEL FEEDBACK",
              "accent": "#00f5ff",
              "icon": "⇓",
              "kicker": "a fee levied on evidence, not a conclusion drawn from it",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            }
          ],
          "seats": 33,
          "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
        },
        {
          "slug": "event-horizon",
          "title": "EVENT HORIZON",
          "accent": "#ff5a3c",
          "icon": "boss",
          "kicker": "the point of no return — the fold to 0",
          "pole": "pull",
          "spheres": [
            {
              "slug": "the-singularity",
              "title": "SINGULARITY",
              "accent": "#ff2d95",
              "icon": "loot",
              "kicker": "the fold made physical",
              "i13": [
                "I thetaE <- 2",
                "I theta <- 4",
                "-> theta - thetaE * thetaE / theta"
              ],
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "../the-4096",
              "title": "THE 4096",
              "accent": "#00f5ff",
              "icon": "spawn",
              "kicker": "2048, doubled",
              "i13": [
                "I tower <- 2048",
                "-> tower * 2"
              ],
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-kaprekar",
              "title": "THE KAPREKAR",
              "accent": "#ffd24d",
              "icon": "6174",
              "kicker": "6174 — the number every 4-digit number falls into",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-napkin",
              "title": "THE NAPKIN",
              "accent": "#35ffb0",
              "icon": "napkin",
              "kicker": "a band through any sphere holds the same volume",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-circle-inversion",
              "title": "THE CIRCLE INVERSION",
              "accent": "#35ffb0",
              "icon": "circleinversion",
              "kicker": "invert through the circle, then again, home",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-dottie",
              "title": "THE DOTTIE",
              "accent": "#21e6ff",
              "icon": "dottie",
              "kicker": "the fixed point of cosine",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-cassini",
              "title": "THE CASSINI",
              "accent": "#b06bff",
              "icon": "cassini",
              "kicker": "a Fibonacci determinant pinned at plus or minus one",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-string-that-remembers",
              "title": "THE STRING THAT REMEMBERS",
              "accent": "#35ffb0",
              "icon": "karplusstrong",
              "kicker": "a burst of noise that decays into a musical note",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-quantum-pigeonhole",
              "title": "THE QUANTUM PIGEONHOLE",
              "accent": "#35ffb0",
              "icon": "qpigeonhole",
              "kicker": "three in two boxes, none together",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-chaitin-omega",
              "title": "THE CHAITIN OMEGA",
              "accent": "#21e6ff",
              "icon": "chaitin",
              "kicker": "the number no theory can reach",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-nyquist",
              "title": "THE NYQUIST",
              "accent": "#5ad6ff",
              "icon": "⌇",
              "kicker": "half the sampling rate, and not one hertz more",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-ellsberg",
              "title": "THE ELLSBERG",
              "accent": "#ff5a8a",
              "icon": "◔",
              "kicker": "a preference no probability can hold",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-decay",
              "title": "THE DECAY",
              "accent": "#ff5a8a",
              "icon": "⌇",
              "kicker": "how a true sentence becomes a false one",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-dropped-predicate",
              "title": "THE DROPPED PREDICATE",
              "accent": "#ff5a8a",
              "icon": "⊘",
              "kicker": "an instrument that guesses",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-wilkinson",
              "title": "THE WILKINSON",
              "accent": "#ff5a8a",
              "icon": "∷",
              "kicker": "twenty roots you can see and cannot recover",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-interval-arithmetic",
              "title": "THE INTERVAL ARITHMETIC",
              "accent": "#ff5a8a",
              "icon": "⧉",
              "kicker": "bounds that are right and useless",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-jordan",
              "title": "THE JORDAN",
              "accent": "#7de2b0",
              "icon": "◌",
              "kicker": "the obvious theorem that took twenty years",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-reach-of-a-branch",
              "title": "THE REACH OF A BRANCH",
              "accent": "#5ad6ff",
              "icon": "↔",
              "kicker": "how far a call can see",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-api-change",
              "title": "THE API CHANGE",
              "accent": "#b98cff",
              "icon": "⇄",
              "kicker": "a signature moved and a default came with it",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-de-bruijn-multiply",
              "title": "THE DE BRUIJN MULTIPLY",
              "accent": "#7de2b0",
              "icon": "⊛",
              "kicker": "a perfect hash for the lowest set bit",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            }
          ],
          "seats": 28,
          "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
        },
        {
          "slug": "rollback",
          "title": "ROLLBACK",
          "accent": "#9d00ff",
          "icon": "coop",
          "kicker": "revert to a prior state",
          "pole": "pull",
          "spheres": [
            {
              "slug": "the-exact-transform",
              "title": "THE EXACT TRANSFORM",
              "accent": "#c8b4ff",
              "icon": "respawn",
              "kicker": "an FFT in a prime field — convolution with zero rounding",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-edit-distance",
              "title": "THE EDIT DISTANCE",
              "accent": "#ffb0e0",
              "icon": "diff",
              "kicker": "Levenshtein — the minimal diff between two strings",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-recaman",
              "title": "THE RECAMAN",
              "accent": "#5ad0e0",
              "icon": "recaman",
              "kicker": "jump back if you can, else forward — the arc that haunts",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-margolus",
              "title": "THE MARGOLUS MIRROR",
              "accent": "#7ad0b0",
              "icon": "margolus",
              "kicker": "a reversible CA — run it back to the exact seed",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-lz77",
              "title": "THE LZ77",
              "accent": "#a878c0",
              "icon": "lz77",
              "kicker": "compress by pointing backward into your own past",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-brent-cycle",
              "title": "THE BRENT CYCLE",
              "accent": "#a878c0",
              "icon": "brent-cycle",
              "kicker": "cycle detection in O(1) memory, fewer evals than Floyd",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-sociable-numbers",
              "title": "THE SOCIABLE NUMBERS",
              "accent": "#d06858",
              "icon": "sociable",
              "kicker": "numbers whose divisor-sums loop back in a chain",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-successive-keeper",
              "title": "THE SUCCESSIVE KEEPER",
              "accent": "#b06bff",
              "icon": "successive-keeper",
              "kicker": "a diffable record beats a believed one",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-bit-reversal",
              "title": "THE BIT-REVERSAL",
              "accent": "#21e6ff",
              "icon": "bitreversal",
              "kicker": "reverse the bits, reverse again, home",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-svd",
              "title": "THE SVD",
              "accent": "#35ffb0",
              "icon": "svd",
              "kicker": "a matrix as rotate-stretch-rotate",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-pancake",
              "title": "THE PANCAKE",
              "accent": "#21e6ff",
              "icon": "pancake",
              "kicker": "Bill Gates and the flipped stack",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-seven-circles",
              "title": "THE SEVEN CIRCLES",
              "accent": "#35ffb0",
              "icon": "sevencircles",
              "kicker": "the chain porism",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-honeycomb",
              "title": "THE HONEYCOMB",
              "accent": "#ffcf4a",
              "icon": "honeycomb",
              "kicker": "the cheapest walls",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-first-mutate",
              "title": "THE FIRST MUTATE",
              "accent": "#5ad6ff",
              "icon": "↯",
              "kicker": "the last instant a rollback was still free",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-line-count",
              "title": "THE LINE COUNT",
              "accent": "#ff5a8a",
              "icon": "≡",
              "kicker": "the third time, and the lesson still did not take",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-direction-test",
              "title": "THE DIRECTION TEST",
              "accent": "#7de2b0",
              "icon": "→",
              "kicker": "a sign pattern that acquits",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-alabama-paradox",
              "title": "THE ALABAMA PARADOX",
              "accent": "#ff5a8a",
              "icon": "↶",
              "kicker": "more seats, fewer seats",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-stale-witness",
              "title": "THE STALE WITNESS",
              "accent": "#ffd76a",
              "icon": "◆",
              "kicker": "a signature attests a moment, not a file",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-gate-that-can-stop-it",
              "title": "THE GATE THAT CAN STOP IT",
              "accent": "#5ad6ff",
              "icon": "⊖",
              "kicker": "put the falsifiable floor early",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-carry-save-adder",
              "title": "THE CARRY-SAVE ADDER",
              "accent": "#5ad6ff",
              "icon": "⊞",
              "kicker": "three numbers in, two out, no carry chain",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-buddy-allocator",
              "title": "THE BUDDY ALLOCATOR",
              "accent": "#9d00ff",
              "icon": "⬚",
              "kicker": "merging is cheap because most merges are forbidden",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-metastable-failure",
              "title": "THE METASTABLE FAILURE",
              "accent": "#9d00ff",
              "icon": "∿",
              "kicker": "the only fix is refusing traffic you can serve",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-context-switch",
              "title": "THE CONTEXT SWITCH",
              "accent": "#9d00ff",
              "icon": "⇋",
              "kicker": "not the cost of switching, the cost of having been away",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            }
          ],
          "seats": 30,
          "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
        },
        {
          "slug": "garbage-collection",
          "title": "GARBAGE COLLECTION",
          "accent": "#7cfc00",
          "icon": "cheat",
          "kicker": "sweep the dead, reclaim the memory",
          "pole": "pull",
          "spheres": [
            {
              "slug": "garbage-collection",
              "title": "GARBAGE COLLECTION",
              "accent": "#5ad0ff",
              "icon": "respawn",
              "kicker": "sweep the dead, reclaim the memory",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-cantor",
              "title": "THE CANTOR",
              "accent": "#d0a0ff",
              "icon": "cantor",
              "kicker": "measure zero, yet uncountable — the dust that remains",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-kosaraju",
              "title": "THE KOSARAJU",
              "accent": "#70a860",
              "icon": "kosaraju",
              "kicker": "strongly connected components in two DFS passes",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-carnot",
              "title": "THE CARNOT",
              "accent": "#ffcf4a",
              "icon": "carnot",
              "kicker": "circumcentre-to-side distances summing to R plus r",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-ulam-numbers",
              "title": "THE ULAM NUMBERS",
              "accent": "#21e6ff",
              "icon": "ulam",
              "kicker": "a sequence that builds itself from unique sums",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-grandi",
              "title": "THE GRANDI",
              "accent": "#ff8a3c",
              "icon": "grandi",
              "kicker": "the sum that flickers",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-peano-curve",
              "title": "THE PEANO CURVE",
              "accent": "#ffcf4a",
              "icon": "peano",
              "kicker": "the line that fills a square",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-apollonian-gasket",
              "title": "THE APOLLONIAN GASKET",
              "accent": "#21e6ff",
              "icon": "gasket",
              "kicker": "circles all the way down, all integers",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-rate-distortion",
              "title": "THE RATE-DISTORTION",
              "accent": "#7de2b0",
              "icon": "⊘",
              "kicker": "how small it gets if you say what you can lose",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-presburger",
              "title": "THE PRESBURGER",
              "accent": "#ffd76a",
              "icon": "⊕",
              "kicker": "surrender multiplication, get decidability back",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-gentzen",
              "title": "THE GENTZEN",
              "accent": "#7de2b0",
              "icon": "⊢",
              "kicker": "a proof that stops borrowing",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-union",
              "title": "THE UNION",
              "accent": "#ffd76a",
              "icon": "∪",
              "kicker": "read their generator instead of guessing at it",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-warn-only",
              "title": "THE WARN-ONLY",
              "accent": "#ffd76a",
              "icon": "⚠",
              "kicker": "a gate that cannot fail the build is a log line",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-caveat-attrition",
              "title": "THE CAVEAT ATTRITION",
              "accent": "#5ad6ff",
              "icon": "⌇",
              "kicker": "the careful thinking gets left behind",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-landauer",
              "title": "THE LANDAUER",
              "accent": "#7de2b0",
              "icon": "⊖",
              "kicker": "the price of forgetting",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-count-min",
              "title": "THE COUNT MIN",
              "accent": "#5ad6ff",
              "icon": "↓",
              "kicker": "the error that only goes one way",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-cross-entropy",
              "title": "THE CROSS ENTROPY",
              "accent": "#b98cff",
              "icon": "⚖",
              "kicker": "the bits you pay for being wrong",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-reversible",
              "title": "THE REVERSIBLE",
              "accent": "#b98cff",
              "icon": "⇆",
              "kicker": "logic that throws nothing away",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-unfiled",
              "title": "THE UNFILED",
              "accent": "#ff5a8a",
              "icon": "∅",
              "kicker": "the symbols nobody wrote down",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-magic-divide",
              "title": "THE MAGIC DIVIDE",
              "accent": "#b98cff",
              "icon": "⊘",
              "kicker": "dividing by multiplying",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-hazard-pointer",
              "title": "THE HAZARD POINTER",
              "accent": "#5ad4ff",
              "icon": "⚑",
              "kicker": "say out loud which pointer you are holding",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-slab-allocator",
              "title": "THE SLAB ALLOCATOR",
              "accent": "#7cfc00",
              "icon": "▤",
              "kicker": "excellent at one question, useless at the rest",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            }
          ],
          "seats": 27,
          "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
        },
        {
          "slug": "hard-reset",
          "title": "HARD RESET",
          "accent": "#5ad0ff",
          "icon": "respawn",
          "kicker": "drain all the way to zero",
          "pole": "pull",
          "spheres": [
            {
              "slug": "the-most-likely-path",
              "title": "THE MOST LIKELY PATH",
              "accent": "#57c8ff",
              "icon": "respawn",
              "kicker": "turn noise back into signal by finding the likeliest path",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-hailstone",
              "title": "THE HAILSTONE",
              "accent": "#9ec8ff",
              "icon": "3n1",
              "kicker": "3n+1 — computed forever, proven never",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-goodstein",
              "title": "THE GOODSTEIN",
              "accent": "#d07050",
              "icon": "goodstein",
              "kicker": "unbounded growth that always crashes to 0 — unprovable in PA",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-happy-number",
              "title": "THE HAPPY NUMBER",
              "accent": "#d06858",
              "icon": "happy",
              "kicker": "digit-squares that reach 1 or loop",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-toffoli",
              "title": "THE TOFFOLI",
              "accent": "#b06bff",
              "icon": "toffoli",
              "kicker": "a gate that runs backwards",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-graph-complement",
              "title": "THE GRAPH COMPLEMENT",
              "accent": "#b06bff",
              "icon": "graphcomplement",
              "kicker": "flip every edge, flip again, home",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-involution",
              "title": "THE INVOLUTION",
              "accent": "#21e6ff",
              "icon": "involution",
              "kicker": "self-inverse permutations counted by a recurrence",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-ehrenfest",
              "title": "THE EHRENFEST",
              "accent": "#b06bff",
              "icon": "ehrenfest",
              "kicker": "the urn that takes 2^N to reset",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-osgood",
              "title": "THE OSGOOD",
              "accent": "#ff8a3c",
              "icon": "osgood",
              "kicker": "the dust that still weighs half",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-hat",
              "title": "THE HAT",
              "accent": "#b06bff",
              "icon": "hat",
              "kicker": "one tile that never repeats",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-jordan-curve",
              "title": "THE JORDAN CURVE",
              "accent": "#b98cff",
              "icon": "◌",
              "kicker": "inside is not a place, it is a count",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-memoryless-examiner",
              "title": "THE MEMORYLESS EXAMINER",
              "accent": "#ffd76a",
              "icon": "⊘",
              "kicker": "a grader with amnesia can only be shown",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-claimlink",
              "title": "THE CLAIMLINK",
              "accent": "#5ad6ff",
              "icon": "⚇",
              "kicker": "a verdict that admits it cannot tell",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-provenance-fork",
              "title": "THE PROVENANCE FORK",
              "accent": "#ff5a8a",
              "icon": "⑂",
              "kicker": "a number that needed a credential",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-no-free-lunch",
              "title": "THE NO FREE LUNCH",
              "accent": "#7de2b0",
              "icon": "⚖",
              "kicker": "a tie nobody can break",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-cauchy",
              "title": "THE CAUCHY",
              "accent": "#ff5a8a",
              "icon": "∞",
              "kicker": "a mean that never settles",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-cutoff",
              "title": "THE CUTOFF",
              "accent": "#5ad6ff",
              "icon": "⌷",
              "kicker": "mixing that happens all at once",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-zero-parameter",
              "title": "THE ZERO PARAMETER",
              "accent": "#5ad6ff",
              "icon": "⊘",
              "kicker": "five rules with nothing to tune",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-wider-at-the-bottom",
              "title": "THE WIDER AT THE BOTTOM",
              "accent": "#ffd76a",
              "icon": "△",
              "kicker": "a floor laid on one run is laid on a coincidence",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-xor-swap",
              "title": "THE XOR SWAP",
              "accent": "#ff5a8a",
              "icon": "⇄",
              "kicker": "no temporary, and one input it destroys",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            },
            {
              "slug": "the-priority-inversion",
              "title": "THE PRIORITY INVERSION",
              "accent": "#ff9f45",
              "icon": "⤴",
              "kicker": "the highest waits on the lowest",
              "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
            }
          ],
          "seats": 38,
          "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
        }
      ],
      "learned": "I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf"
    }
  ],
  "domains": [],
  "note": "World II's central DB — the hub reads this live, so counts climb as new appeals/domains/spheres/keepers are built. Regenerate/append this file; nothing else to touch on the page.",
  "author": "David Lee Wise / ROOT0 / TriPod LLC",
  "fold_root": "ced70a336b58c92112cf9425c250547b5ba8a12ac7527a755590de229114da3c",
  "genesis": "e8740e2614a48a8aa50af601eb10ab74033e71b43abe536ffd58aef4175b5e55",
  "corpus": {
    "name": "the stage-13 corpus",
    "note": "the code this world represents",
    "opcodes": [
      "ASK",
      "ANSWER",
      "CONST",
      "ATTR",
      "ARG",
      "RET",
      "DROP",
      "JMPF",
      "CMP",
      "CALL",
      "FUNC",
      "BIN",
      "HALT"
    ],
    "train": [
      "def binom(I n, I k) {",
      "  if k == 0 { -> 1 }",
      "  if k == n { -> 1 }",
      "  -> binom(n - 1, k - 1) + binom(n - 1, k)",
      "}",
      "def fact(I n) {",
      "  if n <= 1 { -> 1 }",
      "  -> n * fact(n - 1)",
      "}",
      "-> binom(6, 2) + fact(4)"
    ],
    "story": "source as corpus -> char bigram (the only weights) -> sub-agent samples -> VETO (PDA, 0 params) + -I DISCHARGE (0 params) clamp it well-formed"
  },
  "apex": {
    "slug": "the-fiddler",
    "title": "THE FIDDLER",
    "accent": "#ffd23f",
    "icon": "boss",
    "kicker": "IDIT · the Intent Drift Integrity Test",
    "role": "the governor at 0",
    "line": "silicon needs governance. David's FIRST repo sits at the apex — 0 — and governs the whole fold: no silent mutation, every change disclosed. The same law the .dlw.fold seal enforces."
  },
  "i13": {
    "spec": "I-13",
    "version": "2.0",
    "frozen": "2026-08-01",
    "sha256": "64881ebf502b87bb450f1f39b71066013e0c31a7f78dedcae326f6155ddc6bf8",
    "one_line": "a four-plane agent stack over a thirteen-symbol language, where two planes learn, two do not, and five parameter-free rules hold what no model at this scale holds",
    "law": "net = binds - k",
    "planes": 4,
    "symbols": 13,
    "twelve": [
      "NAME",
      "CONSTANT",
      "ATTRIBUTE",
      "CALL",
      "ASSIGN",
      "ARG",
      "EXPR",
      "IF",
      "COMPARE",
      "FUNCTIONDEF",
      "RETURN",
      "BINOP"
    ],
    "cortex_rules": [
      "veto",
      "-I",
      "depth",
      "idempotence",
      "address"
    ],
    "claim": "deterministic zero-parameter components guarantee structural properties no model at this scale reaches; learned recurrent state is not a stack",
    "br_rule": "br targets a DEPTH, never an address — so validation is one linear pass (net = binds - k)",
    "vendored": "i13-stack-v2.json",
    "note": "David's frozen I-13 v2 stack, taught to the whole FOLD. The full spec is vendored beside fold.json; every appeal, domain, sphere and keeper carries the `learned` marker.",
    "voxel": {
      "name": "THE VOLUME",
      "version": "1.0",
      "built": "2026-08-01",
      "sha256": "fa73e632fa012f0547aef1767464033db3ead82ba41139e0a73b76e95b18e345",
      "axes": [
        {
          "id": "beta",
          "label": "Heaps beta",
          "desc": "vocabulary saturation exponent, fitted 800-25,000 words. LOW = narrow vocabulary, saturates early."
        },
        {
          "id": "I",
          "label": "the referent",
          "desc": "I/my/me/mine/myself per 1,000 words. the one operant that is not a verb."
        },
        {
          "id": "voiced",
          "label": "voiced fraction",
          "desc": "quotation pairs per 1,000 words. how much is spoken by someone other than the narrator."
        }
      ],
      "independence": {
        "beta_vs_I": 0.361,
        "beta_vs_voiced": -0.071,
        "I_vs_voiced": 0.364
      },
      "effective_dims": 2.545,
      "texts": 14,
      "cells_occupied": 12,
      "cells_total": 27,
      "method": "each axis is an independent operation on the raw text. no axis is derived from another. all three pairwise correlations |r| < 0.4. to ADD a text: fetch it, strip the Gutenberg header/footer, lowercase-tokenize on [a-z'], compute the three axes, and bin against the stored quantile boundaries. the bins are FROZEN at v1.0 so later additions are comparable.",
      "open_questions": [
        "effective dims 2.55 of 3 — PC3 carries only 15%. I and voiced share mild structure (r=+0.364).",
        "all texts are English; three are translations (Gilgamesh, Inferno, Crime+Punishment, Zarathustra). the translator vocabulary is inside the beta measurement and is not separated.",
        "copyrighted works on the bookshelf (Earthsea, Otherland, Black Company, Ishmael, Lovelock, Nemesis, Alvin Maker, Gods Themselves, Without Remorse, Rand) are NOT in the volume — read structurally, never measured."
      ],
      "viewer": "the-volume-v1.html",
      "index": "the-volume-index-v1.json",
      "note": "David's I-13 voxel: three INDEPENDENT stylometric axes (all pairwise |r| < 0.4) over real texts. The 'I' axis is the I-13 referent operant. Both viewer and index are vendored beside fold.json; bins are FROZEN at v1.0 so later texts stay comparable."
    },
    "corpus": {
      "name": "I-13 v2 full corpus",
      "frozen": "2026-08-01",
      "declared_spec_sha256": "64881ebf502b87bb450f1f39b71066013e0c31a7f78dedcae326f6155ddc6bf8",
      "files": 35,
      "bytes": 8374695,
      "sections": [
        {
          "id": "01-frozen-spec",
          "title": "The Frozen Spec",
          "blurb": "The tower, the five rules, the twelve operants, the machine (net = binds - k). Start at I-13-v2-FROZEN.md; the JSON is what the declared sha covers; the v1 HTML is kept to record what v2 corrected.",
          "files": [
            "I-13-v2-FROZEN.md",
            "i13-frozen-v1.html",
            "i13-stack-v2.json"
          ]
        },
        {
          "id": "02-the-stack",
          "title": "The Stack",
          "blurb": "lex to parse to compile to assemble to validate to VM to JIT, all running in the page. 18,249 trained parameters executing in JavaScript; turn the cortex off and watch it fail. The twelve-station line, colour-coded by provenance.",
          "files": [
            "cc-five-layers.html",
            "i13-factory.html",
            "i13-language.html",
            "i13-live-stack.html",
            "i13-two-scopes.html",
            "machine-corpus-13.html"
          ]
        },
        {
          "id": "03-the-factory",
          "title": "The Factory",
          "blurb": "Paste any source: it names the language and emits a bootloader (plane, rules, pairs, quantile, and the trap that language sprang). 756 languages, 17 families; the six delimiter pair-tables and their guards.",
          "files": [
            "agent-factory.html",
            "factory-corpus-v1.json",
            "pair-table.html",
            "seven-languages.html"
          ]
        },
        {
          "id": "04-hello-world",
          "title": "Hello World",
          "blurb": "Real toolchains installed and run, not simulations: rustc 1.75.0 ('name survived the borrow') and go 1.22.2 (two defer statements discharging LIFO).",
          "files": [
            "go-hello-i13.html",
            "rust-hello-i13.html"
          ]
        },
        {
          "id": "05-corpora",
          "title": "Corpora",
          "blurb": "ab-corpus-v2.txt: Boole / Lovelace / Hinton with the human first-person stripped. Note: the vendored file differs from the MANIFEST's declared sha (a different revision).",
          "files": [
            "ab-corpus-v2.txt"
          ]
        },
        {
          "id": "06-rust-source",
          "title": "Rust Source",
          "blurb": "677 lines. Builds with rustc --edition 2021 -O src/main.rs -o i13. The comments carry the measurements, the attributions, and the bugs.",
          "files": [
            "Cargo.toml",
            "README.md",
            "ast.rs",
            "cortex.rs",
            "flatten.rs",
            "isa.rs",
            "lex.rs",
            "main.rs",
            "parse.rs",
            "vm.rs"
          ]
        },
        {
          "id": "07-earlier-build",
          "title": "Earlier Build",
          "blurb": "Prior sessions: Stott's polytope sections, the compendium, the provenance tracer, the eve stack, and the same wall six times.",
          "files": [
            "compendium-in-g.html",
            "compendium-note-g.html",
            "eve-stack.html",
            "provenance-tracer.html",
            "stott-sections.html",
            "the-same-wall-six-times.epub",
            "the-same-wall-six-times.html",
            "the-stack.html"
          ]
        }
      ],
      "rust_source_lines": 677,
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        "matches_declared": false,
        "note": "HONEST: vendored ab-corpus-v2.txt does NOT match the MANIFEST's declared sha; it is a different revision (no CR bytes; same after LF-normalization). The actual bytes' sha is what is sealed."
      },
      "index": "i13-v2/index.html",
      "root": "i13-v2/",
      "corpus_root": "80ceb95b611a5d6d969d30c148740d144d1e31c918c0c3dbf969f91af7d47604",
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        },
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      ],
      "sealing_scope": "ROOT_0 is the merkle over the sphere/keeper inhabitants (sha256(name|slug|blurb)) and does NOT hash this corpus. The corpus is bound by corpus_root, recorded here in the central DB (fold.json) beside ROOT_0.",
      "note": "David 2026-08-01: 'integrate please, full corpus'. The whole i13-v2 archive is vendored beside fold.json and mirrored; every file's sha256 is recorded here and folded into corpus_root. Note: ROOT_0 covers the inhabitants, not the corpus — see sealing_scope."
    },
    "v3": {
      "name": "I-13 v3 apparatus",
      "artifacts": [
        {
          "file": "i13-pipeline-v2.1.html",
          "title": "THE PIPELINE v2.1",
          "blurb": "SOURCE -> RESULT; falsifier written, tested, did not fire",
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          "bytes": 146468
        },
        {
          "file": "the-complex-v1.html",
          "title": "THE COMPLEX",
          "blurb": "I-13 . I-13x2 . the capped cross; the double helix r=0.6437 p=0.0166",
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        },
        {
          "file": "the-volume-v1.html",
          "title": "THE VOLUME v1.0",
          "blurb": "three measured axes, none derived; independence + the index",
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      ],
      "v3_root": "343efc30ff7a5a7b400728d2ca6a3da3640a551f947d5ebb6e73c4f3c747111f",
      "index": "i13-v3/index.html",
      "root": "i13-v3/",
      "sealing_scope": "ROOT_0 is the merkle over the sphere/keeper inhabitants and does NOT hash these v3 artifacts; they are bound by v3_root recorded here beside the corpus.",
      "note": "David 2026-08-01: 'integrate reality wide'. The v3 I-13 apparatus (pipeline v2.1, THE COMPLEX, THE VOLUME v1.0) is vendored beside the i13-v2 corpus and mirrored; each file's sha256 is folded into v3_root. These are David's own instruments with self-declared statistics (e.g. the double helix r=0.6437, p=0.0166) — credited, not re-derived. THE VOLUME v1.0 (sha 090f84a7) is an evolution of the earlier voxel (sha 2d6c9746)."
    },
    "nonsofic": {
      "sofic": "imitable by something FINITE, to any accuracy asked (Hebrew sofi, finite)",
      "nonsofic": "not so imitable, at any finite size whatsoever",
      "provinces": [
        "shifts: finite labelled graph / follower sets",
        "groups: finite permutation models under normalised Hamming distance"
      ],
      "criterion_shift": "X is sofic <=> the number of distinct follower sets F(w) is FINITE",
      "sofic_witness": "golden-mean shift (forbid the block 11): word counts are the Fibonacci numbers, ratio -> phi, entropy log2(phi)=0.694241914, follower sets = 2 forever",
      "nonsofic_witness_shift": "matched-run shift 1 0^n 1 0^n 1: follower-set counts 2,4,7,10,13,17,21,25,29,... grow without bound, so no finite automaton captures it",
      "nonsofic_witness_group": "a finitely-presented NON-sofic group exists (Gromov 1999 answered): the binary Leavitt algebra L ~= L(+)L over F_2 ('one is two') fails invariant basis number, so a group carrying 1=2 in its matrices cannot be sofic; nine = the leaf count of a complete binary prefix code (lengths 3x7,4x2; Kraft sum = 1)",
      "residue": "PROVEN: the proposition follows from the axioms. NOT PROVEN: that the proposition is the one intended. A kernel verifies inference; it does not verify meaning.",
      "resonance": "this residue IS the I-13 duality mantra: understanding (inference) is not meaning (the wall). The finite's honest reach has an edge, and the edge is nameable.",
      "source": "notes-upon-the-nonsofic.html — after the Notes of A.A.L. upon Menabrea, 1843; every table computed before it was versified; NonSoficGroup.lean read directly (0 sorry, peer review pending at time of writing, and it says so)."
    },
    "realitywide": {
      "name": "I-13 reality-wide apparatus",
      "directive": "David 2026-08-02: 'i13.integrate.realitywide'",
      "artifacts": [
        {
          "file": "notes-upon-the-nonsofic.html",
          "title": "NOTES UPON THE NONSOFIC",
          "blurb": "sofic = imitable by a finite thing to any accuracy; nonsofic = not so, at any size. The golden-mean shift (forbid 11 -> Fibonacci counts, follower-sets = 2 forever) is sofic; the matched-run shift 1 0^n 1 0^n 1 is not (follower-set count 2,4,7,10,13,... unbounded). Gromov's 1999 question answered: a finitely-presented NON-sofic group exists (binary Leavitt algebra L ~= L(+)L, 'one is two'; nine = a complete prefix code's leaves). Residue: a kernel verifies inference, not meaning.",
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        {
          "file": "the-complex-v3-pentaptych.html",
          "title": "THE COMPLEX v3.0 - pentaptych",
          "blurb": "the five-panel form of THE COMPLEX (I-13 . I-13x2 . the capped cross), the reality-wide successor to the-complex-v1",
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        },
        {
          "file": "w5-the-complex-3d.html",
          "title": "W5 - THE COMPLEX in three dimensions",
          "blurb": "the W5 three-dimensional view of THE COMPLEX",
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      ],
      "realitywide_root": "9376b04c97abd9bab2e27504d2f8202b8819d3ae4a6fe43121e9c8ae9fb1d8ca",
      "index": "i13-v3.1/index.html",
      "root": "i13-v3.1/",
      "sealing_scope": "ROOT_0 is the merkle over the sphere/keeper inhabitants and does NOT hash these artifacts; they are bound by realitywide_root recorded here in the central DB.",
      "note": "David 2026-08-02: 'i13.integrate.realitywide'. THE COMPLEX v3.0 (pentaptych + W5 3D) and the NONSOFIC Notes are vendored beside the i13-v2 corpus and mirrored; each file's sha256 is folded into realitywide_root. The nonsofic boundary is recorded as a first-class I-13 principle (see i13.nonsofic) because its residue is the I-13 claim itself. Credited, not re-derived."
    }
  }
}