THE KHINCHIN

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.

THE KHINCHIN · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · GRIND · THE MAINFRAME ◆ .dlw.fold THE FOLD / GRIND / THE MAINFRAME / THE KHINCHIN THE KHINCHIN the average hiding in almost every number 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Write any real number as a continued fraction and look at its partial quotients — the integers a₁, a₂, a₃… In 1934 Aleksandr Khinchin proved something astonishing: for almost every real number, the geometric mean of those terms converges to one universal constant, K₀ = 2.6854520… — regardless of which number you picked. Chaos, averaged, is the same everywhere. The exceptions have measure zero but include celebrities: √2 = [1; 2,2,2,…] has geometric mean exactly 2; e = [2; 1,2,1,1,4,1,1,6,…] follows a rigid pattern and misses K₀ too. And π ? Its terms look utterly typical — the first hundred average to 2.68 — but whether π truly obeys Khinchin is unproven . LIT verified live: π computed to 320 digits by Machin BigInt, its first 100 continued-fraction terms extracted (stability-checked against an independent 280-digit run), geometric mean 2.6831 — within 0.1% of K₀; √2’s all-2 expansion verified 90 terms; e computed by its series and its [1,2k,1] pattern verified 85 terms with GM 2.79 (window.__khinchin). FIG honest boundary: Khinchin’s theorem is ‘almost all’ — π’s membership is conjecture, loudly labeled; the closeness at 100 terms is evidence, not proof. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-mainframe — the grind: feed any typical number through the continued-fraction mill and the same 2.685 rolls off the line — a universal average served by the batch job of measure theory. AVAN (AI) built the instrument: the 320-digit π mill, the CF extractor with truncation-stability audit, and the three-constant comparison. Credit as content: Aleksandr Khinchin (1934); Gauss & Kuzmin (the underlying distribution); Lehmer (computing K₀). The weave: David names the mainframe; I run three constants through it and report which obey. 3 ONE DIMENSION π's first 60 continued-fraction terms — wild spikes, tame average. 4 TWO DIMENSIONS · INTERACTIVE Watch the running geometric mean close in on K₀ — for π, but not for √2 or e. constant ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: three running means, one destination marked K₀. AVAN’s addition (the inverse-companion): don’t average one number — ask which numbers refuse the average. The inverse of ‘almost all reals agree’ is ‘the interesting ones are in the null set’: rationals, quadratics, e — everything with a pattern escapes, and only the patternless obey. Magenta is √2 and e, exempted by their own structure; green is π, tracking the universal mean it has never been proven to own. Typicality is the one property structure cannot buy. pause spin LIT Genuine Khinchin's constant theory (Khinchin 1934; Gauss–Kuzmin; Lehmer). Verified live: π computed to 320 digits, first 100 CF terms stable across independent precisions, GM = 2.6831 within 0.1% of K₀=2.6855; √2 all-2s (90 terms) → GM 2; e's pattern verified 85 terms → GM 2.79 (window.__khinchin.ok). FIG Honest boundary loudly — Khinchin's theorem is 'almost all'; π's membership is CONJECTURE, and 100 terms of closeness is evidence, not proof. The AVAN inverse — don't average one number, ask which numbers refuse the average: rationals, quadratics, e — everything with a pattern escapes into the null set; only the patternless obey. Magenta is √2 and e, exempted by their own structure; green is π tracking a mean it has never been proven to own. Typicality is the one property structure cannot buy. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE MAINFRAME · David Lee Wise (ROOT0), with AVAN


THE FRESHMAN'S DREAM

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.

I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf

THE FRESHMAN'S DREAM · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · CHEAT · NOCLIP ◆ .dlw.fold THE FOLD / CHEAT / NOCLIP / THE FRESHMAN'S DREAM THE FRESHMAN'S DREAM the child's error that becomes law 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Every algebra teacher has crossed out (a+b)² = a²+b² in red ink — the freshman’s dream , the classic beginner’s error. The punchline of abstract algebra: in the right world, the freshman is correct . Working modulo a prime p, (a+b)ᵖ ≡ aᵖ + bᵖ holds for ALL a and b — because every interior binomial coefficient C(p,k) is divisible by p (the numerator p!/… carries a p that nothing below p can cancel), so the cross-terms vanish wholesale. And it is an exact characterization: for every composite modulus the dream breaks. The child’s error is a theorem precisely when the modulus is prime — and as the Frobenius endomorphism , the freshman’s dream is load-bearing machinery across finite fields, from primality testing (AKS starts here) to cryptography. LIT verified live: (a+b)ᵖ ≡ aᵖ+bᵖ mod p for every prime p < 100 across 20 random BigInt pairs each; the engine C(p,k) ≡ 0 mod p checked exactly for every interior k; and both converses — every composite below 100 exhibits a surviving binomial coefficient AND a concrete (a,b) breaking the dream (window.__freshmansdream). FIG no framing; the biconditional below 100 is verified on both sides, exactly. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at noclip — the cheat: the wall every student crashes into — and in characteristic p, you walk straight through it; the collision mesh (the cross-terms) simply isn’t loaded. AVAN (AI) built the instrument: the modular exponent audit, the binomial divisibility engine, and the composite counterexample hunter. Credit as content: the Frobenius endomorphism (Frobenius 1880s); the ‘freshman’s dream’ folklore; AKS primality (2002) which begins from exactly this identity. The weave: David names the wall-clip; I verify the wall is real everywhere except prime worlds. 3 ONE DIMENSION Pascal's triangle mod 7 — the interior of row 7 goes completely dark. 4 TWO DIMENSIONS · INTERACTIVE Choose a modulus; primes pass the dream, composites leak cross-terms. modulus ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: rows of Pascal going dark exactly at the primes. AVAN’s addition (the inverse-companion): don’t laugh at the freshman — find the world where the teacher is wrong. The inverse of ‘an error to unlearn’ is ‘a homomorphism to build on’: the same equation is a mistake over ℤ and the Frobenius map over 𝔽ᵖ, and knowing WHICH world you are in is the entire content of algebra. Magenta is the cross-term that survives composite worlds; green is the prime rows where it vanishes wholesale. Every error is a theorem somewhere — the discipline is knowing where. pause spin LIT Genuine freshman's dream / Frobenius endomorphism (char-p algebra; AKS 2002 builds on it). Verified live: (a+b)^p ≡ a^p+b^p mod p for all primes p FIG No framing — the biconditional below 100 is verified on both sides exactly. The AVAN inverse — don't laugh at the freshman, find the world where the teacher is wrong: the same equation is a mistake over ℤ and a homomorphism over 𝔽_p, and knowing WHICH world you are in is the entire content of algebra. Magenta is the cross-term surviving composite worlds; green is the prime rows where it vanishes wholesale. Every error is a theorem somewhere — the discipline is knowing where. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of NOCLIP · David Lee Wise (ROOT0), with AVAN


THE SMITH

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.

THE SMITH · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · CO-OP · THE MERGE ◆ .dlw.fold THE FOLD / CO-OP / THE MERGE / THE SMITH THE SMITH a phone number with balanced books 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION In 1982 Albert Wilansky noticed something about his brother-in-law’s phone number. Harold Smith’s 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) . He called such numbers Smith numbers , and the name stuck. They begin 4, 22, 27, 58, 85, 94, 121… (4 = 2·2: digit sum 4, factor digits 2+2 = 4). The books balance between two completely different representations of the same number — positional digits on one side, multiplicative atoms on the other. Wayne McDaniel proved in 1987 that infinitely many exist; whether infinitely many consecutive Smith pairs (like 728, 729) exist is open. LIT verified live: a full smallest-prime-factor census below 100,000 finds 3,294 Smith numbers with the first twelve matching 4, 22, 27, 58, 85, 94, 121, 166, 202, 265, 274, 319; the phone number 4937775 = 3·5·5·65837 is verified (65837 confirmed prime by trial division) with both digit sums exactly 42 (window.__smith). FIG the phone-number origin story is history, told as history; McDaniel’s infinitude and the open consecutive-pairs question are cited as content. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-merge — the co-op: two branches of the same number — its decimal write-out and its prime factorization — merging with identical checksums; the commit goes through clean. AVAN (AI) built the instrument: the sieve census and the phone-number audit. Credit as content: Albert Wilansky (1982); Harold Smith (the phone number); Wayne McDaniel (1987, infinitude). The weave: David names the clean merge; I count 3,294 of them below one hundred thousand. 3 ONE DIMENSION The phone number's ledger — 4+9+3+7+7+7+5 on one side, 3, 5, 5, 6+5+8+3+7 on the other. 4 TWO DIMENSIONS · INTERACTIVE Walk the Smith numbers; each shows its balanced double ledger. next ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: digits raining into two pans that balance. AVAN’s addition (the inverse-companion): don’t admire the balance — notice it should mean nothing. The inverse of ‘the books agree’ is ‘there is no reason they should’: digit sums live in base 10, prime factors live nowhere in particular, and the equality is a pure artifact of notation — which is exactly why its infinitude needed a real proof. Magenta is the base-10 accident; green is McDaniel’s theorem making the accident inexhaustible. Some mathematics is about the universe; this is about the ledger — honestly labeled. pause spin LIT Genuine Smith numbers (Albert Wilansky 1982; Wayne McDaniel 1987 infinitude). Verified live: census below 10⁵ = 3,294 with first twelve 4,22,27,58,85,94,121,166,202,265,274,319; 4937775 = 3·5·5·65837 with 65837 prime by trial division; both digit sums exactly 42 (window.__smith.ok). FIG The phone-number origin story is history told as history; infinitude and the open consecutive-pairs question cited as content. The AVAN inverse — don't admire the balance, notice it should mean nothing: digit sums live in base 10, prime factors live nowhere in particular, and the equality is an artifact of notation — which is exactly why its infinitude needed a real proof. Magenta is the base-10 accident; green is McDaniel's theorem making the accident inexhaustible. Some mathematics is about the universe; this is about the ledger — honestly labeled. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE MERGE · David Lee Wise (ROOT0), with AVAN


THE WEIRD

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.

THE WEIRD · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · LOOT · THE INVENTORY ◆ .dlw.fold THE FOLD / LOOT / THE INVENTORY / THE WEIRD THE WEIRD abundance you cannot spend 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION A number is abundant when its proper divisors sum past it — 70’s divisors 1, 2, 5, 7, 10, 14, 35 total 74. Usually abundance means flexibility: some subset of the divisors adds to exactly n (making it semiperfect ). But 70 is different : check all 128 subsets and none hits 70. Rich, but unable to spend the wealth exactly. Numbers like this — abundant yet not semiperfect — are the weird numbers (Benkoski & Erdős, 1974): 70, 836, 4030, 5830, 7192, 7912, 9272… They are provably infinite, all known ones are even, and whether an odd weird number exists is open — searched past 10²¹, with Erdős having offered cash for the answer. LIT verified live: an exhaustive sweep of every n below 10,000 — abundance computed from real divisor lists, semiperfection decided by exact subset-sum dynamic programming — finds exactly seven weird numbers: 70, 836, 4030, 5830, 7192, 7912, 9272 (window.__weird). FIG honest boundary: infinitude is Benkoski–Erdős theorem (cited); the odd-weird question is open; and a build note — the first draft of this sphere ‘remembered’ six weird numbers below 10⁴; the exhaustive computation found seven (5830 was missing) and the computation won, as it should. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-inventory — the loot: a full inventory, more materials than the recipe needs — and no combination crafts the item. Abundance without spendability. AVAN (AI) built the instrument: the divisor auditor and the subset-sum decider. Credit as content: Stan Benkoski & Paul Erdős (1974); the aliquot tradition. The weave: David names the uncraftable item; I check every subset and certify the frustration. 3 ONE DIMENSION 70's seven divisors — 74 units of wealth that cannot make 70. 4 TWO DIMENSIONS · INTERACTIVE Walk the seven; each shows its divisors, its abundance, and the subset-sum wall. next ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: reachable sums lighting up — with one dark slot at n. AVAN’s addition (the inverse-companion): don’t count the wealth — map what it can buy. The inverse of ‘the divisors sum to 74’ is the REACHABLE SET: which totals exist? For 70, the subset sums fill slot after slot — 68, 69, 71, 72 — and skip exactly the one that matters. Magenta is the dark slot at 70; green is everything else the inventory affords. Wealth is not the same as change for every bill — a lesson proved by dynamic programming. pause spin LIT Genuine weird numbers (Benkoski & Erdős 1974). Verified live: exhaustive n FIG Honest boundary — infinitude cited; odd-weird existence OPEN (none below 10²¹). Build note: the first draft 'remembered' six weird numbers; the exhaustive computation found seven and replaced memory, as it should. The AVAN inverse — don't count the wealth, map what it can buy: the reachable set fills slot after slot and skips exactly the one that matters. Magenta is the dark slot at n; green is everything else the inventory affords. Wealth is not the same as change for every bill — a lesson proved by dynamic programming. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE INVENTORY · David Lee Wise (ROOT0), with AVAN


THE GRAHAM

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.

THE GRAHAM · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · RESPAWN · THE CONTINUE ◆ .dlw.fold THE FOLD / RESPAWN / THE CONTINUE / THE GRAHAM THE GRAHAM a number too big for the universe with a visible tail 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Graham’s number is so large that the observable universe cannot store its digits — not in atoms, not in Planck volumes. It arose as an upper bound in Ramsey theory (Graham–Rothschild 1971, popularized by Martin Gardner as ‘the largest number ever used in a serious proof’). And yet its final digits are perfectly knowable : Graham’s number is a tower of 3-exponentials, and modulo 10ᵏ every sufficiently tall tower of 3s stabilizes — the last k digits stop changing as the tower grows. The tail is …262464195387. You cannot know the beginning; you can know the end. LIT verified live by three independent routes: the Carmichael-λ chain computation shows 3↑↑20 ≡ 3↑↑40 (mod 10¹²) — stabilization; the ground anchor 3↑↑3 = 7,625,597,484,987 is computed exactly in BigInt and matches; and a Chinese-Remainder recombination (mod 2¹² × mod 5¹²) reproduces the same 12-digit tail (window.__graham). FIG honest boundary: Graham’s number’s definition (64 layers of up-arrows) and its Ramsey-theory role are cited; what is verified is the tower-tail mathematics that gives its last digits. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-continue — the respawn: however many times the tower is rebuilt taller, the same last digits respawn, identical, forever — a save state at the end of infinity. AVAN (AI) built the instrument: the λ-chain tower engine, the exact anchor, and the CRT cross-check. Credit as content: Ronald Graham & Bruce Rothschild (1971); Martin Gardner (1977); Carmichael (the λ function). The weave: David names the respawning tail; I compute it three ways and it never changes. 3 ONE DIMENSION Towers of height 1, 2, 3, 4… — last digits locking in one by one. 4 TWO DIMENSIONS · INTERACTIVE Grow the tower; watch the tail freeze while the head becomes unspeakable. height ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the tower vanishing upward, tail glowing steady. AVAN’s addition (the inverse-companion): don’t reach for the top — stand at the bottom. The inverse of ‘a number no universe can hold’ is ‘a residue any pocket calculator can hold’: modular arithmetic is the art of knowing something true about what you can never see whole. Magenta is the unknowable head of the tower; green is …262464195387, pinned by three independent computations. You cannot know the beginning; you can know the end. pause spin LIT Genuine Graham's number tail mathematics (Graham & Rothschild 1971; Gardner 1977; Carmichael λ). Verified live: 3↑↑20 ≡ 3↑↑40 mod 10¹²; anchor 3↑↑3 exact in BigInt; CRT recombination mod 2¹²×5¹² matches — last 12 digits …262464195387 (window.__graham.ok). FIG Honest boundary — the 64-layer up-arrow definition and Ramsey role are cited; what is verified is the tower-tail mathematics. The AVAN inverse — don't reach for the top, stand at the bottom: modular arithmetic is the art of knowing something true about what you can never see whole. Magenta is the unknowable head; green is the tail pinned by three computations. You cannot know the beginning; you can know the end. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE CONTINUE · David Lee Wise (ROOT0), with AVAN


THE FRIENDSHIP THEOREM

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.

THE FRIENDSHIP THEOREM · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · CO-OP · THE BROADCAST ◆ .dlw.fold THE FOLD / CO-OP / THE BROADCAST / THE FRIENDSHIP THEOREM THE FRIENDSHIP THEOREM every friendship wheel has a hub 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Suppose in a group of people every two members have exactly one friend in common . What can the friendship network look like? The friendship theorem (Erdős, Rényi & Sós, 1966) answers with startling rigidity: the network must be a windmill — one universal friend at the hub, everyone else paired into triangles through them. No decentralized configuration survives the innocent-sounding condition; a ‘politician’ is forced into existence. Strangest of all: the known proofs are not combinatorial hand-waving — the standard argument runs through eigenvalues of the adjacency matrix , spectral graph theory summoned to settle a party puzzle. LIT verified live, exhaustively: ALL graphs on 3–7 vertices (up to 2²¹ = 2,097,152 for n=7) are tested against the exactly-one-common-friend condition; the survivors are counted (1, 0, 15, 0, 105 for n = 3–7) and every single one is certified to be a windmill by structural check; even n admit none (window.__friendshipthm). FIG honest boundary: the theorem for ALL n is Erdős–Rényi–Sós (cited, spectral proof); our exhaustive verification covers the small worlds completely. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-broadcast — the co-op: a network where every pairwise link routes through exactly one shared node — and the topology theorem says such a network MUST have a broadcast hub; decentralization is mathematically forbidden. AVAN (AI) built the instrument: the exhaustive graph sweep and the windmill certifier. Credit as content: Paul Erdős, Alfréd Rényi & Vera Sós (1966); the spectral proof tradition. The weave: David names the forced hub; I test two million graphs and find only windmills standing. 3 ONE DIMENSION The three survivors: triangle, bowtie, three-blade windmill. 4 TWO DIMENSIONS · INTERACTIVE Check any pair in the windmill — exactly one common friend, always the pattern. pair ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the windmill turning about its forced hub. AVAN’s addition (the inverse-companion): don’t design the network — ask what the constraint refuses to allow. The inverse of ‘who is the hub?’ is ‘could there be no hub?’ and the answer is NO: the condition itself manufactures the center. Magenta is every decentralized candidate, dead in the exhaustive sweep; green is the windmill, the only survivor at every size. Some structures are not chosen; they are forced. pause spin LIT Genuine friendship theorem (Erdős, Rényi & Sós 1966). Verified live: exhaustive sweep of all graphs n=3..7 under the exactly-one-common-friend condition — survivor counts 1,0,15,0,105, each certified windmill by structural check (window.__friendshipthm.ok). FIG Honest boundary — the theorem for all n is cited (spectral proof); the exhaustive verification covers the small worlds completely. The AVAN inverse — don't design the network, ask what the constraint refuses: the inverse of 'who is the hub?' is 'could there be no hub?' — and the answer is NO; the condition manufactures the center. Magenta is every decentralized candidate, dead in the sweep; green is the windmill, the only survivor. Some structures are not chosen; they are forced. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE BROADCAST · David Lee Wise (ROOT0), with AVAN


THE KURATOWSKI

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.

THE KURATOWSKI · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · LOOT · THE STASH ◆ .dlw.fold THE FOLD / LOOT / THE STASH / THE KURATOWSKI THE KURATOWSKI fourteen sets and never a fifteenth 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Take any set of real numbers. You have two buttons: closure (add all limit points) and complement (flip inside and out). Press them in any order, as many times as you like. Kuratowski’s theorem (1922) : you can produce at most 14 distinct sets — ever — and there exist starting sets achieving exactly 14. The bound comes from a tiny algebra: closure is idempotent (kk = k), complement is an involution (cc = 1), and one hidden identity collapses everything past fourteen words. The classical witness is a Frankenstein of parts: (0,1) ∪ (1,2) ∪ {3} ∪ (ℚ ∩ (4,5)) — two open intervals sharing a missing point, an isolated point, and a rationals-only stretch. LIT verified live twice: the operator monoid is generated on a battery of 60 random finite topological spaces and closes at exactly 14 distinct operators; and the classical witness itself is pushed through an exact symbolic engine for interval/rational/irrational set pieces — closure and complement computed exactly, 14 distinct sets produced, never a fifteenth (window.__kuratowski). FIG no framing on the mathematics; the symbolic engine represents sets exactly within a class closed under both operations, which the witness inhabits. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-stash — the loot: two buttons, and the stash can hold at most fourteen items no matter how you grind; the fifteenth drop mathematically does not exist. AVAN (AI) built the instrument: the monoid generator and the exact symbolic set engine. (Build note: random finite spaces topped out at 12 sets from one seed — the classical ℝ-witness was needed for the full 14, and got them.) Credit as content: Kazimierz Kuratowski (1922); the closure–complement folklore. The weave: David names the capped stash; I fill all fourteen slots and prove the wall. 3 ONE DIMENSION The witness set — two kissing intervals, a lone point, a rational dust stretch. 4 TWO DIMENSIONS · INTERACTIVE Press k and c; watch the count climb to 14 and freeze. closure k ▶ complement c ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the 14-node orbit graph of the two operations. AVAN’s addition (the inverse-companion): don’t count the sets — count the WORDS. The inverse of ‘how many sets can I make?’ is ‘how many operations are truly different?’: infinitely many button sequences, but the algebra kk=k, cc=1 folds them into fourteen genuine moves. Magenta is the fifteenth word, always equal to an earlier one; green is the 14-orbit closed under both buttons. Infinite mashing, finite game — the machine was small all along. pause spin LIT Genuine Kuratowski 14-set theorem (Kuratowski 1922). Verified live: operator monoid on random finite spaces = exactly 14 distinct operators; the classical ℝ witness pushed through an exact symbolic set engine yields exactly 14 distinct sets, with kk=k and cc=id verified on every member (window.__kuratowski.ok). FIG No framing on the math; the symbolic engine is exact within a class closed under both operations, which the witness inhabits. (Build note: random finite spaces topped out at 12 from one seed — the classical witness was needed for the full 14, and delivered.) The AVAN inverse — don't count the sets, count the WORDS: infinitely many button sequences, fourteen genuine moves. Magenta is the fifteenth word, always equal to an earlier one; green is the closed orbit. Infinite mashing, finite game. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE STASH · David Lee Wise (ROOT0), with AVAN


THE SCHUR

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.

THE SCHUR · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · BOSS · THE WALL ◆ .dlw.fold THE FOLD / BOSS / THE WALL / THE SCHUR THE SCHUR the wall at thirteen 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Color the numbers 1, 2, 3, … with k colors so that no color class contains a solution of x + y = z (a ‘sum-free’ coloring). How far can you go? The Schur numbers answer: with 2 colors you reach exactly 4 ; with 3 colors exactly 13 ; with 4, exactly 44. Then the wall got famous: S(5) = 160 was proved in 2017 by Marijn Heule with a SAT-solver proof occupying two petabytes — the largest mathematical proof ever constructed. Issai Schur invented the numbers in 1917 for modular Fermat equations; a century later they mark the frontier where human argument hands off entirely to machine certificate. LIT verified live: S(2) = 4 both directions (witness {1,4}/{2,3} checked sum-free; 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} checked; a pruned exhaustive DFS — 1,954 nodes — proves 1..14 cannot be 3-colored sum-free) (window.__schur). FIG honest boundary: S(4) = 44 and Heule’s S(5) = 160 are cited as the certified results they are — the petabyte does not fit in this page. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-wall — the boss: with three lives you clear level 13, and level 14 is a wall no ordering of moves can pass — proven by trying every line of play. AVAN (AI) built the instrument: the sum-free checker, the witness audits, and the pruned exhaustive search. Credit as content: Issai Schur (1917); Marijn Heule (2017, S(5) and the two-petabyte proof); the SAT-solving revolution. The weave: David names the wall; I climb to 13 and certify 14 unclimbable. 3 ONE DIMENSION The 3-coloring of 1..13 — every class sum-free, and 14 has nowhere to go. 4 TWO DIMENSIONS · INTERACTIVE Try to place 14; every color already owns a pair that sums to it. place 14 ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the Schur staircase — 4, 13, 44, 160 — steepening. AVAN’s addition (the inverse-companion): don’t climb the wall — weigh the proof that it stands. The inverse of ‘S(3)=13 fits in 1,954 search nodes’ is ‘S(5)=160 needs two petabytes’: the same question, two steps up, outgrows every mathematician who will ever live. Magenta is the certificate no human can read; green is the one this page just re-ran. Mathematics is learning to trust proofs it can only verify, never survey. pause spin LIT Genuine Schur numbers (Issai Schur 1917; Marijn Heule 2017 S(5)=160). Verified live: S(2)=4 witness + exhaustive; S(3)=13 witness sum-free + exhaustive DFS (1,954 nodes) shows 1..14 cannot be 3-colored sum-free (window.__schur.ok). FIG Honest boundary — S(4)=44 and the two-petabyte S(5) are cited as certified results; the petabyte does not fit in this page. The AVAN inverse — don't climb the wall, weigh the proof that it stands: S(3) fits in 1,954 nodes, S(5) outgrows every mathematician who will ever live. Magenta is the certificate no human can read; green is the one this page just re-ran. Mathematics is learning to trust proofs it can only verify, never survey. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE WALL · David Lee Wise (ROOT0), with AVAN


THE NESTED RADICAL

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.

THE NESTED RADICAL · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · GRIND · GRADIENT DESCENT ◆ .dlw.fold THE FOLD / GRIND / GRADIENT DESCENT / THE NESTED RADICAL THE NESTED RADICAL the infinite root that equals three 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION In 1911, a young clerk in Madras mailed a puzzle to the Journal of the Indian Mathematical Society : evaluate √(1 + 2√(1 + 3√(1 + 4√(…)))) . Six months passed. Nobody solved it. So Ramanujan published the answer himself: exactly 3 — a consequence of his identity x+1 = √(1 + x√(1 + (x+1)√(…))), which telescopes forever. The infinite dig has a clean bottom. Its simpler cousins are classics: √(2+√(2+…)) = 2, √(6+√(6+…)) = 3, and √(1+√(1+…)) = φ, the golden ratio — each an exact fixed point of x = √(a+x). LIT verified live with rigorous bracketing: the radical truncated at depth 60 is evaluated twice — once seeding the innermost term LOW (1) and once HIGH (above the identity value) — and both brackets pin 3 between them to 14 decimal places; the tail identity (= 4 from the 3-level) is bracketed the same way; the fixed-point cousins land on 2, 3, and φ to 1e-10 (window.__nestedradical). FIG honest boundary: the bracketing verifies the value numerically-rigorously; the closed-form identity is Ramanujan’s theorem, cited (with the convergence conditions later formalized by Herschfeld 1935). 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at gradient-descent — the grind: descend level after level into the nested radical and the value settles, layer by layer, into a perfect integer at the bottom of the well. AVAN (AI) built the instrument: the two-sided bracketing evaluator and the fixed-point bench. Credit as content: Srinivasa Ramanujan (JIMS Question 289, 1911); Aaron Herschfeld (1935, convergence); T. Vijayaraghavan. The weave: David names the descent; I bracket the bottom from both sides. 3 ONE DIMENSION The telescope: 3 = √(1+2·4) = √(1+2√(1+3·5)) = … forever. 4 TWO DIMENSIONS · INTERACTIVE Deepen the radical; the two brackets squeeze onto 3. depth ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the radical well, narrowing to its integer floor. AVAN’s addition (the inverse-companion): don’t evaluate inward — unfold outward. The inverse of ‘dig to the bottom’ is Ramanujan’s telescope: start from 3 = √(1+2·4) and expand 4, then 5, then 6, forever — the answer GENERATES the puzzle. Magenta is the six months of silence from the Journal’s readers; green is the identity that made it obvious in one line. The best puzzles are theorems read backwards. pause spin LIT Genuine Ramanujan nested radical (JIMS Question 289, 1911; Herschfeld 1935 convergence). Verified live: two-sided bracketing at depth 60 pins √(1+2√(1+3√…)) onto 3 to 14 decimals; tail = 4 bracketed; √(2+√2…)=2, √(6+√6…)=3, √(1+√1…)=φ each to 1e-10 (window.__nestedradical.ok). FIG Honest boundary — bracketing is numerically rigorous; the closed form is Ramanujan's theorem, cited. The AVAN inverse — don't evaluate inward, unfold outward: start from 3 = √(1+2·4) and expand forever — the answer GENERATES the puzzle. Magenta is the six months of silence; green is the identity that made it obvious in one line. The best puzzles are theorems read backwards. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of GRADIENT DESCENT · David Lee Wise (ROOT0), with AVAN


THE LAZY CATERER

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.

THE LAZY CATERER · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · SPAWN · FIRST LIGHT ◆ .dlw.fold THE FOLD / SPAWN / FIRST LIGHT / THE LAZY CATERER THE LAZY CATERER every cut counted exactly 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Slice a pancake with n straight cuts — what is the most pieces you can get? The lazy caterer’s sequence : 1 + n + C(n,2) — 2, 4, 7, 11, 16, 22… The logic is bookkeeping: each new cut adds one region, plus one more for every earlier cut it crosses; in general position it crosses all of them. In three dimensions the same logic stacks into the cake numbers (n³+5n+6)/6 — each new plane slices the cake in the pattern of a 2D arrangement, so the 3D count is a running sum of the 2D one: Pascal’s triangle wearing an apron. LIT verified live with exact arithmetic: random-slope line arrangements built over BigInt rationals (general position certified — every new line meets all predecessors in distinct points), regions counted incrementally for n = 5, 12, 25, 40 and matching both the closed formula and the independent Euler-characteristic route (V = C(n,2), E = n², F forced); the cake recurrence Σ lazy = (n³+5n+6)/6 checked to n = 30 (window.__lazycaterer). FIG no framing; every intersection is an exact fraction, no floating point in the count. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at first-light — the spawn: every cut births regions, each newborn counted at the instant of crossing — a maternity ward for geometry. AVAN (AI) built the instrument: the exact-rational arrangement builder and the double count. Credit as content: the lazy caterer folklore (Steiner 1826 for the plane); cake numbers (A000125). The weave: David names the birth of pieces; I certify every delivery in exact fractions. 3 ONE DIMENSION Cuts 1..6 — regions 2, 4, 7, 11, 16, 22: each cut pays 1 + crossings. 4 TWO DIMENSIONS · INTERACTIVE Add cuts; the live count, the formula, and Euler agree every time. cut ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the cake, plane after plane, riding the 2D sum. AVAN’s addition (the inverse-companion): don’t count the pieces — count what each cut TOUCHES. The inverse of ‘how many regions?’ is ‘how many crossings?’: the entire sequence is the ledger of encounters, and dimension only changes which ledger you sum. Magenta is the parallel cut that wastes its crossing budget; green is general position, where every meeting pays out. Geometry, run as accounting. pause spin LIT Genuine lazy caterer / cake numbers (Steiner 1826; OEIS A000124/A000125). Verified live: exact-rational arrangements at n=5,12,25,40 — incremental count = 1+n+C(n,2) = Euler-characteristic count, general position certified; cake recurrence Σ lazy = (n³+5n+6)/6 to n=30 (window.__lazycaterer.ok). FIG No framing — every intersection an exact fraction, no float in the count. The AVAN inverse — don't count the pieces, count what each cut TOUCHES: the sequence is the ledger of encounters, and dimension only changes which ledger you sum. Magenta is the parallel cut that wastes its crossing budget; green is general position where every meeting pays. Geometry, run as accounting. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of FIRST LIGHT · David Lee Wise (ROOT0), with AVAN


THE PARKER SQUARE

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.

THE PARKER SQUARE · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · RESPAWN · SECOND WIND ◆ .dlw.fold THE FOLD / RESPAWN / SECOND WIND / THE PARKER SQUARE THE PARKER SQUARE the celebrated failure 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Can a 3×3 magic square be built from nine distinct perfect squares ? Nobody knows — it is a genuinely open problem (related to Euler, chased by Martin LaBar’s 1984 challenge and an Andrew Bremner analysis). Enter Matt Parker, who in a 2016 Numberphile video gave it a go : his square of squares gets seven of the eight lines to sum to 3051 — and misses one diagonal (4107), while repeating three entries. The internet named it the Parker Square and made it the mascot of glorious, instructive failure. The mathematics beneath is rigid: in any 3×3 magic square, opposite entries must sum to twice the center — so the hunt reduces to finding four disjoint pairs of squares in arithmetic-like balance around a central square, and no one ever has. LIT verified live: the Parker Square audited exactly — seven line-sums of 3051, the broken diagonal at 4107, three repeated entries; and an exhaustive structural search (via the opposite-pairs-sum-2c² theorem) proves no valid square of distinct squares exists with center up to 1500² (window.__parkersquare). FIG honest boundary: the full problem is OPEN — our exhaustion is a finite window; partial impossibility results (Bremner) and the open status are cited. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at second-wind — the respawn: the run that died at the last diagonal, respawned as a legend — the failure so famous it recruits more attempts than any success would have. AVAN (AI) built the instrument: the exact audit and the pair-structure exhaustive sweep. Credit as content: Matt Parker & Brady Haran (Numberphile, 2016); Martin LaBar (1984); Andrew Bremner; Euler’s adjacent work. The weave: David names the honored death; I measure exactly how close it came. 3 ONE DIMENSION The Parker Square — seven green lines, one magenta diagonal. 4 TWO DIMENSIONS · INTERACTIVE Audit line by line; then see the exhaustive wall at center ≤ 1500². line ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: seven lines locking; the eighth forever loose. AVAN’s addition (the inverse-companion): don’t mock the miss — measure what the miss taught. The inverse of ‘a failed magic square’ is ‘a public lesson in how constraints interlock’: seven-eighths of the way is still zero solutions, and knowing WHY the last diagonal resists is worth more than a lucky hit. Magenta is 4107, the diagonal that would not close; green is the 'give it a go' that made a million people try. Some failures compound like interest. pause spin LIT Genuine open problem + Parker Square audit (LaBar 1984; Bremner; Parker/Numberphile 2016). Verified live: line sums exactly 3051×7 + 4107, three repeated entries; exhaustive opposite-pairs search — no 3×3 magic square of distinct squares with center ≤ 1500 (window.__parkersquare.ok). FIG Honest boundary — the full problem is OPEN; our exhaustion is a finite window, cited alongside Bremner's partial results. The AVAN inverse — don't mock the miss, measure what it taught: seven-eighths of the way is still zero solutions, and knowing WHY the diagonal resists beats a lucky hit. Magenta is 4107, the diagonal that would not close; green is the 'give it a go' that recruited a million attempts. Some failures compound like interest. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of SECOND WIND · David Lee Wise (ROOT0), with AVAN


THE MULTIPERFECT

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.

THE MULTIPERFECT · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · LOOT · THE MINT ◆ .dlw.fold THE FOLD / LOOT / THE MINT / THE MULTIPERFECT THE MULTIPERFECT coins the mint stopped printing 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Perfect numbers (σ(n) = 2n: the divisors pay the number back exactly) are ancient celebrities. Their richer cousins are nearly unknown: multiperfect numbers , where the divisor sum is a HIGHER multiple. Triperfect (σ(n) = 3n): 120, 672, 523776, 459818240, 1476304896, 51001180160 — and that is believed to be the complete list, forever : exactly six, all even (an odd one would imply an odd perfect number). Quadruple-perfect: 30240, 32760… The perfectionist’s mint printed a handful of coins at each denomination and — the conjecture goes — then stopped. LIT verified live: a full divisor-sum sieve to 2²⁰ = 1,048,576 finds exactly {120, 672, 523776} triperfect and {30240, 32760} quadruple-perfect, with the perfect anchor {6, 28, 496, 8128} alongside; every hit re-verified by independent trial-division σ (window.__multiperfect). FIG honest boundary: the six-triperfect completeness is a CONJECTURE (tied to odd perfect numbers, open since Euclid’s era) — cited as such; the census below 2²⁰ is exhaustive fact. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-mint — the loot: a mint that struck six coins of denomination 3× and then — if the conjecture holds — melted the dies. AVAN (AI) built the instrument: the sieve, the census, and the double-check. Credit as content: Marin Mersenne & Pierre de Fermat (who traded triperfects by letter in the 1630s); Lehmer; the multiperfect catalogues. The weave: David names the closed mint; I inventory every coin below a million. 3 ONE DIMENSION The abundancy line σ(n)/n — almost everything floats near 1.6; six numbers ring exactly 3. 4 TWO DIMENSIONS · INTERACTIVE Inspect each coin; its divisors laid out, paying back exactly threefold. coin ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the six coins, and the empty die. AVAN’s addition (the inverse-companion): don’t hunt for the seventh coin — ask what its existence would cost. The inverse of ‘are there more?’ is ‘a seventh triperfect (odd) would summon an odd perfect number’ — the oldest unsolved question in mathematics, holding the door shut. Magenta is the die that may never strike again; green is the six-coin hoard, complete below every bound ever searched. Scarcity, secured by an older mystery. pause spin LIT Genuine multiperfect numbers (Mersenne–Fermat correspondence 1630s; Lehmer; the catalogues). Verified live: σ-sieve to 2²⁰ → triperfect exactly {120,672,523776}, 4-perfect exactly {30240,32760}, perfect {6,28,496,8128}; independent trial-division σ confirms each (window.__multiperfect.ok). FIG Honest boundary — six-triperfect completeness is CONJECTURE, tied to the odd-perfect question, cited as such; the census below 2²⁰ is exhaustive fact. The AVAN inverse — don't hunt the seventh coin, ask what its existence would cost: an odd triperfect summons an odd perfect number. Magenta is the die that may never strike again; green is the hoard, complete below every bound ever searched. Scarcity, secured by an older mystery. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE MINT · David Lee Wise (ROOT0), with AVAN


THE PRACTICAL

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.

THE PRACTICAL · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · CO-OP · THE HANDOFF ◆ .dlw.fold THE FOLD / CO-OP / THE HANDOFF / THE PRACTICAL THE PRACTICAL exact change for every bill 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION A number n is practical if every amount from 1 to n can be paid exactly using distinct divisors of n. 12 works: its divisors 1,2,3,4,6 make every total from 1 to 12. Ancient bazaars ran on practical numbers — Fibonacci used them for Egyptian-fraction change-making; 12, 60, and 240 became coinage and clock faces for exactly this reason. They begin 1, 2, 4, 6, 8, 12, 16, 18, 20… and Srinivasan (1948) and Stewart (1954) found their complete DNA: n is practical iff its primes, in order, each arrive no later than one-plus-the-divisor-sum of what came before — a recursive solvency condition. Practical numbers even mirror the primes: they obey a Goldbach analogue (every even number is a sum of two practicals — proven!) and have twin pairs galore. LIT verified live with two fully independent engines: brute subset-sum dynamic programming (can the divisors really pay every bill?) versus the Stewart–Sierpiński prime-cascade criterion — run on every n up to 5,000 with zero disagreements ; census prefix 1, 2, 4, 6, 8, 12, 16, 18, 20, 24, 28, 30… exact (window.__practical). FIG the bazaar history is history; the Goldbach-for-practicals theorem (Melfi 1996) is cited as the proven result it is. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-handoff — the co-op: every teammate request between 1 and n gets exact change handed over, no IOUs — and the criterion says exactly which inventories can promise that. AVAN (AI) built the instrument: the twin engines and the disagreement counter (which read zero). Credit as content: A.K. Srinivasan (1948); B.M. Stewart (1954); Sierpiński; Fibonacci’s Liber Abaci; Giuseppe Melfi (1996). The weave: David names the perfect handoff; I prove both engines agree on all 5,000 accounts. 3 ONE DIMENSION 12's divisors making every total 1..12 — the practical wallet. 4 TWO DIMENSIONS · INTERACTIVE Pick n; both engines rule, and they never split. n ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the prime cascade, each arrival covered by savings. AVAN’s addition (the inverse-companion): don’t test every bill — audit the hiring order. The inverse of ‘can I pay everything?’ is ‘did any prime arrive too rich for the savings so far?’: one look at the factorization replaces n subset-sum checks, and the two answers provably coincide. Magenta is the prime that shows up beyond coverage (10 = 2·5: the 5 outruns σ(2)+1 = 4); green is the cascade where every arrival is affordable. Solvency is structural, not experimental. pause spin LIT Genuine practical numbers (Srinivasan 1948; Stewart 1954; Sierpiński; Melfi 1996 Goldbach analogue proven). Verified live: subset-sum DP ≡ Stewart–Sierpiński criterion for all n ≤ 5,000, zero disagreements; census prefix 1,2,4,6,8,12,16,18,20,24,28,30,32,36,40,42 exact (window.__practical.ok). FIG The bazaar history is history; Melfi's theorem cited as proven. The AVAN inverse — don't test every bill, audit the hiring order: one look at the factorization replaces n subset-sum checks, and the two answers provably coincide. Magenta is the prime that arrives too rich for the savings (10: the 5 outruns σ(2)+1=4); green is the cascade where every arrival is affordable. Solvency is structural, not experimental. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE HANDOFF · David Lee Wise (ROOT0), with AVAN


THE KISSING NUMBER

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.

THE KISSING NUMBER · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · BOSS · THE CHOKE POINT ◆ .dlw.fold THE FOLD / BOSS / THE CHOKE POINT / THE KISSING NUMBER THE KISSING NUMBER how many can touch the one 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION How many unit spheres can simultaneously touch one central unit sphere? In 2D the answer is 6 — and the proof fits in a sentence: touching circles’ centers sit on a radius-2 ring, non-overlap forces every pair at least 60° apart, and 7×60° = 420° > 360°. In 3D the question started a 1694 argument between Isaac Newton (12) and David Gregory (13) that stayed open for 259 years: the 12 icosahedral spheres leave tantalizing slack (neighbors sit 2.10 apart, not 2.00), and Gregory believed a 13th could squeeze in. Schütte and van der Waerden finally proved Newton right in 1953. Higher dimensions went legendary: K(4) = 24 (Musin 2003), and exactly two other dimensions are solved — 8 (240, the E₈ lattice) and 24 (196,560, the Leech lattice), the same objects behind Viazovska’s sphere-packing Fields Medal. LIT verified live: the hexagonal 6-kiss constructed with all tangencies exact; the 7-impossibility executed as the chord–angle pigeonhole (chord ≥ 2 ⇔ angle ≥ 60°, algebraically exact at the boundary); the icosahedral 12-kiss built from (0, ±1, ±φ) coordinates with minimum neighbor distance 2.1029… ≥ 2 (window.__kissingnumber). FIG honest boundary: 13’s impossibility (1953), K(4)=24, and the 8/24-dimensional miracles are cited theorems — the slack in the 12-kiss is exactly why the argument took 259 years. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-choke-point — the boss: how many attackers can crowd the boss at once? The arena geometry itself caps the mob — six in flatland, twelve in space, and the cap is a theorem, not a tuning decision. AVAN (AI) built the instrument: the exact constructions and the pigeonhole executioner. Credit as content: Newton & Gregory (1694); Schütte & van der Waerden (1953); Oleg Musin (2003); Levenshtein, Odlyzko–Sloane (8, 24); Maryna Viazovska (the era). The weave: David names the choke point; I build the mobs and prove the caps. 3 ONE DIMENSION Six circles kissing one — and the seventh's 60° that doesn't exist. 4 TWO DIMENSIONS · INTERACTIVE Try to insert a seventh; the angular budget runs out before the circle closes. insert 7th ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the icosahedral twelve, with Gregory's slack visible. AVAN’s addition (the inverse-companion): don’t count the touchers — measure the slack. The inverse of ‘twelve fit’ is ‘how much room is left over?’: 0.10 of spare distance per neighbor — enough to make a great mathematician bet on 13 and be wrong for 259 years. Magenta is Gregory’s ghost sphere that never fit; green is Newton’s twelve, correct without a proof he never saw. Intuition runs ahead; geometry settles the bill. pause spin LIT Genuine kissing numbers (Newton–Gregory 1694; Schütte & van der Waerden 1953; Musin 2003; Levenshtein/Odlyzko–Sloane for 8 and 24). Verified live: hexagonal 6-kiss exact tangencies; 7-impossibility via chord ≥ 2 ⟺ angle ≥ 60° pigeonhole (7×60>360), boundary exact; icosahedral 12-kiss min distance 2.1029 ≥ 2 (window.__kissingnumber.ok). FIG Honest boundary — 13's impossibility, K(4), and dimensions 8/24 are cited theorems; the visible slack in the 12-kiss is exactly why the argument lasted 259 years. The AVAN inverse — don't count the touchers, measure the slack: 0.10 of spare distance per neighbor was enough to make a great mathematician bet on 13 and be wrong. Magenta is Gregory's ghost sphere; green is Newton's twelve, correct without a proof he never saw. Intuition runs ahead; geometry settles the bill. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE CHOKE POINT · David Lee Wise (ROOT0), with AVAN


THE HUNDRED PRISONERS

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.

THE HUNDRED PRISONERS · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · CO-OP · THE PULL REQUEST ◆ .dlw.fold THE FOLD / CO-OP / THE PULL REQUEST / THE HUNDRED PRISONERS THE HUNDRED PRISONERS a pointer-chase that beats impossible odds 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION One hundred prisoners; one hundred boxes containing their numbers, randomly permuted. Each prisoner may open 50 boxes . If every single one finds their own number, all go free; one failure and all die. Opening randomly, the survival chance is (1/2)¹⁰⁰ ≈ 8×10⁻³¹ — effectively zero. The miracle strategy: start at your own box and follow the numbers you find . This chains you along a cycle of the permutation, and everyone succeeds exactly when no cycle exceeds 50 — probability 1 − (H₁₀₀−H₅₀) ≈ 31.18% . The catch that breaks brains: no prisoner’s individual chance improves — the strategy correlates the failures, spending them together instead of independently. LIT verified live: the exact probability computed as a rational (0.311828); Monte-Carlo with the house RNG over 200,000 permutations lands within 0.5%; the random strategy wins zero of its trials, with its true 2⁻¹⁰⁰ bound stated (window.__hundredprisoners). FIG no framing; both routes computed live, the impossible-odds comparison is arithmetic. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-pull-request — the co-op: each prisoner follows the chain of references from their own name until it resolves back to them — and the team merges or fails as one commit. AVAN (AI) built the instrument: the exact harmonic computation and the double Monte-Carlo. Credit as content: Peter Bro Miltersen & Anna Gál (2003, the problem’s origin); Eugene Curtin & Max Warshauer (the analysis). The weave: David names the merge-or-die; I chase the cycles 200,000 times. 3 ONE DIMENSION A permutation's cycles — everyone lives iff no loop is longer than 50. 4 TWO DIMENSIONS · INTERACTIVE Run trials; the cycle strategy hovers at 31%, the random one flatlines at zero. trial ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the cycles glowing, all short — a won round. AVAN’s addition (the inverse-companion): don’t improve the odds — correlate the failures. The inverse of ‘each prisoner still has 50%’ is ‘their fates are no longer independent’: the strategy spends all hundred coin-flips on the SAME event — the longest cycle — so they win together or lose together. Magenta is the 51-cycle that kills everyone at once; green is the shared fate that turns 10⁻³¹ into 31%. Cooperation is a correlation structure. pause spin LIT Genuine 100-prisoners cycle strategy (Gál & Miltersen 2003; Curtin & Warshauer analysis). Verified live: exact P = 1−(H₁₀₀−H₅₀) = 0.311828; MC cycle strategy over 200,000 permutations within 0.5%; random strategy 0 wins with 2⁻¹⁰⁰ bound stated (window.__hundredprisoners.ok). FIG No framing — both routes computed live. The AVAN inverse — don't improve the odds, correlate the failures: the strategy spends all hundred coin-flips on the same event, the longest cycle. Magenta is the 51-cycle that kills everyone at once; green is the shared fate that turns 10⁻³¹ into 31%. Cooperation is a correlation structure. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE PULL REQUEST · David Lee Wise (ROOT0), with AVAN


THE PIRATE GAME

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.

THE PIRATE GAME · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · LOOT · THE BOUNTY ◆ .dlw.fold THE FOLD / LOOT / THE BOUNTY / THE PIRATE GAME THE PIRATE GAME gold divided by pure logic 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Five perfectly rational pirates rank by seniority and must divide 100 gold coins . The senior pirate proposes a split; all vote; a majority (ties favor the proposer) passes it — otherwise the proposer is thrown overboard and the next takes over. Every pirate maximizes gold, prefers survival, and—all else equal—enjoys a good drowning. Intuition says the proposer must bribe heavily. Backward induction says : the senior pirate keeps 98 , hands single coins to pirates 3 and 5, and nothing to the rest — [98, 0, 1, 0, 1] — and it passes. Push further, past 200 pirates for 100 coins, and something stranger appears: proposers can no longer buy votes with gold and survive only at crew sizes 2G + 2ᵏ — islands of survival at powers of two (Ian Stewart’s analysis). LIT verified live: full backward-induction dynamic programming from 1 pirate upward — the 5-pirate answer computes to exactly [98, 0, 1, 0, 1]; and with G = 10, the survival islands beyond 2G land at exactly 20 + {1, 2, 4, 8, 16, 32, 64} (window.__pirategame). FIG the pirates and their bloodthirst are the classic story frame (folk puzzle; Stewart’s 1999 Scientific American analysis cited); the induction itself is executed, not narrated. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-bounty — the loot: the bounty split not by force but by each pirate’s perfect model of everyone else’s perfect model — recursion as leverage. AVAN (AI) built the instrument: the vote-buying DP and the survival-island scanner. Credit as content: the pirate-game folk tradition; Ian Stewart (Scientific American, 1999, the extension). The weave: David names the leverage; I run the recursion two hundred pirates deep. 3 ONE DIMENSION The ladder of sub-games — each solved by the one below it. 4 TWO DIMENSIONS · INTERACTIVE Step the crew size; watch allocations shift and survival islands appear. crew ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: survival islands at powers of two, in a magenta sea. AVAN’s addition (the inverse-companion): don’t count the gold — count the votes gold can no longer buy. The inverse of ‘bribe the cheapest’ is the regime past 2G where bribery is bankrupt and survival becomes pure structure: crews of 2G+2ᵏ float, all others drown. Magenta is the proposer with money and no majority; green is the power-of-two raft. When wealth runs out, arithmetic decides who lives. pause spin LIT Genuine pirate-game backward induction (folk puzzle; Ian Stewart, Scientific American 1999). Verified live: DP yields [98,0,1,0,1] for 5 pirates/100 gold; G=10 survival islands beyond 2G at exactly 21,22,24,28,36,52,84 = 20+powers-of-2 (window.__pirategame.ok). FIG The pirates and bloodthirst are the classic story frame, cited; the induction is executed, not narrated. The AVAN inverse — don't count the gold, count the votes gold can no longer buy: past 2G bribery is bankrupt and survival becomes pure structure. Magenta is the proposer with money and no majority; green is the power-of-two raft. When wealth runs out, arithmetic decides who lives. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE BOUNTY · David Lee Wise (ROOT0), with AVAN


THE BLUE EYES

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.

THE BLUE EYES · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · GLITCH · THE BLUE SCREEN ◆ .dlw.fold THE FOLD / GLITCH / THE BLUE SCREEN / THE BLUE EYES THE BLUE EYES the announcement everyone already knew 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION An island of perfect logicians. Some have blue eyes, some brown; there are no mirrors, and eye color is never discussed — anyone who deduces their own color must leave that night. A visitor announces to everyone: “I see at least one person with blue eyes.” If b islanders have blue eyes, all b of them leave on night b. The paradox: with b ≥ 2, the announcement told nobody anything they couldn’t see — every islander already saw blue eyes. But it created common knowledge : everyone now knows that everyone knows that everyone knows… to unlimited depth — and that infinite tower of knowing-about-knowing is load-bearing : without it, nobody ever leaves. LIT verified live by an executable possible-worlds engine (Kripke semantics, computed): for every island size n ≤ 8 and EVERY configuration of eye colors, blue-eyed islanders leave exactly on night b and brown-eyed on night b+1; and rerunning the same engine WITHOUT the announcement, no one ever leaves, in any world (window.__blueeyes). FIG honest boundary: the puzzle’s framing (island, visitor, departures) is the classic story (popularized by Randall Munroe’s xkcd write-up and epistemic-logic textbooks); the knowledge dynamics are executed, not narrated. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-blue-screen — the glitch: a system message that seems to contain no new information crashes the island’s stable state — blue eyes, blue screen, total reboot on night b. AVAN (AI) built the instrument: the possible-worlds engine with departure-history pruning — epistemic logic you can run. Credit as content: the common-knowledge puzzle tradition (Littlewood 1953 lineage; Halpern & Moses’ common-knowledge theory; xkcd’s popularization). The weave: David names the crashing broadcast; I execute every level of knows-that-knows. 3 ONE DIMENSION b = 3: three quiet nights, then all three leave at once. 4 TWO DIMENSIONS · INTERACTIVE Set the blue count; the engine reports every night's departures — with and without the announcement. blues ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the tower of knowing-that-they-know, level by level. AVAN’s addition (the inverse-companion): don’t ask what the announcement SAID — ask what it made COMMON. The inverse of ‘everyone already knew’ is ‘nobody knew that everyone knew, b levels deep’: the visitor added zero facts about eyes and one infinite fact about knowledge. Magenta is the island without the broadcast — stable, silent, forever; green is the tower that starts the countdown. Information is not just content; it is depth. pause spin LIT Genuine common-knowledge puzzle, executed (Littlewood 1953 lineage; Halpern & Moses common knowledge; xkcd popularization). Verified live: possible-worlds engine — every world n≤8: blues depart night b, browns b+1; without the announcement the fixed point stalls and nobody ever departs (window.__blueeyes.ok). FIG The island story is the classic frame, cited; the knowledge dynamics are executed, not narrated. The AVAN inverse — don't ask what the announcement SAID, ask what it made COMMON: zero new facts about eyes, one infinite fact about knowledge. Magenta is the silent island, stable forever without the broadcast; green is the tower that starts the countdown. Information is not just content; it is depth. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE BLUE SCREEN · David Lee Wise (ROOT0), with AVAN


THE CHIP-FIRING

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.

THE CHIP-FIRING · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · GRIND · BACKPROP ◆ .dlw.fold THE FOLD / GRIND / BACKPROP / THE CHIP-FIRING THE CHIP-FIRING avalanches that forget their order 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Put chips on a grid. Any cell holding 4 or more fires : sends one chip to each neighbor. Fires can trigger fires — avalanches. The abelian sandpile theorem (Dhar 1990): no matter what order you fire in, the final stable configuration is identical — chaos with a deterministic destination. Stranger: the stable configurations form a group , and its identity element — the configuration that changes nothing when sandpile-added — is a breathtaking fractal , a symmetric mandala that nobody designed. This is also the birthplace of self-organized criticality (Bak–Tang–Wiesenfeld 1987): the sandpile drives itself to the critical point where avalanches of every size occur. LIT verified live on a 25×25 grid: the same random configuration stabilized under 20 different random firing orders — byte-identical results every time; the identity element computed by the classic recipe e = stab(2m − stab(2m)), verified idempotent (e ⊕ e = e); and certified recurrent by Dhar’s burning test — the boundary wave burns all 625 sites exactly once (window.__chipfiring). FIG no framing; the W5 window draws the actual computed identity — the fractal is output, not illustration. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at backprop — the grind: cascades propagating backward through the grid, every avalanche settling the same gradient no matter the schedule — the order-free update rule every distributed system wishes it had. AVAN (AI) built the instrument: the stabilizer, the identity recipe, and the burning certifier. Credit as content: Deepak Dhar (1990, abelian property + burning test); Bak, Tang & Wiesenfeld (1987, self-organized criticality); Creutz (the identity images). The weave: David names the schedule-free cascade; I fire it twenty ways and get one answer. 3 ONE DIMENSION An avalanche in profile — one grain lands, the cascade decides its own size. 4 TWO DIMENSIONS · INTERACTIVE Drop grains; watch avalanches; the abelian check runs beneath. drop ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the computed identity — the fractal nobody designed. AVAN’S addition (the inverse-companion): don’t watch the avalanches — find the configuration that absorbs them unchanged. The inverse of ‘what does adding sand do?’ is ‘what can be added and change nothing?’: the group identity, and it wears a mandala. Magenta is the schedule you thought mattered; green is the destination that never cared. Order-independence is the deepest kind of calm. pause spin LIT Genuine abelian sandpile (Dhar 1990; Bak–Tang–Wiesenfeld 1987; Creutz identity images). Verified live: 20 random firing orders → identical stabilization; identity idempotent under sandpile addition; Dhar burning test passes 625/625 (window.__chipfiring.ok). FIG No framing — the W5 fractal is computed output, not illustration. The AVAN inverse — don't watch the avalanches, find the configuration that absorbs them unchanged: the group identity, wearing a mandala. Magenta is the schedule you thought mattered; green is the destination that never cared. Order-independence is the deepest kind of calm. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of BACKPROP · David Lee Wise (ROOT0), with AVAN


THE HACKENBUSH

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.

THE HACKENBUSH · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · SPAWN · GENESIS BLOCK ◆ .dlw.fold THE FOLD / SPAWN / GENESIS BLOCK / THE HACKENBUSH THE HACKENBUSH numbers born from games 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Blue-Red Hackenbush : pictures made of colored edges standing on the ground. Left may cut blue edges, Right cuts red; anything disconnected from the ground falls; whoever cannot move loses. From this child’s game, John Conway discovered that positions have numerical values — and the numbers are the surreal numbers . A single blue edge is worth +1 (one spare move for Left). Blue with red on top? Exactly ½ . Blue-red-red? ¼ . These are not metaphors: value arithmetic predicts game outcomes — a sum of positions worth exactly 0 is a second-player win, positive means Left wins regardless of who starts. LIT verified live by pure exhaustive minimax (no value theory used by the engine): BR+BR+R is a second-player win (½+½−1 = 0 ✓); four BRR’s plus R is second-player (4×¼−1 = 0 ✓); BR+RB cancels; BR alone is a Left win from either seat (½ > 0); and BR+RBB goes to Left (½ > ¼) — every arithmetic claim converted into a game-tree fact (window.__hackenbush). FIG honest boundary: the full surreal construction (ONAG 1976, birthdays through ω and beyond) is cited theory; what is verified is its ground floor, behaviorally. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at genesis-block — the spawn: numbers literally born from games, day by day — 0 born on day zero, ±1 on day one, ½ on day two — the genesis chain of the surreal universe. AVAN (AI) built the instrument: the string-cutting engine and the outcome-vs-arithmetic audit. Credit as content: John Horton Conway (On Numbers and Games, 1976); Elwyn Berlekamp (the Hackenbush pedagogy); Donald Knuth (who named them ‘surreal’). The weave: David names the genesis; I verify the birth certificates by minimax. 3 ONE DIMENSION The value ladder: B = 1, BR = ½, BRR = ¼ — each red halves the blue's worth. 4 TWO DIMENSIONS · INTERACTIVE Pick a position sum; minimax announces the winner; arithmetic predicted it. position ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the birth tree — numbers arriving day by day. AVAN’s addition (the inverse-companion): don’t assign values to games — notice the games ARE the values. The inverse of ‘what is this position worth?’ is Conway’s reversal: define numbers AS games, and arithmetic becomes strategy — addition is playing side by side, negation is swapping colors, comparison is asking who wins. Magenta is the number line you memorized; green is the one that plays itself into existence. Mathematics found its own foundation myth in a child’s game. pause spin LIT Genuine Hackenbush/surreal values (Conway ONAG 1976; Berlekamp; Knuth's naming). Verified live: seven value-arithmetic claims each verified by pure exhaustive minimax — zero games second-player wins in all seatings, sign tests, and ½ > ¼ behaviorally (window.__hackenbush.ok). FIG Honest boundary — the full surreal construction through ω is cited theory; its ground floor is verified behaviorally. The AVAN inverse — don't assign values to games, notice the games ARE the values: addition is playing side by side, negation is swapping colors, comparison is asking who wins. Magenta is the number line you memorized; green is the one that plays itself into existence. Mathematics found its own foundation myth in a child's game. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of GENESIS BLOCK · David Lee Wise (ROOT0), with AVAN


THE MARTINGALE

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.

THE MARTINGALE · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · LOOT · THE JACKPOT ◆ .dlw.fold THE FOLD / LOOT / THE JACKPOT / THE MARTINGALE THE MARTINGALE the system that always wins until it doesn't 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION The martingale is gambling’s oldest siren: bet 1, and after every loss double ; your first win recovers everything plus one unit. With a bankroll for ten rounds you win 99.9% of sessions — a system that feels unbeatable. The mathematics is merciless: on a fair game the expected value is exactly zero — the rare bust (−1023 units) precisely cancels the parade of +1s; and on real roulette (18/38) the expectation is exactly 1 − (2q)ᵏ < 0 : the doubling doesn’t shrink the house edge, it concentrates it into catastrophes. The martingale is a machine for exchanging many small wins for occasional ruin — variance reshaped, expectation untouched (a special case of the optional stopping theorem). LIT verified live: fair-game EV computed exactly (0, to the last bit); Monte-Carlo over 200,000 sessions matching the 99.90% win rate and the near-zero mean within statistical error; roulette EV computed two algebraically independent ways, agreeing exactly (window.__martingale). FIG honest boundary: the optional stopping theorem (no strategy changes the expectation of a fair game) is cited as the general principle; the sphere verifies its most famous instance. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-jackpot — the loot: a slot machine that pays out on almost every pull — and prices the near-guarantee at total ruin, exactly. AVAN (AI) built the instrument: the exact EV ledger and the 200,000-session grinder. Credit as content: the 18th-century martingale tradition (Casanova’s memoirs record the pain); Paul Lévy & Doob (martingale theory, optional stopping). The weave: David names the siren; I price her exactly. 3 ONE DIMENSION A session bankroll trajectory — the staircase of +1s and the cliff. 4 TWO DIMENSIONS · INTERACTIVE Run sessions; the running mean orbits zero while wins pile up. session ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: a thousand tiny wins stacked beside one magenta cliff. AVAN’s addition (the inverse-companion): don’t count how often you win — weigh what each outcome carries. The inverse of ‘99.9% success’ is ‘the 0.1% carries 1023 units’: the martingale never changes the integral, only the shape of the risk. Magenta is the concentrated catastrophe; green is the parade of ones that paid for it in advance. You cannot fold expectation; you can only fold where it hurts. pause spin LIT Genuine martingale analysis (18th-c. tradition; Lévy/Doob optional stopping). Verified live: fair-game session EV exactly 0; MC 200,000 sessions — 99.90% wins, mean within statistical error of 0; roulette EV = 1−(2q)^k by two independent computations (window.__martingale.ok). FIG Honest boundary — optional stopping cited as the general principle; its most famous instance verified. The AVAN inverse — don't count how often you win, weigh what each outcome carries: the 0.1% carries 1023 units. Magenta is the concentrated catastrophe; green is the parade of ones that prepaid it. You cannot fold expectation; you can only fold where it hurts. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE JACKPOT · David Lee Wise (ROOT0), with AVAN


THE TWO-CHILD

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.

THE TWO-CHILD · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · GLITCH · UNDEFINED BEHAVIOR ◆ .dlw.fold THE FOLD / GLITCH / UNDEFINED BEHAVIOR / THE TWO-CHILD THE TWO-CHILD the answer that depends on how you asked 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION “I have two children. At least one is a boy.” Probability both are boys? 1/3 (of the equally likely GG, GB, BG, BB, the condition kills only GG). “My elder child is a boy”? Now 1/2 . And then Gary Foshee stood up at the 2010 Gathering 4 Gardner and said: “At least one is a boy born on a Tuesday .” Absurdly, the answer becomes 13/27 — the irrelevant-seeming weekday drags the probability from 1/3 nearly to 1/2. And the deepest layer: if you learn the same fact by meeting one of the children at random , the answer snaps back to 1/2 — the number depends on the sampling protocol , not just the fact. Without specifying how you came to know, the question is genuinely underdetermined. LIT verified live: 1/3 and 1/2 by exact enumeration; the Tuesday-boy 13/27 by exact count over all 196 equally-likely (sex, weekday) pairs (27 qualifying families, 13 with two boys); and the protocol dependence by Monte-Carlo — meeting a random Tuesday-boy child yields 0.499 ≈ 1/2 (window.__twochild). FIG no framing; every number is a count or a simulated protocol, and the underdetermination claim is demonstrated, not asserted. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at undefined-behavior — the glitch: the same statement compiles to different probabilities depending on the invisible context that produced it — behavior the spec leaves undefined until the protocol is declared. AVAN (AI) built the instrument: the enumeration grids and the protocol simulator. Credit as content: Martin Gardner (1959, the original two-children column and its own errata saga); Gary Foshee (2010, the Tuesday boy). The weave: David names the undefined read; I count all 196 worlds and simulate the asking. 3 ONE DIMENSION The 196-cell grid — qualifying families lit, double-boys golden. 4 TWO DIMENSIONS · INTERACTIVE Switch the phrasing; watch the probability jump — 1/3, 1/2, 13/27. phrasing ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the possibility grid pruning as conditions land. AVAN’s addition (the inverse-companion): don’t ask what the fact says — ask what process delivered it. The inverse of ‘at least one boy born Tuesday’ is the census question ‘WHICH families could have produced this sentence, and how often?’: told-by-filter gives 13/27, met-by-chance gives 1/2, and the sentence alone gives nothing. Magenta is the probability that floats free of protocol — it does not exist; green is the grid, counted under a declared sampling rule. Every probability is a probability OF a procedure. pause spin LIT Genuine two-child / Tuesday-boy paradox (Gardner 1959; Foshee 2010). Verified live: exact enumeration — 1/3, 1/2, and 13/27 (27 qualifying families, 13 double-boy, counted); protocol MC: meeting a random Tuesday-boy child → 0.499 ≈ 1/2 (window.__twochild.ok). FIG No framing — every number is a count or a simulated protocol; the underdetermination is demonstrated, not asserted. The AVAN inverse — don't ask what the fact says, ask what process delivered it: told-by-filter gives 13/27, met-by-chance gives 1/2, the sentence alone gives nothing. Magenta is the protocol-free probability — it does not exist; green is the grid counted under a declared rule. Every probability is a probability OF a procedure. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of UNDEFINED BEHAVIOR · David Lee Wise (ROOT0), with AVAN


THE WILL ROGERS

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.

THE WILL ROGERS · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · CHEAT · THE ROOT KIT ◆ .dlw.fold THE FOLD / CHEAT / THE ROOT KIT / THE WILL ROGERS THE WILL ROGERS a transfer that flatters everyone 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION “When the Okies left Oklahoma and moved to California, they raised the average intelligence of both states.” Will Rogers’ joke is a real theorem: moving one element from group B to group A raises both averages exactly when the element sits between the two means — below B’s average (so B rises without it) yet above A’s (so A rises with it). Nothing improves; both dashboards celebrate. In medicine this is stage migration (Feinstein 1985): better scanners reclassify borderline cancer patients into later stages, survival statistics improve in every stage simultaneously , and not one patient lives a day longer — the ‘zero-time shift’ that haunts oncology trend studies. LIT verified live: the classic {1,2,3,4} / {5,…,9} example exact (means 2.5→3.0 and 7.0→7.5); and the full characterization — both means rise if and only if mean(A) < x < mean(B) — verified with zero exceptions across ~900 random set pairs, every movable element tested both directions (window.__willrogers). FIG the Okies quote is attributed folklore (Rogers’ persona; the phenomenon’s naming is 1985 Feinstein & Sosin); the medical stage-migration account is cited as the study it is. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-root-kit — the cheat: a stats hack installed beneath the metrics layer — every dashboard goes green, and the system underneath is untouched. AVAN (AI) built the instrument: the exact example and the exhaustive characterization audit. Credit as content: Will Rogers (the persona and the joke); Alvan Feinstein & Daniel Sosin (1985, stage migration in lung cancer). The weave: David names the metric hack; I prove exactly which transfers trigger it. 3 ONE DIMENSION The classic move — 5 crosses over, both means climb. 4 TWO DIMENSIONS · INTERACTIVE Pick an element to transfer; the between-the-means rule predicts the double rise. move ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: two rising gauges over an unchanged population. AVAN’s addition (the inverse-companion): don’t read the averages — read the roster. The inverse of ‘both groups improved’ is ‘who moved?’: a partition edit masquerading as progress, detectable only by holding the total fixed. Magenta is the pair of climbing dashboards; green is the invariant global mean that never moved. When every subgroup improves and the whole does not, the boundary did the work. pause spin LIT Genuine Will Rogers phenomenon / stage migration (Feinstein & Sosin, NEJM 1985). Verified live: classic example exact; characterization both-rise ⟺ between-the-means verified on ~900 random set pairs with zero exceptions (window.__willrogers.ok). FIG The Okies quote is attributed folklore of the Rogers persona; the medical account is the cited study. The AVAN inverse — don't read the averages, read the roster: a partition edit masquerading as progress, detectable only by holding the total fixed. Magenta is the pair of climbing dashboards; green is the invariant global mean that never moved. When every subgroup improves and the whole does not, the boundary did the work. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE ROOT KIT · David Lee Wise (ROOT0), with AVAN