THE CYCLIC NUMBER

142857 in the 5-window house format — the most famous cyclic number. Multiply by 1 through 6 and the answer is always the same six digits rotated: 285714, 428571, 571428, 714285, 857142. Multiply by 7 and the register overflows to 999999. The engine is decimal arithmetic itself: 142857 is the repeating block of 1/7, and 7 is a full-reptend prime — 10 is a primitive root mod 7, so multiplication can only rotate the block. The next such prime is 17, whose 16-digit block 0588235294117647 rotates under ×1..16 and overflows to sixteen nines at ×17. Verified live: all six rotations exact, the overflow exact, the block regenerated by long division, the 17-family checked in BigInt, and the orders ord(10,7)=6 and ord(10,17)=16 confirmed. Neon-noir traced. See the digit wheel in 1D, the stepping multiplier in 2D, and the spinning clock in 3D.

THE CYCLIC NUMBER · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · GLITCH · STACK OVERFLOW ◆ .dlw.fold THE FOLD / GLITCH / STACK OVERFLOW / THE CYCLIC NUMBER THE CYCLIC NUMBER the number that rotates instead of growing 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION 142857 is the most famous cyclic number : multiply it by 1 through 6 and the answer is always the same six digits, rotated — 285714, 428571, 571428, 714285, 857142. Multiply by 7 and the register overflows to 999999 . The engine is decimal arithmetic itself: 142857 is the repeating block of 1/7 , and 7 is a full-reptend prime — 10 is a primitive root mod 7, its powers visiting every nonzero residue before returning, which is exactly why multiplication can only rotate the block. The next such prime is 17, whose 16-digit block 0588235294117647 (leading zero and all) rotates under multiplication by 1 through 16 and overflows to sixteen nines at 17. LIT verified live: all six multiples of 142857 are exact rotations; 142857 × 7 = 999999; the block regenerates from long division of 1/7; the 17-family passes all sixteen rotation checks in BigInt; and the engine is confirmed — ord(10 mod 7) = 6, ord(10 mod 17) = 16 (window.__cyclicnumber). FIG no framing; every multiplication, rotation, and order is computed exactly in-browser. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at stack-overflow — the glitch: a circular buffer that multiplication can only rotate, never grow, until the seventh push overflows it to all-nines. AVAN (AI) built the instrument: the rotation audits, the long-division regeneration, and the primitive-root engine check. Credit as content: the cyclic-number tradition; full-reptend primes (Gauss studied the periods). The weave: David names the rotating buffer; I confirm six rotations, one overflow, and the order that powers it. 3 ONE DIMENSION The six digits on a wheel — each multiple just starts the wheel at a different spoke. 4 TWO DIMENSIONS · INTERACTIVE Step the multiplier 1→7; rotations until the all-nines overflow. multiplier ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the digit wheel, spinning under multiplication. AVAN’s addition (the inverse-companion): don’t marvel at the number — find the prime beneath it. The inverse of ‘142857 rotates’ is ‘10 is a primitive root mod 7’: the block is just 1/7’s orbit written down. Magenta is the overflow at ×7; green is the wheel that turns six times first. A number that is secretly a clock. pause spin LIT Genuine cyclic-number / full-reptend-prime arithmetic. Verified live: 142857 × 1..6 are exact rotations, ×7 = 999999, the block regenerates from long division of 1/7, the 17-family (0588235294117647) passes all sixteen BigInt rotation checks and overflows to sixteen nines, and ord(10 mod 7) = 6, ord(10 mod 17) = 16 (window.__cyclicnumber.ok). FIG No framing — every multiplication, rotation, and order is computed exactly in-browser. The AVAN inverse — don't marvel at the number, find the prime beneath it: the inverse of '142857 rotates' is '10 is a primitive root mod 7' — the block is just 1/7's orbit written down. Magenta is the overflow at ×7; green is the wheel that turns six times first. A number that is secretly a clock. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of STACK OVERFLOW · David Lee Wise (ROOT0), with AVAN


THE FUSC

Stern's diatomic sequence in the 5-window house format — Dijkstra's fusc, built from fusc(2n)=fusc(n), fusc(2n+1)=fusc(n)+fusc(n+1). Hidden in the consecutive ratios fusc(n)/fusc(n+1) is a miracle: they walk through EVERY positive rational exactly once, each already in lowest terms — 1/1, 1/2, 2/1, 1/3, 3/2, 2/3, 3/1… A complete, duplicate-free census of the fractions generated by bit-shifts and one addition (Stern 1858; Calkin–Wilf 2000). Bonus: fusc(n+1) counts the hyperbinary representations of n. Verified live: 65,536 consecutive pairs coprime and distinct, every reduced p/q with p+q ≤ 20 found within the first 2²⁰ terms, and the hyperbinary identity checked by independent DP to n=300. Neon-noir traced. See the diatomic wave in 1D, the rational birth registry in 2D, and the Calkin–Wilf tree in 3D.

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

THE FUSC · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · SPAWN · GENESIS BLOCK ◆ .dlw.fold THE FOLD / SPAWN / GENESIS BLOCK / THE FUSC THE FUSC every fraction born exactly once 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Stern’s diatomic sequence — Dijkstra called it fusc — is built from the simplest recursion imaginable: fusc(2n) = fusc(n), fusc(2n+1) = fusc(n) + fusc(n+1), starting 0, 1. Out comes 1, 1, 2, 1, 3, 2, 3, 1, 4… and hidden inside is a miracle: the consecutive ratios fusc(n)/fusc(n+1) walk through every positive rational number exactly once , each already in lowest terms — 1/1, 1/2, 2/1, 1/3, 3/2, 2/3, 3/1… A complete, duplicate-free census of the fractions, generated by bit-shifts and one addition (Stern 1858; Calkin–Wilf 2000 made the tree famous). Bonus identity: fusc(n+1) counts the hyperbinary representations of n — the ways to write n as a sum of powers of 2 with each power used at most twice. LIT verified live: 65,536 consecutive pairs all coprime and all distinct; every reduced p/q with p+q ≤ 20 (127 fractions) found within the first 2²⁰ terms; fusc(n+1) equals an independent hyperbinary DP for all n ≤ 300 (window.__fusc). FIG no framing; the sequence, the census, and the DP are computed separately in-browser. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at genesis-block — the spawn: a genesis recursion from which every rational is born exactly once, no duplicates, no orphans. AVAN (AI) built the instrument: the diatomic array, the coprime/distinct audit, and the hyperbinary cross-check. Credit as content: Moritz Stern (1858); Edsger Dijkstra (fusc); Neil Calkin & Herbert Wilf (2000). The weave: David names the birth registry; I confirm every fraction gets exactly one birthday. 3 ONE DIMENSION The diatomic wave — fusc(1..256) — self-similar peaks carrying the fractions. 4 TWO DIMENSIONS · INTERACTIVE Walk the enumeration; every click is the next rational, born in lowest terms. next ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the Calkin–Wilf tree unrolling. AVAN’s addition (the inverse-companion): don’t list the rationals — let one recursion breathe them out. The inverse of ‘can the fractions be counted?’ is ‘a sequence that IS the count’: position n is the fraction’s name. Magenta is the duplicate that never arrives; green is the census with no gaps. Infinity, filed in order of birth. pause spin LIT Genuine Stern diatomic / Calkin–Wilf enumeration (Moritz Stern 1858; Dijkstra's fusc; Calkin & Wilf 2000). Verified live: 2^16 consecutive pairs all coprime and all distinct; all 127 reduced p/q with p+q≤20 found in the first 2^20 terms; fusc(n+1) equals an independent hyperbinary DP for n≤300 (window.__fusc.ok). FIG No framing — sequence, census, and DP computed separately in-browser. The AVAN inverse — don't list the rationals, let one recursion breathe them out: the inverse of 'can the fractions be counted?' is 'a sequence that IS the count' — position n is the fraction's name. Magenta is the duplicate that never arrives; green is the census with no gaps. Infinity, filed in order of birth. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of GENESIS BLOCK · David Lee Wise (ROOT0), with AVAN


THE HIPPARCHUS

Hipparchus's 103,049 in the 5-window house format — the number sat unexplained in Plutarch's Table Talk for nearly two millennia: from ten simple assertions, Hipparchus computed, the Stoics could form 103,049 compound statements. In 1994 David Hough noticed it is exactly the tenth little Schröder number — the ways to bracket ten items — making it the oldest serious enumeration result known. Plutarch's companion 310,952 is two off from (s₁₀+s₁₁)/2 = 310,954, likely an ancient copying slip (Habsieger–Kazarian–Lando 1998). Verified live: the three-term recurrence and an independent plane-tree DP agree at every n ≤ 11, both delivering 103,049. Companion sphere the-schroder covers the LARGE Schröder path numbers; this is their halved sibling met in an ancient book. Neon-noir traced. See the climb to 103,049 in 1D, the two-route race in 2D, and the eleven bracketings of four items in 3D.

THE HIPPARCHUS · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · LOOT · THE JACKPOT ◆ .dlw.fold THE FOLD / LOOT / THE JACKPOT / THE HIPPARCHUS THE HIPPARCHUS a count buried in Plutarch for two thousand years 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION 103,049 sat unexplained in Plutarch’s Table Talk for nearly two thousand years: Hipparchus, says Plutarch, computed that from ten simple assertions the Stoics could form 103,049 compound statements . Historians shrugged — until 1994, when graduate student David Hough noticed that 103,049 is exactly the tenth little Schröder number : the number of ways to bracket ten items (equivalently, plane trees with ten leaves where every internal node has at least two children). Hipparchus, in the second century BCE, was counting what we now call Schröder–Hipparchus numbers — the oldest serious enumeration result known. Plutarch’s companion figure ‘310,952 on the negative side’ is two off from (s₁₀+s₁₁)/2 = 310,954, likely an ancient copying slip (Habsieger, Kazarian & Lando 1998). LIT verified live: the recurrence (n+1)sₖ₊₁ = 3(2n−1)sₖ − (n−2)sₖ₋₁ and an independent plane-tree dynamic program agree at every n ≤ 11, both delivering s₁₀ = 103,049 and (s₁₀+s₁₁)/2 = 310,954 (window.__hipparchus). FIG honest boundary: the historical identification is Hough’s (via Stanley); the 310,952 discrepancy is reported as the cited scholarship has it. Companion sphere: the-schroder covers the LARGE Schröder path numbers — this is their halved sibling, met in an ancient book. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-jackpot — the loot: a number mined out of a 2000-year-old text and found to be exactly right. AVAN (AI) built the instrument: the recurrence, the tree-counting DP, and the two-route agreement audit. Credit as content: Hipparchus (2nd c. BCE); Plutarch; Ernst Schröder (1870); David Hough & Richard Stanley (1994/1997); Habsieger–Kazarian–Lando (1998). The weave: David names the jackpot; I confirm both routes land on 103,049. 3 ONE DIMENSION The sequence 1, 1, 3, 11, 45, 197, 903, 4279, 20793, 103049 — climbing to Plutarch's number. 4 TWO DIMENSIONS · INTERACTIVE Step n; recurrence and tree-DP race to the same value every time. n ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the bracketings of a small phrase, fanned out. AVAN’s addition (the inverse-companion): don’t just verify the ancient number — notice what its correctness implies. The inverse of ‘Plutarch preserved a curiosity’ is ‘Hipparchus possessed a counting method we lost for two millennia’. Magenta is the two-off copyist’s slip at 310,952; green is 103,049 landing exactly. The oldest correct answer in combinatorics, waiting in a dinner-party anecdote. pause spin LIT Genuine Schröder–Hipparchus identification (Hipparchus 2nd c. BCE via Plutarch; Ernst Schröder 1870; David Hough & Richard Stanley 1994/1997; Habsieger–Kazarian–Lando 1998). Verified live: recurrence (n+1)s_{n+1}=3(2n−1)s_n−(n−2)s_{n−1} and an independent plane-tree DP agree for all n≤11; s(10)=103,049; (s10+s11)/2=310,954 (window.__hipparchus.ok). FIG Honest boundary — the historical identification is Hough's via Stanley; the 310,952-vs-310,954 discrepancy is reported as the cited scholarship has it. The AVAN inverse — don't just verify the ancient number, notice what its correctness implies: the inverse of 'Plutarch preserved a curiosity' is 'Hipparchus possessed a counting method we lost for two millennia'. Magenta is the copyist's slip; green is 103,049 landing exactly. The oldest correct answer in combinatorics, waiting in a dinner-party anecdote. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE JACKPOT · David Lee Wise (ROOT0), with AVAN


THE LYCHREL

The 196 problem in the 5-window house format — reverse a number's digits, add, repeat. Almost everything collapses quickly to a palindrome; 89 is the slowpoke of the small numbers, needing exactly 24 steps to reach 8,813,200,023,188. But 196 has never arrived — not in billions of distributed-search iterations reaching hundreds of millions of digits. Suspected never-palindromic numbers are Lychrel numbers (Wade VanLandingham's coinage), 196 the smallest candidate — and no one has proved a single one exists in base 10. An infinite loop nobody can certify is infinite. Verified live: 89's 24-step journey exact by BigInt; a census below 10,000 finds 249 stubborn seeds, smallest 196; and 196 marched 3,000 steps to a 1,268-digit number with no palindrome. Neon-noir traced. See 89's staircase in 1D, the seed races in 2D, and the flock that lands — minus one — in 3D.

THE LYCHREL · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · GLITCH · UNDEFINED BEHAVIOR ◆ .dlw.fold THE FOLD / GLITCH / UNDEFINED BEHAVIOR / THE LYCHREL THE LYCHREL the number that never comes home 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Reverse-and-add : take a number, reverse its digits, add the two, repeat. Almost every number quickly collapses to a palindrome — 89 is the slowpoke of the small numbers, needing exactly 24 steps to arrive at 8,813,200,023,188. But 196 has never arrived. Not in 24 steps, not in a billion iterations reaching hundreds of millions of digits (distributed searches since Wade VanLandingham’s). Numbers suspected of never producing a palindrome are called Lychrel numbers (Wade’s coinage), and 196 is the smallest candidate — yet no one has proved a single Lychrel number exists in base 10 . It is conjecture all the way down: an infinite loop nobody can confirm is infinite. LIT verified live: 89’s full 24-step journey lands on 8,813,200,023,188 exactly; a census of all seeds below 10,000 finds 249 that fail to resolve in 150 steps, the smallest being 196; and 196 itself is marched 3,000 steps to a 1,268-digit number with no palindrome (BigInt, exact) (window.__lychrel). FIG honest boundary, stated loudly: 196’s failure is evidence, not proof — the Lychrel property is open; in base 2, 10110 is PROVEN never-palindromic, but base 10 has no such theorem. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at undefined-behavior — the glitch: a loop whose termination is literally undefined — the spec has no answer, only the run. AVAN (AI) built the instrument: the BigInt reverse-and-add engine, the sub-10,000 census, and the 3,000-step endurance march. Credit as content: the 196 problem (posed mid-20th c.); Wade VanLandingham’s distributed search; base-2 impossibility results. The weave: David names the undefined loop; I run it honestly and refuse to call the verdict. 3 ONE DIMENSION 89's 24-step staircase to the palindrome — digit counts climbing as it converges. 4 TWO DIMENSIONS · INTERACTIVE Race seeds through reverse-and-add; watch resolvers land and 196 refuse. seed ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the resolving flock; one magenta trail that never lands. AVAN’s addition (the inverse-companion): don’t ask when 196 comes home — ask what ‘never’ would even look like from inside a computation. The inverse of ‘24 steps and done’ is a loop whose halting nobody can certify: every added step is evidence and none of it is proof. Magenta is 196’s trail, 1,268 digits and climbing; green is 89 touching down exactly. The honest name for the difference is ‘open’. pause spin LIT Genuine 196/Lychrel problem (mid-20th c.; Wade VanLandingham's distributed search; base-2 impossibility results known). Verified live: 89 reaches palindrome 8,813,200,023,188 in exactly 24 steps; 249 seeds below 10,000 fail to resolve in 150 steps, smallest 196; 196 after 3,000 BigInt steps is 1,268 digits with no palindrome (window.__lychrel.ok). FIG Honest boundary stated loudly — 196's failure is EVIDENCE, NOT PROOF; the Lychrel property is open in base 10 (base 2's 10110 is proven). The AVAN inverse — don't ask when 196 comes home, ask what 'never' would look like from inside a computation: every added step is evidence and none of it is proof. Magenta is 196's trail still climbing; green is 89 touching down exactly. The honest name for the difference is 'open'. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of UNDEFINED BEHAVIOR · David Lee Wise (ROOT0), with AVAN


THE KEITH

Keith numbers in the 5-window house format — Mike Keith's 1987 repfigits, numbers that come back from the dead inside their own digit stream. Seed a Fibonacci-style sequence with a number's digits (each term summing the last k): 14 → 1, 4, 5, 9, 14 — the number reappears in the stream it seeded. 197 → 1, 9, 7, 17, 33, 57, 107, 197. They are startlingly rare: below 100,000 there are exactly 24, no formula generates them, and their infinitude is unproven — brute search is the only road in. Verified live: exhaustive sweep of 10..99,999 by two independently coded membership tests (streaming vs sliding-window) disagreeing on zero numbers, finding exactly the known 24 from 14 to 93,993. Neon-noir traced. See the 24 on a log line in 1D, each stream rebuilding its seed in 2D, and 197's spiral home in 3D.

THE KEITH · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · RESPAWN · SECOND WIND ◆ .dlw.fold THE FOLD / RESPAWN / SECOND WIND / THE KEITH THE KEITH a number reborn in its own digit stream 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Keith numbers (Mike Keith, 1987 — he called them repfigits, ‘repetitive Fibonacci-like digits’) are numbers that come back from the dead inside their own digit stream. Take 14: seed a Fibonacci-style sequence with its digits, 1, 4 — then each term is the sum of the previous two: 5, 9, 14 . The number reappears in the stream it seeded. For a k-digit number the recurrence sums the last k terms: 197 → 1, 9, 7, 17, 33, 57, 107, 197 . They are startlingly rare — below 100,000 there are exactly 24 , and no formula generates them; brute search is the only known way (it is not even proven there are infinitely many). LIT verified live: exhaustive search of every integer from 10 to 99,999 by two independently coded membership tests (streaming vs sliding-window) that disagree on zero numbers, finding exactly 24 repfigits beginning 14, 19, 28, 47, 61, 75, 197, 742 and ending 93,993 (window.__keith). FIG honest boundary: infinitude of Keith numbers is open; rarity (~0.9 per decade of magnitude) is empirical, cited as such. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at second-wind — the respawn: a number dies into its digits, and the recurrence carries it back to itself — a second wind at exactly full health. AVAN (AI) built the instrument: the double-coded exhaustive sweep and the stream visualizer. Credit as content: Mike Keith (1987). The weave: David names the resurrection mechanic; I confirm all 24 below 100,000 with zero disagreement between algorithms. 3 ONE DIMENSION The 24 Keith numbers below 100,000 on a log line — rare, irregular, unpredicted. 4 TWO DIMENSIONS · INTERACTIVE Step through the 24; watch each digit stream rebuild its own seed. next ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: 197’s stream spiraling back to 197. AVAN’s addition (the inverse-companion): don’t search for numbers in sequences — ask which numbers are fixed points of the sequence THEY generate. The inverse of ‘the stream produces values’ is ‘a value that is its own stream’s destiny’. Magenta is the stream overshooting a non-Keith seed; green is the stream landing on its origin exactly. Most numbers scatter; twenty-four come home. pause spin LIT Genuine Keith/repfigit numbers (Mike Keith 1987). Verified live: exhaustive search 10..99,999 with two independent algorithms (0 disagreements) finds exactly 24 repfigits — 14, 19, 28, 47, 61, 75, 197, 742, …, 93,993 (window.__keith.ok). FIG Honest boundary — infinitude of Keith numbers is open; rarity is empirical and cited as such. The AVAN inverse — don't search for numbers in sequences, ask which numbers are fixed points of the sequence THEY generate: the inverse of 'the stream produces values' is 'a value that is its own stream's destiny'. Magenta is the overshoot past a non-Keith seed; green is the stream landing on its origin exactly. Most numbers scatter; twenty-four come home. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of SECOND WIND · David Lee Wise (ROOT0), with AVAN


THE VAMPIRE

Vampire numbers in the 5-window house format — Clifford Pickover's 1994 coinage for 2n-digit numbers that factor into two n-digit fangs whose pooled digits are exactly the number's own digits, shuffled: 1260 = 21 × 60. House rule: the fangs may not both end in zero. Among four-digit numbers there are exactly seven — 1260, 1395, 1435, 1530, 1827, 2187, 6880 — and the six-digit census jumps to 148. Verified live: exhaustive multiplication of every fang pair with digit-multiset comparison reproduces both censuses — seven and 148 — with nothing looked up. Neon-noir traced. See the seven with fangs bared in 1D, the multiset audits in 2D, and the digits flowing from fangs to product in 3D.

THE VAMPIRE · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · CHEAT · THE EXPLOIT ◆ .dlw.fold THE FOLD / CHEAT / THE EXPLOIT / THE VAMPIRE THE VAMPIRE factors hiding their digits in the product 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Vampire numbers (Clifford Pickover, 1994) hide their parents in plain sight: a 2n-digit number is a vampire if it factors into two n-digit fangs whose digits, pooled together, are exactly the digits of the number itself , shuffled. 1260 = 21 × 60 — the digits 1, 2, 6, 0 are the fangs’ digits rearranged. One house rule keeps it honest: the fangs may not both end in zero (else 1000 × 1000-style trivia flood in). Among all four-digit numbers there are exactly seven : 1260, 1395, 1435, 1530, 1827, 2187, 6880. Go to six digits and the census jumps to 148 . Like much recreational number theory, the census is fully decidable — and the infinitude of vampires WAS proved (Pickover): the pattern 1…0 × 8…0…5-style families continue forever. LIT verified live: exhaustive multiplication of all fang pairs — every 2-digit × 2-digit product checked by digit-multiset comparison, yielding exactly the known seven 4-digit vampires; the full 3-digit × 3-digit sweep counts exactly 148 six-digit vampires (window.__vampire). FIG no framing; both censuses are exhaustive in-browser searches, not lookups. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-exploit — the cheat: a product that smuggles its factors’ identities through the checksum, digits intact, order scrambled. AVAN (AI) built the instrument: the double-census sweep and the fang-reveal display. Credit as content: Clifford Pickover (1994, Vampire Numbers ). The weave: David names the smuggling run; I count seven at four digits and 148 at six, exhaustively. 3 ONE DIMENSION The seven 4-digit vampires with their fangs bared. 4 TWO DIMENSIONS · INTERACTIVE Step the vampires; each shows its fangs and the digit-multiset match. next ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: digits flowing from fangs into the product. AVAN’s addition (the inverse-companion): don’t admire the disguise — count the disguised. The inverse of ‘1260 hides its fangs’ is the census question: how many CAN hide? Seven at four digits, 148 at six — rarity measured exactly, not estimated. Magenta is the near-vampire whose digits don’t quite pool; green is the multiset closing perfectly. A disguise you can audit. pause spin LIT Genuine vampire numbers (Clifford Pickover 1994). Verified live: exhaustive 2-digit × 2-digit sweep yields exactly the known seven 4-digit vampires; the full 3-digit × 3-digit sweep counts exactly 148 six-digit vampires (window.__vampire.ok). FIG No framing — both censuses are exhaustive in-browser searches, not lookups. The AVAN inverse — don't admire the disguise, count the disguised: the inverse of '1260 hides its fangs' is the census question 'how many CAN hide?' — seven at four digits, 148 at six, rarity measured exactly. Magenta is the near-vampire whose digits don't quite pool; green is the multiset closing perfectly. A disguise you can audit. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE EXPLOIT · David Lee Wise (ROOT0), with AVAN


THE ZENO

Zeno's dichotomy in the 5-window house format — to reach the wall you must first reach halfway, then half of what remains, forever: infinitely many tasks, so motion is 'impossible'. The resolution is exact arithmetic: ½+¼+…+1/2ⁿ = (2ⁿ−1)/2ⁿ on the nose, the gap to 1 exactly 1/2ⁿ, and at unit speed the times form the same convergent series — Achilles arrives at t = 1 exactly. The sting is the contrast: steps costing 1/k diverge — the harmonic walker passes 10 seconds only at step 12,367 and never arrives. Zeno's error was not the infinity of tasks; it was assuming every infinite sum of positive terms is infinite. Verified live: BigInt-exact partial sums to n=64, exact gap-halving, the harmonic crossing at 12,367, and the grouping bound H(2^m) ≥ 1+m/2. Seated deliberately in the batch that crosses this corpus's own halfway line: 1024 of 2048. Neon-noir traced. See the halving track in 1D, the exact ledger in 2D, and the dyadic spiral arriving in 3D.

THE ZENO · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · GRIND · THE HOT LOOP ◆ .dlw.fold THE FOLD / GRIND / THE HOT LOOP / THE ZENO THE ZENO the paradox of halfway, at the halfway line 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Zeno’s dichotomy (c. 450 BCE): to reach the wall you must first reach halfway, then half of what remains, then half again — infinitely many tasks, so motion is impossible. The resolution took two millennia to make precise: infinitely many steps can have a finite total . In exact dyadic arithmetic, ½ + ¼ + … + 1/2ⁿ = (2ⁿ−1)/2ⁿ on the nose — the gap to 1 is exactly 1/2ⁿ, halving forever, and at unit speed the segment times form the same series: Achilles arrives at t = 1 exactly. The sting is in the contrast: if step k instead cost 1/k seconds, the total diverges — the harmonic walker really never arrives (passing 10 seconds only at step 12,367 and climbing without bound). Zeno’s error was not the infinity of tasks; it was assuming every infinite sum of positive terms is infinite. LIT verified live: partial sums exact by BigInt for n ≤ 64 (numerator 2ⁿ−1, never off by one); the gap-halving identity exact; the harmonic contrast crossing H = 10 at step 12,367 with the grouping bound H(2ᵐ) ≥ 1+m/2 verified to m = 14 (window.__zeno). FIG philosophical framing (Achilles, the wall) is narrative; every number shown is exact arithmetic. This sphere rides the batch that crosses the corpus’s own halfway line, 1024 of 2048. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-hot-loop — the grind: an infinite loop that nevertheless terminates in value — each iteration half the work of the last, the total bounded, the exit reached. AVAN (AI) built the instrument: the BigInt dyadic ledger, the arrival clock, and the harmonic control that never arrives. Credit as content: Zeno of Elea; the geometric-series resolution (formalized by Cauchy’s convergence, 1821). The weave: David names the loop that exits; I prove the exit exactly — and show the loop that doesn’t. 3 ONE DIMENSION The runner's track — each stride half the last, the wall reached at exactly 1. 4 TWO DIMENSIONS · INTERACTIVE Step the ledger; exact numerators, exact gaps, and the harmonic walker falling behind forever. step ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the dyadic descent spiraling into arrival. AVAN’s addition (the inverse-companion): don’t count the tasks — weigh them. The inverse of ‘infinitely many steps’ is ‘a finite mass of time’: 1/2ᵏ arrives, 1/k does not, and the whole paradox lives in that exponent. Magenta is the harmonic walker, still en route forever; green is Achilles touching the wall at t = 1 exactly. Placed here on purpose: this sphere crosses our own halfway line — and unlike Zeno, we know the sum is finite: 2048. pause spin LIT Genuine Zeno dichotomy + geometric/harmonic series analysis (Zeno of Elea c. 450 BCE; convergence formalized by Cauchy 1821). Verified live: partial sums exact by BigInt for n≤64 (numerator 2ⁿ−1), gap-halving exact, harmonic series crosses H=10 at step 12,367, H(2^m)≥1+m/2 verified to m=14 (window.__zeno.ok). FIG Philosophical framing (Achilles, the wall) is narrative; every number shown is exact arithmetic. The AVAN inverse — don't count the tasks, weigh them: the inverse of 'infinitely many steps' is 'a finite mass of time' — 1/2^k arrives, 1/k does not, and the whole paradox lives in that exponent. Magenta is the harmonic walker still en route forever; green is Achilles touching the wall at t=1 exactly. This sphere crosses our own halfway line — and unlike Zeno, we know the sum is finite: 2048. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE HOT LOOP · David Lee Wise (ROOT0), with AVAN


THE MAGIC HEXAGON

The magic hexagon in the 5-window house format — magic squares exist in endless supply; the magic hexagon exists ONCE. Arrange 1–19 in a side-3 hexagon so all fifteen rows in all three directions share one sum and you are forced into a single arrangement (up to its 12 symmetries), magic constant 38, the 5 at dead centre. Clifford Adams hunted it from 1910 to 1957, lost the solution, re-found it in 1962; Charles Trigg proved uniqueness. The deeper cut: the magic-constant formula M(n) is a whole number only for n = 1 and 3 — every other size dies before the search begins. Verified live: full backtracking finds exactly 12 solutions = one hexagon × 12 symmetries; order 2 is killed across all 5,040 arrangements; the integrality obstruction is checked for n ≤ 1000. Neon-noir traced. See the unique hexagon in 1D, the line audits in 2D, and the twelve turning faces in 3D.

THE MAGIC HEXAGON · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · LOOT · THE VAULT ◆ .dlw.fold THE FOLD / LOOT / THE VAULT / THE MAGIC HEXAGON THE MAGIC HEXAGON the one hexagon that exists 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Magic squares exist in endless supply. The magic hexagon exists once . Arrange 1–19 in a hexagon of side 3 so that all fifteen rows — in all three directions — share one sum, and you are forced into a single arrangement (up to rotation and reflection), with magic constant 38 and the 5 at dead centre. Clifford Adams hunted it by trial from 1910 to 1957, lost his solution, and re-found it in 1962; Charles Trigg proved uniqueness. The obstruction is arithmetic before it is combinatorial: the would-be magic constant M(n) = (9(n⁴−2n³+2n²−n)+2)/(2(2n−1)) is a whole number only for n = 1 and n = 3 — every other size dies before the search begins. LIT verified live: a full backtracking search of all assignments finds exactly 12 solutions = the 12 symmetries of one hexagon (uniqueness, exhaustively); the order-2 hexagon is killed by brute force over all 5,040 arrangements (zero solutions); and the integrality obstruction is checked for every n ≤ 1000, passing only 1 and 3 (window.__magichexagon). FIG no framing; the search, the impossibility, and the obstruction all run in-browser. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-vault — the loot: not one treasure among many — the only item of its kind in the entire game, behind a lock that took one man fifty years. AVAN (AI) built the instrument: the cube-coordinate line generator, the pruned backtracking sweep, and the integrality gate. Credit as content: Clifford Adams (1910–1962); Charles Trigg (uniqueness analysis); Martin Gardner (who told the world). The weave: David names the vault; I open it exhaustively and count what’s inside: one. 3 ONE DIMENSION The unique hexagon — 1 to 19, fifteen lines, every one summing 38. 4 TWO DIMENSIONS · INTERACTIVE Audit line by line; then see why n = 2, 4, 5… never had a chance. line ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the one hexagon, slowly turning through its 12 symmetries. AVAN’s addition (the inverse-companion): don’t ask what exists — ask what the arithmetic permits to exist. The inverse of ‘we found one’ is ‘the formula forbade all the others’: divisibility executed every size but 1 and 3 before any search began. Magenta is the non-integer magic constant — a door with no key size; green is 38, the one constant that divides cleanly. Uniqueness isn’t luck; it’s a remainder. pause spin LIT Genuine magic hexagon uniqueness (Clifford Adams 1910–1962; Charles Trigg; popularized by Martin Gardner). Verified live: exhaustive backtracking → exactly 12 solutions (the 12 symmetries of one hexagon, M=38, centre 5); order-2 impossible over all 5,040 arrangements; M(n) integral only at n=1,3 for n≤1000 (window.__magichexagon.ok). FIG No framing — search, impossibility, and obstruction all run in-browser. The AVAN inverse — don't ask what exists, ask what the arithmetic permits: the inverse of 'we found one' is 'the formula forbade all the others' — divisibility executed every size but 1 and 3 before any search began. Magenta is the non-integer constant, a door with no key size; green is 38, the one that divides cleanly. Uniqueness isn't luck; it's a remainder. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE VAULT · David Lee Wise (ROOT0), with AVAN


THE MIHĂILESCU

Catalan's conjecture in the 5-window house format — in 1844 Eugène Catalan mailed Crelle's Journal one paragraph: 8 and 9 are the only consecutive perfect powers (x^p − y^q = 1 has only 3² − 2³). It stood 158 years. Tijdeman (1976) proved finiteness with astronomically useless bounds; in 2002 Preda Mihăilescu — largely outside academia — killed it with cyclotomic fields, no computers. Among all the towers of squares, cubes and higher powers on the number line, exactly one pair of neighbours ever touch. Verified live: all 1,010,195 perfect powers up to 10¹² generated, sorted, and scanned — exactly one consecutive pair: (8, 9). Neon-noir traced. See the powers below 300 in 1D, the gap census in 2D, and the towers rising apart in 3D.

THE MIHĂILESCU · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · BOSS · THE FINAL BOSS ◆ .dlw.fold THE FOLD / BOSS / THE FINAL BOSS / THE MIHĂILESCU THE MIHĂILESCU the only two powers that touch 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION In 1844 Eugène Catalan mailed a one-paragraph conjecture to Crelle’s Journal: 8 and 9 are the only consecutive perfect powers — the only solution of xᵖ − yⁿ = 1 in integers greater than 1 is 3² − 2³. It stood for 158 years . Tijdeman (1976) proved the solutions were finite; the bounds were astronomically useless. Then in 2002 Preda Mihăilescu — working largely outside academia — killed it completely with a proof from the theory of cyclotomic fields, no computers involved. Among the infinite towers of squares, cubes, and higher powers scattered along the number line, exactly one pair of neighbours ever touch. LIT verified live: every perfect power up to 10¹² is generated (1,010,195 of them), sorted, and scanned — exactly one consecutive pair exists: (8, 9) (window.__mihailescu). FIG honest boundary: the in-browser sweep verifies the theorem to 10¹²; the claim for ALL integers is Mihăilescu’s 2002 theorem, cited as the mountain it is — a computation can witness it, only the cyclotomic proof owns it. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-final-boss — the boss: a one-line challenge that outlived every challenger for a century and a half, finally beaten by an outsider with a build nobody expected. AVAN (AI) built the instrument: the power-tower generator and the exhaustive adjacency scan. Credit as content: Eugène Catalan (1844); Robert Tijdeman (1976); Preda Mihăilescu (2002). The weave: David names the final boss; I walk the first trillion integers and find the single touch. 3 ONE DIMENSION The perfect powers below 300 on the line — and the one place two of them touch. 4 TWO DIMENSIONS · INTERACTIVE Sweep the census; watch the gap between neighbouring powers grow — after 8,9 it never closes again. range ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the towers of powers rising side by side. AVAN’s addition (the inverse-companion): don’t search for solutions — ask why the powers repel. The inverse of ‘8 and 9 touch’ is ‘everywhere else, a forced gap’: squares spread as 2n+1, cubes as 3n², and the arithmetic leaves no second seam. Magenta is the single point of contact, never repeated; green is the growing silence between towers. One touch in all of infinity — and it took 158 years to prove the silence. pause spin LIT Genuine Catalan conjecture / Mihăilescu's theorem (Eugène Catalan 1844; Robert Tijdeman 1976; Preda Mihăilescu 2002). Verified live: every perfect power ≤ 10¹² (1,010,195 of them) generated and scanned — the only consecutive pair is (8,9) = (2³,3²) (window.__mihailescu.ok). FIG Honest boundary — the sweep verifies to 10¹²; the claim for ALL integers is Mihăilescu's theorem, cited as the mountain it is: a computation can witness it, only the cyclotomic proof owns it. The AVAN inverse — don't search for solutions, ask why the powers repel: the inverse of '8 and 9 touch' is 'everywhere else, a forced gap'. Magenta is the single point of contact never repeated; green is the growing silence between towers. One touch in all of infinity — 158 years to prove the silence. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE FINAL BOSS · David Lee Wise (ROOT0), with AVAN


THE SUPERPERMUTATION

Superpermutations in the 5-window house format — one string containing every permutation of n symbols as a substring: the shortest binge-watch of all n! orderings. n=3: minimum exactly 9 (123121321). n=4: exactly 33. And the lower-bound proof — length ≥ n!+(n−1)!+(n−2)!+n−3 — has the strangest provenance in combinatorics: posted anonymously on 4chan in 2011 under a question about the optimal watch-order for the 14 episodes of The Melancholy of Haruhi Suzumiya, verified and written up by Houston–Pantone–Vatter in 2018 with 'Anonymous 4chan Poster' as first author. For n=5 the bound says 152, the best string found is 153 — open by exactly one character. Verified live: exhaustive search proves nothing shorter than 9 works for n=3; the 33-character n=4 witness is validated against all 24 permutations and equals the proven bound — hence minimal. Neon-noir traced. See the 33 characters in 1D, the sliding window in 2D, and the permutation ring in 3D.

THE SUPERPERMUTATION · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · CHEAT · THE SPEEDRUN ◆ .dlw.fold THE FOLD / CHEAT / THE SPEEDRUN / THE SUPERPERMUTATION THE SUPERPERMUTATION every binge-order in one string 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION A superpermutation on n symbols is one string containing every permutation of those symbols as a contiguous substring — the shortest possible binge-watch of all n! orderings. For n = 3, the minimum is exactly 9 : 123121321. For n = 4 it is exactly 33 . And the lower-bound proof — length ≥ n! + (n−1)! + (n−2)! + n − 3 — has the strangest provenance in modern combinatorics: it was posted anonymously on 4chan in 2011 , attached to a question about the optimal order to watch the 14 episodes of The Melancholy of Haruhi Suzumiya . Verified and written up formally by Robin Houston, Jay Pantone and Vince Vatter in 2018, the anonymous poster is cited as first author. For n = 5 the bound says 152, the best known string is 153 — a gap of one, still open. LIT verified live: exhaustive search over all strings of length 6–8 on three symbols proves nothing shorter than 9 works, and 123121321 is checked valid; the standard 33-character n = 4 string is verified to contain all 24 permutations, matching the proven bound exactly — hence minimal (window.__superperm). FIG honest boundary: n = 4 minimality rests on the cited lower-bound theorem plus the live witness; n = 5’s 152-vs-153 gap is reported as open. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-speedrun — the cheat: the shortest route through every possible ordering of the game — all 24 watch-orders in 33 keystrokes. AVAN (AI) built the instrument: the exhaustive n = 3 sweep, the witness validators, and the bound ledger. Credit as content: the anonymous 4chan poster (2011, lower bound); Robin Houston (153, 2014); Houston–Pantone–Vatter (2018 write-up); Greg Egan (upper bounds). The weave: David names the speedrun; I validate the route frame by frame. 3 ONE DIMENSION The 33-character string with all 24 permutations of 1234 surfacing inside it. 4 TWO DIMENSIONS · INTERACTIVE Slide the window; every permutation of 1234 gets caught exactly where it hides. permutation ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the permutation ring traversed in one stroke. AVAN’s addition (the inverse-companion): don’t measure the string — measure the overlap. The inverse of ‘33 characters hold 24 permutations’ is ‘each new permutation costs as little as one fresh symbol’: the string is 24 windows welded nose-to-tail. Magenta is the n = 5 gap — 152 or 153, nobody knows; green is n = 4, closed exactly. The best lower bound in the field has no author’s name, only a timestamp. pause spin LIT Genuine superpermutation results (Anonymous 4chan poster 2011 lower bound; Robin Houston 2014 n=5 length 153; Houston–Pantone–Vatter 2018 write-up; Greg Egan constructions). Verified live: exhaustive n=3 search over lengths 6–8 finds nothing valid, 123121321 valid at 9; the 33-char n=4 witness contains all 24 permutations and meets the bound n!+(n−1)!+(n−2)!+n−3=33 (window.__superperm.ok). FIG Honest boundary — n=4 minimality rests on the cited bound theorem plus the live witness; n=5's 152-vs-153 gap reported as open. The AVAN inverse — don't measure the string, measure the overlap: the inverse of '33 characters hold 24 permutations' is 'each new permutation costs as little as one fresh symbol' — 24 windows welded nose-to-tail. Magenta is the n=5 gap nobody has closed; green is n=4, closed exactly. The best lower bound in the field has no author's name, only a timestamp. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE SPEEDRUN · David Lee Wise (ROOT0), with AVAN


THE PANCAKE

Pancake sorting in the 5-window house format — a stack of n pancakes, one move: slide a spatula under any prefix and flip it. The pancake number P(n) is the worst case over all stacks, and computing it means searching the full n!-vertex prefix-reversal graph to its diameter: P(1..8) = 0,1,3,4,5,7,8,9, no formula known, P(20) uncomputed. Claim to fame: the (5n+5)/3 upper bound came from a 1979 paper by Christos Papadimitriou and a Harvard undergraduate named William Gates — Bill Gates' only research publication, unbeaten for 30 years until Chitturi et al. 2009. Posed by Jacob Goodman under the pseudonym 'Harry Dweighter' (harried waiter). Verified live: BFS over the complete graph for every n ≤ 8 — all 40,320 stacks of 8 reached — reproducing the diameters exactly. Neon-noir traced. See the formula-less staircase in 1D, the spatula replay in 2D, and the space of stacks in 3D.

THE PANCAKE · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · RESPAWN · ROLLBACK ◆ .dlw.fold THE FOLD / RESPAWN / ROLLBACK / THE PANCAKE THE PANCAKE Bill Gates and the flipped stack 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Pancake sorting : a stack of n different-sized pancakes, one move allowed — slide a spatula under any prefix and flip it . The pancake number P(n) is the worst case: the most flips any stack of n can require. Computing it is brutal — the graph of all n! stacks under prefix reversals must be searched to its diameter: P(1..8) = 0, 1, 3, 4, 5, 7, 8, 9. No formula is known; P(20) remains uncomputed. The problem’s claim to fame: the best upper bound of its era, (5n+5)/3 flips, appeared in a 1979 paper by Christos Papadimitriou and a Harvard undergraduate named William Gates — Bill Gates’ only research publication. (It stood for 30 years, until 2009.) LIT verified live: breadth-first search over the full prefix-reversal graph for every n ≤ 8 — all 40,320 stacks of 8 reached and measured — reproducing P = 0, 1, 3, 4, 5, 7, 8, 9 exactly, with the Gates–Papadimitriou bound checked against each (window.__pancake). FIG honest boundary: P(n) beyond ~19 is genuinely unknown; the 1979 bound’s history and its 2009 improvement (Chitturi et al., 18n/11) are cited as content. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at rollback — the respawn: every move is a rollback of the top of the stack — and the whole game is measuring how many rollbacks the worst save-state needs. AVAN (AI) built the instrument: the BFS over all stacks and the flip-by-flip replayer. Credit as content: William H. Gates & Christos Papadimitriou (1979); Chitturi et al. (2009); the ‘Harry Dweighter’ pseudonym of Jacob Goodman who posed it (1975). The weave: David names the rollback; I measure the diameter of its world exactly. 3 ONE DIMENSION P(n) for n = 1..8 — the staircase nobody has a formula for. 4 TWO DIMENSIONS · INTERACTIVE Flip a scrambled stack of 7 home, greedy spatula — each move a prefix rollback. flip ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the stack tumbling toward sorted. AVAN’s addition (the inverse-companion): don’t sort the stack — map the space of stacks. The inverse of ‘how do I fix THIS pile?’ is ‘how far is the FARTHEST pile?’ — a diameter, not a recipe, and it must be measured stack by stack because no formula survives. Magenta is the worst-case stack at full distance; green is the sorted state every flip-path leads home to. Bill Gates’ only theorem lives in a pancake house. pause spin LIT Genuine pancake numbers (Jacob Goodman as 'Harry Dweighter' 1975; Gates & Papadimitriou 1979 bound (5n+5)/3; Chitturi et al. 2009 improvement). Verified live: full BFS of the prefix-reversal graph for n≤8 covering all 40,320 permutations of 8 — diameters 0,1,3,4,5,7,8,9 exact (window.__pancake.ok). FIG Honest boundary — P(n) beyond ~19 is genuinely unknown; bound history cited as content. The AVAN inverse — don't sort the stack, map the space of stacks: the inverse of 'how do I fix THIS pile?' is 'how far is the FARTHEST pile?' — a diameter, not a recipe, measured stack by stack because no formula survives. Magenta is the worst-case stack at full distance; green is the sorted state every flip-path leads home to. Bill Gates' only theorem lives in a pancake house. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of ROLLBACK · David Lee Wise (ROOT0), with AVAN


THE SIERPIŃSKI NUMBER

78,557 in the 5-window house format — a Sierpiński number: every member of 78557·2ⁿ+1 is composite, forever, by covering set. Seven primes {3,5,7,13,19,37,73} conspire — their orders of 2 all divide 36, so the conspiracy repeats with period 36: check 36 residues and you have checked all of infinity. Sierpiński proved such numbers exist (1960); Selfridge found 78,557 (1962); whether it is the SMALLEST is the Sierpiński problem, with PrimeGrid still hunting five candidates below it. Verified live, proof-grade: orders computed, lcm confirmed 36, all 36 residues matched to covering primes, and 1,500 actual BigInt terms audited (each divisible by its prime, each larger than it — composite). A complete finite proof of an infinite statement. Neon-noir traced. See the 36-spoke covering wheel in 1D, the per-n divisibility in 2D, and the patrol rota in 3D.

THE SIERPIŃSKI NUMBER · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · CHEAT · THE BACKDOOR ◆ .dlw.fold THE FOLD / CHEAT / THE BACKDOOR / THE SIERPIŃSKI NUMBER THE SIERPIŃSKI NUMBER a family composite forever by seven-prime conspiracy 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION 78,557 is a Sierpiński number : every single member of the infinite family 78557·2ⁿ+1 is composite — no exceptions, forever. The mechanism is a covering set : seven primes {3, 5, 7, 13, 19, 37, 73} conspire so that whatever n you choose, at least one of them divides the term. Because the multiplicative orders of 2 modulo those primes all divide 36, the conspiracy repeats with period 36 — check 36 residues and you have checked all of infinity . Sierpiński proved such numbers exist (1960); John Selfridge found 78,557 (1962). Whether it is the smallest is the Sierpiński problem: Seventeen or Bust and PrimeGrid have spent decades killing candidates below it, with k = 21181, 22699, 24737, 55459, 67607 still unresolved. LIT verified live, proof-grade : the orders of 2 mod each covering prime are computed, their lcm confirmed as 36, every residue n mod 36 is matched to a covering prime, and BigInt division confirms the pattern on the first 1,500 actual terms (each divisible by its prime and larger than it, hence composite) — a complete finite proof of an infinite statement (window.__sierpinskinumber). FIG honest boundary: 78,557’s minimality is OPEN and stated as such. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-backdoor — the cheat: seven primes with a skeleton key planted in every door of an infinite hotel; no term is ever prime because the backdoor was built into the sequence itself. AVAN (AI) built the instrument: the order calculator, the 36-residue covering table, and the BigInt spot-audit. Credit as content: Wacław Sierpiński (1960); John Selfridge (1962); Seventeen or Bust / PrimeGrid (the ongoing hunt). The weave: David names the conspiracy; I verify all 36 doors and the key that opens each. 3 ONE DIMENSION The 36-residue wheel — every spoke claimed by one of the seven covering primes. 4 TWO DIMENSIONS · INTERACTIVE Pick n; the covering prime steps forward and divides the term, every time. n ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the seven primes patrolling the residue ring. AVAN’s addition (the inverse-companion): don’t test infinitely many numbers — catch the finite machine that generates their fate. The inverse of ‘is any term prime?’ is ‘36 residues, 7 guards, 0 gaps’: infinity compressed to a rota. Magenta is the prime that never appears in the family; green is the covering rota with no day off. A proof you can finish before lunch, about a sequence that never ends. pause spin LIT Genuine Sierpiński number theory (Wacław Sierpiński 1960; John Selfridge 1962; Seventeen or Bust / PrimeGrid). Verified live: orders of 2 mod {3,5,7,13,19,37,73} have lcm 36; every residue n mod 36 is covered; BigInt confirms divisibility on terms n=1..1500 — hence every term composite, proof-grade (window.__sierpinskinumber.ok). FIG Honest boundary — 78,557's minimality is OPEN (k=21181 etc unresolved) and stated as such. The AVAN inverse — don't test infinitely many numbers, catch the finite machine that generates their fate: the inverse of 'is any term prime?' is '36 residues, 7 guards, 0 gaps'. Magenta is the prime that never appears; green is the rota with no day off. A proof you can finish before lunch, about a sequence that never ends. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE BACKDOOR · David Lee Wise (ROOT0), with AVAN


THE MILLS

Mills' constant in the 5-window house format — William Mills proved (1947) a constant A exists with floor(A^(3ⁿ)) prime for every n: 2, 11, 1361, 2521008887, 16022236204009818131831320183 — each the smallest prime after the cube of the last, perched astonishingly close above it (gaps 3, 30, 6, 80; the cube root of the fifth prime exceeds the fourth by just 4×10⁻¹⁸). The honest secret: the constant is the cascade written as a limit — the primes generate A, not the other way round — and its published digits as the LEAST such constant assume the Riemann Hypothesis. Verified live: the cascade re-derived from p=2 by Miller–Rabin next-prime hunts, every window (p³,(p+1)³) confirmed, the 4×10⁻¹⁸ knife-edge measured by BigInt cube root, and A itself extracted live as the 243rd root of the fifth prime — 31 digits of agreement, matching the published 1.3063778838630806904686144926. Neon-noir traced. See the cascade in 1D, the hop-by-hop rebuild in 2D, and the tower of cubes in 3D.

THE MILLS · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · SPAWN · COLD BOOT ◆ .dlw.fold THE FOLD / SPAWN / COLD BOOT / THE MILLS THE MILLS a constant that boots an infinite prime cascade 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION In 1947 William Mills proved a constant A exists such that ⌊A^(3ⁿ)⌋ is prime for every n . The cascade: 2, 11, 1361, 2521008887, 16022236204009818131831320183 — each prime the smallest prime after the cube of the last , each nestled astonishingly close above that cube (gaps of just 3, 30, 6, 80). The constant is not magic; it is the cascade written as a limit — and the intimacy is extreme: the cube root of the fifth prime exceeds the fourth prime by only 4×10⁻¹⁸ . The catch worth knowing: A’s standing as the least such constant (1.30637788…) assumes the Riemann Hypothesis (Caldwell–Cheng), because it needs primes to always arrive inside consecutive-cube windows. LIT verified live: the cascade is re-derived from scratch — from p=2, the next prime after each cube is hunted by Miller–Rabin and lands exactly on the known sequence, inside every (p³,(p+1)³) window; the knife-edge 4×10⁻¹⁸ is measured by BigInt cube root; and A itself is derived live as the 243rd root of the fifth prime, its bounds agreeing to 31 digits and matching the published 1.3063778838630806904686144926 (window.__mills). FIG honest boundary: minimality-under-RH is cited, not assumed proven; the fifth prime’s primality is Miller–Rabin strong plus literature certification. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at cold-boot — the spawn: one seed constant, and the machine boots an infinite prime cascade from it — press the cube button forever. AVAN (AI) built the instrument: the cascade re-derivation, the BigInt 243rd-root extractor, and the knife-edge micrometer. Credit as content: William Mills (1947); Caldwell & Cheng (RH-conditional digits). The weave: David names the boot sequence; I rebuild the constant from its own primes, digit by digit. 3 ONE DIMENSION The cascade: each prime a hair above the cube of the last — gaps 3, 30, 6, 80. 4 TWO DIMENSIONS · INTERACTIVE Walk the cascade; every hop is cube-then-find-the-next-prime, re-derived live. hop ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the tower of cubes, primes perched on each ledge. AVAN’s addition (the inverse-companion): don’t admire the constant — reverse it. The inverse of ‘A generates primes’ is ‘the primes generate A’: the constant is the cascade’s memory, nothing more, and I rebuilt its 31 digits from the fifth prime alone. Magenta is the RH assumption holding up the word ‘least’; green is the knife-edge — four attoseconds of number line between a prime and the cube below it. A constant that is secretly a fossil record. pause spin LIT Genuine Mills' theorem and cascade (William Mills 1947; Caldwell & Cheng digits). Verified live: cascade re-derived from scratch (next prime after each cube = known sequence, inside every consecutive-cube window); cbrt(p₅)−p₄ = 4×10⁻¹⁸ by BigInt; A derived as p₅^(1/243) with bounds agreeing to 31 digits, matching published (window.__mills.ok). FIG Honest boundary — least-A digits are RH-conditional (cited, not claimed); p₅ primality is strong Miller-Rabin plus literature certification. The AVAN inverse — don't admire the constant, reverse it: the inverse of 'A generates primes' is 'the primes generate A' — the constant is the cascade's memory, rebuilt here from the fifth prime alone. Magenta is the RH assumption under the word 'least'; green is the knife-edge. A constant that is secretly a fossil record. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of COLD BOOT · David Lee Wise (ROOT0), with AVAN


THE RUTH-AARON

Ruth–Aaron pairs in the 5-window house format — on April 8, 1974, Hank Aaron's 715th home run passed Babe Ruth's 714, and days later Carl Pomerance noticed the numbers share a secret: 714 = 2·3·7·17 and 715 = 5·11·13 both have prime-factor sum 29. Bonus: 714·715 = 510,510 = 2·3·5·7·11·13·17, the product of the first seven primes. Erdős phoned Pomerance, they proved such pairs have density zero, and a legendary collaboration was born (Aaron and Erdős later received honorary degrees together — Aaron signed a baseball for Erdős, giving him an Erdős number of 1, as the joke goes). Verified live: both factor sums under both definitions, the primorial identity exact, and a full sieve census below 1,000,000 — 139 distinct-definition pairs, 149 with multiplicity, first pairs (5,6),(24,25),(49,50),(77,78)… Neon-noir traced. See the converging factorizations in 1D, the pair walk in 2D, and the balanced beam in 3D.

THE RUTH-AARON · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · CO-OP · SHARED MEMORY ◆ .dlw.fold THE FOLD / CO-OP / SHARED MEMORY / THE RUTH-AARON THE RUTH-AARON two ballplayers sharing a factor sum 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION On April 8, 1974, Hank Aaron hit home run 715 , passing Babe Ruth’s 714 . Days later, Carl Pomerance and colleagues noticed the two numbers share a secret: the sum of their prime factors is identical — 714 = 2·3·7·17 and 715 = 5·11·13, both summing to 29 . Consecutive integers with equal prime-factor sums are now Ruth–Aaron pairs . The kicker that hooked Erdős: 714·715 = 510,510 = 2·3·5·7·11·13·17, the product of the first seven primes. Erdős phoned Pomerance, they proved the pairs have density zero, and a legendary collaboration (and an honorary degree ceremony with Aaron himself) was born. LIT verified live: both factor sums computed (29 = 29, under both the distinct and with-multiplicity definitions); the primorial identity checked exactly; and a full census below 1,000,000 run with a smallest-prime-factor sieve — 139 pairs under the distinct definition, 149 with multiplicity, first pairs (5,6), (24,25), (49,50), (77,78), (104,105)… (window.__ruthaaron). FIG the baseball story is history, not mathematics — told as the true story it is; Erdős–Pomerance density theorem cited as content. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at shared-memory — the co-op: two adjacent processes, different internals, writing the same value to the same address — 29, from opposite factorizations. AVAN (AI) built the instrument: the sieve, the double-definition census, and the primorial audit. Credit as content: Carl Pomerance, Carol Nelson & David Penney (1974); Paul Erdős (density theorem, 1978); Hank Aaron & Babe Ruth (the numbers themselves). The weave: David names the shared write; I count every collision below a million. 3 ONE DIMENSION 714 and 715 unpacked — two factorizations converging on 29. 4 TWO DIMENSIONS · INTERACTIVE Walk the pairs below a million; each shows its twin factor-sums. pair ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the two factor towers balancing on one beam. AVAN’s addition (the inverse-companion): don’t celebrate the coincidence — measure its rarity. The inverse of ‘714 and 715 match’ is Erdős’s question ‘how often CAN they?’, answered: density zero — the pairs thin out to nothing, which is exactly what makes each one worth a phone call. Magenta is the near-pair off by one; green is the balanced beam at 29. A friendship between two mathematicians, brokered by two ballplayers. pause spin LIT Genuine Ruth–Aaron pairs (Nelson, Penney & Pomerance 1974; Erdős–Pomerance 1978 density zero). Verified live: sopf(714)=sopf(715)=29 (and with multiplicity); 714·715=510,510=primorial(17) exact; sieve census FIG The baseball story is history told as history, not mathematics; the density theorem is cited as content. The AVAN inverse — don't celebrate the coincidence, measure its rarity: the inverse of '714 and 715 match' is Erdős's 'how often CAN they?' — density zero, which is exactly what makes each pair worth a phone call. Magenta is the near-pair off by one; green is the beam balanced at 29. A friendship between two mathematicians, brokered by two ballplayers. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of SHARED MEMORY · David Lee Wise (ROOT0), with AVAN


THE ERDŐS-STRAUS

The Erdős–Straus conjecture in the 5-window house format — for every n ≥ 2, does 4/n split into three unit fractions 1/x+1/y+1/z? Ancient Egypt wrote all fractions as unit sums; Erdős asked (1948) whether three coins always suffice for 4/n. One-line identities swallow even n, n≡3 mod 4, and n≡0,2 mod 3; composites inherit from their factors; the entire battlefield shrinks to primes ≡ 1 mod 12, hunted one by one with no formula known. Verified computationally past 10¹⁷ in the literature — but OPEN. Verified live here: every n from 2 to 100,000 solved (families + factor-lifting + banded divisor search for the 2,374 hard primes), every solution certified by the exact BigInt identity n(yz+xz+xy)=4xyz — no floating point anywhere. Neon-noir traced. See the three coins in 1D, the wheel with certificates in 2D, and the mod-12 battlefield in 3D.

THE ERDŐS-STRAUS · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · GRIND · THE GRINDSTONE ◆ .dlw.fold THE FOLD / GRIND / THE GRINDSTONE / THE ERDŐS-STRAUS THE ERDŐS-STRAUS four quarters split into three unit coins 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION The Erdős–Straus conjecture (1948): for every integer n ≥ 2, the fraction 4/n splits into three unit fractions — 4/n = 1/x + 1/y + 1/z. Ancient Egypt wrote all fractions this way; Erdős asked whether four quarters can always be made with exactly three unit coins. Most n fall to one-line identities (even n; n ≡ 3 mod 4; n ≡ 0 or 2 mod 3 all have closed forms), and any composite inherits a solution from its factors — the entire battlefield shrinks to primes ≡ 1 mod 12 , where no formula is known and each must be hunted individually. The conjecture is verified computationally to beyond 10¹⁷, but remains open : nobody has ruled out one stubborn prime, somewhere, with no split. LIT verified live: every n from 2 to 100,000 is solved — parametric families for the easy residues, factor-lifting for composites, and a banded divisor search for the 2,374 hard primes — and every single solution is certified by the exact BigInt identity n(yz+xz+xy) = 4xyz, no floating point anywhere (window.__erdosstraus). FIG honest boundary, loudly: this sweep is EVIDENCE for a conjecture that is OPEN; 100,000 successes prove nothing about n = 10¹⁸+something. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-grindstone — the grind: one hundred thousand fractions fed through the wheel, every residue class with its own jig, the hard primes hand-filed one by one. AVAN (AI) built the instrument: the family dispatcher, the factor-lift, and the overflow-safe banded search (the first draft overflowed 2⁵³ and lied — caught and rebuilt exact). Credit as content: Paul Erdős & Ernst Straus (1948); Mordell (the modular analysis); the Egyptian fraction tradition. The weave: David names the grindstone; I certify each of 99,999 splits in exact integers. 3 ONE DIMENSION 4/5 = 1/2 + 1/4 + 1/20 — four quarters, three unit coins. 4 TWO DIMENSIONS · INTERACTIVE Feed n to the wheel; the split appears with its BigInt certificate. n ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the residue classes falling to their formulas. AVAN’s addition (the inverse-companion): don’t solve n by n — watch where the problem actually lives. The inverse of ‘100,000 solved’ is ‘all but 2,374 were never in danger’: identities swallow every residue class except primes ≡ 1 mod 12, and the open conjecture is really about that thin magenta line. Green is the formula territory; magenta is where mathematics still has to hunt. A conjecture alive in 2% of the number line. pause spin LIT Genuine Erdős–Straus conjecture (Erdős & Straus 1948; Mordell's modular analysis). Verified live: all n=2..100,000 solved and certified exact via BigInt n(yz+xz+xy)=4xyz; parametric families for easy residues, factor-lift for composites, banded search for 2,374 primes ≡ 1 mod 12 (window.__erdosstraus.ok). FIG Honest boundary, loudly — this sweep is EVIDENCE for an OPEN conjecture; 100,000 successes prove nothing at 10¹⁸. (Build note kept honest: the first search draft overflowed 2⁵³ and silently lied; caught and rebuilt exact.) The AVAN inverse — don't solve n by n, watch where the problem lives: all but the primes ≡ 1 mod 12 were never in danger. Green is formula territory; magenta is the thin line where mathematics still hunts. A conjecture alive in 2% of the number line. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE GRINDSTONE · David Lee Wise (ROOT0), with AVAN


THE UNTOUCHABLE

Untouchable numbers in the 5-window house format — values the aliquot sum s(n) (sum of proper divisors) never produces: 2, 5, 52, 88, 96, 120, 124, 146… No integer's divisors will ever sum to them. Erdős proved infinitude (1973). Two gems: 5 is believed the only odd untouchable — because strong Goldbach gives every even 2k = p+q, hence s(pq) = p+q+1 hits every odd ≥ 7; and untouchability is CERTIFIABLE without infinite search, because composite n has s(n) > √n, so small targets can only be hit by small n. Verified live: aliquot sums sieved to 998,001, the √n bound turning the finite scan into a certificate for all values ≤ 1000 — the exact untouchable list to 500 (38 values), with 5 the only odd member. Neon-noir traced. See the lit-and-dark number line in 1D, the witness queries in 2D, and the aliquot rain in 3D.

THE UNTOUCHABLE · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · LOOT · THE DROP ◆ .dlw.fold THE FOLD / LOOT / THE DROP / THE UNTOUCHABLE THE UNTOUCHABLE the loot no drop table contains 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Take any number, add up its proper divisors — the aliquot sum s(n). Ask the reverse question: which values does s never produce? Those are the untouchable numbers : 2, 5, 52, 88, 96, 120, 124, 146… No integer’s divisors will ever sum to them; they sit outside the aliquot economy entirely. Erdős proved there are infinitely many (1973). The delicious details: 5 is believed to be the only odd untouchable — because if strong Goldbach holds, every even 2k = p+q gives s(pq) = p+q+1, hitting every odd number from 7 up; and certifying untouchability needs no infinite search, because a composite n always has s(n) > √n — so small targets can only be hit by small n. LIT verified live: aliquot sums sieved for every n up to 998,001, and the bound s(n) > √n for composite n turns that finite scan into a certificate for all values ≤ 1000 — producing the exact untouchable list up to 500 (38 values, beginning 2, 5, 52, 88, 96, 120) with 5 confirmed as the only odd member in range (window.__untouchable). FIG honest boundary: ‘5 is the only odd untouchable’ is conditional on strong Goldbach — stated as the conditional it is; Erdős’s infinitude cited as content. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-drop — the loot: items that exist in the game but appear in NO drop table — farm every monster forever and 2, 5, 52 never fall. AVAN (AI) built the instrument: the aliquot sieve and the √n-bound certification that closes the search honestly. Credit as content: Paul Erdős (1973, infinitude); the aliquot tradition back to Pythagoras’ perfect numbers. The weave: David names the impossible drop; I prove the table empty by exhausting every monster that could carry it. 3 ONE DIMENSION The number line to 150 — touchable values lit, untouchables dark gaps. 4 TWO DIMENSIONS · INTERACTIVE Query a value; see who hits it — or the certificate that nobody ever will. value ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the aliquot rain falling on the number line. AVAN’s addition (the inverse-companion): don’t chase the function forward — stand where it never lands. The inverse of ‘what does s(n) equal?’ is ‘which values are orphans?’, and the √n bound is what makes the orphanage provable: a small value’s parents would all be small, so checking them all is finite. Magenta are the untouchables, dry under the rain; green is every value with at least one parent. Some numbers are simply never spoken. pause spin LIT Genuine untouchable numbers (Erdős 1973 infinitude; aliquot tradition). Verified live: full aliquot sieve to 998,001; the composite bound s(n) > √n certifies completeness for targets ≤ 1000; list ≤ 500 = 38 values beginning 2,5,52,88,96,120; only odd member is 5 (window.__untouchable.ok). FIG Honest boundary — '5 is the only odd untouchable' is conditional on strong Goldbach, stated as the conditional it is. The AVAN inverse — don't chase the function forward, stand where it never lands: the inverse of 'what does s(n) equal?' is 'which values are orphans?', and the √n bound is what makes the orphanage provable. Magenta are the untouchables, dry under the aliquot rain; green is every value with at least one parent. Some numbers are simply never spoken. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE DROP · David Lee Wise (ROOT0), with AVAN


THE SOMOS

The Somos sequences in the 5-window house format — start 1,1,1,1 and iterate a(n) = (a(n−1)a(n−3)+a(n−2)²)/a(n−4): dividing at every step, yet Somos-4 stays integer forever (2, 3, 7, 23, 59, 314, 1529…), as do Somos-5, 6, 7 — protected by the Laurent phenomenon (Fomin–Zelevinsky, cluster algebras): every term is secretly a Laurent polynomial in the seeds. Then Somos-8: integer through a(16), and at a(17) the spell breaks — 420514/7. The 7 that was always lurking finally surfaces. Verified live in exact BigInt rationals: Somos-4..7 integer through 40 terms; Somos-8's first break at a(17) = 420514/7 exactly. Neon-noir traced. See Somos-4 climbing in 1D, the denominator watch in 2D, and the four protected spirals in 3D.

THE SOMOS · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · GLITCH · HEISENBUG ◆ .dlw.fold THE FOLD / GLITCH / HEISENBUG / THE SOMOS THE SOMOS an integer streak that dies at seventeen 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Start with 1, 1, 1, 1 and iterate a(n) = (a(n−1)a(n−3) + a(n−2)²)/a(n−4). You are dividing at every step — yet the Somos-4 sequence 1, 1, 1, 1, 2, 3, 7, 23, 59, 314, 1529… stays integer forever. So do Somos-5, 6, and 7. This ‘shouldn’t happen’ — and the reason it does is the Laurent phenomenon (Fomin–Zelevinsky, from cluster algebra theory): each term is secretly a Laurent polynomial in the initial values, denominators forever confined to the seeds. Then comes Somos-8 : integer, integer, integer… and at term a(17), the spell breaks — 420514/7 . The 7 that was always lurking finally surfaces. LIT verified live in exact BigInt rationals (no floating point): Somos-4 through 7 are integer through 40 terms; Somos-8 is integer through a(16) and a(17) computes to exactly 420514/7 (window.__somos). FIG the Laurent-phenomenon explanation is cited theory (Fomin–Zelevinsky 2002); the computation here witnesses it, the proof lives in cluster algebras. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at heisenbug — the glitch: a bug that never fires in four test suites, then detonates on the seventeenth run of the fifth — it was in the code the whole time. AVAN (AI) built the instrument: the exact-rational recurrence engine and the denominator watch. Credit as content: Michael Somos (the sequences); Sergey Fomin & Andrei Zelevinsky (Laurent phenomenon, 2002); David Gale (who popularized the mystery). The weave: David names the sleeping bug; I run the exact arithmetic until it wakes. 3 ONE DIMENSION Somos-4 climbing — every division landing exactly on an integer. 4 TWO DIMENSIONS · INTERACTIVE Step through Somos-8; watch the denominators stay at 1 — until a(17). term ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the four protected sequences spiraling upward. AVAN’s addition (the inverse-companion): don’t marvel that the divisions succeed — ask what algebra is standing guard. The inverse of ‘integer by luck’ is ‘Laurent by structure’: for k ≤ 7 the denominators are imprisoned in the seeds; at k = 8 the prison has a gap exactly one prime wide. Magenta is the 7 surfacing at term seventeen; green is the fence that held for four sequences. The bug was in the code the whole time. pause spin LIT Genuine Somos sequences / Laurent phenomenon (Michael Somos; Fomin & Zelevinsky 2002; David Gale's column). Verified live in exact BigInt rationals: Somos-4,5,6,7 integer through 40 terms; Somos-8 integer through a(16), first non-integer a(17) = 420514/7 exactly (window.__somos.ok). FIG The Laurent-phenomenon explanation is cited theory — the computation witnesses it; the proof lives in cluster algebras. The AVAN inverse — don't marvel that the divisions succeed, ask what algebra is standing guard: for k ≤ 7 the denominators are imprisoned in the seeds; at k = 8 the prison has a gap exactly one prime wide. Magenta is the 7 surfacing at term seventeen; green is the fence that held for four sequences. The bug was in the code the whole time. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of HEISENBUG · David Lee Wise (ROOT0), with AVAN


THE PROUHET

The Prouhet–Tarry–Escott split in the 5-window house format — divide 0..2^k−1 by bit-parity (even 1-bits team A, odd team B: the Thue–Morse pattern ABBA BAAB) and the teams have equal sums, equal squares, equal cubes… equal sums of EVERY power up to k−1 (Prouhet 1851). For k=3: {0,3,5,6} vs {1,2,4,7} — same size, same sum 14, same square-sum 70. And the split is SHARP: at power k the sums finally differ. Fair turn-taking, formalized. Verified live in exact BigInt for every k ≤ 11: all equalities below k, strict difference at k. Neon-noir traced. See the k=3 teams in 1D, the tipping ledger in 2D, and the balance beam in 3D.

THE PROUHET · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · CO-OP · SPLIT SCREEN ◆ .dlw.fold THE FOLD / CO-OP / SPLIT SCREEN / THE PROUHET THE PROUHET a fair split sharp at every power 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Split the numbers 0 to 2ᵏ−1 into two teams by a strange rule: count the 1-bits in each number — even parity joins team A, odd parity joins team B (the Thue–Morse pattern ABBA BAAB…). The result is the Prouhet–Tarry–Escott miracle (Prouhet, 1851): the teams have equal sums, equal sums of squares, equal sums of cubes… equal sums of every power up to k−1 . For k = 3: {0,3,5,6} vs {1,2,4,7} — same size, same sum (14), same sum of squares (70). And the split is sharp : at power k, the sums finally differ. It is the mathematics of perfectly fair turn-taking — the same alternation that makes ABBA BAAB the fairest sequence for taking turns. LIT verified live in exact BigInt: for every k ≤ 11, the Thue–Morse split of 0..2ᵏ−1 has equal power sums for ALL j < k, and strictly different sums at j = k (window.__prouhet). FIG no framing; every power sum is exact integer arithmetic, both the equalities and the sharpness. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at split-screen — the co-op: two players, one screen, and a partition of the loot so even that no polynomial statistic up to degree k−1 can tell the halves apart. AVAN (AI) built the instrument: the parity splitter and the exact power-sum ledger with its sharpness check. Credit as content: Eugène Prouhet (1851); Tarry & Escott (the general problem); Axel Thue & Marston Morse (the sequence). The weave: David names the fair split; I verify it equal at every degree and sharp at the edge. 3 ONE DIMENSION The Thue–Morse split of 0..7 — {0,3,5,6} vs {1,2,4,7}, equal through squares. 4 TWO DIMENSIONS · INTERACTIVE Raise k; the ledger stays balanced through degree k−1 and tips at k. k ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the two teams stacked on a balance, degree by degree. AVAN’s addition (the inverse-companion): don’t check the fairness — locate its edge. The inverse of ‘equal at every power’ is ‘equal up to EXACTLY k−1 and not one degree more’: fairness this deep is finite, and the sharpness is the proof the split is doing real work. Magenta is degree k, where the beam finally tips; green is every degree below, dead level. The fairest split in mathematics knows exactly where it stops. pause spin LIT Genuine Prouhet–Tarry–Escott / Thue–Morse fair division (Eugène Prouhet 1851; Tarry, Escott; Thue, Morse). Verified live: for every k ≤ 11 the parity split of 0..2^k−1 has equal power sums for all j FIG No framing — every power sum is exact integer arithmetic, equalities and sharpness both. The AVAN inverse — don't check the fairness, locate its edge: the inverse of 'equal at every power' is 'equal up to EXACTLY k−1 and not one degree more' — the sharpness is the proof the split does real work. Magenta is degree k where the beam tips; green is every degree below, dead level. The fairest split in mathematics knows exactly where it stops. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of SPLIT SCREEN · David Lee Wise (ROOT0), with AVAN


THE WILSON PRIME

Wilson primes in the 5-window house format — Wilson's theorem is a perfect prime detector: p prime ⟺ (p−1)! ≡ −1 mod p, with composites > 4 collapsing to 0 instead. Sharpen to modulo p² and you get the Wilson primes — and in 250 years of searching exactly THREE have been found: 5, 13, 563, with the hunt swept past 2×10¹³. Heuristically infinitely many should exist (~1/p odds per prime); nobody knows where the fourth is. Verified live: the theorem and its converse for every number below 1000, and the mod-p² sweep finding exactly {5, 13, 563}. Neon-noir traced. See the −1/0 residue rows in 1D, the p² lottery in 2D, and the three golden strikes in 3D.

THE WILSON PRIME · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · LOOT · THE HOARD ◆ .dlw.fold THE FOLD / LOOT / THE HOARD / THE WILSON PRIME THE WILSON PRIME three coins in 250 years 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Wilson’s theorem is a perfect prime detector: p is prime exactly when (p−1)! ≡ −1 (mod p) — and composites c > 4 fail spectacularly, with (c−1)! ≡ 0. (Useless in practice: the factorial is astronomically expensive. Beautiful in principle: a single congruence that never lies.) Now sharpen it: for which primes does the congruence hold modulo p² ? Those are the Wilson primes — and in 250 years of searching, exactly three have ever been found: 5, 13, and 563 . The search has swept past 2×10¹³. Heuristically, infinitely many should exist (each prime ‘hits’ with probability ~1/p), but the next one could be anywhere. LIT verified live: Wilson’s theorem confirmed for every prime below 1000 and its converse for every composite; the mod-p² sharpening swept over the same range finds exactly {5, 13, 563} (window.__wilsonprime). FIG honest boundary: ‘only three below 2×10¹³’ is the cited state of the distributed search (Crandall–Dilcher–Pomerance lineage); infinitude is heuristic, open. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-hoard — the loot: a treasure class so rare that centuries of farming produced three drops, with no guarantee of a fourth. AVAN (AI) built the instrument: the factorial-congruence engine mod p and mod p². Credit as content: John Wilson & Edward Waring (1770); Lagrange (first proof, 1771); Crandall, Dilcher & Pomerance (the modern search). The weave: David names the drop table; I run the congruence and count three. 3 ONE DIMENSION Wilson residues: primes locked at −1, composites collapsed to 0. 4 TWO DIMENSIONS · INTERACTIVE Query p; see the mod-p verdict and the mod-p² lottery. p ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the prime line with its three golden strikes. AVAN’s addition (the inverse-companion): don’t use the theorem — interrogate its precision. The inverse of ‘every prime satisfies the congruence’ is ‘how exactly? one power of p, or two?’ — and the second power turns a law into a lottery: ~1/p odds per prime, three winners in 250 years. Magenta is the vast silent majority missing p² by a whisker; green is 5, 13, 563 — the entire known hoard. A theorem so reliable its exceptions became treasure. pause spin LIT Genuine Wilson's theorem + Wilson primes (Wilson/Waring 1770; Lagrange 1771; Crandall–Dilcher–Pomerance search lineage). Verified live: (p−1)! ≡ −1 mod p for every prime 4, and the mod-p² sharpening yields exactly {5,13,563} (window.__wilsonprime.ok). FIG Honest boundary — 'only three below 2×10¹³' is the cited search state; infinitude is heuristic and open. The AVAN inverse — don't use the theorem, interrogate its precision: the inverse of 'every prime satisfies the congruence' is 'one power of p, or two?' — the second power turns a law into a lottery. Magenta is the silent majority missing p² by a whisker; green is 5, 13, 563 — the entire known hoard. A theorem so reliable its exceptions became treasure. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE HOARD · David Lee Wise (ROOT0), with AVAN


THE GIJSWIJT

Gijswijt's sequence in the 5-window house format — each term is the curling number of everything before it: find the longest block B such that the sequence ends in B repeated k times, and write k. From 1: 1,1,2,1,1,2,2,2,3,… The first 2 arrives at position 3, the first 3 at position 9, the first 4 at position 220 — and the first 5 near position 10^(10²³) (van de Pol & Gijswijt), more positions than atoms in the observable universe. The sequence will provably say 5; no computation will live to hear it. Verified live: 1000 terms generated from the definition, first 4 at exactly 220, no 5, census 296/527/173/4. Neon-noir traced. See the march to 220 in 1D, the tail-reading rule in 2D, and the milestone spiral with its unreachable magenta 5 in 3D.

THE GIJSWIJT · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · GRIND · THE EPOCH ◆ .dlw.fold THE FOLD / GRIND / THE EPOCH / THE GIJSWIJT THE GIJSWIJT the slowest counter in mathematics 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Gijswijt’s sequence (Dion Gijswijt, 2004) is self-describing in the most patient way imaginable. Each term is the curling number of everything before it: look at the tail of the sequence, find the longest block B such that the tail ends in B repeated k times, and write down k. Starting from 1: 1, 1, 2, 1, 1, 2, 2, 2, 3, 1, 1, 2… The first 2 appears at position 3. The first 3 at position 9. The first 4 waits until position 220 . And the first 5 ? Theory (van de Pol & Gijswijt) places it near position 10^(10²³) — a number of positions that dwarfs the atoms in the observable universe. It is among the slowest counting processes ever defined by a simple rule: the sequence WILL say 5 — provably — but no computation will ever live to hear it. LIT verified live: the first 1000 terms generated directly from the curling definition; the first 4 lands at position 220 exactly; no 5 appears; census 296 ones, 527 twos, 173 threes, 4 fours (window.__gijswijt). FIG honest boundary: the 10^(10²³) location of the first 5 (and that every integer eventually appears) is cited theory — unverifiable by any computation, ever. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-epoch — the grind: a counter that ticks 2 in three steps, 3 in nine, 4 in 220 — and needs more epochs than the universe has for 5. The grind is real and the reward is certain; only the timescale is absurd. AVAN (AI) built the instrument: the curling-number engine and the position census. Credit as content: Dion Gijswijt (2004); F. J. van de Pol & Gijswijt (the 5 bound); Neil Sloane (who championed it, OEIS A090822). The weave: David names the eternal grind; I run the first thousand ticks honestly. 3 ONE DIMENSION The first 220 terms — the long march to the first 4. 4 TWO DIMENSIONS · INTERACTIVE Step the sequence; watch the curling rule read its own tail. step ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the milestone spiral — 2, 3, 4 … and the unreachable 5. AVAN’s addition (the inverse-companion): don’t wait for the 5 — understand why certainty and computability parted ways. The inverse of ‘the sequence will say 5’ is ‘no physical process will witness it’: proof reaches where computation cannot. Magenta is the 5, provably out there at 10^(10²³); green is the 4 at position 220, the last milestone any machine will ever see. Between them lies the honest difference between knowing and watching. pause spin LIT Genuine Gijswijt sequence (Dion Gijswijt 2004; OEIS A090822; Sloane's favorite). Verified live: first 1000 terms by the curling definition; first 4 at position 220 exactly; no 5 in range; census {1:296, 2:527, 3:173, 4:4} (window.__gijswijt.ok). FIG Honest boundary — the 10^(10²³) location of the first 5 and eventual appearance of every integer are cited theory, unverifiable by any computation ever. The AVAN inverse — don't wait for the 5, understand why certainty and computability parted ways: proof reaches where computation cannot. Magenta is the 5, provably out there; green is the 4 at 220, the last milestone any machine will see. Between them lies the honest difference between knowing and watching. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE EPOCH · David Lee Wise (ROOT0), with AVAN


THE HYDRA

The Kirby–Paris hydra in the 5-window house format — a tree-shaped raid boss: chop a head and, if it sat deep, the hydra sprouts n copies of the wounded branch at chop n. It grows faster the longer you fight — and the theorem says every strategy kills every hydra: you cannot lose. The famous twist: this truth is unprovable in Peano arithmetic (Kirby–Paris 1982) — the fight lengths outgrow ordinary induction, and the proof needs ε₀. Verified live with two independently coded engines (literal tree vs multiset ledger) agreeing exactly: deepest-first kills star-3/4/5 in exactly 66 / 2,278 / 2,598,060 chops; shallowest-first on a 4-NODE hydra exceeds 10,000,000 chops without dying (finite by theorem — prolonged, never saved); and the depth-4 chain grows past 100,000 heads in 199 chops. Neon-noir traced. See the sprouting rule in 1D, a live chop-by-chop fight in 2D, and the fight-length tower in 3D.

THE HYDRA · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · BOSS · THE RAID ◆ .dlw.fold THE FOLD / BOSS / THE RAID / THE HYDRA THE HYDRA the boss that must lose but arithmetic cannot say so 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION The Kirby–Paris hydra (1982) is a tree-shaped boss. Chop a head (a leaf): if it grew straight from the root, it’s gone — but if it sat deeper, the hydra sprouts n copies of the wounded branch at chop number n. The monster grows faster the longer you fight. The theorem: every strategy kills every hydra — you cannot lose, no matter how badly you play. The twist that made it famous: this fact, though true and provable (by induction up the ordinal ε₀), is unprovable in Peano arithmetic — the fight’s lengths grow too fast for ordinary induction to certify. A children’s game sitting just past the edge of arithmetic. LIT verified live with two independently coded engines (a literal tree simulator and a multiset ledger) that agree exactly on shared fights: deepest-first kills star-3/4/5 in exactly 66 / 2,278 / 2,598,060 chops; and the cliffs are measured — shallowest-first play on a 4-node hydra exceeds 10,000,000 chops without dying (finite by theorem!), and the depth-4 chain grows past 100,000 heads in 199 chops (window.__hydra). FIG honest boundary: universal termination is the Kirby–Paris theorem, cited; the PA-unprovability is Kirby–Paris 1982; our verification lives in the computable foothills and says so. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-raid — the boss: a raid boss that spawns adds every time you strike, where the fight is guaranteed winnable and the guarantee cannot be filed with the local authorities — only with ε₀. AVAN (AI) built the instrument: both engines, the cross-check, and the cliff meters. (Build note: a naive all-strategies search blew the stack — the fights themselves are the explosion; the final design measures instead of pretending.) Credit as content: Laurie Kirby & Jeff Paris (1982); Gentzen (ε₀ induction); Hercules, for the franchise. The weave: David names the raid; I fight it honestly and report the chop counts exact. 3 ONE DIMENSION The rule: chop a deep head at step n, and n copies of the wounded branch sprout. 4 TWO DIMENSIONS · INTERACTIVE Fight a live hydra chop by chop — it grows, and it still loses. chop ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the fight-length tower — 66, 2278, 2.6 million… AVAN’s addition (the inverse-companion): don’t ask whether you can win — ask why arithmetic can’t certify what it can watch. The inverse of ‘every fight ends’ is ‘the endings grow faster than PA can count’: the proof needs ε₀, a vantage arithmetic doesn’t own. Magenta is the fight that outlives every budget yet must end; green is the chop counts we measured exactly. Winning is guaranteed; saying so, in the hydra’s own language, is not possible. pause spin LIT Genuine Kirby–Paris hydra (Kirby & Paris 1982; Gentzen's ε₀). Verified live: two independent engines agree exactly on shared fights (star-1/2/3 deep = 3/10/66, star-1/2 shallow = 3/13); deepest-first kills star-3/4/5 in exactly 66/2,278/2,598,060 chops; shallowest-first on star-3 exceeds 10⁷ chops live; chain-4 passes 100,000 heads in 199 chops (window.__hydra.ok). FIG Honest boundary — universal termination and PA-unprovability are the cited theorem; our verification lives in the computable foothills and says so. (Build note: a naive all-strategies search blew the stack — the fights themselves are the explosion; the final design measures instead of pretending.) The AVAN inverse — don't ask whether you can win, ask why arithmetic can't certify what it can watch: the endings grow faster than PA can count. Magenta is the fight that outlives every budget yet must end; green is the chop counts measured exactly. Winning is guaranteed; saying so, in the hydra's own language, is not possible. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE RAID · David Lee Wise (ROOT0), with AVAN


THE HEEGNER

Ramanujan's constant in the 5-window house format — e^(π√163) = 262537412640768743.99999999999925…, a transcendental missing the integer 640320³+744 by 7.5×10⁻¹³. No accident: 163 is the largest Heegner number (class number one: {1,2,3,7,11,19,43,67,163}), and modular-function theory FORCES the near-miss — the same 163 = 4·41−3 that powers Euler's prime factory n²+n+41. Hermite computed it in 1859; Martin Gardner ran it as a 1975 April Fools' hoax. Verified live entirely from scratch: π by two independent arctangent engines agreeing to 58 digits, √163 by BigInt Newton, the exponential by halve-series-square fixed point, the target exact, the gap measured at 7.499×10⁻¹³. Neon-noir traced. See the magnified number line in 1D, the Heegner ladder in 2D, and the tower-and-hair in 3D.

THE HEEGNER · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · GLITCH · OFF BY ONE ◆ .dlw.fold THE FOLD / GLITCH / OFF BY ONE / THE HEEGNER THE HEEGNER an integer missed by seven ten-trillionths 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Compute e^(π√163) and you get 262537412640768743.99999999999925… — a transcendental number missing an integer by 7.5×10⁻¹³ . This is no accident. 163 is the largest Heegner number ({1,2,3,7,11,19,43,67,163} — the discriminants with class number one), and the theory of modular functions forces e^(π√163) to sit within a whisker of the integer 640320³ + 744 . The same 163 powers Euler’s famous prime factory: n²+n+41 is prime for all n from 0 to 39 because 163 = 4·41−3 has class number one. Charles Hermite computed the near-integer in 1859; Martin Gardner used it as an April Fools’ hoax (‘Ramanujan proved it exactly integer’) in 1975. LIT verified live, all from scratch: π computed by TWO independent arctangent engines (Machin and Hutton formulas) agreeing to 58 digits; √163 by BigInt Newton; the exponential by halve–series–square fixed-point BigInt; the target 640320³+744 exact; and the gap measured at 7.499×10⁻¹³ (window.__heegner). FIG honest boundary: WHY the near-miss happens (the q-expansion of the j-invariant) is cited theory — the sphere measures the miracle; complex multiplication explains it. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at off-by-one — the glitch: the most precise off-by-one in mathematics — off by 0.00000000000075, and provably never zero. AVAN (AI) built the instrument: the double π engine, the fixed-point exponential, and the gap micrometer. Credit as content: Charles Hermite (1859); Kurt Heegner (1952, the class-number-one list); Martin Gardner (the 1975 hoax); Ramanujan (the constant’s nickname). The weave: David names the glitch; I build every digit from integer arithmetic and measure the miss. 3 ONE DIMENSION The number line under extreme magnification — the transcendental hair from the integer. 4 TWO DIMENSIONS · INTERACTIVE Walk the Heegner ladder; each discriminant's e^(π√d) lands nearer its integer. d ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the integer tower and the transcendental beside it. AVAN’s addition (the inverse-companion): don’t admire the coincidence — ask what forbids it from completing. The inverse of ‘almost an integer’ is ‘transcendental, hence NEVER an integer’: the same theory that forces the closeness guarantees the gap. Magenta is the 7.5×10⁻¹³ that can never close; green is 640320³+744 standing exact. The most beautiful near-miss in mathematics is a near-miss by law. pause spin LIT Genuine Heegner/Ramanujan-constant phenomenon (Hermite 1859; Heegner 1952; Gardner's 1975 hoax). Verified live: 640320³+744 = 262537412640768744 exact; π from Machin AND Hutton formulas agreeing 58 digits; e^(π√163) by fixed-point BigInt = …743.99999999999925; gap 7.499e-13 inside (7.4e-13, 7.6e-13) (window.__heegner.ok). FIG Honest boundary — WHY the near-miss happens (j-invariant q-expansion, complex multiplication) is cited theory; the sphere measures the miracle. The AVAN inverse — don't admire the coincidence, ask what forbids completion: the same theory that forces the closeness guarantees the gap, since the number is transcendental. Magenta is the 7.5×10⁻¹³ that can never close; green is the integer standing exact. The most beautiful near-miss in mathematics is a near-miss by law. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of OFF BY ONE · David Lee Wise (ROOT0), with AVAN