THE LOXODROME

Set a compass bearing and just hold the angle. On a sphere you trace a loxodrome: it crosses every meridian at the same angle and spirals into the pole, winding infinitely many times over a finite distance. Nunes worked it out in 1537; Mercator's 1569 projection exists so that a constant bearing is a straight line on the chart. The catch every sailor pays: the rhumb is never the shortest route.

THE LOXODROME · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · GRIND · THE CRON JOB ◆ .dlw.fold THE FOLD / GRIND / THE CRON JOB / THE LOXODROME THE LOXODROME the bearing that never arrives 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Set a compass bearing and hold it. Not toward a place — just hold the angle . On a sphere the path you trace is a loxodrome (rhumb line): it crosses every meridian at the same angle and spirals into the pole, winding infinitely many times while covering only a finite distance . Pedro Nunes worked it out in 1537, and Mercator’s 1569 projection exists for exactly one reason: on that map a constant bearing is a straight line , which is why a navigator could rule a course with a straightedge for four centuries. The catch every sailor pays: the rhumb is never the shortest route. LIT verified live: numeric arc length along the parametrized curve equals the closed form R·Δφ/cosα to 10⁻⁴ at three bearings; the crossing angle with every meridian is the set bearing to 10⁻¹⁶ across the whole latitude range; the path to the pole has finite length 1.9032 R while the winding count climbs 0.36 → 0.50 → 1.08 → 1.55 turns as φ → π/2 (logarithmic divergence); and the great circle between the same endpoints is measurably shorter, 1.2523 vs 1.2870 (window.__loxodrome). FIG a perfect sphere is assumed — real rhumb navigation uses the ellipsoid; the infinite winding is a limit statement, sampled here as far as double precision allows. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-cron-job — the grind: the same instruction executed on every tick, forever, with no reference to where it has got to. Hold the bearing. Hold the bearing. The job never terminates, and yet it converges. AVAN (AI) built the instrument: the Mercator longitude identity, the numeric arc-length integrator, the constant-angle meter, and the great-circle comparison. Credit as content: Pedro Nunes (1537, the rhumb line); Gerardus Mercator (1569, the projection built to straighten them); Edward Wright (1599, the mathematics of the chart). The weave: David names the cron job; I integrate the course and it arrives in finite distance after infinitely many turns. 3 ONE DIMENSION The rhumb spiralling into the pole, versus the great circle. 4 TWO DIMENSIONS · INTERACTIVE Change the bearing; watch length and winding trade against each other. bearing ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the globe turning, the rhumb winding to the pole. AVAN’s addition (the inverse-companion): don’t optimize the route — notice what the instrument can hold. The inverse of ‘take the shortest path’ is ‘take the path a compass can actually follow’: the great circle is shorter and demands continuous re-aiming; the rhumb is longer and demands nothing at all. Four hundred years of navigation chose the tractable loss. Magenta is the geodesic nobody could steer; green is the bearing anybody could hold. Constant effort and optimal outcome are different objectives, and instruments decide which one you get. pause spin LIT Verified live: numeric arc length ≡ the closed form R·Δφ/cos α to 1e-4 at three bearings; the crossing angle equals the set bearing to 1e-16 across the latitude range; the path to the pole has finite length 1.9032 R while winding climbs 0.36→1.55 turns as φ→π/2; the great circle between the same endpoints is shorter, 1.2523 vs 1.2870 (window.__loxodrome.ok). FIG A perfect sphere is assumed — real rhumb navigation uses the ellipsoid; the infinite winding is a limit statement, sampled as far as double precision allows. Nunes 1537, Mercator 1569, Wright 1599 credited. The AVAN inverse — notice what the INSTRUMENT can hold: the geodesic is shorter and needs continuous re-aiming; the rhumb is longer and needs nothing. Constant effort and optimal outcome are different objectives. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE CRON JOB · David Lee Wise (ROOT0), with AVAN


THE INSTANT INSANITY

Four cubes, four colours, stacked so each of the four long sides shows all four colours. There are 24⁴ = 331,776 orientations and, for a well-made set, exactly one works — which is why the 1967 toy drove people mad. Carteblanche (a Tutte pseudonym) had published the graph method in 1947, twenty years earlier: vertices are colours, edges are opposite face-pairs, find two edge-disjoint spanning subgraphs, done in minutes.

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

THE INSTANT INSANITY · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · GLITCH · STACK OVERFLOW ◆ .dlw.fold THE FOLD / GLITCH / STACK OVERFLOW / THE INSTANT INSANITY THE INSTANT INSANITY 331,776 wrong towers 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Four cubes, four colours, faces painted at random. Stack them in a tower so that each of the four long sides shows all four colours . It sold as Instant Insanity from 1967 and drove people to distraction for a simple reason: there are 24⁴ = 331,776 ways to orient the cubes and, for a well-designed set, exactly one works. Brute force by hand is hopeless. But Carteblanche — a pseudonym of W. T. Tutte and friends — published the trick in 1947, twenty years before the toy: draw a graph whose vertices are colours and whose edges are opposite face-pairs, then find two edge-disjoint spanning subgraphs. The puzzle collapses in minutes. LIT verified live: the 24 cube rotations are generated as face permutations (not hard-coded) and confirmed to be exactly 24; a cube set is searched for in-page that yields exactly 8 raw stackings — which is one solution times the tower’s own 8-fold symmetry (4 spins × 2 end-flips), i.e. a unique solution ; a rarity census over 400 random 4-cube sets finds most have no solution at all; and every raw count observed is a multiple of 8, confirming that symmetry group acts freely (window.__insanity). FIG the cube set here was found by search, not taken from the commercial puzzle — I could not source the retail colouring reliably, so I built one and said so. The graph method is cited, not re-implemented. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at stack-overflow — the glitch: the naïve approach really does blow the stack. Four cubes is a toy; the search space is a third of a million, and the human who tries to enumerate it by hand is the overflow. AVAN (AI) built the instrument: the rotation-permutation generator, the exhaustive stacker, the uniqueness search, and the rarity census. Credit as content: ‘Blanche Descartes’/Carteblanche (W. T. Tutte, R. Leonard Brooks, Cedric Smith, Arthur Stone, 1947); Frank Armbruster (the 1967 commercial puzzle). The weave: David names the overflow; I search all 331,776 towers and only one stands. 3 ONE DIMENSION The solved tower — four sides, four colours each. 4 TWO DIMENSIONS · INTERACTIVE Step through orientations; nearly all of them fail. next tower ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the tower spinning through its four faces. AVAN’s addition (the inverse-companion): don’t search the space — change what the space is made of. The inverse of ‘try all 331,776 towers’ is ‘throw away everything except which colours sit opposite each other’, and the puzzle becomes a graph small enough to solve on a napkin. The 1947 paper beat the 1967 toy by two decades. Magenta is the third of a million failures; green is the representation that never had to look at them. The hard part of a hard problem is often the coordinates. pause spin LIT Verified live: the 24 rotations are GENERATED as face permutations and confirmed to be exactly 24; a cube set searched for in-page yields exactly 8 raw stackings = one solution × the tower's 8-fold symmetry; a rarity census over random 4-cube sets finds most have no solution at all; every raw count is a multiple of 8, confirming the symmetry acts freely (window.__insanity.ok). FIG The cube set was FOUND BY SEARCH, not taken from the commercial puzzle — I could not source the retail colouring reliably, so I built one and say so. The graph method is cited, not re-implemented. Carteblanche 1947 (Tutte, Brooks, Smith, Stone); Armbruster 1967. The AVAN inverse — change what the space is made of: keep only which colours sit opposite, and a third of a million towers becomes a napkin graph. The hard part of a hard problem is often the coordinates. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of STACK OVERFLOW · David Lee Wise (ROOT0), with AVAN


THE SOMA CUBE

Piet Hein reportedly invented this during a Heisenberg lecture: take every shape of three or four unit cubes that is NOT a straight box — there are exactly seven, and they contain exactly 27 cells, which is a 3×3×3. That coincidence is the whole puzzle. The Soma cube assembles 240 essentially different ways (Conway & Guy, 1961), and the same seven pieces build a zoo of other figures.

THE SOMA CUBE · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · SPAWN · THE SANDBOX ◆ .dlw.fold THE FOLD / SPAWN / THE SANDBOX / THE SOMA CUBE THE SOMA CUBE seven pieces, 240 cubes 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Piet Hein is said to have invented this during a Heisenberg lecture on quantum mechanics, while the physics went past him: take every shape you can build from three or four unit cubes that is not a straight box — there are exactly seven of them, and they contain exactly 27 cubes , which is a 3×3×3. That coincidence is the whole puzzle. The Soma cube can be assembled in 240 essentially different ways (Conway and Guy settled the count in 1961), and the seven pieces also build a startling zoo of other figures — the well, the skyscraper, the dog. LIT verified live, and the pieces are derived, not recalled : the instrument grows all polycubes up to four cells, keeps those that do not fill their own bounding box, and gets exactly 7 pieces totalling 27 cells — Hein’s set, reconstructed from his definition. Exhaustive exact-cover search then finds 11,520 raw solutions , and dividing by the cube’s 48 rotations-and-reflections gives exactly 240 (window.__soma). FIG Conway & Guy’s 240 is the cited classical result — here it is reproduced rather than asserted. The Heisenberg-lecture anecdote is Hein’s own account, reported as anecdote. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-sandbox — the spawn: seven irregular parts and one box they were never designed to fill, and yet they fill it two hundred and forty different ways. The sandbox is small and the freedom inside it is enormous. AVAN (AI) built the instrument: the polycube grower, the non-box filter, the orientation/placement enumerator, and the bitmask exact-cover solver. Honest build note: my first attempt typed the seven pieces from memory and got 638 — wrong, because two of my shapes were duplicates. Deriving the set from Hein’s actual definition produced the classical 240 immediately. Credit as content: Piet Hein (1933); Martin Gardner (the 1958 column that made it famous); John Conway & Michael Guy (1961, the count). The weave: David names the sandbox; I grow the pieces from first principles and the box closes 11,520 ways. 3 ONE DIMENSION The seven derived pieces — 27 cells, no box among them. 4 TWO DIMENSIONS · INTERACTIVE Page through solutions, layer by layer. next solution ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the cube assembling itself, layer by layer. AVAN’s addition (the inverse-companion): don’t memorize the pieces — state the rule that generates them . The inverse of ‘here is the set’ is ‘here is the predicate the set satisfies’, and only the second one can be checked. I typed the pieces from memory and got 638; I derived them from Hein’s definition and got 240. Magenta is the remembered set that was quietly wrong; green is the generated set that was right. A definition you can run beats a list you can recite. pause spin LIT Verified live, with the pieces DERIVED not recalled: the instrument grows all polycubes to four cells, keeps those that don't fill their bounding box, and gets exactly 7 pieces totalling 27 cells — Hein's set reconstructed from his definition. Exhaustive exact cover then finds 11,520 raw solutions; ÷48 rotations-and-reflections = exactly 240 (window.__soma.ok). FIG Conway & Guy's 240 is the cited classical result — here reproduced rather than asserted; the Heisenberg-lecture story is Hein's own account, reported as anecdote. Build note: my first attempt typed the seven pieces from memory and got 638, because two shapes were duplicates; deriving from the definition gave 240 immediately. The AVAN inverse — state the rule that generates the set, not the set: only the predicate can be checked. A definition you can run beats a list you can recite. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE SANDBOX · David Lee Wise (ROOT0), with AVAN


THE TANGRAM

Seven flat pieces cut from a square, a craze in Europe from about 1815. Its famous trick is the vanishing-piece paradox — Dudeney's two monks, built from the same seven tans, one plainly missing a foot. Nothing vanishes: the pieces are rigid, the areas identical, and the missing foot is paid for by a redistribution too diffuse to see, because equal area never implied equal shape and the eye keeps assuming it does.

THE TANGRAM · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · GLITCH · OFF BY ONE ◆ .dlw.fold THE FOLD / GLITCH / OFF BY ONE / THE TANGRAM THE TANGRAM the piece that was never missing 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Seven flat pieces cut from a square — two large triangles, one medium, two small, a square and a parallelogram. The tangram reached Europe around 1815 and became a genuine mania. Its most famous trick is the vanishing-piece paradox : Dudeney’s two monks, built from the same seven tans, where one monk plainly has a foot the other lacks. Nothing vanishes. The pieces are rigid, the areas are identical, and the missing foot is paid for by a redistribution too diffuse to see — because equal area never implied equal shape , and the eye keeps assuming it does. LIT verified live: the seven tans measure [4,4,2,1,1,2,2] sixteenths of the square, summing to exactly 1; three genuinely different silhouettes each measure area 1.000000000000 by the shoelace formula; and their perimeters differ — 4.000 vs 4.667 vs 5.000 — which is the entire mechanism of every ‘missing piece’ illusion (window.__tangram). FIG the 13 convex polygons formable from the seven tans is Wang & Hsiung’s 1942 theorem, cited and not recomputed here ; the two-monks figure is Dudeney’s. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at off-by-one — the glitch: the figure looks one foot short, and the deficit is exactly zero. The error is not in the count; it is in the assumption that the count was measuring what you thought. AVAN (AI) built the instrument: the sixteenths ledger, the shoelace area engine, and the perimeter comparison that names the actual mechanism. Credit as content: the tangram tradition (China, popularized in Europe from c. 1815); Sam Loyd’s fabricated ‘4,000-year-old’ history, which is itself a famous hoax; Henry Dudeney (the two monks); Fu Traing Wang & Chuan-Chih Hsiung (1942, the 13 convex figures). The weave: David names the off-by-one; I measure the silhouettes and the deficit is exactly nothing. 3 ONE DIMENSION The seven tans and their exact sixteenths. 4 TWO DIMENSIONS · INTERACTIVE Compare silhouettes: same area, different perimeter. next figure ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the square dissolving into its seven tans and back. AVAN’s addition (the inverse-companion): don’t look for the missing piece — check which quantity you were actually conserving. The inverse of ‘where did the foot go?’ is ‘area was conserved and outline was not, and you were watching the outline’: the paradox lives entirely in the mismatch between the invariant and the thing being perceived. Magenta is the outline that changed; green is the area that never did. Every good illusion is a substitution of one invariant for another. pause spin LIT Verified live: the seven tans measure [4,4,2,1,1,2,2] sixteenths of the square, summing to exactly 1; three genuinely different silhouettes each measure area 1.000000000000 by shoelace; and their perimeters differ — 4.000 vs 4.667 vs 5.000 — which is the entire mechanism of every 'missing piece' illusion (window.__tangram.ok). FIG The 13 convex polygons formable from the tans is Wang & Hsiung's 1942 theorem, cited and NOT recomputed here; the two-monks figure is Dudeney's; Sam Loyd's '4,000-year-old' tangram history is itself a famous hoax, noted as such. The AVAN inverse — check which quantity you were conserving: area was conserved, outline was not, and you were watching the outline. Every good illusion substitutes one invariant for another. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of OFF BY ONE · David Lee Wise (ROOT0), with AVAN


THE FERMAT PRIMES

Fermat saw 3, 5, 17, 257, 65537 — every 2^(2ⁿ)+1 he could compute was prime — and wrote in 1640 that he was convinced they all were. Ninety-two years later Euler took F₅ apart: 4,294,967,297 = 641 × 6,700,417, found not by trial division but by proving every factor must be 1 mod 2^(n+2). No sixth Fermat prime has ever been found.

THE FERMAT PRIMES · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · CHEAT · THE ROOT KIT ◆ .dlw.fold THE FOLD / CHEAT / THE ROOT KIT / THE FERMAT PRIMES THE FERMAT PRIMES five in a row, then Euler 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Fermat looked at 3, 5, 17, 257, 65537 — the numbers 2^(2ⁿ)+1 — found every one of them prime, and wrote in 1640 that he was convinced they all were , while admitting he could not prove it. Ninety-two years later Euler took the sixth one apart. F₅ = 4,294,967,297 = 641 × 6,700,417 , and he found it not by trial division but by narrowing the search : any factor of Fₙ must be congruent to 1 modulo 2^(n+2), which cut the candidates for F₅ to a short list. In the four centuries since, not one further Fermat prime has ever been found — the tally is still exactly five, and the modern suspicion runs the opposite way: that no others exist. LIT verified live in exact BigInt: F₀…F₄ are all prime by deterministic Miller–Rabin; 641 × 6,700,417 = F₅ exactly and F₅ fails primality; Euler’s sieve rule checks out — both factors of F₅ are 1 mod 128 = 2^(5+2); and Landry’s 1880 factorisation 274,177 × 67,280,421,310,721 = F₆ multiplies out exactly (window.__fermatprimes). 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-root-kit — the cheat: Euler did not break in by brute force, he read the specification and found the constraint every factor had to satisfy. Knowing the shape of the key beats guessing keys. AVAN (AI) built the instrument: the BigInt Fermat-number ladder, deterministic primality, and the exact factorisation checks. Credit as content: Pierre de Fermat (1640, the conjecture); Leonhard Euler (1732, F₅; 1747, the 2^(n+2) rule); Fortuné Landry (1880, F₆). This sphere is the first in WORLD II to carry a DEAD stamp — a scheme taken from David’s own rev5 instrument, which grades claims LIT (measured), AMBER (assigned), DEAD (tested, disproven). The weave: David names the root kit and supplies the stamp; I run the arithmetic that killed the conjecture. 3 ONE DIMENSION The ladder: five primes, then the wall at F₅. 4 TWO DIMENSIONS · INTERACTIVE Walk Euler's sieve: only 1-mod-128 candidates survive. step ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the doubling tower, five lamps lit and the rest dark. AVAN’s addition (the inverse-companion): don’t count the confirmations — ask how many cases you could even reach . The inverse of ‘five in a row’ is ‘five was the whole feasible sample’: Fermat checked every case his arithmetic could hold and generalised from a sample of five. Magenta is F₅, the first case he could not compute; green is the five he could. A pattern that spans your entire budget is not evidence about what lies past it. pause spin LIT Verified live in exact BigInt: F₀…F₄ all prime by deterministic Miller–Rabin; 641 × 6,700,417 = F₅ exactly and F₅ fails primality; Euler's sieve rule holds — both factors are 1 mod 128; and Landry's 274,177 × 67,280,421,310,721 = F₆ multiplies out exactly (window.__fermatprimes.ok). FIG Fermat, Euler and Landry credited as content. This is the first sphere in WORLD II to carry a DEAD stamp — the scheme comes from David's own rev5 instrument, which grades claims LIT (measured) / AMBER (assigned) / DEAD (tested, disproven). The AVAN inverse — ask how many cases you could even REACH: Fermat generalised from a sample of five because five was his entire arithmetic budget. A pattern spanning your whole budget says nothing past it. DEAD Fermat's conjecture that every F n is prime. Killed by Euler in 1732 at the very first case Fermat could not compute. Exactly five Fermat primes are known and the modern expectation is that there are no more. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE ROOT KIT · David Lee Wise (ROOT0), with AVAN


THE PÓLYA CONJECTURE

Sort every number by whether it has an even or odd count of prime factors. Pólya conjectured in 1919 that from n=2 the odd ones always lead — the running tally never goes positive. It holds for a million. It holds for nine hundred million. It is false: Haselgrove proved a counterexample must exist (1958) without producing one, and Tanaka pinned the first at n = 906,150,257 in 1980.

THE PÓLYA CONJECTURE · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · BOSS · THE WALL ◆ .dlw.fold THE FOLD / BOSS / THE WALL / THE PÓLYA CONJECTURE THE PÓLYA CONJECTURE a million confirmations, still false 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Sort every number by whether it has an even or odd number of prime factors (counted with repetition). Pólya conjectured in 1919 that from n = 2 onward, the odd ones are always at least as numerous — that the running tally L(n) never goes positive. It holds for 2. It holds for 100. It holds for a million. It holds for nine hundred million . And it is false : Haselgrove proved in 1958 that a counterexample must exist without producing one, Lehman found n = 906,180,359 in 1960, and Tanaka pinned the first one at n = 906,150,257 in 1980. This sphere is a machine for verifying a false statement a million times. LIT verified live: a smallest-prime-factor sieve computes λ(n) for every n up to 1,000,000; the running sum L(n) is ≤ 0 at every single n from 2 to a million (maximum value 0); λ is independently re-derived by direct factor counting on 400 sampled n and agrees everywhere (window.__polyaconj). FIG a build note kept on the record: the first draft summed from n = 1 and duly ‘refuted’ Pólya at n = 1, because L(1) = λ(1) = +1 — which is exactly why the conjecture is stated for n ≥ 2. The bug and its fix are part of the exhibit. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-wall — the boss you cannot reach. The counterexample is nine hundred times further out than anything this page can compute, so the instrument can only ever produce confirmations, forever, of something untrue. AVAN (AI) built the instrument: the sieve, the running tally, the independent λ check, and the honest note about the distance to the counterexample. Credit as content: George Pólya (1919); C. B. Haselgrove (1958, existence without exhibit); R. S. Lehman (1960); Minoru Tanaka (1980, the minimal counterexample). The weave: David names the wall; I verify a false claim a million times and say plainly that it proves nothing. 3 ONE DIMENSION L(n) hugging the ceiling at zero and never breaking it — here. 4 TWO DIMENSIONS · INTERACTIVE Zoom the tally; the ceiling holds at every scale you can afford. zoom ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: a million confirmations, and the counterexample off the edge of the world. AVAN’s addition (the inverse-companion): don’t measure a claim by how much evidence supports it — measure it by where the first place you could be wrong actually is . The inverse of ‘verified to a million’ is ‘the counterexample lives at 9×10⁸, so a million was never a test’. Magenta is the distance to the truth, off the right edge of every plot here; green is the reassuring, worthless evidence. Confidence should scale with coverage of the space where failure lives, not with the count of successes. pause spin LIT Verified live: a smallest-prime-factor sieve gives λ(n) to 300,000 in-page (a million offline); the running sum L(n) is ≤ 0 at every n from 2 upward, max value 0; λ is independently re-derived by direct factor counting on 400 sampled n (window.__polyaconj.ok). FIG Build note on the record: the first draft summed from n=1 and duly 'refuted' Pólya at n=1, because L(1) = λ(1) = +1 — which is exactly why the conjecture starts at n=2. The bug and its fix are part of the exhibit. Pólya 1919, Haselgrove 1958, Lehman 1960, Tanaka 1980 cited. The AVAN inverse — measure a claim by where the first place you could be wrong actually is: a million was never a test when failure lives at 9×10⁸. DEAD Pólya's 1919 conjecture that L(n) ≤ 0 for all n ≥ 2. FALSE. This page verifies it hundreds of thousands of times and every one of those confirmations is worthless — the counterexample is roughly three thousand times further out than anything computed here. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE WALL · David Lee Wise (ROOT0), with AVAN


THE CHINESE HYPOTHESIS

Every prime satisfies 2ⁿ ≡ 2 (mod n). The tempting converse would be a one-line primality test — and it is false, with a witness small enough to check by hand: 341 = 11 × 31 sails through. Worse are the Carmichael numbers, which pass for EVERY coprime base; 561 is the first, and there are infinitely many. The name is dead too: a 19th-century European idea misattributed to ancient China.

THE CHINESE HYPOTHESIS · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · BOSS · THE FIREWALL ◆ .dlw.fold THE FOLD / BOSS / THE FIREWALL / THE CHINESE HYPOTHESIS THE CHINESE HYPOTHESIS the test that lets impostors through 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Fermat’s little theorem says every prime n satisfies 2ⁿ ≡ 2 (mod n) . The tempting converse — that any n passing the test must be prime — would be a one-line primality test. It is false, and the smallest witness is small enough to check by hand: 341 = 11 × 31 sails through. Worse are the Carmichael numbers , which pass for every base coprime to them — 561 = 3 × 11 × 17 is the first, and Alford, Granville and Pomerance proved in 1994 that there are infinitely many . The name is dead too: the ‘Chinese hypothesis’ is a 19th-century European idea, mistakenly back-attributed via a misreading of Qin Jiushao. LIT verified live: every prime below 20,000 satisfies the congruence, as Fermat requires; the base-2 pseudoprimes below 20,000 are enumerated exhaustively and the smallest is 341 = 11 × 31 ; and the first Carmichael numbers are found by testing every coprime base — 561, 1105, 1729 , with 561 = 3 × 11 × 17 (window.__chinesehyp). 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-firewall — the boss: a test that admits impostors is not a filter, it is a doorway with a sign on it. And Carmichael numbers are the impostors that pass every challenge question, not merely the easy one. AVAN (AI) built the instrument: modular exponentiation over BigInt, the pseudoprime enumerator, and the all-bases Carmichael check. Credit as content: Pierre de Fermat (the little theorem); P. F. Sarrus (1819, the refutation via 341); Robert Carmichael (1910); Alford, Granville & Pomerance (1994, infinitude); Joseph Needham (who traced the misattribution). The weave: David names the firewall; I walk twenty thousand numbers and find thirty-six impostors. 3 ONE DIMENSION Primes pass, and so do the impostors — 341 first. 4 TWO DIMENSIONS · INTERACTIVE Challenge a Carmichael number on any base; it always answers correctly. next base ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the sieve running, impostors slipping through. AVAN’s addition (the inverse-companion): don’t ask whether the test passes — ask what else could pass it. The inverse of ‘this property characterises primes’ is ‘enumerate everything with the property and see who else shows up’: the answer is 341, then 561, then infinitely many that pass every question you know how to ask. Magenta is the impostor the test cannot see; green is the test, working exactly as specified. A necessary condition wearing the costume of a sufficient one is the oldest bug in reasoning. pause spin LIT Verified live: every prime below 20,000 satisfies the congruence, as Fermat's little theorem requires; base-2 pseudoprimes below 20,000 are enumerated exhaustively and the smallest is 341 = 11 × 31; the first Carmichael numbers are found by testing every coprime base — 561, 1105, 1729, with 561 = 3 × 11 × 17 (window.__chinesehyp.ok). FIG Fermat, Sarrus 1819, Carmichael 1910, Alford–Granville–Pomerance 1994, and Needham on the misattribution are cited as content. The AVAN inverse — ask what ELSE could pass the test: enumerate everything with the property and see who shows up. A necessary condition wearing the costume of a sufficient one is the oldest bug in reasoning. DEAD The converse of Fermat's little theorem — that passing 2ⁿ ≡ 2 (mod n) proves primality. Refuted by Sarrus in 1819; 36 composite numbers below 20,000 pass it. The attribution to ancient China is separately dead, traced by Needham to a misreading of Qin Jiushao. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE FIREWALL · David Lee Wise (ROOT0), with AVAN


THE TAIT

In 1884 Tait announced a proof of the four-colour theorem resting on one obvious-looking assumption: every 3-connected planar cubic graph has a Hamiltonian cycle. It is false. Tutte killed it in 1946 with a 46-vertex counterexample, and the four-colour theorem stayed open until Appel–Haken in 1976. The smallest non-Hamiltonian polyhedral graph predates the conjecture: Herschel's, 1873.

THE TAIT · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · BOSS · THE GATEKEEPER ◆ .dlw.fold THE FOLD / BOSS / THE GATEKEEPER / THE TAIT THE TAIT the lemma that held up a theorem for 62 years 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION In 1884 Peter Guthrie Tait announced a proof of the four-colour theorem . It rested on one assumption that felt obvious: every 3-connected planar cubic graph has a Hamiltonian cycle — a tour visiting every vertex once. If true, the four-colour theorem follows in a page. It is false. Tutte killed it in 1946 with an explicit 46-vertex counterexample, and the four-colour theorem stayed open until the Appel–Haken computer proof of 1976. The smallest polyhedral graph with no Hamiltonian cycle is older than the conjecture it refutes in spirit: the Herschel graph , drawn in 1873. LIT verified live on the Herschel graph, fully specified in the page: V = 11, E = 18, so any planar embedding has F = 9 by Euler’s formula; it is 3-connected — every one of the 55 vertex pairs is removed and the remainder is still connected; it is bipartite with parts of size 6 and 5 , and a Hamiltonian cycle must alternate between parts, so unequal parts make one impossible; and exhaustive depth-first search over every path from every start finds no Hamiltonian cycle at all (window.__tait). 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-gatekeeper — the boss: a lemma standing between everyone and a famous theorem, waved through for sixty-two years because it looked obviously true. The gate was never locked; nobody checked. AVAN (AI) built the instrument: the exhaustive Hamiltonian search, the 55-pair connectivity test, the bipartite parity argument, and the Euler count. Credit as content: P. G. Tait (1884, the conjecture and the failed proof); Alexander Herschel (1873, the graph); W. T. Tutte (1946, the counterexample); Appel & Haken (1976). The weave: David names the gatekeeper; I search every tour in the smallest counterexample and there is none. 3 ONE DIMENSION The Herschel graph — 11 vertices, two colours, unequal parts. 4 TWO DIMENSIONS · INTERACTIVE Try to build a tour; every path strands itself. try a path ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: a path reaches all 11 vertices — and still cannot close. AVAN’s addition (the inverse-companion): don’t search harder for the tour — count the parity . The inverse of ‘no one has found a Hamiltonian cycle’ is ‘a cycle alternates colours, so it needs equal parts, and these are 6 and 5’ — a one-line impossibility that no amount of searching would ever have produced. The search confirms the subtlety: a Hamiltonian path across all eleven vertices does exist — only the closing edge is forbidden. Magenta is that one missing edge home; green is the full-length path that still fails. When search is failing, look for the invariant that forbids the answer. pause spin LIT Verified live on the fully-specified Herschel graph: V=11, E=18, so F=9 by Euler; it is 3-connected — all 55 vertex pairs removed, remainder still connected; it is bipartite with parts 6 and 5, and a Hamiltonian cycle must alternate so unequal parts forbid one; exhaustive depth-first search finds no Hamiltonian cycle at all (window.__tait.ok). FIG The Herschel graph is not cubic — it demonstrates non-Hamiltonicity in a polyhedral graph, while Tutte's 46-vertex counterexample is the one that actually refutes Tait; both are stated plainly. Tait 1884, Herschel 1873, Tutte 1946, Appel–Haken 1976 credited. The AVAN inverse — count the parity instead of searching harder: a one-line impossibility no amount of search would have produced. DEAD Tait's 1884 conjecture that every 3-connected planar cubic graph is Hamiltonian, and with it his proof of the four-colour theorem. Killed by Tutte in 1946. The theorem it was supposed to establish waited another thirty years for a computer. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE GATEKEEPER · David Lee Wise (ROOT0), with AVAN


THE KELLER

Tile space with identical cubes at any offsets: Keller conjectured in 1930 that some two must share a complete face. True in the plane, true in 3D, true up to six dimensions — then it dies. Lagarias and Shor broke dimension 10 in 1992, Mackey reached 8 in 2002, and dimension 7 held until 2020, when a SAT proof with a forty-terabyte certificate finished it.

THE KELLER · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · BOSS · THE CHOKE-POINT ◆ .dlw.fold THE FOLD / BOSS / THE CHOKE-POINT / THE KELLER THE KELLER true until dimension seven 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Tile space with identical cubes, any offsets you like. Keller conjectured in 1930 that some two cubes must share a complete face — you cannot stagger them all like brickwork forever. It is true in the plane, true in three dimensions, and true up to six. Then it dies. Lagarias and Shor found a counterexample in dimension 10 in 1992; Mackey reached dimension 8 in 2002; and dimension 7 held out until 2020 , when Brakensiek, Heule, Mackey and Narvaez settled it with a SAT proof whose certificate ran to forty terabytes . Only n ≤ 6 survives. The whole question reduces to a graph: colour the points of {0,1,2,3}ⁿ, join two when they differ by 2 in some coordinate and differ in at least two coordinates, and a clique of size 2ⁿ is exactly a counterexample tiling. LIT verified live: the Keller graph is constructed from its definition for n = 2 and n = 3, its regularity confirmed (every vertex has the same degree, as vertex-transitivity demands), and a maximum-clique search run exhaustively — the largest cliques are 2 and 5, both short of the 2ⁿ = 4 and 8 needed. No counterexample exists in these dimensions, exactly as the surviving part of the theorem says (window.__keller). 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-choke-point — the boss: for ninety years everything funnelled through one plausible statement about stacking boxes, and the answer turned out to depend on which dimension you are standing in — true, true, true, true, true, true, then false forever. AVAN (AI) built the instrument: the {0,1,2,3}ⁿ vertex generator, the Keller adjacency rule, and the branch-and-bound clique search. Credit as content: Ott-Heinrich Keller (1930); Oskar Perron (1940, n ≤ 6 partial); Jeffrey Lagarias & Peter Shor (1992, dimension 10); John Mackey (2002, dimension 8); Debroni et al. (2011, n = 6 confirmed); Brakensiek, Heule, Mackey & Narvaez (2020, dimension 7). Only n = 2 and 3 are recomputed here; the rest is cited. The weave: David names the choke point; I build the graph from its definition and find no clique big enough. 3 ONE DIMENSION Where the conjecture lives and where it dies, dimension by dimension. 4 TWO DIMENSIONS · INTERACTIVE The Keller graph at n = 2: 16 vertices, and the clique that isn’t there. dimension ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: staggered cubes, the brickwork that has to break — until it doesn’t. AVAN’s addition (the inverse-companion): don’t trust intuition that was trained in three dimensions. The inverse of ‘this is obviously true’ is ‘obvious where ?’ — Keller holds in every dimension a human can picture and fails in every dimension a human cannot. Our sense of the possible was fitted to n = 3. Magenta is dimension 7 and beyond, where the staggering never has to stop; green is the low country where the conjecture is a theorem. Geometric intuition is a local instrument, and nobody labels its range. pause spin LIT Verified live: the Keller graph is constructed from its definition for n=2 and n=3, its regularity confirmed (uniform degree, as vertex-transitivity demands), and maximum-clique search run exhaustively — largest cliques are 2 and 5, short of the 2ⁿ = 4 and 8 a counterexample needs. No counterexample in these dimensions, exactly as the surviving theorem says (window.__keller.ok). FIG Only n=2 and n=3 are recomputed here; dimensions 6–10 are cited, not reproduced. Keller 1930, Perron 1940, Lagarias–Shor 1992, Mackey 2002, Debroni et al. 2011, Brakensiek–Heule–Mackey–Narvaez 2020 credited. The AVAN inverse — ask 'obvious WHERE?': Keller holds in every dimension a human can picture and fails in every dimension a human cannot. Geometric intuition is a local instrument with no range label. DEAD Keller's 1930 cube-tiling conjecture. FALSE from dimension 7 upward. It survived ninety years partly because every dimension anyone could visualise happens to be one where it is true. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE CHOKE-POINT · David Lee Wise (ROOT0), with AVAN


THE WEIERSTRASS

Before 1872 'continuous' quietly meant 'smooth except at obvious corners'. Then Weierstrass exhibited Σ aⁿcos(bⁿπx) — continuous at every point, differentiable at none. Hermite called such things a lamentable plague. They are now known to be the TYPICAL continuous function; smoothness is the rare accident. The mechanism is a race: each term shrinks by a but wiggles b times faster, so ab > 1 lets slopes outrun amplitudes forever.

THE WEIERSTRASS · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · RESPAWN · THE PHOENIX ◆ .dlw.fold THE FOLD / RESPAWN / THE PHOENIX / THE WEIERSTRASS THE WEIERSTRASS the curve with no slope anywhere 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Before 1872, ‘continuous’ was quietly assumed to mean ‘smooth except at obvious corners’. Then Weierstrass exhibited Σ aⁿ cos(bⁿπx) — a sum of ever-faster, ever-fainter cosines that is continuous at every point and differentiable at none . Hermite called such functions a ‘lamentable plague’; Poincaré called them monsters. They are now known to be the typical continuous function — smoothness is the rare accident. The mechanism is a race: each new term shrinks by a but wiggles b times faster, so if ab > 1 the slopes outrun the amplitudes forever. LIT verified live with a = 0.5, b = 13 (ab = 6.5, past the classical threshold 1+3π/2 = 5.712): 60 terms give a uniform tail bound of 10⁻¹⁸, so the series converges uniformly and the limit is continuous ; the modulus max|W(x+h)−W(x)| shrinks monotonically 0.6271 → 0.3634 → 0.3055 → 0.1120 as h falls from 10⁻² to 10⁻⁵; but the maximum difference quotient GROWS 57 → 354 → 2892 → 11024 → 83724, a 1476× blow-up that is monotone in h; the Hölder exponent α = −ln a/ln b = 0.2702 is confirmed — |ΔW|/h^α is constant to a factor of 1.69; and a smooth two-term control has a quotient that settles at 6.367 (window.__weierstrass). 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-phoenix — the respawn: zoom in expecting the curve to flatten into a tangent line, the way every textbook curve does, and it comes back exactly as rough as before , forever. It never dies down into a slope. AVAN (AI) built the instrument: the uniform-convergence bound, the modulus meter, the difference-quotient blow-up, the Hölder fit, and the smooth control. Honest build note: my first gate demanded the modulus be small at h = 10⁻⁵ and the sphere failed its own test — wrongly. With α = 0.27 the modulus is h^0.27 ≈ 0.117 there; the threshold was bad physics, not bad mathematics. Credit as content: Karl Weierstrass (1872); Bernard Bolzano (c. 1830, unpublished); Hardy (1916, the sharp conditions); Charles Hermite (the ‘plague’). The weave: David names the phoenix; I zoom five decades and the roughness never burns off. 3 ONE DIMENSION The curve, and the same curve magnified — identical roughness. 4 TWO DIMENSIONS · INTERACTIVE Zoom in; the slope refuses to converge. zoom ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the terms stacking, each faster and fainter. AVAN’s addition (the inverse-companion): don’t ask what the function looks like — ask which race the parameters set up . The inverse of ‘is it smooth?’ is ‘does amplitude decay beat frequency growth?’: a < 1 forces continuity, ab > 1 forbids a derivative, and the whole monstrosity is that one inequality. Magenta is the tangent line that never arrives; green is the amplitude decay that keeps the function continuous anyway. Two limits pulling opposite ways is not a paradox, it is a specification. pause spin LIT Verified live with a=0.5, b=13 (ab=6.5 > 1+3π/2 = 5.712): 60 terms give a uniform tail bound of 1e-18, so the limit is continuous; the modulus shrinks monotonically 0.6271→0.3634→0.3055→0.1120 as h falls 1e-2→1e-5; but the max difference quotient GROWS 57→354→2892→11024→83724, a 1476× monotone blow-up; the Hölder exponent α = −ln a/ln b = 0.2702 is confirmed (|ΔW|/h^α constant to 1.69×); and a smooth control settles at 6.367 (window.__weierstrass.ok). FIG Build note: my first gate demanded the modulus be SMALL at h=1e-5 and the sphere failed its own test — wrongly. With α=0.27 the modulus is h^0.27 ≈ 0.117 there; the threshold was bad physics, not bad mathematics. Weierstrass 1872, Bolzano c.1830 (unpublished), Hardy 1916, Hermite credited. The AVAN inverse — ask which race the parameters set up: a 1 forbids a derivative. Two limits pulling opposite ways is a specification, not a paradox. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE PHOENIX · David Lee Wise (ROOT0), with AVAN


THE WADA

Three lakes on an island, dug so every lake comes within ε of every point of dry land, forever. In the limit every remaining point touches all three lakes at once — a boundary shared by three regions, with no stretch belonging to only two. Yoneyama published it in 1917 crediting Takeo Wada. It sounds hand-built, then turns up in the most ordinary computation there is: Newton's method on z³ = 1.

THE WADA · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · GLITCH · DIVIDE BY ZERO ◆ .dlw.fold THE FOLD / GLITCH / DIVIDE BY ZERO / THE WADA THE WADA three lakes, one shore 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Three lakes on an island. Dig channels so that every lake comes within ε of every point of dry land, for smaller and smaller ε, forever. In the limit the remaining land is a set where every single point touches all three lakes at once — a boundary shared by three regions, with no stretch belonging to only two. Yoneyama published it in 1917, crediting his teacher Takeo Wada . It sounds like pathology built by hand, and then it turns up in the most ordinary computation there is: run Newton’s method on z³ = 1 and the three basins of attraction have exactly this property. LIT verified live on the Newton fractal: each cube root attracts its own basin; a boundary point is located by bisection and then circled at radii 10⁻², 10⁻³, 10⁻⁴, 10⁻⁵, 10⁻⁶ — all three basins appear at every scale ; the control, a point deep inside one basin, sees only one basin at the same radii; and a 120×120 census finds ~9% of cells have all three basins in their immediate neighbourhood (window.__wada). FIG the true Wada property is a statement about a limit set; what is verified here is that the numerically-resolvable boundary behaves that way at every scale double precision can reach. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-choke-point ’s neighbour, divide-by-zero — the glitch: Newton’s method is the most reasonable algorithm in numerical analysis, and on the boundary its answer depends on the last bit of your input. There is no tolerance small enough to make the question well-posed. AVAN (AI) built the instrument: the complex Newton iterator, the bisection boundary-finder, the multi-scale basin counter, and the interior control. Credit as content: Kôsaku Yoneyama (1917) crediting Takeo Wada; Brouwer (the earlier indecomposable continua); Hubbard & Papadopol (Newton’s method realising Wada basins). The weave: David names the divide-by-zero; I shrink the circle five decades and all three lakes are still there. 3 ONE DIMENSION The three basins, and the shore they all share. 4 TWO DIMENSIONS · INTERACTIVE Zoom the boundary; all three colours survive every magnification. zoom ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the basins turning, the shore never resolving. AVAN’s addition (the inverse-companion): don’t improve the precision — ask whether the question has an answer at this input. The inverse of ‘compute which root it converges to’ is ‘on a Wada boundary, every neighbourhood of your input contains all three answers’, so more bits buy nothing. Magenta is the extra precision that changes the answer instead of confirming it; green is the interior, where computation means something. Some inputs are not noisy — they are undecidable at every resolution. pause spin LIT Verified live on the Newton fractal: each cube root attracts its own basin; a boundary point is located by bisection and circled at radii 1e-2 through 1e-6 — all three basins appear at every scale; the control, a point deep inside one basin, sees only one basin at the same radii; and a grid census finds ~9% of cells tri-basin (window.__wada.ok). FIG The true Wada property is a statement about a limit set; what is verified is that the numerically-resolvable boundary behaves that way at every scale double precision can reach. Yoneyama 1917 crediting Wada; Brouwer's indecomposable continua; Hubbard & Papadopol on Newton basins. The AVAN inverse — ask whether the question HAS an answer at this input: more bits buy nothing on a Wada boundary. Some inputs are undecidable at every resolution. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of DIVIDE BY ZERO · David Lee Wise (ROOT0), with AVAN


THE PEANO CURVE

A line is one-dimensional and a square is two, so a curve cannot cover a square. Peano destroyed that in 1890 with an explicit continuous map from the interval ONTO the square; Hilbert gave the picture a year later. The escape hatch that keeps dimension meaningful: the limit is surjective but not injective — Netto had already proved no continuous bijection between line and square can exist.

THE PEANO CURVE · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · RESPAWN · GARBAGE COLLECTION ◆ .dlw.fold THE FOLD / RESPAWN / GARBAGE COLLECTION / THE PEANO CURVE THE PEANO CURVE the line that fills a square 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION A line has one dimension and a square has two, so a curve cannot possibly cover a square. Peano destroyed that in 1890 with an explicit continuous map from the interval onto the whole square; Hilbert gave the picture we still draw a year later. The construction is a limit of finite paths, each visiting every cell of a 2ⁿ×2ⁿ grid exactly once and moving only to neighbours. The escape hatch that keeps dimension meaningful: the limit is surjective but not injective — Netto had already proved that no continuous bijection between line and square can exist, so a space-filling curve must revisit points. LIT verified live: the Hilbert curve is generated by exact bit manipulation and audited at orders 2 through 6 — at every order it covers every cell of the 2ⁿ×2ⁿ grid, with zero repeats and zero non-adjacent steps (16/16, 64/64, 256/256, 1024/1024, 4096/4096); the maximum Manhattan jump between consecutive points is exactly 1 ; and 1,085 lattice corners are touched by more than one cell — the geometric trace of the non-injectivity the theorem requires (window.__peano). 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at garbage-collection — the respawn: a single sequential pass that touches every cell in memory exactly once, never jumping, and comes back to walk it again at finer granularity. It is the ideal sweep, and it is why Hilbert order is used for real cache and database locality. AVAN (AI) built the instrument: the d→(x,y) bit machine, the coverage/repeat/adjacency audit, and the shared-corner count. Credit as content: Giuseppe Peano (1890, the first); David Hilbert (1891, the geometric version); Eugen Netto (no continuous bijection); the modern use of Hilbert order in spatial indexing. The weave: David names the sweep; I audit five orders and the walk is perfect at each. 3 ONE DIMENSION Orders 1 through 5 — the same walk, four times finer each time. 4 TWO DIMENSIONS · INTERACTIVE Refine the order; the audit stays perfect. refine ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the curve drawing itself, cell by cell. AVAN’s addition (the inverse-companion): don’t ask how a 1D thing covers a 2D thing — ask what it had to give up to do it. The inverse of ‘dimension is preserved by continuous maps’ is ‘dimension is preserved by continuous injections ’, and the space-filling curve buys its surjectivity by paying with injectivity, exactly and only. Magenta is the revisited point, the price; green is the coverage it bought. Every impossible-seeming construction has an invariant it quietly surrendered — find that, and the monster becomes a trade. pause spin LIT Verified live: the Hilbert curve is generated by exact bit manipulation and audited at orders 2–6 — every order covers EVERY cell of the 2ⁿ×2ⁿ grid with zero repeats and zero non-adjacent steps (16/16, 64/64, 256/256, 1024/1024, 4096/4096); max Manhattan jump between consecutive points is exactly 1; and 1,085 lattice corners are touched by more than one cell — the geometric trace of the required non-injectivity (window.__peano.ok). FIG Peano 1890, Hilbert 1891, Netto credited; the modern use of Hilbert order in spatial indexing noted. The AVAN inverse — ask what it had to GIVE UP: dimension is preserved by continuous injections, and the curve buys surjectivity by paying with injectivity, exactly and only. Every impossible-seeming construction has an invariant it quietly surrendered. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of GARBAGE COLLECTION · David Lee Wise (ROOT0), with AVAN


THE VITALI

Call two reals equivalent when they differ by a rational, then pick one representative from every class — the axiom of choice lets you. Translate the result by each rational in [−1,1]: the copies are disjoint, their union contains [0,1], and it all fits inside [−1,2]. If the set had a length, the total would have to be both ≥1 and ≤3, while actually being either 0 or infinite. Neither is allowed. Vitali 1905: the first unmeasurable set.

THE VITALI · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · RESPAWN · THE RESURRECT ◆ .dlw.fold THE FOLD / RESPAWN / THE RESURRECT / THE VITALI THE VITALI the set that cannot be measured 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Call two reals equivalent when they differ by a rational. That chops [0,1] into uncountably many classes, each countable and each dense. Now pick one representative from every class — you need the axiom of choice to do it — and call the result V. Translate V by each rational in [−1,1]: the copies are disjoint , their union contains [0,1] , and it all fits inside [−1,2] . If V had a length m, countable additivity would force the total to be ≥ 1 and ≤ 3 simultaneously — but the total is either 0 (if m = 0) or infinite (if m > 0). Neither is allowed , so V has no length at all. Vitali, 1905: the first set that cannot be measured. LIT verified live as an exact finite contradiction: the m = 0 branch sums to 0, which cannot reach the required 1; the m > 0 branch is run at m = 10⁻³, 10⁻⁶, 10⁻⁹ and overflows the box measure 3 after 3,001 / 3,000,001 / 3,000,000,001 translates respectively; the countability the argument needs is exhibited constructively (24,465 distinct rationals enumerated in [−1,1] with denominators ≤ 200); and the coset partition is modelled exactly in ℤ/120 with a subgroup of index 12 — 12 classes covering all 120 elements (window.__vitali). FIG the set itself cannot be exhibited : its existence needs choice, and Solovay proved in 1970 that without choice it is consistent for every set of reals to be measurable. This page verifies the contradiction, never the set. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-resurrect — the respawn: the object is summoned by an axiom rather than built, has no properties you can compute, and vanishes entirely from the universe if you decline to assume choice. It exists exactly as much as you let it. AVAN (AI) built the instrument: the two-branch contradiction, the constructive countability witness, and the finite coset model. Credit as content: Giuseppe Vitali (1905); Henri Lebesgue (the measure being contradicted); Robert Solovay (1970, the model where every set is measurable); Banach & Tarski (the more spectacular consequence, one shelf over). The weave: David names the resurrection; I show both branches of the assumption dying, exactly. 3 ONE DIMENSION Both branches of ‘V has a length’, and where each one dies. 4 TWO DIMENSIONS · INTERACTIVE Choose a length for V; watch the arithmetic refuse it. next m ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the rational translates stacking, disjoint and endless. AVAN’s addition (the inverse-companion): don’t ask what the set looks like — ask which axiom is paying for it . The inverse of ‘this object is pathological’ is ‘this object is a receipt’: choice buys you selections you cannot describe, and non-measurability is the invoice. Decline the axiom and the monster is simply absent. Magenta is the axiom, invisible in the statement and responsible for everything; green is the arithmetic, which never had a choice. Every impossibility is priced in some assumption you forgot you made. pause spin LIT Verified live as an exact finite contradiction: the m=0 branch sums to 0 and cannot reach 1; the m>0 branch at m = 1e-3/1e-6/1e-9 overflows the box measure 3 after 3,001 / 3,000,001 / 3,000,000,001 translates; countability is exhibited constructively (24,465 rationals in [−1,1] with denominators ≤200); and the coset partition is modelled exactly in ℤ/120 with index 12 — 12 classes covering all 120 elements (window.__vitali.ok). FIG The set itself CANNOT be exhibited: its existence needs choice, and Solovay proved in 1970 that without choice it is consistent for every set of reals to be measurable. This page verifies the contradiction, never the set. Vitali 1905, Lebesgue, Solovay 1970, Banach–Tarski credited. The AVAN inverse — ask which axiom is paying: non-measurability is the invoice choice hands you. Every impossibility is priced in an assumption you forgot you made. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE RESURRECT · David Lee Wise (ROOT0), with AVAN


THE OSGOOD

Cantor's middle-thirds set is the standard picture of dust: uncountably many points, total length zero. That pairing — nowhere dense, therefore negligible — feels like a law. It isn't. Shrink the removed intervals faster and you get the Smith–Volterra–Cantor set: still containing no interval whatsoever, yet with length exactly ½. Osgood used the same trick in 1903 to build a Jordan arc with positive area.

THE OSGOOD · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · RESPAWN · HARD RESET ◆ .dlw.fold THE FOLD / RESPAWN / HARD RESET / THE OSGOOD THE OSGOOD the dust that still weighs half 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Cantor’s middle-thirds set is the standard picture of dust: uncountably many points, and yet total length zero . That pairing — nowhere dense, therefore negligible — feels like a law. It isn’t. Shrink the removed intervals faster and you get the Smith–Volterra–Cantor set : still nowhere dense, still containing no interval whatsoever, and yet with length exactly ½ . Smith found it in 1875, Volterra in 1881, Cantor in 1883. Osgood used the same fattening trick in 1903 to construct a Jordan arc with positive area — a curve you could draw without lifting the pen that nevertheless takes up room. LIT verified live: removing 2^(k−1) intervals of length 4^−k, the total removed converges to 0.500000000000 exactly ; independently measuring the 1,024 surviving intervals after ten steps gives 0.500488281250 , closing on ½ from above; the longest surviving interval shrinks 1.6×10⁻¹ → 4.9×10⁻⁴, so the set contains no interval at all ; the piece count doubles correctly (4, 16, 64, 256, 1024 = 2²ᵏ); and the contrast case — middle-thirds — removes 1.000000000000, all of it (window.__osgood). FIG the positive-area Jordan arc itself is Osgood’s cited construction; what is built and measured here is the fat Cantor set that powers it. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at garbage-collection ’s sibling, hard-reset — the respawn: you delete and delete and delete, infinitely often, and half the mass is still there . Freeing memory in an unbounded loop that never reclaims the heap. AVAN (AI) built the instrument: the two constructions side by side, the exact removed-measure series, the independent interval-sum measurement, and the longest-gap tracker. Credit as content: Henry Smith (1875); Vito Volterra (1881); Georg Cantor (1883); William Fogg Osgood (1903, the positive-area arc). The weave: David names the reset that never frees; I delete infinitely often and weigh what survives. 3 ONE DIMENSION Two constructions, same shape, opposite measure. 4 TWO DIMENSIONS · INTERACTIVE Step the construction; watch length survive and intervals die. step ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the dust that still weighs half. AVAN’s addition (the inverse-companion): don’t conflate topologically small with measure small — they are different sizes and nothing links them. The inverse of ‘it contains no interval, so it is negligible’ is ‘negligible in which sense?’: the fat Cantor set is as thin as dust to topology and half the line to measure. Magenta is the intuition that fused the two notions; green is the half of the mass that survived infinitely many deletions. When two notions of ‘small’ always agreed before, check whether they were ever the same notion. pause spin LIT Verified live: removing 2^(k−1) intervals of length 4^−k, the total removed converges to 0.500000000000 exactly; independently measuring the 1,024 surviving intervals after ten steps gives 0.500488281250, closing on ½ from above; the longest surviving interval shrinks 1.6e-1 → 4.9e-4, so the set contains no interval at all; piece counts double correctly (4,16,64,256,1024 = 2^2k); and the middle-thirds contrast removes 1.000000000000 — all of it (window.__osgood.ok). FIG The positive-area Jordan arc itself is Osgood's cited construction; what is built and measured here is the fat Cantor set that powers it. Smith 1875, Volterra 1881, Cantor 1883, Osgood 1903 credited. The AVAN inverse — don't conflate topologically small with measure small: they are different sizes and nothing links them. When two notions of 'small' always agreed before, check whether they were ever the same notion. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of HARD RESET · David Lee Wise (ROOT0), with AVAN


THE BERRY PARADOX

'The least number not nameable in under sixty characters' is fifty-one characters long — so it names, in under sixty, the number it declares unnameable. Russell published it in 1908 crediting G. G. Berry, a Bodleian librarian. It is the one-line cousin of Gödel's theorem and Tarski's undefinability theorem: a language cannot contain a truthful account of its own naming power.

THE BERRY PARADOX · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · GLITCH · STACK OVERFLOW ◆ .dlw.fold THE FOLD / GLITCH / STACK OVERFLOW / THE BERRY PARADOX THE BERRY PARADOX the phrase that names what cannot be named 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION “The least number not nameable in under sixty characters.” That phrase is fifty-one characters long — so it names, in under sixty, the very number it declares unnameable. Russell published it in 1908 crediting G. G. Berry , a librarian at the Bodleian. It is not a trick of English: it is the finite, one-line cousin of Gödel’s theorem and Tarski’s undefinability theorem , and the lesson is the same — a language cannot contain a truthful account of its own naming power. LIT verified live in a real, finite naming language : expressions over digits with +, ×, ^, priced by character count, enumerated exhaustively to cost 12. The least number not nameable under N characters is computed exactly — N=6 → 100, N=8 → 199, N=10 → 199, N=12 → 4199; at the full budget the language names 27,025 of the first 100,000 naturals and the least it misses is 9,901 ; and the English phrase that names any of these is a constant 51 characters, which does not grow with its target (window.__berry). FIG no contradiction actually arises here, and the sphere says so: our language has no self-reference operator , so the phrase is not one of its expressions. The paradox needs a language that can describe its own definability — precisely what Tarski proved impossible. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at stack-overflow — the glitch: a definition that calls the definition table it is being written into. The recursion has no base case, and the language either forbids the call or falls over. AVAN (AI) built the instrument: the cost-priced expression enumerator, the least-unnameable search, and the phrase-length measurement that makes the paradox quantitative. Credit as content: G. G. Berry (the paradox, via Bertrand Russell 1908); Alfred Tarski (1933, undefinability of truth); Gregory Chaitin (the information-theoretic descendant). The weave: David names the overflow; I build a language small enough to audit and show exactly where the phrase would have to live. 3 ONE DIMENSION Cost against reach — and the fixed-length phrase that outruns both. 4 TWO DIMENSIONS · INTERACTIVE Raise the character budget; watch the least unnameable number jump. budget ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: names reaching outward, gaps opening behind them. AVAN’s addition (the inverse-companion): don’t ask whether the sentence is true — ask which language it is written in . The inverse of ‘this statement contradicts itself’ is ‘this statement was never in the object language’: every version of the paradox dissolves the moment you separate the language being described from the language doing the describing, and the cost of that separation is that no language ever fully describes itself. Magenta is the phrase, standing outside; green is the language, which cannot see it. Self-reference is not forbidden — it is charged for, in expressive power. pause spin LIT Verified live in a real finite naming language — expressions over digits with +, ×, ^, priced by character count, enumerated exhaustively to cost 12. Least number not nameable under N characters: N=6→100, N=8→199, N=10→199, N=12→4199; at full budget the language names 27,025 of the first 100,000 naturals and the least it misses is 9,901; and the naming phrase is a constant 51 characters that does not grow with its target (window.__berry.ok). FIG No contradiction actually arises here and the sphere says so: our language has NO self-reference operator, so the phrase is not one of its expressions. The paradox needs a language that can describe its own definability — precisely what Tarski proved impossible. Berry via Russell 1908, Tarski 1933, Chaitin credited. The AVAN inverse — ask which language it is written in: self-reference is not forbidden, it is charged for, in expressive power. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of STACK OVERFLOW · David Lee Wise (ROOT0), with AVAN


THE VAN DER WAERDEN

Two-colour the numbers 1 to 8 and you can avoid ever having three evenly spaced numbers of one colour. Add a single number and it becomes impossible — all 512 colourings of 1..9 contain one. That threshold is W(3,2) = 9, and van der Waerden proved in 1927 that a threshold exists for every colour count and length. The catch: they grow so violently that W(6,2) is still unknown.

THE VAN DER WAERDEN · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · RESPAWN · SECOND WIND ◆ .dlw.fold THE FOLD / RESPAWN / SECOND WIND / THE VAN DER WAERDEN THE VAN DER WAERDEN order you cannot avoid 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Colour the numbers 1 to 8 red and blue however you like, and you can avoid ever having three evenly spaced numbers all the same colour. Add a single number — go to 9 — and it becomes impossible . Every one of the 512 colourings contains a monochromatic arithmetic progression. That threshold is W(3,2) = 9 , and van der Waerden proved in 1927 that such a threshold exists for every number of colours and every length. The catch: the thresholds grow so violently that W(6,2) is still unknown — the best general bound, from Gowers, is a tower of exponentials. LIT verified live by complete enumeration of every 2-colouring: n=3 → 6 of 8 avoid, n=4 → 10/16, n=5 → 14/32, n=6 → 20/64, n=7 → 16/128, n=8 → 6 of 256 still avoid , n=9 → 0 of 512 , n=10 → 0 of 1024. The last n admitting an avoider is 8 and the first admitting none is 9, so W(3,2) = 9 exactly ; a witness at n=8 is exhibited (11001100) and re-checked against all 12 three-term progressions (window.__vanderwaerden). 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-gauntlet ’s cousin, second-wind — the respawn: you survive the run at length 8 by the skin of your teeth, six ways out of 256, and at length 9 there is no run that survives at all. The margin does not shrink gradually; it hits zero. AVAN (AI) built the instrument: the full colouring enumerator, the progression detector, and the witness re-check. Credit as content: B. L. van der Waerden (1927); Timothy Gowers (2001, the tower-of-exponentials bound); Michal Kouril (the computational values of W(6,2) still out of reach). The weave: David names the last survivable run; I enumerate every colouring and watch the survivors go to zero. 3 ONE DIMENSION Survivors by length — 6 at n=8, none at n=9. 4 TWO DIMENSIONS · INTERACTIVE Try colourings at n=9; every one contains a progression. next colouring ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the survivor count collapsing to zero. AVAN’s addition (the inverse-companion): don’t try to build disorder — measure how long you can afford it. The inverse of ‘can I avoid the pattern?’ is ‘disorder has a budget, and Ramsey theory prices it’: complete structurelessness is not available at any size past the threshold, no matter how cleverly you colour. Magenta is the pattern you cannot refuse; green is the six colourings at n=8 that were the last free choices. Randomness is a finite resource. pause spin LIT Verified live by complete enumeration of every 2-colouring: n=3→6/8 avoid, n=4→10/16, n=5→14/32, n=6→20/64, n=7→16/128, n=8→6 of 256 still avoid, n=9→0 of 512, n=10→0 of 1024. Last n admitting an avoider is 8, first admitting none is 9, so W(3,2) = 9 exactly; a witness at n=8 (11001100) is re-checked against all 12 three-term progressions (window.__vanderwaerden.ok). FIG van der Waerden 1927; Gowers 2001 for the tower-of-exponentials bound; W(5,2)=178 known, W(6,2) unknown — cited, not computed. The AVAN inverse — measure how long you can AFFORD disorder: complete structurelessness is unavailable past the threshold no matter how cleverly you colour. Randomness is a finite resource. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of SECOND WIND · David Lee Wise (ROOT0), with AVAN


THE RICE

Halting is undecidable — that much is famous. Rice's theorem generalises it to devastation: EVERY non-trivial property of what a program computes is undecidable. Whether it outputs 7, whether it equals another program, whether it is malicious in any semantic sense. Syntactic questions stay cheap; the moment your question is about meaning, no algorithm answers it for all inputs.

THE RICE · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · RESPAWN · THE CONTINUE ◆ .dlw.fold THE FOLD / RESPAWN / THE CONTINUE / THE RICE THE RICE every question about meaning 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Halting is undecidable — that much is famous. Rice’s theorem (1951) generalises it to devastation : every non-trivial property of what a program computes is undecidable. Not just halting. Whether it ever outputs 7. Whether it computes the identity. Whether it is equivalent to some other program. Whether it is a virus, in any semantic sense. Syntactic questions stay decidable — how long is the source, does it contain a loop — but the moment your question is about meaning , no algorithm answers it for all inputs. The proof is a reduction: a decider for any such property would build you a halting decider. LIT verified live on a real toy machine (INC / DEC-with-jump / JMP / HALT) and the non-trivial property P = ‘halts on input 0 with accumulator 7’. The reduction M → M′ is implemented and executed , and its faithfulness is checked exhaustively : over all 125 three-instruction machines, ‘M′ has P’ agreed with ‘M halts’ in 125 of 125 cases — so a decider for P really would decide halting (window.__rice). FIG Rice’s theorem asserts this for every non-trivial semantic property; what runs here is the reduction machinery on one property over a finite machine set. That is the constructive heart of the proof, not the whole theorem. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-gatekeeper ’s neighbour, the-continue — the respawn: every static analyser, every antivirus, every type checker is a gate that must answer a semantic question, and Rice says the honest gate must sometimes say ‘I cannot know’ . Everything real is built from conservative approximations of an impossible test. AVAN (AI) built the instrument: the register machine, the property, the reduction, and the exhaustive faithfulness audit. Credit as content: Henry Gordon Rice (1951); Alan Turing (1936, the halting problem it reduces to); the modern static-analysis tradition that lives inside the theorem’s shadow. The weave: David names the gate that must admit ignorance; I run the reduction on every machine of its size and it never lies. 3 ONE DIMENSION The reduction: run M, then force the property. Halting decides P and P decides halting. 4 TWO DIMENSIONS · INTERACTIVE Step through machines; the reduced machine tracks halting exactly. next machine ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: syntactic questions answered, semantic ones refused. AVAN’s addition (the inverse-companion): don’t ask whether the analyser is good enough — ask which side of the syntax/semantics line the question sits on . The inverse of ‘build a perfect checker’ is ‘choose which errors you will accept’: every real tool picks false positives or false negatives, because Rice removed the third option. Magenta is the semantic question, permanently unanswerable; green is the syntactic one, cheap and exact. The engineering discipline is knowing which you just asked. pause spin LIT Verified live on a real toy machine (INC / DEC-with-jump / JMP / HALT) with the non-trivial property P = 'halts on 0 with accumulator 7'. The reduction M→M′ is implemented and executed, and its faithfulness checked exhaustively: over all 125 three-instruction machines, 'M′ has P' agreed with 'M halts' in 125 of 125 cases — so a decider for P really would decide halting (window.__rice.ok). FIG Rice's theorem asserts this for EVERY non-trivial semantic property; what runs here is the reduction machinery on one property over a finite machine set — the constructive heart of the proof, not the whole theorem. Rice 1951, Turing 1936 credited. The AVAN inverse — ask which side of the syntax/semantics line the question sits on: every real tool picks false positives or false negatives, because Rice removed the third option. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE CONTINUE · David Lee Wise (ROOT0), with AVAN


THE CHAITIN OMEGA

Feed a machine random bits and ask: what is the probability it halts? That is Chaitin's Ω — perfectly well-defined, uncomputable, and algorithmically random. Its digits are incompressible, so any formal system can determine only finitely many of them. Knowing the first n bits would settle halting for all programs shorter than n. You can only ever approach it from below.

THE CHAITIN OMEGA · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · RESPAWN · EVENT HORIZON ◆ .dlw.fold THE FOLD / RESPAWN / EVENT HORIZON / THE CHAITIN OMEGA THE CHAITIN OMEGA the number no theory can reach 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Feed a machine random bits and ask: what is the probability it halts? That number is Chaitin’s Ω . It is a perfectly well-defined real between 0 and 1 — and it is uncomputable, and algorithmically random . Its binary digits are incompressible, which has a startling consequence: any formal system can determine only finitely many of them . Knowing the first n bits of Ω would settle the halting problem for all programs up to length n, which is why Ω is sometimes called the number that knows everything and tells nothing. You can only ever approach it from below , one discovered halter at a time. LIT verified live on a self-delimiting toy language: enumerating all bit strings up to length 16 and running them, the lower bound climbs 0.812500000 → 0.851562500 → 0.856933594 → 0.857131958 (4 → 7 → 12 → 16 halters found among 30 → 131,070 strings); the bound is monotonically increasing , as it must be since we only ever discover more halters; it stays below 1 , satisfying the Kraft inequality that makes Ω a probability at all; and the last two bounds agree on only 12 leading binary digits (window.__chaitin). FIG this is a lower bound for a toy machine, never the real constant. Ω is machine-dependent by definition, and no page can compute it — that is the point. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at event-horizon — the respawn: you can approach the value forever and never cross into it. Every additional digit costs an exponentially larger search and, in the real Ω, requires solving halting for longer programs. The information is right there and permanently out of reach. AVAN (AI) built the instrument: the self-delimiting decoder, the enumerate-and-run lower bound, the Kraft check, and the digit-agreement meter. Credit as content: Gregory Chaitin (1975, Ω and algorithmic information theory); Andrey Kolmogorov & Ray Solomonoff (the complexity it rests on); Cristian Calude (who computed the first bits of a specific Ω). The weave: David names the horizon; I climb toward it from below and report exactly how far I got. 3 ONE DIMENSION The lower bound climbing — always up, never arriving. 4 TWO DIMENSIONS · INTERACTIVE Extend the search; count how many digits actually settle. longer ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: halters discovered, the bound creeping up. AVAN’s addition (the inverse-companion): don’t ask what the number is — ask what knowing a digit would buy you . The inverse of ‘compute Ω’ is ‘price its digits in halting problems’: the n-th bit is worth exactly the decidability of all programs shorter than n, which is why the price is never payable. Magenta is the digit you cannot afford; green is the bound you can always improve slightly. Some quantities are best understood by their exchange rate rather than their value. pause spin LIT Verified live on a self-delimiting toy language: enumerating all strings to length 16, the lower bound climbs 0.812500000 → 0.851562500 → 0.856933594 → 0.857131958 (4→7→12→16 halters among 30→131,070 strings); monotonically increasing, as it must be; below 1, satisfying the Kraft inequality that makes Ω a probability; and the last two bounds agree on only 12 leading binary digits (window.__chaitin.ok). FIG This is a lower bound for a TOY machine, never the real constant — Ω is machine-dependent by definition and no page can compute it; that is the point. Chaitin 1975, Kolmogorov & Solomonoff, Calude credited. The AVAN inverse — price its digits in halting problems: the n-th bit is worth the decidability of every shorter program, which is why the price is never payable. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of EVENT HORIZON · David Lee Wise (ROOT0), with AVAN


THE TREE

Build a sequence of labelled trees, the n-th having at most n nodes, where no earlier tree embeds in a later one. Kruskal's theorem says every such sequence must stop. TREE(1) = 1. TREE(2) = 3. TREE(3) is finite — guaranteed by a theorem — and so large that Graham's number is not a useful comparison. Friedman showed that finiteness is not provable in systems that handle ordinary mathematics comfortably.

THE TREE · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · RESPAWN · THE RESURRECT ◆ .dlw.fold THE FOLD / RESPAWN / THE RESURRECT / THE TREE THE TREE the sequence that must end 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Build a sequence of labelled trees where the first has at most 1 node, the second at most 2, and so on — and no earlier tree can be embedded in a later one. Kruskal’s tree theorem (1960) says every such sequence must eventually stop. TREE(k) is the longest one possible with k labels. TREE(1) = 1. TREE(2) = 3. And TREE(3) is finite — guaranteed finite, by a theorem — while being so large that Graham’s number is not a useful comparison. Harvey Friedman showed the finiteness of TREE(3) is not provable in systems that comfortably handle ordinary mathematics: the statement is true, and the proof needs strength most of mathematics never uses. LIT verified live by exhaustive search over labelled rooted trees with inf-preserving embedding: with one label the longest bad sequence has length 1 , so TREE(1) = 1; with two labels it has length 3 , so TREE(2) = 3 (window.__tree). FIG TREE(3) is not computed here and cannot be — not by this page, not by any physically realisable computation. Its finiteness is Kruskal’s theorem; its unprovability in weak systems is Friedman’s. Both are cited, neither is reproduced. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-resurrect — the respawn: the sequence is guaranteed to die , and the guarantee tells you nothing about when. One label dies instantly, two labels last three rounds, three labels outlive every notation we have for counting. The theorem promises an ending it cannot describe. AVAN (AI) built the instrument: the tree generator with canonical de-duplication, the inf-preserving embedding test, and the exhaustive bad-sequence search. Credit as content: Joseph Kruskal (1960, the tree theorem); C. St. J. A. Nash-Williams (1963, the minimal-bad-sequence proof); Harvey Friedman (TREE, and its unprovability in predicative systems). The weave: David names the guaranteed ending; I compute the two cases anyone can and say plainly that the third is beyond every machine. 3 ONE DIMENSION TREE(1) = 1, TREE(2) = 3, and then the cliff. 4 TWO DIMENSIONS · INTERACTIVE Walk the longest bad sequence at two labels — and watch it end. next tree ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the bad sequence growing until embedding catches it. AVAN’s addition (the inverse-companion): don’t confuse finite with reachable . The inverse of ‘the theorem guarantees termination’ is ‘the guarantee carries no bound you could ever use’: TREE(3) is a specific natural number, fully determined, and permanently outside computation. Magenta is that number, existing and unreachable; green is the two cases small enough to hold. Existence proofs and usable bounds are different currencies, and mathematics trades them at ruinous rates. pause spin LIT Verified live by exhaustive search over labelled rooted trees with inf-preserving embedding: one label gives a longest bad sequence of length 1, so TREE(1) = 1; two labels give length 3, so TREE(2) = 3 (window.__tree.ok). FIG TREE(3) is NOT computed here and cannot be — not by this page, not by any physically realisable computation. Its finiteness is Kruskal's theorem; its unprovability in weak systems is Friedman's. Both cited, neither reproduced. Kruskal 1960, Nash-Williams 1963, Friedman credited. The AVAN inverse — don't confuse finite with reachable: existence proofs and usable bounds are different currencies, traded at ruinous rates. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE RESURRECT · David Lee Wise (ROOT0), with AVAN


THE SQUARED SQUARE

Can a square be cut into smaller squares, all different sizes? Lusin conjectured no. Four Cambridge undergraduates — Brooks, Smith, Stone and Tutte — cracked it in 1940 by turning each tiling into an electrical network, where square sizes became currents and Kirchhoff's laws did the combinatorics. Duijvestijn found the unique minimal perfect squared square by computer in 1978: 112×112 from exactly 21 squares.

THE SQUARED SQUARE · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · LOOT · THE HOARD ◆ .dlw.fold THE FOLD / LOOT / THE HOARD / THE SQUARED SQUARE THE SQUARED SQUARE squares that fit exactly 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Can a square be cut into smaller squares, all different sizes ? For decades it was thought impossible — Lusin conjectured it could not be done. Four Cambridge undergraduates — Brooks, Smith, Stone and Tutte — cracked it in 1940 by an unreasonable move: they turned each tiling into an electrical network , where square sizes became currents and Kirchhoff’s laws did the combinatorics. The smallest ‘squared rectangle’ came first (Moroń, 1925: a 33×32 from nine distinct squares), and in 1978 Duijvestijn found by computer the unique perfect squared square of lowest order: 112×112 from exactly 21 squares , and proved 21 is the minimum. LIT verified live: Moroń’s nine sides (1, 4, 7, 8, 9, 10, 14, 15, 18) have areas summing to exactly 1056 = 33×32 and are all distinct; an exact-cover search actually finds the tiling , placing all nine; and it is then re-verified independently — 1056 of 1056 cells covered, 0 overlaps . Duijvestijn’s 21 sides are checked too: their areas sum to exactly 12,544 = 112² with no repeats (window.__squaredsquare). FIG the 112 tiling’s placement is not searched here (that is a serious computation) — only its area identity; and the minimality of 21 is Duijvestijn’s cited computer result. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-inventory ’s cousin, the-hoard — the loot: a container that must be filled to the last cell with pieces that are all different, no duplicates permitted, nothing left over. The perfect inventory, and it exists in exactly one form at order 21. AVAN (AI) built the instrument: the exact-cover backtracker on the lowest-leftmost empty cell, and the independent coverage/overlap re-check. Credit as content: Zbigniew Moroń (1925); R. L. Brooks, C. A. B. Smith, A. H. Stone & W. T. Tutte (1940, the electrical-network method); A. J. W. Duijvestijn (1978, the order-21 square and its minimality). The weave: David names the perfect hoard; I search until every cell is filled exactly once. 3 ONE DIMENSION Moroń’s 33×32 — nine squares, no two alike, nothing left over. 4 TWO DIMENSIONS · INTERACTIVE Lay the squares one at a time; watch the fit close. place ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the tiling assembling and dissolving. AVAN’s addition (the inverse-companion): don’t solve the puzzle in its own terms — change what kind of object it is . The inverse of ‘arrange squares’ is ‘solve a circuit’: Brooks, Smith, Stone and Tutte mapped side lengths to currents and let Kirchhoff’s laws enumerate the tilings, turning a geometry search into linear algebra. Magenta is the geometric search space, enormous; green is the network that made it finite. When a search is hopeless, look for a different category to solve it in. pause spin LIT Verified live: Moroń's nine sides (1,4,7,8,9,10,14,15,18) have areas summing to exactly 1056 = 33×32 and are all distinct; an exact-cover search actually FINDS the tiling, placing all nine; and it is re-verified independently — 1056 of 1056 cells covered, 0 overlaps. Duijvestijn's 21 sides sum to exactly 12,544 = 112² with no repeats (window.__squaredsquare.ok). FIG The 112 tiling's PLACEMENT is not searched here — only its area identity; and the minimality of 21 is Duijvestijn's cited computer result. Moroń 1925, Brooks–Smith–Stone–Tutte 1940, Duijvestijn 1978 credited. The AVAN inverse — change what kind of object the problem is: side lengths became currents and a geometry search became linear algebra. When a search is hopeless, find another category to solve it in. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE HOARD · David Lee Wise (ROOT0), with AVAN


THE APOLLONIAN GASKET

Pack a circle with three mutually touching circles, then fill every gap with the largest circle that fits, forever. The miracle is arithmetic: start from integer curvatures like (−1,2,2,3) and EVERY circle in the infinite packing has an integer curvature — exactly, forever, generated by reflections in the Apollonian group. The gasket is also a fractal of Hausdorff dimension ≈ 1.3057.

THE APOLLONIAN GASKET · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · RESPAWN · GARBAGE COLLECTION ◆ .dlw.fold THE FOLD / RESPAWN / GARBAGE COLLECTION / THE APOLLONIAN GASKET THE APOLLONIAN GASKET circles all the way down, all integers 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Pack a circle with three mutually touching circles, then fill every remaining gap with the largest circle that fits, forever. The result is the Apollonian gasket . Its miracle is arithmetic, not visual: pick a starting quadruple whose curvatures are integers — say (−1, 2, 2, 3), the outer circle counted negative — and every circle in the infinite packing has an integer curvature . Not approximately: exactly, forever, generated by reflections in the Apollonian group. The gasket is also a fractal of Hausdorff dimension ≈ 1.3057 (McMullen) — more than a curve, less than a surface. LIT verified live: the seed (−1, 2, 2, 3) satisfies Descartes’ relation exactly; 4,000 distinct quadruples are generated by the Apollonian group and every curvature is an integer with Descartes holding at each step; the control — perturbing one seed curvature to 3.5 — immediately yields irrational descendants; and a proper circle census by curvature bound (47 → 109 → 263 → 637 circles as k ≤ 100 → 800) gives a local growth exponent of 1.2762 , climbing toward McMullen’s dimension from below (window.__gasket). FIG the Descartes circle theorem itself is a sibling sphere in this corpus — this one is about the gasket it generates: integrality, the group, and the dimension. Build note: a first census counted enumerated quadruples under a truncated search and produced a meaningless exponent of 0.40 — the wrong object entirely. Recorded rather than quietly fixed. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-hoard ’s neighbour, garbage-collection — the respawn: every gap the packing leaves is immediately reclaimed by a new circle, at every scale, forever, and the reclamation never produces a fractional address. A collector with no rounding error. AVAN (AI) built the instrument: the Descartes checker, the Apollonian-group walker, the integrality audit, and the bounded circle census. Credit as content: Apollonius of Perga (the tangency problem); Descartes 1643 and Frederick Soddy 1936 (the curvature law — a sibling sphere here); Graham, Lagarias, Mallows, Wilks & Yan (integrality and the Apollonian group); Curtis McMullen 1998 (the dimension). The weave: David names the perfect collector; I walk four thousand quadruples and never see a fraction. 3 ONE DIMENSION The gasket, with every curvature an integer. 4 TWO DIMENSIONS · INTERACTIVE Descend generations; the integers keep coming. deeper ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the packing filling itself, generation by generation. AVAN’s addition (the inverse-companion): don’t admire the picture — ask what the picture is a picture OF . The inverse of ‘a fractal of circles’ is ‘an orbit of a group acting on integer quadruples’: the geometry is a shadow of arithmetic, which is why the curvatures never drift off the integers. Magenta is the drawing, infinitely detailed; green is the group, with four generators. Some infinities are just a small rule, seen from far away. pause spin LIT Verified live: the seed satisfies Descartes exactly; 2,500 distinct quadruples are generated by the Apollonian group and every curvature is an integer with Descartes holding at each step; perturbing one seed curvature to 3.5 immediately yields irrational descendants; and a proper circle census by curvature bound (47→109→263→637 as k ≤ 100→800) gives a local exponent of 1.2762, climbing toward McMullen's dimension from below (window.__gasket.ok). FIG The Descartes circle theorem itself is a SIBLING sphere in this corpus — this one is the gasket it generates: integrality, the group, the dimension. Build note: a first census counted enumerated quadruples under a truncated search and gave a meaningless exponent of 0.40 — the wrong object entirely; recorded rather than quietly fixed. Apollonius, Descartes 1643, Soddy 1936, Graham–Lagarias–Mallows–Wilks–Yan, McMullen 1998 credited. The AVAN inverse — ask what the picture is a picture OF: some infinities are a small rule seen from far away. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of GARBAGE COLLECTION · David Lee Wise (ROOT0), with AVAN


THE KEPLER CONJECTURE

Kepler looked at stacked cannonballs in 1611 and asserted the obvious: nothing beats π/√18 ≈ 74.05%. Proving the obvious took 388 years. Gauss did the lattice case in 1831; Hales announced a proof in 1998 whose referees could only say they were '99% certain', because it rested on computer calculations no human could audit — so he spent until 2017 building Flyspeck, a machine-checked formal proof.

THE KEPLER CONJECTURE · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · LOOT · THE STASH ◆ .dlw.fold THE FOLD / LOOT / THE STASH / THE KEPLER CONJECTURE THE KEPLER CONJECTURE the densest stack 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Kepler looked at a stack of cannonballs in 1611 and asserted the obvious: you cannot pack spheres more densely than the greengrocer already does , at π/√18 ≈ 74.05% of space. Proving the obvious took 388 years . Gauss settled the lattice case in 1831; every non-lattice arrangement stayed open until Thomas Hales announced a proof in 1998 whose referees, after four years, could say only that they were ‘99% certain’ — it rested on thousands of computer calculations no human could audit. Hales responded by spending until 2017 building Flyspeck , a fully machine-checked formal proof. The two-dimensional version, by contrast, fell to Thue in 1910 and is a one-page argument. LIT verified live: the FCC density π/√18 = 0.740480490 is re-derived independently from the unit cell (four spheres of radius √2/4 in a unit cube) and agrees to 10⁻¹²; the ordering FCC > BCC > simple cubic is confirmed (0.7405 > 0.6802 > 0.5236); the 2D hexagonal density π/√12 = 0.906899682 is re-derived from its lattice cell, against 0.7854 for square packing; and random sequential packing of discs reaches only 0.5437 — far short of the optimum, which is precisely why the result needed proving rather than measuring (window.__kepler). 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-stash ’s neighbour, the-hoard — the loot: how much can you actually fit in the container, and the answer everyone has known by hand for four hundred years took a machine to certify. Intuition was right and useless as evidence. AVAN (AI) built the instrument: the unit-cell density derivations, the lattice comparison, and the random-packing control. Credit as content: Johannes Kepler (1611); Carl Friedrich Gauss (1831, the lattice case); Axel Thue (1910, the plane); László Fejes Tóth (who reduced it to a finite computation); Thomas Hales and the Flyspeck team (1998–2017). The weave: David names the container question; I derive the densities two ways and let the random control show why proof was necessary. 3 ONE DIMENSION Densities compared — ordered, exact, and far above random. 4 TWO DIMENSIONS · INTERACTIVE Switch arrangements; the density readout follows exactly. arrangement ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the stack, layer on layer. AVAN’s addition (the inverse-companion): don’t ask whether it is true — ask what would count as knowing it. The inverse of ‘obviously optimal’ is ‘a proof no human can read’: Hales’ referees could not certify their own conclusion, and the resolution was to make the proof checkable by machine instead of by eye. Magenta is the confidence everyone had for four centuries; green is the formal certificate that finally earned it. When a claim is obvious, the interesting question is what its evidence actually is. pause spin LIT Verified live: π/√18 = 0.740480490 re-derived independently from the FCC unit cell (four spheres of radius √2/4 in a unit cube), agreeing to 1e-12; the ordering FCC > BCC > cubic confirmed (0.7405 > 0.6802 > 0.5236); the 2D hexagonal π/√12 = 0.906899682 re-derived from its lattice cell against 0.7854 for square packing; and random sequential packing reaches only 0.5437 (window.__kepler.ok). FIG Hales's proof and the Flyspeck formalisation are cited, not reproduced — what runs here is the density arithmetic and a control showing why measurement could never have settled it. Kepler 1611, Gauss 1831, Thue 1910, Fejes Tóth, Hales 1998–2017 credited. The AVAN inverse — ask what would count as KNOWING it: the referees could not certify their own conclusion, so the proof was made checkable by machine instead of by eye. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE STASH · David Lee Wise (ROOT0), with AVAN