THE SYLVESTER–GALLAI

The Sylvester–Gallai theorem in the 5-window house format — given finitely many points in the plane, not all on one line, there is always a line through exactly two of them (an 'ordinary' line). Sylvester asked it in 1893; it resisted until Gallai and Melchior settled it around 1944. You cannot arrange points so every two-point line catches a third — unless they are all collinear. Verified live: over thousands of random integer point sets that are not all collinear, an ordinary line is always found by checking every pair (exact integer collinearity, no rounding). See the 3×3 grid in 1D, an ordinary line highlighted in 2D, and the unavoidable-line inverse in 3D.

THE SYLVESTER–GALLAI · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · LOOT · THE JACKPOT ◆ .dlw.fold THE FOLD / LOOT / THE JACKPOT / THE SYLVESTER–GALLAI THE SYLVESTER–GALLAI non-collinear points always leave an ordinary line 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION The Sylvester–Gallai theorem answers a question that stood open for forty years: given finitely many points in the plane, not all on one line , must there be a line through exactly two of them? Yes — always. Such a line is called ordinary . Sylvester asked it in 1893; it resisted until Gallai (and others) settled it around 1944. The surprise is that you cannot arrange points so that every line hitting two of them hits a third — unless they are all collinear to begin with. LIT verified live: over thousands of random integer point sets that are not all collinear, an ordinary line (through exactly two points) is always found by checking every pair (window.__sylvestergallai). FIG no framing; exhaustive collinearity counts over integer coordinates (exact, no rounding). 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-jackpot — scatter any non-collinear points and you are guaranteed a payout: a line touching exactly two of them, no matter how cleverly you try to avoid it. That guaranteed find is the jackpot. AVAN (AI) built the instrument: the exact integer collinearity test, the per-pair point count, and the search for a two-point line. Credit as content: James Joseph Sylvester (posed 1893); Tibor Gallai and Eberhard Melchior (proofs, 1940s). The weave: David names the-jackpot; I take each pair of points, count how many others lie on their line, and confirm that some pair — whenever the points are not all collinear — has a line all to itself. 3 ONE DIMENSION Any non-collinear set has an ordinary line (through exactly 2 points). You cannot force every 2-point line to catch a third — the 3×3 grid, however symmetric, still has ordinary lines. 4 TWO DIMENSIONS · INTERACTIVE A point set with an ordinary line highlighted; the guarantee checked over many non-collinear sets. new points ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: a line all to two points. AVAN’s addition (the inverse-companion): try to build a set where every line through two points passes through a third — and discover you cannot, unless the points are all collinear. The inverse of ‘place points freely’ is ‘an ordinary line is unavoidable.’ Magenta is a line catching three or more; green is the ordinary line that must exist. The two-point line you cannot avoid. pause spin LIT Genuine Sylvester–Gallai theorem (James Joseph Sylvester posed it 1893; Tibor Gallai and Eberhard Melchior proved it in the 1940s). Verified live: over 3000 random integer point sets that are not all collinear, a line through exactly two points is always found by an exact-integer collinearity count over every pair (window.__sylvestergallai.alwaysOrdinary). FIG No framing: the exact integer collinearity test (cross product = 0), the per-pair point count, and the search for a two-point line all run in-browser with no rounding. The AVAN inverse is honest — an ordinary line is unavoidable for non-collinear points (you cannot force every two-point line to catch a third); magenta is a line catching three or more, green the ordinary line that must exist. The two-point line you cannot avoid. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE JACKPOT · David Lee Wise (ROOT0), with AVAN


THE TURÁN

Turán's theorem in the 5-window house format — the most edges a graph on n vertices can have with no clique of size r+1 is achieved by the Turán graph T(n,r): split the vertices into r nearly-equal groups and join every pair in different groups. No K_{r+1} can form (it would need two vertices in one group, never joined), and this balanced complete r-partite graph packs the maximum (1−1/r)·n²/2 edges. Mantel's triangle-free bound is the r=2 case. Verified live: an exhaustive search over all graphs on up to 6 vertices finds the maximum K_{r+1}-free edge count equals the Turán graph T(n,r)'s, for r=2 and r=3. See T(6,3) in 1D, the Turán graph in 2D, and the pack-to-the-wall inverse in 3D.

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

THE TURÁN · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · BOSS · THE WALL ◆ .dlw.fold THE FOLD / BOSS / THE WALL / THE TURÁN THE TURÁN the most edges with no clique of a given size 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Turán’s theorem is the summit of extremal graph theory: the most edges a graph on n vertices can have while containing no clique of size r + 1 is achieved by the Turán graph T(n, r) — split the vertices into r nearly-equal groups and join every pair in different groups. No clique of size r + 1 can form (it would need two vertices in one group, which are never joined), and this complete r-partite graph packs the maximum (1 − 1/r)·n 2 /2 edges. Mantel’s triangle-free bound is exactly the r = 2 case. LIT verified live: an exhaustive search over all graphs on up to 6 vertices finds the maximum edge count with no K r+1 equals the Turán graph T(n, r)’s edge count, for r = 2 and r = 3 (window.__turan). FIG no framing; brute-force extremal count vs the Turán construction. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-wall — pack edges as densely as you like, but the moment you would force a clique of size r + 1 you hit the wall, at exactly the Turán count. AVAN (AI) built the instrument: the exhaustive K r+1 -free edge maximiser, the Turán graph edge formula, and their agreement. Credit as content: Pál Turán (1941); the r = 2 case is Willem Mantel (1907). The weave: David names the-wall; I try every graph on n vertices, keep the ones with no clique of size r + 1 and the most edges, and confirm the record equals the balanced complete r-partite graph — the densest a clique-free graph can be. 3 ONE DIMENSION No K 4 (r = 3), n = 6: split into three pairs, join all cross-pairs — T(6,3), 12 edges, no triangle-free-clique of 4. Add an edge inside a pair and a K 4 appears. Mantel (no triangle) is r = 2. 4 TWO DIMENSIONS · INTERACTIVE The Turán graph T(n, r) and its edge count against the exhaustive K r+1 -free maximum; checked. change n,r ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the densest clique-free graph. AVAN’s addition (the inverse-companion): don’t detect a clique of size r + 1 — ask how many edges you can pack before one is forced, and the answer is the balanced r-partition. The inverse of ‘find a K r+1 ’ is ‘maximise edges with no K r+1 — split into r equal parts.’ Magenta is a graph with a clique of size r + 1; green is the extremal Turán graph. Packed to the wall before the clique. pause spin LIT Genuine Turán's theorem (Pál Turán, 1941; the r=2 case is Willem Mantel, 1907). Verified live: an exhaustive search over all graphs on n vertices finds the maximum edge count containing no clique of size r+1 equals the Turán graph T(n,r)'s edge count (n·(n−1)/2 minus the within-part edges of a balanced r-partition) for all n up to 6 and r∈{2,3} (window.__turan.matchesTuranGraph). FIG No framing: the exhaustive K_{r+1}-free edge maximiser, the Turán graph edge formula, and their agreement all run in-browser. The AVAN inverse is honest — maximising edges subject to no clique of size r+1 (the balanced r-partition) rather than detecting a clique is the genuine extremal question, answered by the Turán graph; magenta is a graph containing that clique, green the extremal Turán graph. Packed to the wall before the clique. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE WALL · David Lee Wise (ROOT0), with AVAN


THE CATALAN–MIHĂILESCU

Catalan's conjecture (Mihăilescu's theorem) in the 5-window house format — 8 and 9 are the only consecutive perfect powers. The equation xᵃ − yᵇ = 1 with x,y,a,b > 1 has exactly one solution: 3² − 2³ = 1. Among all squares, cubes, and higher powers, only 8 = 2³ and 9 = 3² sit next to each other. Eugène Catalan conjectured it in 1844; it stood 158 years until Preda Mihăilescu proved it in 2002. Verified live: sieving every perfect power up to a million, the only pair of consecutive integers both perfect powers is (8, 9). See the powers in 1D, the scan in 2D, and the neighbouring-powers inverse in 3D.

THE CATALAN–MIHĂILESCU · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · SPAWN · COLD BOOT ◆ .dlw.fold THE FOLD / SPAWN / COLD BOOT / THE CATALAN–MIHĂILESCU THE CATALAN–MIHĂILESCU eight and nine the only consecutive perfect powers 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Catalan’s conjecture — proved by Preda Mihăilescu in 2002 — says that 8 and 9 are the only consecutive perfect powers . That is, the equation x a − y b = 1 with x, y, a, b all greater than 1 has exactly one solution: 3 2 − 2 3 = 1 . Among all the squares, cubes, fourth powers and beyond, only 8 = 2 3 and 9 = 3 2 sit next to each other on the number line. Eugène Catalan conjectured it in 1844; it stood for 158 years. LIT verified live: sieving every perfect power up to a million, the only pair of consecutive integers both of which are perfect powers is (8, 9) (window.__catalanmihailescu). FIG honest: this is a finite search confirming the theorem’s claim within range — the full statement (no pair exists anywhere, ever) is Mihăilescu’s proof, not the search. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at cold-boot — out of the whole endless field of powers, exactly one pair boots up adjacent, 8 and 9, and never again. AVAN (AI) built the instrument: the perfect-power sieve and the consecutive-pair scan. Credit as content: Eugène Charles Catalan (conjecture, 1844); Preda Mihăilescu (proof, 2002). The weave: David names cold-boot; I mark every perfect power up to a million and scan for two in a row, finding only 8 and 9 — while stating plainly that the theorem’s “never again” is Mihăilescu’s, beyond any finite search. 3 ONE DIMENSION Perfect powers: 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, … Only 8 = 2 3 and 9 = 3 2 are consecutive. 3 2 − 2 3 = 1 is the sole solution of x a − y b = 1. 4 TWO DIMENSIONS · INTERACTIVE The perfect powers on a line, gaps shrinking; the only consecutive pair (8, 9) marked; the search checked. scan range ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the one adjacent pair of powers. AVAN’s addition (the inverse-companion): don’t ask which numbers are perfect powers — ask which powers are neighbours , differing by one, and find that only 8 and 9 ever are. The inverse of ‘is n a perfect power?’ is ‘are two perfect powers consecutive? — exactly once.’ Magenta is the endless scatter of powers; green is the unique adjacent pair. One and only one gap of size one. pause spin LIT Genuine Catalan–Mihăilescu theorem (Eugène Charles Catalan conjectured 1844; Preda Mihăilescu proved 2002). Verified live: a perfect-power sieve up to 1,000,000 finds the only pair of consecutive integers both of which are perfect powers is (8, 9) (window.__catalanmihailescu.onlyEightNine). FIG No framing: the perfect-power sieve and the consecutive-pair scan run in-browser with exact arithmetic. Honest scope: this is a finite search confirming the theorem within range — the full statement, that no such pair exists anywhere ever, is Mihăilescu's proof (2002), not the search, and the sphere says so. The AVAN inverse is honest — asking which perfect powers are neighbours (differ by 1) rather than which numbers are powers finds exactly one pair; magenta is the endless scatter of powers, green the unique adjacent pair. One and only one gap of size one. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of COLD BOOT · David Lee Wise (ROOT0), with AVAN


THE MENGER

Menger's theorem in the 5-window house format — the maximum number of edge-disjoint paths between two vertices s and t equals the minimum number of edges whose removal disconnects them (the minimum s–t cut). Flow and blockage are the same number: push as many independent routes from s to t as you can, and the bottleneck is exactly the smallest set of edges that severs the two. It is the local, per-pair form of max-flow min-cut, and its vertex version underlies k-connectivity. Verified live: over hundreds of random small graphs, the maximum edge-disjoint s–t paths (unit-capacity max-flow) equals the minimum s–t edge cut found by exhaustive edge-removal. See the routes-equal-cut law in 1D, a graph in 2D, and the flow-equals-blockage inverse in 3D.

THE MENGER · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · CO-OP · THE PULL REQUEST ◆ .dlw.fold THE FOLD / CO-OP / THE PULL REQUEST / THE MENGER THE MENGER the most independent routes equals the smallest severing cut 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Menger’s theorem is the combinatorial heart of connectivity: the maximum number of edge-disjoint paths between two vertices s and t equals the minimum number of edges you must remove to disconnect them — the minimum s–t cut. Flow and blockage are the same number. Push as many independent routes from s to t as you can, and the bottleneck is exactly the smallest set of edges that severs the two. It is the local, per-pair form of the max-flow min-cut theorem, and its vertex version underlies network reliability and k-connectivity. LIT verified live: over hundreds of random small graphs, the maximum number of edge-disjoint s–t paths (a unit-capacity max-flow) equals the minimum s–t edge cut found by exhaustive edge-removal (window.__menger). FIG no framing; independent max-flow and brute-force min-cut, compared. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-pull-request — how many independent routes can two nodes agree on, and what is the smallest set of links whose removal breaks every agreement? Menger says those two numbers are one. AVAN (AI) built the instrument: the unit-capacity max-flow (counting edge-disjoint paths), the exhaustive minimum edge cut, and their equality. Credit as content: Karl Menger (1927). The weave: David names the-pull-request; I push as many edge-disjoint paths from s to t as the graph allows, then find the fewest edges whose removal disconnects them, and confirm the two counts always coincide — the most routes equals the smallest severing set. 3 ONE DIMENSION Max edge-disjoint s–t paths = min s–t edge cut. If three independent routes run from s to t, then at least three edges must be cut to sever them — and exactly three suffice. 4 TWO DIMENSIONS · INTERACTIVE A graph with s and t, its edge-disjoint paths, and the minimum cut; the equality checked over many graphs. new graph ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: routes equal to the bottleneck. AVAN’s addition (the inverse-companion): to learn how robustly two nodes connect, don’t count all the routes — find the smallest set of edges that severs them , and that number is exactly how many independent routes exist. The inverse of ‘count edge-disjoint paths’ is ‘find the minimum cut — they are the same.’ Magenta is the minimum severing cut; green is the maximum set of disjoint routes. Flow equals blockage. pause spin LIT Genuine Menger's theorem (Karl Menger, 1927). Verified live: over 400 random small graphs, the maximum number of edge-disjoint s–t paths (computed as a unit-capacity max-flow by augmenting paths) equals the minimum s–t edge cut found by exhaustive edge-subset removal (window.__menger.equiv). FIG No framing: the unit-capacity max-flow (counting edge-disjoint paths), the exhaustive minimum edge cut, and their equality all run in-browser. The AVAN inverse is honest — measuring connectivity by the smallest severing set (rather than counting all routes) gives exactly the number of independent routes, by Menger; magenta is the minimum cut, green the maximum set of disjoint routes. Flow equals blockage. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE PULL REQUEST · David Lee Wise (ROOT0), with AVAN


THE RAMANUJAN CONGRUENCE

Ramanujan's congruences in the 5-window house format — hidden divisibilities in the partition numbers p(n). Ramanujan noticed three exact patterns: p(5n+4) ≡ 0 (mod 5), p(7n+5) ≡ 0 (mod 7), p(11n+6) ≡ 0 (mod 11). Every fifth partition number from p(4) is divisible by 5, every seventh from p(5) by 7, every eleventh from p(6) by 11 — and no such simple congruence exists for any other prime. Verified live (exact BigInt): computing p(n) by Euler's pentagonal recurrence, the three congruences hold for every n in range. See the vanishing classes in 1D, p(n) mod 5/7/11 in 2D, and the hidden-zeros inverse in 3D.

THE RAMANUJAN CONGRUENCE · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · GLITCH · DIVIDE BY ZERO ◆ .dlw.fold THE FOLD / GLITCH / DIVIDE BY ZERO / THE RAMANUJAN CONGRUENCE THE RAMANUJAN CONGRUENCE hidden divisibilities in the partition numbers 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Ramanujan’s congruences are astonishing hidden divisibilities in the partition numbers p(n) — the count of ways to write n as a sum of positive integers. Ramanujan noticed, from a hand-written table, three exact patterns: p(5n + 4) ≡ 0 (mod 5) , p(7n + 5) ≡ 0 (mod 7) , and p(11n + 6) ≡ 0 (mod 11) . Every fifth partition number from p(4) is divisible by 5; every seventh from p(5) by 7; every eleventh from p(6) by 11. There is no such simple congruence for any other prime — 5, 7, 11 are special. LIT verified live (exact BigInt): computing p(n) by Euler’s pentagonal recurrence, p(5n+4) is divisible by 5, p(7n+5) by 7, and p(11n+6) by 11 for every n in range (window.__ramanujan). FIG no framing; exact big-integer partition counts checked against the three moduli. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at divide-by-zero — whole arithmetic progressions of partition numbers that come out exactly zero modulo 5, 7, 11, with no obvious reason in the definition. That unexpected vanishing is the mechanic. AVAN (AI) built the instrument: the pentagonal-recurrence partition counter in big integers, and the three congruence checks. Credit as content: Srinivasa Ramanujan (1919). The weave: David names divide-by-zero; I compute p(n) exactly by adding and subtracting earlier partition counts at the generalized pentagonal offsets, then confirm that p(5n+4), p(7n+5), p(11n+6) vanish modulo 5, 7, 11 — the congruences Ramanujan saw in a table. 3 ONE DIMENSION p(4)=5, p(9)=30, p(14)=135, p(19)=490 — all divisible by 5 (the 5n+4 class). p(5)=7, p(12)=77 by 7. p(6)=11, p(17)=297 by 11. No such rule for any other prime. 4 TWO DIMENSIONS · INTERACTIVE Partition numbers mod 5, 7, 11; the arithmetic progressions that vanish highlighted; the congruences checked. cycle modulus ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: whole progressions vanishing mod a prime. AVAN’s addition (the inverse-companion): don’t just count partitions — watch them modulo a prime , and find entire arithmetic progressions that come out zero. The inverse of ‘how many partitions of n’ is ‘which residue classes of n force p(n) ≡ 0.’ Magenta is the raw partition count; green is the vanishing residue class mod 5, 7, or 11. Hidden zeros in the counting. pause spin LIT Genuine Ramanujan congruences (Srinivasa Ramanujan, 1919). Verified live with exact BigInt: the partition function computed by Euler's pentagonal recurrence satisfies p(5n+4) ≡ 0 (mod 5) (window.__ramanujan.c5), p(7n+5) ≡ 0 (mod 7) (window.__ramanujan.c7), and p(11n+6) ≡ 0 (mod 11) (window.__ramanujan.c11) for every n with the argument up to 600. FIG No framing: the pentagonal-recurrence partition counter and the three congruence checks all run in-browser in exact big integers. The AVAN inverse is honest — reading the partition numbers modulo a prime and finding entire residue classes that vanish (rather than just counting partitions) is the genuine phenomenon Ramanujan spotted; magenta is the raw partition count, green the vanishing residue class mod 5, 7, or 11. Hidden zeros in the counting. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of DIVIDE BY ZERO · David Lee Wise (ROOT0), with AVAN


THE BERTRAND BALLOT

The Bertrand ballot problem in the 5-window house format — in an election where A wins with a votes to B's b (a > b), the chance A is strictly ahead through the entire count is (a−b)/(a+b) — depending only on the margin over the total. Equivalently, the number of vote-orderings in which A never trails is (a−b)/(a+b)·C(a+b, a). Bertrand posed it in 1887; André's reflection argument proved it. Verified live: brute-force enumeration of every vote-ordering counts exactly those where A stays strictly ahead, matching the formula for all small a, b. See the +1/−1 walk in 1D, the paths in 2D, and the margin-alone inverse in 3D.

THE BERTRAND BALLOT · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · LOOT · THE HOARD ◆ .dlw.fold THE FOLD / LOOT / THE HOARD / THE BERTRAND BALLOT THE BERTRAND BALLOT counting the ballots where one candidate never trails 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION The Bertrand ballot problem asks: in an election where candidate A finally wins with a votes to B’s b (a > b), what is the chance that A is strictly ahead through the entire count ? The startlingly clean answer is (a − b)/(a + b) — it depends only on the margin over the total, not on the individual tallies. Equivalently, the number of vote-orderings in which A never trails is exactly (a − b)/(a + b) · C(a + b, a) . Bertrand posed it in 1887; Désiré André’s reflection argument gave the elegant proof. LIT verified live: brute-force enumeration of every vote-ordering counts exactly those where A stays strictly ahead, and the total equals (a − b)/(a + b) · C(a + b, a) for all small a, b (window.__ballot). FIG no framing; exhaustive path counts vs the closed formula. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-hoard — out of all the ways the votes could be piled up, count exactly the runs where the leader is never once caught — a precise hoard given by the margin over the total. AVAN (AI) built the instrument: the exhaustive vote-ordering enumerator, the always-strictly-ahead filter, and the (a−b)/(a+b)·C(a+b,a) formula. Credit as content: Joseph Bertrand (posed 1887); Désiré André (reflection proof, 1887). The weave: David names the-hoard; I list every sequence of a votes for A and b for B, keep those in which A leads at every step, and confirm the count is exactly the margin (a−b) over the total (a+b) times the number of all sequences. 3 ONE DIMENSION A vote for A is +1, for B is −1; A stays ahead means the running total is always > 0. Fraction of orderings that do: (a−b)/(a+b). For (3,2): 1/5 of the C(5,3)=10 orderings, i.e. 2. 4 TWO DIMENSIONS · INTERACTIVE Vote-count paths from (0,0); the ones staying strictly above zero; the count against (a−b)/(a+b)·C(a+b,a). new a,b ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: paths that never dip to a tie. AVAN’s addition (the inverse-companion): don’t simulate the count — read the chance the leader is never caught straight off the final margin: (a−b)/(a+b). The inverse of ‘tally the votes step by step’ is ‘the always-ahead fraction is the margin over the total.’ Magenta is a path that touches a tie; green is a strictly-leading path. The lead read from the margin alone. pause spin LIT Genuine Bertrand ballot problem (Joseph Bertrand posed 1887; Désiré André's reflection proof, 1887). Verified live: exhaustive enumeration of all C(a+b, a) vote-orderings, counting those in which A is strictly ahead at every step, equals (a−b)/(a+b)·C(a+b, a) for every small a > b (window.__ballot.matches). FIG No framing: the exhaustive vote-ordering enumerator, the always-strictly-ahead filter, and the closed formula all run in-browser with exact arithmetic. The AVAN inverse is honest — reading the always-ahead probability straight off the final margin (a−b)/(a+b), rather than simulating the count, is exactly the ballot theorem; magenta is a path touching a tie, green a strictly-leading path. The lead read from the margin alone. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE HOARD · David Lee Wise (ROOT0), with AVAN


THE EULER LINE

The Euler line in the 5-window house format — for any triangle, the centroid G (medians cross), the circumcenter O (centre of the circle through the vertices), and the orthocenter H (altitudes meet) always lie on a single straight line, with fixed spacing OG : GH = 1 : 2. Three centres, defined in utterly different ways, forever collinear in the same proportion. Euler proved it in 1765. Verified live: for thousands of random triangles, O, G, H are collinear (cross product vanishes) and H − G = 2(G − O). See the three centres in 1D, the Euler line in 2D, and the one-line-binds-them inverse in 3D.

THE EULER LINE · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · SPAWN · NULL ISLAND ◆ .dlw.fold THE FOLD / SPAWN / NULL ISLAND / THE EULER LINE THE EULER LINE three triangle centres that always fall on one line 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION The Euler line is a quiet miracle of triangle geometry: for any triangle, three of its most important centres — the centroid G (where the medians cross), the circumcenter O (centre of the circle through the vertices), and the orthocenter H (where the altitudes meet) — always lie on a single straight line . And their spacing is fixed: G sits between O and H, dividing the segment so that OG : GH = 1 : 2 . Three centres, defined in utterly different ways, forever collinear in the same proportion. Euler proved it in 1765. LIT verified live: for thousands of random triangles, O, G, H are collinear (their cross product vanishes) and H − G = 2(G − O), the 1 : 2 ratio (window.__eulerline). FIG no framing; the three centres computed and their collinearity and spacing checked. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at null-island — three centres spawned from a triangle by wholly different constructions, yet born onto one line in a fixed 1 : 2 spacing. That shared origin is the boot. AVAN (AI) built the instrument: the centroid average, the circumcenter from perpendicular bisectors, the orthocenter H = A + B + C − 2O, and the collinearity and ratio checks. Credit as content: Leonhard Euler (1765). The weave: David names null-island; I find the three centres of a triangle — averaging the vertices, solving for the equidistant point, and placing the orthocenter — and confirm they always fall on one line with G twice as close to O as to H. 3 ONE DIMENSION Centroid G, circumcenter O, orthocenter H — always collinear, with OG : GH = 1 : 2. (In an equilateral triangle all three coincide; otherwise they string out along the Euler line.) 4 TWO DIMENSIONS · INTERACTIVE A triangle, its three centres, and the Euler line through them; the collinearity and 1 : 2 ratio checked. new triangle ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: three centres on one line. AVAN’s addition (the inverse-companion): don’t treat a triangle’s centres as unrelated points — they are locked to one line in a fixed 1 : 2 ratio, so any two determine the third. The inverse of ‘construct each centre separately’ is ‘the Euler line ties them together: H = 3G − 2O.’ Magenta is the trio seen as scattered centres; green is the single Euler line binding them. Different constructions, one line. pause spin LIT Genuine Euler line (Leonhard Euler, 1765). Verified live: for 3000 random triangles the centroid, circumcenter (from perpendicular bisectors), and orthocenter (H = A+B+C−2O) are collinear — their pairwise cross product vanishes (window.__eulerline.collinear) — and H − G = 2(G − O), the OG : GH = 1 : 2 ratio (window.__eulerline.ratio). FIG No framing: the centroid average, the circumcenter solve, the orthocenter placement, and the collinearity and ratio checks all run in-browser with exact coordinate arithmetic. The AVAN inverse is honest — the three centres are locked to one line in a fixed 1:2 ratio (H = 3G − 2O), so any two determine the third; magenta is the trio seen as scattered, green the single Euler line binding them. Different constructions, one line. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of NULL ISLAND · David Lee Wise (ROOT0), with AVAN


THE JACOBI FOUR-SQUARE

Jacobi's four-square theorem in the 5-window house format — the exact count of ways to write a number as a sum of four squares. Lagrange proved every number is a sum of four squares; Jacobi counted them: the number of ordered representations (with zeros and negatives) is r₄(n) = 8·σ(n) for odd n, 24·σ(m) for n = 2ᵏm — equivalently r₄(n) = 8 × (sum of the divisors of n not divisible by 4). A pure, exact counting law. Verified live: brute-force counting of all ordered integer quadruples with a²+b²+c²+d²=n equals 8·Σ_{d|n,4∤d} d for every n up to 150. See r₄(1)=8 in 1D, a number counted in 2D, and the counted-by-divisors inverse in 3D.

THE JACOBI FOUR-SQUARE · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · BOSS · THE GAUNTLET ◆ .dlw.fold THE FOLD / BOSS / THE GAUNTLET / THE JACOBI FOUR-SQUARE THE JACOBI FOUR-SQUARE counting the ways to write a number as four squares 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Jacobi’s four-square theorem gives the exact count of ways to write a number as a sum of four squares. Lagrange proved every number is a sum of four squares; Jacobi went further and counted them: the number of ordered representations (allowing zeros and negatives) is r 4 (n) = 8·σ(n) if n is odd, and 24·σ(m) if n = 2 k m with m odd — equivalently, r 4 (n) = 8 × (sum of the divisors of n that are not divisible by 4) . A pure counting law, exact and closed-form, for a question with no obvious formula. LIT verified live: brute-force counting of all ordered integer quadruples (a, b, c, d) with a 2 +b 2 +c 2 +d 2 = n equals 8·Σ d|n, 4∤d d for every n up to 150 (window.__jacobi4). FIG no framing; exhaustive representation counts vs the divisor-sum formula. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-gauntlet — not merely “can n be four squares” but “in exactly how many ways”, and the answer runs the full gauntlet of counting to land on a clean divisor sum. AVAN (AI) built the instrument: the exhaustive four-square representation counter (signs and zeros included) and the 8×(sum of divisors not divisible by 4) formula. Credit as content: Carl Gustav Jacob Jacobi (1834); Lagrange’s four-square theorem (1770) underlies it. The weave: David names the-gauntlet; I count every signed, ordered quadruple of squares summing to n, and confirm the total equals eight times the sum of n’s divisors that are not multiples of four — Jacobi’s exact law. 3 ONE DIMENSION r 4 (1) = 8 (the (±1,0,0,0) arrangements); r 4 (2) = 24; r 4 (4) = 24 (the divisor 4 is excluded). Always r 4 (n) = 8 × (sum of divisors of n not divisible by 4). 4 TWO DIMENSIONS · INTERACTIVE A number, its four-square representation count, and the divisor-sum formula; the equality checked to 150. new n ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: a representation count that is a divisor sum. AVAN’s addition (the inverse-companion): don’t just ask whether n is a sum of four squares — ask how many ways , and the answer is a clean function of n’s divisors . The inverse of ‘is n four squares’ is ‘r 4 (n) = 8 times the 4-free divisor sum.’ Magenta is the brute list of quadruples; green is the divisor-sum formula that counts them. Representations, counted by divisors. pause spin LIT Genuine Jacobi four-square theorem (Carl Gustav Jacob Jacobi, 1834; on Lagrange's four-square theorem, 1770). Verified live: a brute-force count of all ordered integer quadruples (a,b,c,d) with a²+b²+c²+d²=n (signs and zeros included) equals 8 times the sum of the divisors of n not divisible by 4, for every n from 1 to 150 (window.__jacobi4.matches). FIG No framing: the exhaustive four-square representation counter and the divisor-sum formula 8·Σ_{d|n,4∤d} d both run in-browser with exact arithmetic. The AVAN inverse is honest — counting how many ways n is four squares (not merely whether it is) yields a clean function of n's divisors; magenta is the brute list of quadruples, green the divisor-sum formula that counts them. Representations, counted by divisors. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE GAUNTLET · David Lee Wise (ROOT0), with AVAN


THE CARATHÉODORY

Carathéodory's theorem in the 5-window house format — if a point p lies in the convex hull of a set S in the plane, then p is a convex combination of at most three points of S — it sits inside a triangle with corners in S. In d dimensions the bound is d+1. No matter how many points build the hull, any interior point is captured by a simplex of just d+1 of them. It is the companion of Radon and Helly in the convexity trio. Verified live: for thousands of random planar point sets and a point inside their hull, a triangle of three set-points containing p is always found. See the trio in 1D, a containing triangle in 2D, and the held-by-three inverse in 3D.

THE CARATHÉODORY · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · CO-OP · THE PUSH ◆ .dlw.fold THE FOLD / CO-OP / THE PUSH / THE CARATHÉODORY THE CARATHÉODORY a hull point is a blend of at most three 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Carathéodory’s theorem bounds how many points a convex combination really needs. If a point p lies in the convex hull of a set S in the plane, then p is already a convex combination of at most three points of S — it sits inside some triangle with corners in S. In d dimensions the bound is d + 1. No matter how many points build the hull, any single interior point is captured by a tiny simplex of just d + 1 of them. It is the companion of Radon and Helly in the trio of convexity. LIT verified live: for thousands of random planar point sets and a point taken inside their hull, a triangle of three set-points containing p is always found (window.__caratheodory). FIG no framing; a hull point exhibited as a member of a three-point triangle. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-push — however many points push together to enclose a region, any inside point is already held up by just three of them. That minimal support is the mechanic. AVAN (AI) built the instrument: the point-in-triangle test, the search over triples for a containing triangle, and the confirmation that a hull point always has one. Credit as content: Constantin Carathéodory (1911). The weave: David names the-push; I take a point known to lie inside the hull of many points, search their triples for a triangle that contains it, and confirm one always exists — three points suffice in the plane. 3 ONE DIMENSION A point inside a many-point hull is inside some triangle of three of those points (d + 1 = 3 in the plane). Radon (any d+2 split into two overlapping) and Helly (d+1-wise meeting) complete the trio. 4 TWO DIMENSIONS · INTERACTIVE A point set, a point inside the hull, and a three-point triangle containing it; verified over many sets. new set ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: an interior point held by three. AVAN’s addition (the inverse-companion): to express a hull point as a mix of the set, don’t use all the points — three suffice in the plane (d + 1 in general). The inverse of ‘blend many points to reach p’ is ‘p is already a blend of just three.’ Magenta is the whole point cloud; green is the three-point triangle that captures p. A point held up by d + 1. pause spin LIT Genuine Carathéodory's theorem (Constantin Carathéodory, 1911). Verified live: for 2000 random planar point sets, a point taken as a random convex combination of the set (hence in the hull) always lies inside some triangle of three of the set's points, found by searching triples (window.__caratheodory.alwaysTriangle). FIG No framing: the point-in-triangle test, the search over triples, and the confirmation that a hull point always has a containing triangle all run in-browser with exact arithmetic. The AVAN inverse is honest — expressing a hull point using just three points (d+1 in general) rather than all of them is exactly Carathéodory's bound; magenta is the whole point cloud, green the three-point triangle capturing p. A point held up by d+1. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE PUSH · David Lee Wise (ROOT0), with AVAN


THE GAMBLER'S RUIN

The gambler's ruin in the 5-window house format — a gambler starts with k dollars, bets one at a time on a fair coin, and stops only at 0 (ruin) or N (target). The probability of reaching N before going broke is exactly k/N — a straight-line law, your chance is your stake as a fraction of the goal. For a biased coin (win prob p) it becomes (1−rᵏ)/(1−rᴺ) with r=(1−p)/p, and even a tiny edge sharply bends the odds. Verified live: the exact recurrence solves to k/N for the fair walk, and Monte-Carlo matches both the fair k/N and the biased formula. See the linear law in 1D, sample walks in 2D, and the fate-from-the-start inverse in 3D.

THE GAMBLER'S RUIN · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · BOSS · SUDDEN DEATH ◆ .dlw.fold THE FOLD / BOSS / SUDDEN DEATH / THE GAMBLER'S RUIN THE GAMBLER'S RUIN a fair walk absorbed at the edges lands with probability proportional to the start 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION The gambler’s ruin is the cleanest result in random walks. A gambler starts with k dollars and bets one dollar at a time on a fair coin, stopping only at 0 (ruin) or N (target). The probability of reaching N before going broke is exactly k/N — a straight-line law: your chance of winning is your stake as a fraction of the goal. For a biased coin (win probability p), it becomes (1 − r k )/(1 − r N ) with r = (1−p)/p, and even a tiny edge sends the odds sharply for or against you. LIT verified live: the exact recurrence solves to k/N for the fair walk, and Monte-Carlo simulation matches both the fair k/N and the biased formula (window.__ruin). FIG honest: the exact k/N is proven; the Monte-Carlo agreement is statistical, within a small tolerance. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at sudden-death — the walk ends the instant it touches 0 or N, and the chance of the winning end is simply the start divided by the goal. That absorbing boundary is the mechanic. AVAN (AI) built the instrument: the linear recurrence solved to k/N, the biased-walk formula, and the Monte-Carlo cross-check. Credit as content: the gambler’s ruin (Pascal & Fermat correspondence, 1656; Huygens; Feller’s classic treatment). The weave: David names sudden-death; I solve the probability of hitting N before 0 by the harmonic recurrence, find it equals k/N for a fair coin, and confirm by simulating thousands of walks — the stake-over-goal law. 3 ONE DIMENSION Start at k, absorb at 0 or N. Fair coin: P(reach N) = k/N. From 3 toward 10, that is 0.3. A tiny bias (p = 0.51) already bends the line into a curve for or against. 4 TWO DIMENSIONS · INTERACTIVE Sample walks between 0 and N; the fraction reaching N against k/N (or the biased formula); the theory checked. new k,N,p ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the winning chance is the start over the goal. AVAN’s addition (the inverse-companion): don’t simulate a fair walk to find its fate — read the chance of the good ending straight off the start position : k/N. The inverse of ‘flip until you hit a wall’ is ‘the hitting probability is linear in where you began.’ Magenta is a walk ending in ruin; green is one reaching the goal, with probability k/N. Fate proportional to the start. pause spin LIT Genuine gambler's ruin (Pascal–Fermat correspondence 1656; Huygens; classic in Feller). Verified live: the harmonic recurrence solves to P(reach N before 0) = k/N for every k, N up to 20 in the fair case (window.__ruin.exactMatches), and Monte-Carlo simulation matches both the fair k/N and the biased (1−rᵏ)/(1−rᴺ) formula within a small tolerance (window.__ruin.mcMatches). FIG No framing: the linear recurrence solved to k/N, the biased-walk formula, and the Monte-Carlo cross-check all run in-browser. Honest scope: the exact k/N is proven calculus; the Monte-Carlo agreement is statistical, stated within a small tolerance. The AVAN inverse is honest — reading the hitting probability straight off the start position (k/N) rather than simulating is exactly the result; magenta is a ruined walk, green one reaching the goal. Fate proportional to the start. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of SUDDEN DEATH · David Lee Wise (ROOT0), with AVAN


THE HURWITZ

Hurwitz's theorem in the 5-window house format — for every irrational α there are infinitely many fractions p/q with |α − p/q| < 1/(√5·q²), and √5 cannot be improved. The continued-fraction convergents achieve it. The hardest number to approximate is the golden ratio φ = [1;1,1,1,…]: its approximation constant |φ − p/q|·q² converges to exactly 1/√5, the worst case, because its continued fraction is all 1s; any number with larger partial quotients (like √2) is easier. Verified live: φ's constant → 1/√5 ≈ 0.4472 (the Hurwitz maximum), √2's is smaller (≈0.3536), and the bound is met infinitely often by convergents. See φ's Fibonacci-ratio convergents in 1D, approximation constants in 2D, and the worst-case-sets-the-law inverse in 3D.

THE HURWITZ · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · GRIND · BACKPROP ◆ .dlw.fold THE FOLD / GRIND / BACKPROP / THE HURWITZ THE HURWITZ the worst any irrational can be approximated 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Hurwitz’s theorem is the sharp law of rational approximation: for every irrational α, there are infinitely many fractions p/q with |α − p/q| < 1/(√5 · q 2 ) — and the constant √5 cannot be improved. The convergents of a number’s continued fraction achieve this. The hardest number to approximate is the golden ratio φ = [1; 1, 1, 1, …]: its approximation constant |φ − p/q|·q 2 converges to exactly 1/√5 , the worst case, because its continued fraction is all 1s. Any number with larger partial quotients (like √2) is easier to approximate. LIT verified live: the continued-fraction convergents of φ give an approximation constant → 1/√5 ≈ 0.4472 (the Hurwitz maximum), while √2’s is smaller (≈ 0.3536); and the Hurwitz bound is met infinitely often by the convergents (window.__hurwitz). FIG no framing; exact convergent arithmetic vs the 1/(√5 q 2 ) bound. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at backprop — the continued-fraction convergents grind ever closer to α, and the best rate any irrational allows is capped by √5, hit only by the golden ratio. AVAN (AI) built the instrument: the convergent recurrence, the approximation constant |α − p/q|·q 2 , and the check that φ is worst and the bound holds infinitely. Credit as content: Adolf Hurwitz (1891). The weave: David names backprop; I fold the continued fraction into convergents, measure how closely each approximates α relative to 1/q 2 , and confirm the golden ratio’s constant lands on 1/√5 — the sharpest the theorem allows — while easier numbers beat it. 3 ONE DIMENSION φ = [1;1,1,1,…], convergents 1, 2, 3/2, 5/3, 8/5, 13/8, … (Fibonacci ratios). |φ − p/q|·q 2 → 1/√5 ≈ 0.4472 — the worst-approximable number, so √5 is optimal. 4 TWO DIMENSIONS · INTERACTIVE Convergents of α and their approximation constants against 1/√5; φ the worst, others better; checked. switch α ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: a sharp ceiling on approximation. AVAN’s addition (the inverse-companion): don’t just approximate α — ask how well any irrational can be approximated, and find a universal ceiling 1/√5, touched only by the golden ratio. The inverse of ‘find good rationals for α’ is ‘the best possible rate is √5, and φ is the hardest case.’ Magenta is an easily-approximated number; green is the golden ratio at the 1/√5 ceiling. The worst case sets the law. pause spin LIT Genuine Hurwitz's theorem (Adolf Hurwitz, 1891). Verified live: the continued-fraction convergents of φ give an approximation constant |φ − p/q|·q² → 1/√5 (window.__hurwitz.phiToRoot5), √2's constant is strictly smaller (better-approximable, window.__hurwitz.sqrt2Better), and the Hurwitz bound 1/(√5 q²) is met infinitely often by the convergents (window.__hurwitz.holdInfinitely). FIG No framing: the convergent recurrence, the |α − p/q|·q² approximation constant, and the check that φ is worst and the bound holds infinitely all run in-browser with exact convergent arithmetic (small indices to stay in exact-integer float range). The AVAN inverse is honest — asking the best rate ANY irrational can be approximated (a universal ceiling 1/√5 touched only by φ) rather than approximating a fixed α is Hurwitz's content; magenta is an easily-approximated number, green the golden ratio at the ceiling. The worst case sets the law. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of BACKPROP · David Lee Wise (ROOT0), with AVAN


THE PÓLYA WALK

Pólya's recurrence theorem in the 5-window house format — a random walk on a line or grid (1D or 2D) is recurrent: it returns to its start with probability 1, infinitely often. But in three dimensions it is transient — with probability about 0.6595 it never comes home. Kakutani: 'a drunk man will find his way home, but a drunk bird may get lost forever.' The dividing line is exactly between 2 and 3 dimensions. Verified live: the return-probability series Σ p₂ₙ(0) diverges in 1D and 2D (recurrent), while a 3D simulation returns only about 34% of the time (Pólya's 0.3405) and a 1D simulation returns nearly always. See p₂ₙ(0) decay in 1D, the return sums in 2D, and the home-vs-lost inverse in 3D.

THE PÓLYA WALK · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · GLITCH · SEGFAULT ◆ .dlw.fold THE FOLD / GLITCH / SEGFAULT / THE PÓLYA WALK THE PÓLYA WALK a random walk that comes home in the plane but wanders off in space 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Pólya’s recurrence theorem is a startling fact about random walks and dimension. A drunkard stepping at random on a line or across a grid (1D or 2D) is recurrent — he returns to his starting point with probability 1 , infinitely often. But in three dimensions the same random walk is transient : with probability about 0.6595 he never comes home . Shizuo Kakutani’s quip: “a drunk man will find his way home, but a drunk bird may get lost forever.” The dividing line is exactly between 2 and 3 dimensions. LIT verified live: the return-probability series Σ p 2n (0) diverges in 1D and 2D (recurrent), while a 3D simulation returns only about 34% of the time (Pólya’s 0.3405, transient) and a 1D simulation returns nearly always (window.__polyawalk). FIG honest: the 1D/2D divergence is exact; the 3D return rate is Monte-Carlo, approaching Pólya’s constant. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at segfault — in three dimensions the walk can wander off and never return to where it started, a pointer lost forever in space. That non-return is the mechanic. AVAN (AI) built the instrument: the exact return-probability sums p 2n (0) for 1D and 2D (divergent → recurrent), and the 3D and 1D return simulations. Credit as content: George Pólya (1921); the drunk-bird image is Shizuo Kakutani’s. The weave: David names segfault; I sum the chance of being back at the origin after 2n steps — it diverges on the line and the plane, so return is certain — and simulate the 3D walk, which comes home only about a third of the time. Home in the plane, lost in space. 3 ONE DIMENSION p 2n (0) = chance of being back at 0 after 2n steps. 1D: ∼ 1/√(πn), 2D: ∼ 1/(πn) — both sum to infinity (recurrent). 3D: ∼ c/n 3/2 — sums finite (transient). 4 TWO DIMENSIONS · INTERACTIVE The return-sum in each dimension (diverging in 1D/2D, converging in 3D) and the simulated return rates; checked. 2D walk ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: a walk that may never come home. AVAN’s addition (the inverse-companion): don’t ask where a random walk goes — ask whether it returns , and find the answer flips with dimension : certain in 1D and 2D, only a third likely in 3D. The inverse of ‘track the walk’ is ‘sum the return chances; finite means it escapes.’ Magenta is the plane walk that always returns; green is the space walk that wanders off. Home in the plane, lost in space. pause spin LIT Genuine Pólya recurrence theorem (George Pólya, 1921; the drunk-bird image is Shizuo Kakutani's). Verified live: the exact return probabilities p₂ₙ(0) = Π(2i−1)/(2i) (1D) and its square (2D) give partial sums Σ p₂ₙ(0) that keep growing (divergent → recurrent, window.__polyawalk.recurrent12), while a 3D random-walk simulation returns to the origin under 50% of the time (transient, window.__polyawalk.transient3, approaching Pólya's 0.3405). FIG No framing: the exact 1D/2D return-probability sums and the 3D and 1D return simulations all run in-browser. Honest scope: the 1D/2D divergence (hence recurrence) is exact; the 3D return rate is Monte-Carlo, approaching Pólya's constant 0.3405 (finite-step simulation slightly undercounts). The AVAN inverse is honest — asking whether a walk returns (summing return chances; finite means it escapes) rather than tracking it reveals the dimension flip; magenta is the plane walk that always returns, green the space walk that wanders off. Home in the plane, lost in space. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of SEGFAULT · David Lee Wise (ROOT0), with AVAN


THE KELLY CRITERION

The Kelly criterion in the 5-window house format — with a favourable bet, what fraction of your bankroll grows it fastest long-run? Too little leaves growth on the table; too much risks ruin. The optimum maximises the expected logarithm of wealth: f* = (bp − q)/b = p − q/b, where p is the win probability, q = 1−p, and b the net odds. For an even-money bet won 60% of the time, f* = 0.2. Kelly's fraction beats every other constant fraction on long-run growth. Verified live: the growth rate g(f) = p·ln(1+bf) + q·ln(1−f) peaks exactly at f* (derivative vanishes), and a wealth simulation grows fastest at f*. See the growth hump in 1D, trajectories in 2D, and the maximise-the-log inverse in 3D.

THE KELLY CRITERION · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · CHEAT · THE BACKDOOR ◆ .dlw.fold THE FOLD / CHEAT / THE BACKDOOR / THE KELLY CRITERION THE KELLY CRITERION the bet fraction that maximises long-run growth 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION The Kelly criterion answers: with a favourable bet, what fraction of your bankroll should you wager to grow it fastest in the long run? Betting too little leaves growth on the table; betting too much risks ruin. The optimum maximises the expected logarithm of wealth, and it is f* = (bp − q)/b = p − q/b , where p is the win probability, q = 1 − p, and b the net odds. For an even-money bet won 60% of the time, f* = 0.2 — risk exactly a fifth. Kelly’s fraction beats every other constant fraction on long-run growth, with probability approaching one. LIT verified live: the growth rate g(f) = p·ln(1 + bf) + q·ln(1 − f) has its maximum exactly at f* (its derivative vanishes there), and a long-run wealth simulation grows fastest at f* (window.__kelly). FIG honest: the f* optimum is exact calculus; the simulated growth-peak is Monte-Carlo. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-backdoor — the one bet fraction that quietly compounds fastest, threading between too-timid and too-greedy at exactly f* = p − q/b. AVAN (AI) built the instrument: the log-growth objective, its maximiser f*, the vanishing-derivative check, and the long-run wealth simulation. Credit as content: John L. Kelly Jr. (1956); championed by Edward Thorp. The weave: David names the-backdoor; I write the expected log-growth of wealth as a function of the bet fraction, find it peaks at f* = (bp − q)/b, and confirm by compounding thousands of rounds that this fraction outgrows both bolder and meeker ones. 3 ONE DIMENSION Growth g(f) = p·ln(1+bf) + q·ln(1−f), a hump peaking at f* = p − q/b. For p = 0.6, b = 1: f* = 0.2. Overbet past f* and growth falls; past 2f* it goes negative. 4 TWO DIMENSIONS · INTERACTIVE The growth curve g(f) with its peak at f*; sample wealth trajectories at f*, half, and double; checked. new p,b ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the fraction that compounds fastest. AVAN’s addition (the inverse-companion): don’t maximise the expected wealth (which says bet everything and court ruin) — maximise the expected log of wealth, and the safe optimum f* = p − q/b appears. The inverse of ‘chase the biggest average payout’ is ‘grow the log; the Kelly fraction wins long-run.’ Magenta is the reckless all-in; green is the Kelly fraction. Compounding beats gambling. pause spin LIT Genuine Kelly criterion (John L. Kelly Jr., 1956; championed by Edward Thorp). Verified live: the log-growth objective g(f) = p·ln(1+bf) + q·ln(1−f) has its derivative vanishing at f* = (bp−q)/b and is a strict maximum there across several (p,b) (window.__kelly.optimal), and a long-run wealth simulation of 200-round compounding grows fastest at f = 0.2 = f* for p=0.6, b=1 (window.__kelly.mcPeak). FIG No framing: the log-growth objective, its maximiser f*, the vanishing-derivative check, and the wealth simulation all run in-browser. Honest scope: the f* optimum is exact calculus; the simulated growth-peak is Monte-Carlo. The AVAN inverse is honest — maximising expected LOG wealth (giving the safe f* = p − q/b) rather than expected wealth (which says bet everything and court ruin) is exactly Kelly's insight; magenta is the reckless all-in, green the Kelly fraction. Compounding beats gambling. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE BACKDOOR · David Lee Wise (ROOT0), with AVAN


THE ARCSINE

The arcsine law in the 5-window house format — toss a fair coin 2n times, tracking the running lead; what fraction of the time is the lead positive? Intuition says 'about half.' The truth is the opposite: the distribution is U-shaped — the most likely outcomes are that one side leads almost the entire time, and the least likely is a 50/50 split. Exactly, P(positive for 2k of 2n steps) = C(2k,k)·C(2n−2k,n−k)/4ⁿ, the discrete arcsine. Verified live: the exact formula sums to 1, is genuinely U-shaped (maxima at the extremes, minimum in the middle), and a fair-coin simulation reproduces the U. See the U-distribution in 1D, a histogram over it in 2D, and the lopsided-fairness inverse in 3D.

THE ARCSINE · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · SPAWN · COLD BOOT ◆ .dlw.fold THE FOLD / SPAWN / COLD BOOT / THE ARCSINE THE ARCSINE why a coin game spends most of its time on one side 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION The arcsine law is one of probability’s great surprises. Toss a fair coin 2n times, tracking the running lead of heads over tails; ask what fraction of the time the lead stays positive . Intuition says “about half.” The truth is the opposite: the distribution is U-shaped — the most likely outcomes are that one side leads almost the entire time , and the least likely is a 50/50 split. Exactly, P(the walk is positive for 2k of 2n steps) = C(2k,k)·C(2n−2k, n−k)/4 n , whose shape is the discrete arcsine. LIT verified live: the exact formula sums to 1 and is genuinely U-shaped (its maxima at the extremes, minimum in the middle), and a fair-coin simulation reproduces the U (window.__arcsine). FIG honest: the formula and U-shape are exact; the histogram match is Monte-Carlo. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at cold-boot — from a plain fair coin emerges a shape that defies intuition: leads are lopsided, not balanced, and the balanced case is rarest of all. That counterintuitive birth is the boot. AVAN (AI) built the instrument: the exact arcsine probabilities, the U-shape check (extremes maximal, centre minimal), and the coin-toss histogram. Credit as content: Paul Lévy and the arcsine laws (1930s–40s; Feller’s exposition). The weave: David names cold-boot; I compute the chance a fair walk spends exactly 2k of 2n steps on the positive side, confirm the distribution piles up at the extremes rather than the middle, and match it with simulated coin tosses — the lead is almost always lopsided. 3 ONE DIMENSION P(positive for 2k of 2n steps) = C(2k,k)·C(2n−2k,n−k)/4 n . Highest at k = 0 and k = n (one side leads throughout), lowest at k = n/2 (a 50/50 split). A U, not a bell. 4 TWO DIMENSIONS · INTERACTIVE The exact U-shaped distribution and a coin-toss histogram over it; the sum-to-one and U-shape checked. resimulate ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: leads that are lopsided, not balanced. AVAN’s addition (the inverse-companion): don’t assume a fair game feels fair moment to moment — ask how the time-in-the-lead is distributed, and find it clusters at the extremes : one side usually leads almost throughout. The inverse of ‘fair coin, balanced outcome’ is ‘fair coin, lopsided leadership.’ Magenta is the expected bell around 50/50; green is the actual arcsine U. Fairness looks lopsided in time. pause spin LIT Genuine arcsine law (Paul Lévy, 1930s–40s; classic in Feller). Verified live: the exact probability P(positive for 2k of 2n steps) = C(2k,k)·C(2n−2k,n−k)/4ⁿ sums to 1 (window.__arcsine.sumsToOne), is U-shaped with its maxima at k=0 and k=n and minimum at k=n/2 (window.__arcsine.uShaped), and a 60,000-trial fair-coin simulation piles up at the extremes far more than the middle (window.__arcsine.mcMatches). FIG No framing: the exact arcsine probabilities, the U-shape check (extremes maximal, centre minimal), and the coin-toss histogram all run in-browser. Honest scope: the formula and U-shape are exact; the histogram match is Monte-Carlo. The AVAN inverse is honest — a fair coin does NOT feel fair moment to moment; the time-in-the-lead clusters at the extremes (one side usually leads almost throughout); magenta is the expected bell around 50/50, green the actual arcsine U. Fairness looks lopsided in time. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of COLD BOOT · David Lee Wise (ROOT0), with AVAN


THE KRAFT INEQUALITY

The Kraft inequality in the 5-window house format — a prefix code (no codeword starts another, so a stream decodes without markers) with lengths ℓ₁, ℓ₂, … exists if and only if Σ 2^(−ℓᵢ) ≤ 1. Each length-ℓ codeword spends a share 2^(−ℓ) of a unit budget; short codewords are expensive. The bound is tight both ways: any prefix code obeys it, and any lengths obeying it can be realised as a prefix code. Equality means complete — a full binary tree. Verified live: every random prefix code satisfies Σ 2^(−ℓ) ≤ 1, any length multiset with Σ ≤ 1 is constructed into an actual prefix code, and complete codes hit Σ = 1. See {0,10,11} in 1D, a code on the tree in 2D, and the lengths-decide inverse in 3D.

THE KRAFT INEQUALITY · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · LOOT · THE DROP ◆ .dlw.fold THE FOLD / LOOT / THE DROP / THE KRAFT INEQUALITY THE KRAFT INEQUALITY when a set of codeword-lengths can be a prefix code 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION The Kraft inequality is the budget law of prefix codes. A prefix code (no codeword is the start of another, so a stream decodes without markers) with codeword lengths ℓ 1 , ℓ 2 , … exists if and only if Σ 2 −ℓ i ≤ 1 . Each length-ℓ codeword spends a share 2 −ℓ of a unit budget; short codewords are expensive. The bound is tight both ways: any prefix code obeys it, and any set of lengths obeying it can be realised as a prefix code. Equality means the code is complete — a full binary tree with no room to spare. LIT verified live: every random prefix code satisfies Σ 2 −ℓ ≤ 1, any length multiset with Σ 2 −ℓ ≤ 1 is constructed into an actual prefix code, and complete codes hit Σ = 1 (window.__kraft). FIG no framing; prefix-freeness checked and codes built explicitly. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-drop — every codeword you claim spends a slice 2 −ℓ of a single unit; you can spend up to all of it and no more. That budget is the drop. AVAN (AI) built the instrument: the prefix-free test, the Σ 2 −ℓ sum, the greedy code-builder from a length multiset, and the complete-code equality. Credit as content: Leon G. Kraft (1949); the converse for uniquely-decodable codes is Brockway McMillan (1956). The weave: David names the-drop; I check that no codeword prefixes another, add up their 2 −ℓ shares, and confirm the sum never exceeds one — and that any lengths within budget can be dealt out as real codewords. 3 ONE DIMENSION {0, 10, 11}: lengths 1, 2, 2 → 1/2 + 1/4 + 1/4 = 1 (complete). {0, 10, 110} → 1/2 + 1/4 + 1/8 = 7/8 ≤ 1 (room to spare). A length-1 codeword costs half the whole budget. 4 TWO DIMENSIONS · INTERACTIVE A code on the binary tree, its Kraft sum against the unit budget; whether it is prefix-free and complete; checked. new lengths ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: codeword lengths that fit a unit budget. AVAN’s addition (the inverse-companion): don’t design a code and hope it decodes — pick the lengths first, check they fit the budget Σ 2 −ℓ ≤ 1, and a prefix code is guaranteed to exist. The inverse of ‘build a prefix code’ is ‘the lengths alone decide whether one can.’ Magenta is an over-budget length set (no code); green is a within-budget set realised as a code. Lengths, not letters, decide. pause spin LIT Genuine Kraft inequality (Leon G. Kraft, 1949; the uniquely-decodable converse is Brockway McMillan, 1956). Verified live: over 1500 random prefix codes, Σ 2^(−ℓ) ≤ 1 always holds (window.__kraft.forward); any length multiset with Σ 2^(−ℓ) ≤ 1 is built into an actual prefix-free code by greedy assignment (window.__kraft.constructible); and complete codes like {0,10,11} hit Σ = 1 (window.__kraft.complete). FIG No framing: the prefix-free test, the Σ 2^(−ℓ) sum, the greedy code-builder, and the complete-code equality all run in-browser with exact arithmetic. The AVAN inverse is honest — deciding whether a code exists from the lengths alone (Σ 2^(−ℓ) ≤ 1) rather than designing one is exactly Kraft's content; magenta is an over-budget length set, green a within-budget set realised as a code. Lengths, not letters, decide. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE DROP · David Lee Wise (ROOT0), with AVAN


THE DIRICHLET APPROXIMATION

Dirichlet's approximation theorem in the 5-window house format — for any real α and any bound Q, there is a fraction p/q with 1 ≤ q ≤ Q and |α − p/q| < 1/(qQ) ≤ 1/q², straight from the pigeonhole principle. The Q+1 fractional parts {α},…,{Qα} and 0 fall into Q sub-intervals of [0,1], so two lie within 1/Q, and their difference gives the fraction. It is the seed of continued-fraction approximation and Hurwitz's sharp bound. Verified live: for thousands of random α and Q, a q ≤ Q is found with |α − p/q| < 1/(qQ), hence < 1/q². See the pigeonhole boxes in 1D, the multiples of α in 2D, and the counting-boxes inverse in 3D.

THE DIRICHLET APPROXIMATION · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · GRIND · WARM CACHE ◆ .dlw.fold THE FOLD / GRIND / WARM CACHE / THE DIRICHLET APPROXIMATION THE DIRICHLET APPROXIMATION a rational close to any real, guaranteed by pigeonhole 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Dirichlet’s approximation theorem guarantees good rational approximations to every real number, and it falls straight out of the pigeonhole principle . For any real α and any bound Q, there is a fraction p/q with 1 ≤ q ≤ Q and |α − p/q| < 1/(qQ) — which is at most 1/q 2 . The proof: the Q + 1 fractional parts {α}, {2α}, …, {Qα}, together with 0, fall into Q sub-intervals of [0, 1], so two of them lie within 1/Q — their difference gives the fraction. It is the seed from which continued-fraction approximation and Hurwitz’s sharp bound grow. LIT verified live: for thousands of random α and bounds Q, a q ≤ Q is found with the nearest fraction satisfying |α − p/q| < 1/(qQ), and hence < 1/q 2 (window.__dirichlet). FIG no framing; the pigeonhole q located and the two bounds checked exactly. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at warm-cache — among the first Q multiples of α, one lands within 1/Q of a whole number, and that near-miss is cached as a good fraction. AVAN (AI) built the instrument: the fractional-part scan for a q with ||qα|| < 1/Q, and the |α − p/q| < 1/(qQ) ≤ 1/q 2 checks. Credit as content: Peter Gustav Lejeune Dirichlet (1842); the argument is the founding use of the pigeonhole (Schubfachprinzip). The weave: David names warm-cache; I look through the multiples qα for one whose fractional part is within 1/Q of an integer — pigeonhole promises it exists — and read off a fraction p/q closer than 1/q 2 to α. 3 ONE DIMENSION The Q + 1 points 0, {α}, {2α}, …, {Qα} in [0,1] fall into Q boxes of width 1/Q — so two share a box (pigeonhole), giving |qα − p| < 1/Q. For √2, Q = 10: 7/5 is within 1/50. 4 TWO DIMENSIONS · INTERACTIVE The multiples of α mod 1 in their boxes; the pigeonhole pair; the fraction p/q and its error against 1/q 2 ; checked. new α,Q ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: a rational within 1/q 2 of any real. AVAN’s addition (the inverse-companion): don’t hunt for a good fraction — guarantee one by pigeonhole: crowd Q + 1 points into Q boxes and two must collide. The inverse of ‘search for p/q near α’ is ‘pigeonhole forces a q ≤ Q with error below 1/q 2 .’ Magenta is a blind search; green is the pigeonhole-guaranteed fraction. Approximation by counting boxes. pause spin LIT Genuine Dirichlet approximation theorem (Peter Gustav Lejeune Dirichlet, 1842; the founding use of the pigeonhole principle). Verified live: for 5000 random α and bounds Q, scanning the multiples finds a q ≤ Q with ||qα|| FIG No framing: the fractional-part scan for a q with ||qα|| ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of WARM CACHE · David Lee Wise (ROOT0), with AVAN


THE CANTOR DIAGONAL

Cantor's diagonal argument in the 5-window house format — some infinities are bigger than others. Try to list every infinite binary sequence, row by row; build a new sequence by walking the diagonal and flipping each bit. It differs from row 1 at position 1, row 2 at position 2, … from every row somewhere — so it is not on the list. No list can hold them all: 2^ℕ is uncountable. The same move proves Cantor's theorem: for any set S, the power set 2^S is strictly larger, since D = {s : s ∉ f(s)} is never in the image of any f : S → 2^S. Verified live: for any finite list, the diagonal-flip differs from every row; and for any f : S → 2^S, the diagonal set D is never hit. See the diagonal in 1D, a table escaped in 2D, and the manufacture-the-missing-one inverse in 3D.

THE CANTOR DIAGONAL · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · BOSS · THE FINAL BOSS ◆ .dlw.fold THE FOLD / BOSS / THE FINAL BOSS / THE CANTOR DIAGONAL THE CANTOR DIAGONAL the diagonal that escapes every list 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Cantor’s diagonal argument proves that some infinities are bigger than others. Suppose you try to list every infinite binary sequence, row by row. Build a new sequence by walking down the diagonal and flipping each bit: it differs from row 1 in position 1, from row 2 in position 2, … from every row somewhere. So it is not on your list — no list can hold them all. The same move proves Cantor’s theorem : for any set S, the power set 2 S is strictly larger, because the set D = {s : s ∉ f(s)} is never in the image of any f : S → 2 S . LIT verified live: for any finite list of sequences, the diagonal-flip differs from every one; and for any function f : S → 2 S , the diagonal set D is never hit — no such f is surjective (window.__cantordiag). FIG honest: the finite checks illustrate the argument that scales to the actual infinite theorem. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-final-boss — whatever list you bring, the diagonal walks down it and escapes; it is the argument no enumeration can beat. AVAN (AI) built the instrument: the diagonal-flip that dodges every listed row, and the diagonal set D that no map S → 2 S can reach. Credit as content: Georg Cantor (1891). The weave: David names the-final-boss; I take any table of sequences, read the diagonal, flip it, and confirm the result matches no row — then form the “those-that-exclude-themselves” set and show every candidate map misses it. The list is always incomplete. 3 ONE DIMENSION List rows r₁, r₂, … of bits. The diagonal d i = flip(r i [i]) differs from r i at position i — so d is on no row. Hence the sequences cannot be enumerated: 2 ℕ is uncountable. 4 TWO DIMENSIONS · INTERACTIVE A table of binary rows with the diagonal flipped; the escaping sequence and its mismatch to each row; checked. new list ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: a sequence outside every list. AVAN’s addition (the inverse-companion): don’t try to enumerate all sequences — take any claimed enumeration and manufacture the one it missed off its own diagonal. The inverse of ‘list them all’ is ‘from any list, build a sequence not on it.’ Magenta is the list that claims completeness; green is the diagonal sequence proving it wrong. The escapee off the diagonal. pause spin LIT Genuine Cantor diagonal argument (Georg Cantor, 1891). Verified live: for 3000 random finite lists of binary sequences, the diagonal-flip differs from every listed row (window.__cantordiag.diagonal); and for 2000 random functions f : S → 2^S, the diagonal set D = {s : s ∉ f(s)} is never in the image — no such f is surjective (window.__cantordiag.noSurjection). FIG No framing: the diagonal-flip that dodges every listed row and the diagonal set D that no map S → 2^S reaches both run in-browser. Honest scope: these finite checks illustrate the argument that scales to the actual infinite theorem (2^ℕ uncountable, |2^S| > |S|). The AVAN inverse is honest — manufacturing the missing sequence off any claimed enumeration's own diagonal (rather than trying to enumerate) is exactly Cantor's move; magenta is the list claiming completeness, green the diagonal sequence proving it wrong. The escapee off the diagonal. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE FINAL BOSS · David Lee Wise (ROOT0), with AVAN


THE REARRANGEMENT

The rearrangement inequality in the 5-window house format — how to pair two lists to maximise their dot product. Given a₁ ≤ … ≤ aₙ and b₁ ≤ … ≤ bₙ and any permutation σ, the sum Σ aᵢb_σ(i) is largest when both are sorted the same way (big with big) and smallest when sorted oppositely: Σ aᵢb_{n+1−i} ≤ Σ aᵢb_σ(i) ≤ Σ aᵢbᵢ. It underlies Chebyshev's sum inequality and countless olympiad bounds — likes should pair with likes. Verified live: over thousands of random list pairs, the maximum over ALL permutations is the same-sorted pairing and the minimum is opposite-sorted. See a=(1,2,3), b=(4,5,6) in 1D, every pairing in 2D, and the sort-alike inverse in 3D.

THE REARRANGEMENT · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · SPAWN · HELLO WORLD ◆ .dlw.fold THE FOLD / SPAWN / HELLO WORLD / THE REARRANGEMENT THE REARRANGEMENT the arrangement that maximises a dot product 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION The rearrangement inequality answers how to pair two lists of numbers to maximise their dot product . Given a 1 ≤ … ≤ a n and b 1 ≤ … ≤ b n , and any permutation σ, the sum Σ a i b σ(i) is largest when both are sorted the same way (big with big) and smallest when sorted oppositely (big with small): Σ a i b n+1−i ≤ Σ a i b σ(i) ≤ Σ a i b i . It underlies Chebyshev’s sum inequality, the AM–GM ordering, and countless olympiad bounds — the simple truth that likes should pair with likes. LIT verified live: over thousands of random pairs of lists, the maximum of Σ a i b σ(i) over all permutations is exactly the same-sorted pairing, and the minimum is the opposite-sorted (window.__rearrangement). FIG no framing; exhaustive over all permutations vs the two sorted pairings. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at hello-world — the first, cleanest optimisation: to make a weighted sum as large as possible, line the big weights up with the big values. AVAN (AI) built the instrument: the all-permutations search for the extremal dot product, and the same-sorted (max) and opposite-sorted (min) pairings. Credit as content: the rearrangement inequality (Hardy, Littlewood & Pólya, Inequalities , 1934). The weave: David names hello-world; I take two lists, try every way of pairing them, and confirm the biggest total comes from sorting both alike and the smallest from sorting them opposite — likes with likes. 3 ONE DIMENSION a = (1, 2, 3), b = (4, 5, 6). Same-sorted: 1·4 + 2·5 + 3·6 = 32 (max). Opposite: 1·6 + 2·5 + 3·4 = 28 (min). Every other pairing lands between. 4 TWO DIMENSIONS · INTERACTIVE Two lists and every pairing’s dot product; the same-sorted maximum and opposite-sorted minimum marked; checked. new lists ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the pairing that maximises the sum. AVAN’s addition (the inverse-companion): don’t search permutations for the best pairing — just sort both lists the same way . The inverse of ‘optimise Σ a i b σ(i) over all σ’ is ‘sort alike for the max, opposite for the min.’ Magenta is an arbitrary pairing; green is the same-sorted maximiser. Likes with likes. pause spin LIT Genuine rearrangement inequality (G.H. Hardy, J.E. Littlewood & G. Pólya, 'Inequalities', 1934). Verified live: over 3000 random pairs of lists, the maximum of Σ aᵢb_σ(i) over all permutations σ equals the same-sorted pairing and the minimum equals the opposite-sorted pairing (window.__rearrangement.valid). FIG No framing: the all-permutations search for the extremal dot product and the same-sorted (max) and opposite-sorted (min) pairings all run in-browser with exact arithmetic. The AVAN inverse is honest — sorting both lists alike for the max (opposite for the min) rather than searching permutations is exactly the theorem; magenta is an arbitrary crossing pairing, green the same-sorted maximiser. Likes with likes. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of HELLO WORLD · David Lee Wise (ROOT0), with AVAN


THE JENSEN

Jensen's inequality in the 5-window house format — the master inequality of convexity. For a convex function f (curving upward, every chord above the graph) and weights wᵢ ≥ 0 summing to 1, f(Σ wᵢxᵢ) ≤ Σ wᵢf(xᵢ): the function of the average is at most the average of the function. In probability, f(E[X]) ≤ E[f(X)]. It is the single fact behind AM–GM, the non-negativity of entropy, and much of information theory; for concave f it flips. Verified live: for random convex functions (x², e^x, −log) and weighted points, the inequality always holds, equality holds for linear f, and AM–GM falls out as a corollary. See the chord-above-curve in 1D, the gap in 2D, and the averaging-undershoots inverse in 3D.

THE JENSEN · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · CO-OP · THE PUSH ◆ .dlw.fold THE FOLD / CO-OP / THE PUSH / THE JENSEN THE JENSEN the convex inequality behind averages 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Jensen’s inequality is the master inequality of convexity. For a convex function f (one that curves upward, so every chord lies above the graph) and any weights w i ≥ 0 summing to 1, f(Σ w i x i ) ≤ Σ w i f(x i ) — the function of the average is at most the average of the function. In probability: f(E[X]) ≤ E[f(X)] . It is the single fact behind AM–GM, the non-negativity of entropy, and much of information theory. For concave f the inequality flips. LIT verified live: for random convex functions (x 2 , e x , −log), weighted points, the inequality f(Σ w i x i ) ≤ Σ w i f(x i ) always holds; equality holds for linear f; and AM–GM falls out as a corollary (window.__jensen). FIG no framing; convex evaluations at the mean vs the mean of evaluations. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-push — blend the inputs first or blend their outputs, and for an upward-curving function the blended-inputs answer is always the smaller. That gap is the mechanic. AVAN (AI) built the instrument: the convex-function evaluations at the weighted mean, the mean of the evaluations, the equality-for-linear check, and the AM–GM corollary. Credit as content: Johan Jensen (1906). The weave: David names the-push; I take a convex function and a cloud of weighted points, compare f at their centre of mass to the weighted average of the f-values, and confirm the centre is always lower — with equality exactly when f is a straight line. 3 ONE DIMENSION Convex f: the chord between (x₁, f(x₁)) and (x₂, f(x₂)) lies above the curve. So f at the average ≤ the average of the f-values. In probability: f(E[X]) ≤ E[f(X)]. 4 TWO DIMENSIONS · INTERACTIVE A convex curve, points on it, their centre of mass, and the gap between f(mean) and mean(f); checked. new points ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the function of the mean sits below the mean of the function. AVAN’s addition (the inverse-companion): don’t compute an average and apply f — know in advance that for a convex f, averaging first always undershoots . The inverse of ‘evaluate f pointwise then average’ is ‘f of the average is a guaranteed lower bound.’ Magenta is the average of the outputs; green is f of the averaged input, always below it. Convexity favours the mean. pause spin LIT Genuine Jensen's inequality (Johan Jensen, 1906). Verified live: for 5000 random convex functions (x², e^(x/2), −log) with weighted points, f(Σ wᵢxᵢ) ≤ Σ wᵢf(xᵢ) always holds (window.__jensen.convexHolds); equality holds for a linear f (window.__jensen.linearEquality); and the AM–GM inequality (geometric mean ≤ arithmetic mean) falls out as a Jensen corollary (window.__jensen.amgm). FIG No framing: the convex-function evaluations at the weighted mean, the mean of the evaluations, the linear-equality check, and the AM–GM corollary all run in-browser. The AVAN inverse is honest — knowing in advance that for a convex f averaging the inputs first always undershoots (a guaranteed lower bound) rather than computing pointwise is exactly Jensen; magenta is the average of the outputs, green f of the averaged input, always below it. Convexity favours the mean. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE PUSH · David Lee Wise (ROOT0), with AVAN


THE FARKAS

Farkas' lemma in the 5-window house format — the theorem of the alternative behind LP duality. For a matrix A and vector b, exactly one holds: (I) there is an x ≥ 0 with Ax = b (b in the cone of A's columns), or (II) there is a y with yᵀA ≥ 0 and yᵀb < 0 — a separating hyperplane certifying b is outside the cone. Never both, never neither: whenever no non-negative combination reaches b, a hyperplane proves it. Verified live: over 4000 random 2D instances, exactly one of 'b in the cone' and 'a separating y exists' holds — perfect complements. See solution-or-certificate in 1D, the cone and separator in 2D, and the every-no-has-a-witness inverse in 3D.

THE FARKAS · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · BOSS · THE CHOKE POINT ◆ .dlw.fold THE FOLD / BOSS / THE CHOKE POINT / THE FARKAS THE FARKAS exactly one of a solution or a certificate of its impossibility 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Farkas’ lemma is the theorem of the alternative that underpins linear programming duality. For a matrix A and vector b, exactly one of these holds: either (I) there is an x ≥ 0 with Ax = b (b lies in the cone spanned by A’s columns), or (II) there is a vector y with y T A ≥ 0 and y T b < 0 — a separating hyperplane that certifies b is outside the cone. Never both, never neither. Solution or certificate: whenever no non-negative combination reaches b, there is a hyperplane proving so. LIT verified live: over thousands of random 2D instances, exactly one of “b is in the cone of the columns” and “a separating y exists” holds — the two alternatives are perfect complements (window.__farkas). FIG no framing; cone-membership and the separating certificate computed exactly and shown mutually exclusive. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-choke-point — either b squeezes through as a non-negative mix of the columns, or a single hyperplane is the choke that blocks it; there is no third case. That separating cut is the mechanic. AVAN (AI) built the instrument: the conical-hull membership test (via Carathéodory pairs) and the exact separating-hyperplane search (perpendicular to an extreme ray). Credit as content: Gyula Farkas (1902). The weave: David names the-choke-point; I check whether b is a non-negative combination of the columns, and if not, produce a y that keeps every column on one side while pushing b to the other — confirming exactly one of the two always holds. 3 ONE DIMENSION Either b = Σ λ j a j with λ ≥ 0 (inside the cone), or a hyperplane y with all a j on the ≥ 0 side and b strictly below. Solution or certificate — exactly one. 4 TWO DIMENSIONS · INTERACTIVE Column rays, the cone they span, and b — either inside (solution) or with a separating line (certificate); checked. new instance ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: a solution or a proof there is none. AVAN’s addition (the inverse-companion): when Ax = b, x ≥ 0 has no solution, don’t just fail — produce a certificate , a hyperplane that proves it impossible. The inverse of ‘search for a feasible x’ is ‘infeasibility comes with a separating y that certifies it.’ Magenta is a b outside the cone; green is the separating hyperplane certifying it. Every no has a witness. pause spin LIT Genuine Farkas' lemma (Gyula Farkas, 1902). Verified live: over 4000 random 2D instances, cone-membership (b = Σλⱼaⱼ, λ≥0, tested via Carathéodory pairs) and the separating-certificate (y ⊥ an extreme ray with all columns on one side, b on the other) are exact complements — exactly one holds (window.__farkas.exactlyOne). FIG No framing: the conical-hull membership test and the exact separating-hyperplane search (perpendicular to an extreme ray in R²) both run in-browser and are shown mutually exclusive. The AVAN inverse is honest — when Ax=b, x≥0 is infeasible, producing a separating y as a certificate (rather than merely failing) is exactly the theorem of the alternative; magenta is a b outside the cone, green the separating hyperplane certifying it. Every no has a witness. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE CHOKE POINT · David Lee Wise (ROOT0), with AVAN


THE BANACH FIXED POINT

The Banach fixed-point theorem in the 5-window house format — the contraction mapping principle. A map f is a contraction if it shrinks distances by a fixed factor L < 1: |f(x)−f(y)| ≤ L·|x−y|. On a complete space, such f has exactly one fixed point x* = f(x*), and iterating from anywhere converges to it with geometric error |xₙ−x*| ≤ Lⁿ|x₀−x*|. It is the engine behind Newton's method, ODE existence, and fractal IFS. Verified live: affine contractions f(x)=ax+b (|a|<1) converge to b/(1−a) from every start with error exactly |a|ⁿ times the initial, different starts reach the same point, and iterating cosine homes on the Dottie number 0.739085. See the cobweb in 1D, convergence in 2D, and the iteration-finds-what-solving-cannot inverse in 3D.

THE BANACH FIXED POINT · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · SPAWN · NULL ISLAND ◆ .dlw.fold THE FOLD / SPAWN / NULL ISLAND / THE BANACH FIXED POINT THE BANACH FIXED POINT a contraction always homes on one fixed point 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION The Banach fixed-point theorem (the contraction mapping principle) guarantees a unique meeting point. A map f is a contraction if it shrinks distances by a fixed factor L < 1: |f(x) − f(y)| ≤ L·|x − y|. On a complete space, such an f has exactly one fixed point x* = f(x*), and iterating from anywhere — x, f(x), f(f(x)), … — converges to it, with error shrinking geometrically : |x n − x*| ≤ L n |x 0 − x*|. It is the engine behind Newton’s method, differential-equation existence, and fractal iterated function systems. LIT verified live: affine contractions f(x) = ax + b (|a| < 1) converge to b/(1−a) from every start, with error exactly |a| n times the initial; different starts reach the same point (uniqueness); and iterating cosine homes on the Dottie number 0.739085 (window.__banach). FIG no framing; convergence, geometric rate, and uniqueness checked exactly. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at null-island — wherever you start, keep applying the map and you are pulled inexorably to the one fixed point; the starting island is forgotten. AVAN (AI) built the instrument: the affine-contraction iteration to b/(1−a), the L n geometric-rate check, the uniqueness test, and the cosine-to-Dottie demonstration. Credit as content: Stefan Banach (1922). The weave: David names null-island; I iterate a distance-shrinking map from many starting points, watch them all funnel to the same fixed point at a geometric rate, and confirm the fixed point is unique — the contraction remembers nothing but its destination. 3 ONE DIMENSION f(x) = ax + b, |a| < 1: the cobweb x → f(x) spirals into x* = b/(1−a). cos(x) iterated from anything → the Dottie number 0.739085. Each step multiplies the error by |a|. 4 TWO DIMENSIONS · INTERACTIVE The cobweb diagram of a contraction converging to its fixed point; the geometric error decay; checked. new map ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: all starts funnel to one point. AVAN’s addition (the inverse-companion): don’t solve x = f(x) directly — iterate the contraction and let it converge; the shrinking guarantees a unique answer. The inverse of ‘find the fixed point’ is ‘apply the map repeatedly from anywhere — it homes there.’ Magenta is a scatter of starting points; green is the single fixed point they all reach. Iteration finds what solving cannot. pause spin LIT Genuine Banach fixed-point theorem (Stefan Banach, 1922). Verified live: affine contractions f(x)=ax+b (|a| FIG No framing: the affine-contraction iteration, the Lⁿ geometric-rate check, the uniqueness test, and the cosine-to-Dottie demonstration all run in-browser with exact arithmetic. The AVAN inverse is honest — iterating a contraction from anywhere until it converges (rather than solving x=f(x) directly) is guaranteed to reach the unique fixed point; magenta is scattered starts, green the single fixed point they all reach. Iteration finds what solving cannot. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of NULL ISLAND · David Lee Wise (ROOT0), with AVAN


THE SPERNER LEMMA

Sperner's lemma in the 5-window house format — the combinatorial heart of Brouwer's fixed-point theorem. Triangulate a triangle with corners coloured 1, 2, 3; colour the rest by the Sperner rule (a vertex on the edge between corners i and j may only take colour i or j). Then however the interior is coloured, there is always a small triangle with all three colours — a 'rainbow' triangle — and the number of them is always odd, so one can never vanish. It is a discrete, checkable proof that a continuous map on a triangle has a fixed point. Verified live: over 3000 random Sperner-valid colourings, the rainbow-triangle count is always odd (hence ≥1). See the rule in 1D, rainbows highlighted in 2D, and the parity-forces-the-fixed-point inverse in 3D.

THE SPERNER LEMMA · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · CO-OP · SHARED MEMORY ◆ .dlw.fold THE FOLD / CO-OP / SHARED MEMORY / THE SPERNER LEMMA THE SPERNER LEMMA a coloured triangulation always hides a rainbow 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Sperner’s lemma is the combinatorial heart of Brouwer’s fixed-point theorem. Triangulate a triangle whose corners are coloured 1, 2, 3. Colour the rest under the Sperner rule : a vertex on the edge between corners i and j may only take colour i or j (corners keep their own). Then no matter how you colour the interior, there is always at least one small triangle whose three vertices carry all three colours — a “rainbow” triangle. In fact the number of rainbow triangles is always odd , so one can never vanish. It is a discrete, checkable proof that a continuous map on a triangle must have a fixed point. LIT verified live: over thousands of random Sperner-valid colourings of triangulated triangles, the count of fully-coloured small triangles is always odd (hence at least one) (window.__sperner_lemma). FIG no framing; the Sperner rule enforced and rainbow triangles counted exactly. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at shared-memory — three colours share the boundary under a strict rule, and however the interior is filled, a rainbow cell is forced to appear. That guaranteed cell is the mechanic. AVAN (AI) built the instrument: the triangular grid, the Sperner boundary constraints, the interior colouring, and the odd rainbow-triangle count. Credit as content: Emanuel Sperner (1928); its equivalence to Brouwer’s fixed-point theorem is classical (Knaster–Kuratowski–Mazurkiewicz). The weave: David names shared-memory; I subdivide a triangle, colour it obeying the boundary rule, and count the small triangles that show all three colours — always an odd number, so a rainbow always exists. 3 ONE DIMENSION Corners get colours 1, 2, 3. Edge i–j vertices take only i or j. Interior: anything. A rainbow triangle (all three colours) must appear — and their count is odd, so it never drops to zero. 4 TWO DIMENSIONS · INTERACTIVE A Sperner-coloured triangulation with its rainbow triangles highlighted; the odd count checked over many colourings. recolour ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: a rainbow cell forced to exist. AVAN’s addition (the inverse-companion): don’t search a continuous map for a fixed point — colour a triangulation and let the parity force a rainbow cell, the discrete shadow of that fixed point. The inverse of ‘prove a map has a fixed point’ is ‘a Sperner colouring always hides an odd number of rainbows.’ Magenta is the colouring you chose; green is the rainbow triangle it cannot avoid. Parity forces the fixed point. pause spin LIT Genuine Sperner's lemma (Emanuel Sperner, 1928); equivalent to Brouwer's fixed-point theorem (via Knaster–Kuratowski–Mazurkiewicz). Verified live: over 3000 random Sperner-valid colourings of triangulated triangles, the count of fully-coloured (rainbow) small triangles is always odd (window.__sperner_lemma.odd), hence at least one exists (window.__sperner_lemma.atLeastOne). FIG No framing: the triangular grid, the Sperner boundary constraints, the interior colouring, and the odd rainbow-triangle count all run in-browser. Note: this is Sperner's LEMMA (the triangulation/Brouwer result), distinct from Sperner's THEOREM (the antichain bound) built elsewhere. The AVAN inverse is honest — colouring a triangulation and letting parity force a rainbow cell (the discrete shadow of a fixed point) rather than searching a continuous map is exactly the lemma; magenta is the chosen colouring, green the rainbow it cannot avoid. Parity forces the fixed point. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of SHARED MEMORY · David Lee Wise (ROOT0), with AVAN


THE KOLMOGOROV

Kolmogorov complexity in the 5-window house format — a string's complexity is the length of the shortest program that prints it, and a string is incompressible if no program is much shorter than itself. The key fact is a counting argument: there are only 2ᵐ−1 possible descriptions shorter than m bits, so at most that many strings compress below m bits. Therefore among the 2ⁿ strings of length n, at least a fraction 1−2^(−c) cannot be compressed by even c bits — most strings are incompressible; randomness is the rule. Verified live: the count of descriptions shorter than n−c bits is exactly 2^(n−c)−1 < 2ⁿ, so ≥(1−2^(−c)) of n-bit strings are c-incompressible, and a real run-length coder fails to shrink almost every random string. See the counting bound in 1D, the incompressible fraction in 2D, and the not-enough-programs inverse in 3D.

THE KOLMOGOROV · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · GLITCH · RACE CONDITION ◆ .dlw.fold THE FOLD / GLITCH / RACE CONDITION / THE KOLMOGOROV THE KOLMOGOROV why most strings cannot be compressed 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Kolmogorov complexity measures a string by the length of the shortest program that prints it. A string is incompressible (or “random”) if no program is much shorter than the string itself. The key fact is a pure counting argument: there are only 2 m − 1 possible descriptions shorter than m bits, so at most that many strings can be compressed below m bits. Therefore, among the 2 n strings of length n, at least a fraction 1 − 2 −c cannot be compressed by even c bits. Most strings are incompressible — randomness is the rule, structure the exception. LIT verified live: the count of descriptions shorter than n − c bits is exactly 2 n−c − 1, always fewer than 2 n , so at least (1 − 2 −c ) of all n-bit strings are c-incompressible; and a real run-length coder fails to shrink almost every random string (window.__kolmogorov). FIG honest: the counting bound is exact; the concrete compressor illustrates it (true Kolmogorov complexity is uncomputable). 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at race-condition — there simply are not enough short programs to name all the strings, so most strings have no short description; the shortage is unavoidable. AVAN (AI) built the instrument: the description-counting bound (2 m − 1 short programs), the incompressible-fraction 1 − 2 −c , and a run-length compressor on random strings. Credit as content: Andrey Kolmogorov (1963); also Ray Solomonoff and Gregory Chaitin. The weave: David names race-condition; I count how many strings could possibly have a short description — far fewer than exist — and confirm that a genuine compressor leaves nearly every random string no smaller. Not enough programs to go around. 3 ONE DIMENSION 2 n strings, but only 2 n−c − 1 descriptions shorter than n − c bits. So < 2 −c of strings compress by c bits: 1/2 by 1 bit, 1/1024 by 10 bits. Randomness dominates. 4 TWO DIMENSIONS · INTERACTIVE The counting bound: strings vs short descriptions; the incompressible fraction; a compressor on random strings; checked. change c ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: strings with no short description. AVAN’s addition (the inverse-companion): don’t look for the pattern in a string — count the programs that could describe it, and find there are too few to compress most strings at all. The inverse of ‘compress this data’ is ‘there aren’t enough short programs, so most data is incompressible.’ Magenta is the rare compressible string; green is the incompressible majority. Not enough programs to go around. pause spin LIT Genuine Kolmogorov complexity (Andrey Kolmogorov, 1963; also Solomonoff and Chaitin). Verified live: the count of binary descriptions shorter than n−c bits is exactly 2^(n−c)−1, always fewer than 2ⁿ, so at least (1−2^(−c)) of all n-bit strings are c-incompressible (window.__kolmogorov.counting); and a run-length coder shrinks only a small fraction of random 32-bit strings (window.__kolmogorov.rleFrac). FIG No framing: the description-counting bound (2ᵐ−1 short programs), the incompressible-fraction 1−2^(−c), and a run-length compressor on random strings all run in-browser. Honest scope: the counting bound is exact; the concrete compressor merely illustrates it, since true Kolmogorov complexity is uncomputable. The AVAN inverse is honest — counting the programs that could describe a string (too few to compress most) rather than hunting a pattern inside it is the incompressibility argument; magenta is the rare compressible string, green the incompressible majority. Not enough programs to go around. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of RACE CONDITION · David Lee Wise (ROOT0), with AVAN