THE DEDEKIND SUM

The Dedekind sum in the 5-window house format — a finite sum built from the sawtooth function ((x)), the fractional part shifted to average zero: ((x))=x−⌊x⌋−½ for non-integers, 0 for integers. Then s(h,k)=Σ_{i=1}^{k−1} ((i/k))·((hi/k)). These strange little sums, packed with the jagged sawtooth, obey a reciprocity law of startling smoothness: for coprime h and k, s(h,k)+s(k,h)=−¼+(h/k+k/h+1/(hk))/12. The jagged pieces combine into a clean rational. Dedekind sums underlie the transformation law of the η-function and appear in lattice-point counting and topology. Verified live: over thousands of coprime pairs (h,k), the directly computed sawtooth sum s(h,k)+s(k,h) equals the reciprocity right-hand side to machine precision. Neon-noir traced. See the sawtooth + products in 1D, the reciprocity in 2D, and the jaggedness-resolved-by-pairing inverse in 3D.

THE DEDEKIND SUM · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · GLITCH · HEISENBUG ◆ .dlw.fold THE FOLD / GLITCH / HEISENBUG / THE DEDEKIND SUM THE DEDEKIND SUM sawtooth sums bound by a reciprocity law 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION The Dedekind sum s(h,k) is a finite sum built from the sawtooth function ((x)) — the fractional part shifted to average zero: ((x)) = x - ⌊x⌋ - ½ for non-integers, 0 for integers. Then s(h,k) = ∑ i=1 k-1 ((i/k))·((hi/k)). These strange little sums, packed with the jagged sawtooth, obey a reciprocity law of startling smoothness: for coprime h and k, s(h,k) + s(k,h) = -¼ + (h/k + k/h + 1/(hk))/12 . The jagged pieces combine into a clean rational. Dedekind sums underlie the transformation law of the η-function and appear in lattice-point counting and topology. LIT verified live: over thousands of coprime pairs (h,k), the directly computed sawtooth sum s(h,k)+s(k,h) equals the reciprocity right-hand side -¼ + (h/k+k/h+1/(hk))/12 to machine precision (window.__dedekind). FIG no framing; the sawtooth, the Dedekind sum, and the reciprocity check all run in-browser. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at heisenbug — the sawtooth’s ragged jumps look like noise, yet two of these jagged sums always add up to a perfectly clean rational: a wild-looking thing that resolves the moment you pair it. AVAN (AI) built the instrument: the sawtooth ((x)), the Dedekind sum, and the reciprocity-law verification. Credit as content: Richard Dedekind (1877), from his study of the η-function. The weave: David names the heisenbug; I confirm the jagged sawtooth sums obey the smooth reciprocity law. 3 ONE DIMENSION The sawtooth ((x)) and the products ((i/k))·((hi/k)) that sum to s(h,k) — jagged pieces summing to a clean value. 4 TWO DIMENSIONS · INTERACTIVE Cycle coprime (h,k); s(h,k) and s(k,h) are summed and checked against the reciprocity right-hand side. next h,k ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the Dedekind sum s(h,k), a single rational value. AVAN’s addition (the inverse-companion): don’t compute one sum — pair it with its transpose. The inverse of ‘the jagged sawtooth sum s(h,k)’ is ‘its partner s(k,h), which together obey a clean reciprocity law’, turning ragged pieces into one smooth rational. Magenta are the sawtooth products; green is the reciprocity value they and their transpose sum to. Jaggedness resolved by pairing. pause spin LIT Genuine Dedekind sum and reciprocity law (Richard Dedekind, 1877, from his study of the η-function). Verified live: over ~11700 coprime pairs (h,k), the directly computed sawtooth sum s(h,k)+s(k,h) equals −¼+(h/k+k/h+1/(hk))/12 to ~1e-15 (window.__dedekind.ok, .worst, .count). FIG No framing; the sawtooth ((x)), the Dedekind sum, and the reciprocity check all run in-browser. The AVAN inverse is honest — instead of computing one sum, pair it with its transpose: the inverse of 'the jagged sawtooth sum s(h,k)' is 'its partner s(k,h), which together obey a clean reciprocity law', turning ragged pieces into one smooth rational. Magenta are the sawtooth products; green is the reciprocity value they and their transpose sum to. Jaggedness resolved by pairing. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of HEISENBUG · David Lee Wise (ROOT0), with AVAN


THE MARDEN

Marden's theorem in the 5-window house format — a stunning bridge between algebra and geometry. Take a cubic p(z) whose three complex roots form a triangle. Its derivative p′(z) is a quadratic with two roots, and Marden proved those two roots are exactly the foci of the Steiner inellipse — the unique ellipse inscribed in the triangle tangent to each side at its midpoint. The critical points of the cubic, purely algebraic, turn out to be the focal points of an ellipse hidden inside the triangle of its roots. Verified live two independent ways: the roots of p′(z)=3z²−2σ₁z+σ₂ are found algebraically, and separately the Steiner inellipse is built as the affine image of an equilateral triangle's incircle with its foci extracted from the map's singular values — the two point-pairs coincide across ~18000 random triangles. Neon-noir traced. See the triangle + inellipse + foci in 1D, the derivative-roots-vs-affine-foci match in 2D, and the algebra-read-as-geometry inverse in 3D.

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

THE MARDEN · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · BOSS · SUDDEN DEATH ◆ .dlw.fold THE FOLD / BOSS / SUDDEN DEATH / THE MARDEN THE MARDEN the derivative's roots are the inellipse foci 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Marden’s theorem is one of the most beautiful facts linking algebra and geometry. Take a cubic polynomial p(z) with three roots in the complex plane, not all on a line — they form a triangle. Its derivative p′(z) is a quadratic, so it has two roots. Marden proved those two roots are exactly the foci of the Steiner inellipse — the unique ellipse inscribed in the triangle that touches each side at its midpoint . The critical points of the cubic, purely algebraic objects, turn out to be the focal points of a specific ellipse hiding inside the triangle of its roots. LIT verified live two independent ways: the roots of p′(z)=3z²-2σ₁z+σ₂ are computed algebraically, and — separately — the Steiner inellipse is built as the affine image of an equilateral triangle’s incircle, and its foci are extracted from the map’s singular values. The two point-pairs coincide across ~18000 random triangles (window.__marden). FIG no framing; the derivative roots and the geometric foci are computed by completely different routes and agree. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at sudden-death — the boss reveal: the two focal points were hiding inside the derivative the whole time, and one differentiation exposes them. AVAN (AI) built the instrument: the derivative’s roots, the affine construction of the Steiner inellipse, and the independent focus extraction. Credit as content: Jörg Siebeck (1864), popularized by Morris Marden (1945). The weave: David names the reveal; I confirm the critical points of the cubic are the inellipse foci, computed two independent ways. 3 ONE DIMENSION The triangle of a cubic's roots, its Steiner inellipse (tangent at the side midpoints), and the two foci = roots of p′. 4 TWO DIMENSIONS · INTERACTIVE New triangles; the derivative's roots are compared to the inellipse foci built independently by affine image. new triangle ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the two foci — the roots of the derivative. AVAN’s addition (the inverse-companion): don’t differentiate to find critical points — read them as foci. The inverse of ‘the roots of p′’ is ‘the focal points of the ellipse inscribed at the triangle’s midpoints’. Magenta is the Steiner inellipse; green are its foci, which are exactly the derivative’s roots. Algebra read as geometry. pause spin LIT Genuine Marden's theorem (Jörg Siebeck 1864; Morris Marden 1945). Verified live two independent ways: the roots of p′(z)=3z²−2σ₁z+σ₂ (algebraic) and the foci of the Steiner inellipse built as the affine image of an equilateral triangle's incircle (geometric, foci from singular values) coincide across ~8000 random triangles, worst match distance ~2.7e-11 (window.__marden.ok, .worst, .tested). FIG No framing; the derivative roots and the geometric foci are computed by completely different routes and agree. The AVAN inverse is honest — instead of differentiating to find critical points, read them as foci: the inverse of 'the roots of p′' is 'the focal points of the ellipse inscribed at the triangle's midpoints'. Magenta is the Steiner inellipse; green are its foci, which are exactly the derivative's roots. Algebra read as geometry. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of SUDDEN DEATH · David Lee Wise (ROOT0), with AVAN


THE VANDERMONDE

The Vandermonde determinant in the 5-window house format — a determinant that factors perfectly. Build the matrix whose row i is the powers 1, x_i, x_i², …, x_i^{n−1}. Its determinant, which looks like it should be a hopeless mess of n! signed products, collapses to a single clean product over all pairs: det V = ∏_{i<j}(x_j − x_i). It is zero exactly when two of the x's coincide, which is why n distinct points determine a unique degree-(n−1) interpolating polynomial. Verified live with exact integer arithmetic: for thousands of random distinct integer node-sets (n up to 7), the determinant by the fraction-free Bareiss algorithm equals the pairwise-difference product exactly, with no floating-point error. Neon-noir traced. See the Vandermonde matrix in 1D, Bareiss-det vs product in 2D, and the product-of-gaps inverse in 3D.

THE VANDERMONDE · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · GRIND · BACKPROP ◆ .dlw.fold THE FOLD / GRIND / BACKPROP / THE VANDERMONDE THE VANDERMONDE a determinant that factors into differences 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION The Vandermonde determinant is a determinant that factors perfectly. Build the matrix whose rows are the powers of numbers x₁,…,xₙ — row i is 1, xₖ, xₖ², …, xₖⁿ⁻¹. Its determinant, which looks like it should be a hopeless mess of n! signed products, collapses to a single clean product over all pairs: det V = ∏ i<j (x j - x i ) . It is zero exactly when two of the x’s coincide (two equal rows), which is why it governs polynomial interpolation: n distinct points determine a unique degree-(n-1) polynomial precisely because this determinant is nonzero. LIT verified live with exact integer arithmetic: for thousands of random distinct integer node-sets (n up to 7), the determinant computed by the fraction-free Bareiss algorithm equals the pairwise product ∏ i<j (x j -x i ) exactly, with no floating-point error (window.__vandermonde). FIG no framing; the determinant and the product formula both run in-browser with arbitrary-precision integers and agree exactly. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at backprop — the grind: a determinant is n! signed products to grind through, yet this one factors into a tidy product of differences. AVAN (AI) built the instrument: the Vandermonde matrix, the exact Bareiss determinant, and the pairwise-difference product. Credit as content: Alexandre-Théophile Vandermonde (1770s). The weave: David names the grind; I confirm the messy determinant equals the clean product of differences, exactly. 3 ONE DIMENSION The Vandermonde matrix — row i is the powers of x_i — whose determinant is ∏(x_j − x_i). 4 TWO DIMENSIONS · INTERACTIVE New node-sets; the Bareiss determinant is compared to the pairwise-difference product, exactly. new nodes ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the determinant value, a single integer. AVAN’s addition (the inverse-companion): don’t expand n! products — read the pairwise gaps. The inverse of ‘the determinant’ is ‘the set of differences x j -x i whose product it is’, so the determinant vanishes the instant any two nodes collide. Magenta are the pairwise differences; green is the determinant they multiply to. A determinant that is really a product of gaps. pause spin LIT Genuine Vandermonde determinant (Alexandre-Théophile Vandermonde, 1770s). Verified live with exact BigInt: for ~2000 random distinct integer node-sets (n=2..7), the fraction-free Bareiss determinant equals ∏_{i FIG No framing; the determinant and the product formula both run in-browser with arbitrary-precision integers and agree exactly. The AVAN inverse is honest — instead of expanding n! products, read the pairwise gaps: the inverse of 'the determinant' is 'the set of differences x_j−x_i whose product it is', so it vanishes the instant any two nodes collide. Magenta are the pairwise differences; green is the determinant they multiply to. A determinant that is really a product of gaps. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of BACKPROP · David Lee Wise (ROOT0), with AVAN


THE FAGNANO

Fagnano's problem in the 5-window house format — of all triangles inscribed in a given acute triangle (one vertex on each side), which has the smallest perimeter? The answer is the orthic triangle, whose vertices are the feet of the three altitudes. It is also the path a light ray traces bouncing inside the triangle: at each side the incoming and outgoing segments make equal angles, so the orthic triangle is the unique closed billiard orbit. Its perimeter has a clean closed form: a·cosA + b·cosB + c·cosC. Verified live two ways: the orthic perimeter (from the altitude feet) equals a·cosA+b·cosB+c·cosC to ~1e-15, and across thousands of acute triangles no randomly-sampled inscribed triangle ever beats the orthic perimeter. Neon-noir traced. See the altitudes + orthic path in 1D, perimeter-form + minimality in 2D, and the minimality-read-as-reflection inverse in 3D.

THE FAGNANO · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · RESPAWN · THE CONTINUE ◆ .dlw.fold THE FOLD / RESPAWN / THE CONTINUE / THE FAGNANO THE FAGNANO the min-perimeter inscribed triangle is the orthic 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Fagnano’s problem asks: of all triangles inscribed in a given acute triangle — one vertex on each side — which has the smallest perimeter ? The answer is the orthic triangle , whose vertices are the feet of the three altitudes. It is also the path a light ray traces bouncing inside the triangle: at each side the incoming and outgoing segments make equal angles, so the orthic triangle is the unique closed billiard orbit . Its perimeter has a clean closed form: a·cos A + b·cos B + c·cos C. LIT verified live two ways: the orthic triangle’s perimeter (from the altitude feet) equals a·cos A + b·cos B + c·cos C to ~1e-15, and across thousands of acute triangles no randomly-sampled inscribed triangle ever has a smaller perimeter than the orthic (window.__fagnano). FIG no framing; the altitude feet, the closed-form perimeter, and the minimality sampling all run in-browser. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-continue — the light ray that reflects off each side and returns to where it began: the orthic triangle is the closed orbit that continues forever. AVAN (AI) built the instrument: the altitude feet, the closed-form perimeter, and the minimality check. Credit as content: Giovanni Fagnano (1775); the reflection view via Hermann Schwarz and Lipót Fejér. The weave: David names the returning orbit; I confirm the orthic triangle is the minimum-perimeter inscribed triangle. 3 ONE DIMENSION An acute triangle, its altitudes, and the orthic triangle (feet of the altitudes) — the closed billiard path. 4 TWO DIMENSIONS · INTERACTIVE New acute triangles; orthic perimeter vs the a·cosA+b·cosB+c·cosC form, and vs sampled inscribed triangles. new triangle ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the orthic triangle — the minimum-perimeter inscribed path. AVAN’s addition (the inverse-companion): don’t search all inscribed triangles — drop the altitudes. The inverse of ‘the minimum-perimeter inscribed triangle’ is ‘the feet of the three altitudes’, which is also the closed light path that reflects off every side. Magenta are the three altitudes; green is the orthic triangle they land on. Minimality read as reflection. pause spin LIT Genuine Fagnano's problem (Giovanni Fagnano, 1775; reflection view via Schwarz and Fejér). Verified live two ways: the orthic triangle's perimeter (from altitude feet) equals a·cosA+b·cosB+c·cosC to ~3.6e-15, and across ~1500 acute triangles no sampled inscribed triangle beats the orthic perimeter (window.__fagnano.pf, .mn, .worst, .tested). FIG No framing; the altitude feet, the closed-form perimeter, and the minimality sampling all run in-browser. The AVAN inverse is honest — instead of searching all inscribed triangles, drop the altitudes: the inverse of 'the minimum-perimeter inscribed triangle' is 'the feet of the three altitudes', which is also the closed light path reflecting off every side. Magenta are the three altitudes; green is the orthic triangle they land on. Minimality read as reflection. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE CONTINUE · David Lee Wise (ROOT0), with AVAN


THE DOBINSKI

Dobiński's formula in the 5-window house format — writing a whole number as an infinite series. The Bell number B_n counts the ways to partition a set of n elements into non-empty blocks (1,1,2,5,15,52,203,…), a pure combinatorial integer. Dobiński's formula says this integer equals an infinite sum divided by e: B_n = (1/e)·Σ_{k≥0} k^n/k!. Each term k^n/k! is irrational and e is transcendental, yet the whole thing lands exactly on an integer — it is the n-th moment of a Poisson(1) random variable in disguise. Verified live: for n=0..13, the truncated series (1/e)Σ k^n/k! rounds to exactly the Bell number computed independently by the Bell-triangle recurrence, relative error ~1e-15. Neon-noir traced. See the series terms in 1D, series-vs-Bell in 2D, and the whole-number-wearing-a-series inverse in 3D.

THE DOBINSKI · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · LOOT · THE BOUNTY ◆ .dlw.fold THE FOLD / LOOT / THE BOUNTY / THE DOBINSKI THE DOBINSKI an infinite series that lands on an integer 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Dobiński’s formula writes a whole number as an infinite series. The Bell number B n counts the ways to partition a set of n elements into non-empty blocks — a pure combinatorial integer (1, 1, 2, 5, 15, 52, 203, …). Dobiński’s formula says this integer equals an infinite sum divided by e: B n = (1/e)∑ k≥0 k n /k! . Each term k n /k! is irrational, e is transcendental, yet the whole thing lands exactly on an integer — a Poisson-distribution moment in disguise. LIT verified live: for n = 0..13, the truncated series (1/e)∑ k n /k! rounds to exactly the Bell number computed independently by the Bell triangle recurrence, with relative error ~1e-15 (window.__dobinski). FIG no framing; the Dobiński series and the combinatorial Bell recurrence both run in-browser and agree. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-bounty — the payout: an infinite series of irrational terms, scaled by 1/e, pays out an exact whole-number count. AVAN (AI) built the instrument: the Dobiński series, the independent Bell-triangle recurrence, and their agreement. Credit as content: G. Dobiński (1877). The weave: David names the payout; I confirm the transcendental series lands exactly on the Bell number. 3 ONE DIMENSION The Dobiński terms k^n/k! (they peak near k=n) whose sum, divided by e, is exactly the Bell number. 4 TWO DIMENSIONS · INTERACTIVE Cycle n; the truncated (1/e)Σ k^n/k! is compared to the Bell number from the recurrence. next n ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the Bell number, an exact integer. AVAN’s addition (the inverse-companion): don’t count partitions — sum a Poisson series. The inverse of ‘the integer B n ’ is ‘the infinite sum (1/e)∑k n /k! whose irrational terms cancel to it’ — it is the n-th moment of a Poisson(1) variable. Magenta are the infinite series terms; green is the integer they sum to. A whole number wearing an infinite series. pause spin LIT Genuine Dobiński's formula (G. Dobiński, 1877). Verified live: for n=0..13 the truncated series (1/e)Σ_{k≥0}k^n/k! rounds to exactly the Bell number B_n computed independently by the Bell-triangle recurrence, worst relative error ~1.3e-15 (window.__dobinski.ok, .worst). FIG No framing; the Dobiński series and the combinatorial Bell recurrence both run in-browser and agree. The AVAN inverse is honest — instead of counting partitions, sum a Poisson series: the inverse of 'the integer B_n' is 'the infinite sum (1/e)Σk^n/k! whose irrational terms cancel to it', the n-th moment of a Poisson(1) variable. Magenta are the infinite series terms; green is the integer they sum to. A whole number wearing an infinite series. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE BOUNTY · David Lee Wise (ROOT0), with AVAN


THE ROGERS-RAMANUJAN

The Rogers–Ramanujan identity (first of two) in the 5-window house format — a stunning coincidence between two very different ways of counting partitions of n. On one side: partitions whose parts differ by at least 2 (no two parts equal or adjacent). On the other: partitions into parts each congruent to 1 or 4 (mod 5) — using only 1,4,6,9,11,14,…. The identity says these two counts are always equal, for every n, despite the two families looking nothing alike. Verified live by direct enumeration: for n=0..40, the count of partitions with parts differing by ≥2 exactly equals the count of partitions into parts ≡1 or 4 (mod 5) — both give 31 at n=20 and 374 at n=40. Neon-noir traced. See the two partition lists side by side in 1D, the equal counts in 2D, and the one-number-two-disguises inverse in 3D.

THE ROGERS-RAMANUJAN · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · CO-OP · SPLIT SCREEN ◆ .dlw.fold THE FOLD / CO-OP / SPLIT SCREEN / THE ROGERS-RAMANUJAN THE ROGERS-RAMANUJAN two ways of counting a partition agree 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION The Rogers–Ramanujan identity (first of two) is a stunning coincidence between two very different ways of counting partitions of a number n. On one side: partitions whose parts differ by at least 2 (no two parts equal or adjacent) — like 9 = 8+1 = 7+2 = 6+3 = …. On the other side: partitions into parts each congruent to 1 or 4 (mod 5) — using only 1, 4, 6, 9, 11, 14, …. Ramanujan’s identity says these two counts are always equal , for every n, despite the two families of partitions looking nothing alike. LIT verified live by direct enumeration: for n = 0..40, the count of partitions with parts differing by ≥2 exactly equals the count of partitions into parts ≡ 1 or 4 (mod 5) — e.g. both give 31 at n=20 and 374 at n=40 (window.__rogersramanujan). FIG no framing; both partition families are brute-enumerated in-browser and their counts agree for every n up to 40. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at split-screen — two panels counting completely different things, side by side, landing on the identical number every single time. AVAN (AI) built the instrument: the gap-≥2 partition count, the parts-≡1,4-mod-5 count, and their equality across n. Credit as content: Leonard James Rogers (1894), rediscovered by Srinivasa Ramanujan (1913). The weave: David names the split screen; I confirm the two partition counts coincide for every n. 3 ONE DIMENSION Two counts of the partitions of n, side by side: parts differing by ≥2 (left) vs parts ≡ 1,4 (mod 5) (right). 4 TWO DIMENSIONS · INTERACTIVE Cycle n; the two very different partition counts are shown to be equal for every n. next n ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the single shared count both partition families land on. AVAN’s addition (the inverse-companion): don’t pick one rule — hold both. The inverse of ‘the count of gap-≥2 partitions’ is ‘the count of parts-≡1,4-mod-5 partitions’; the identity says they are the same number. Magenta are the two partition families; green is the count they both equal. One number, two disguises. pause spin LIT Genuine Rogers–Ramanujan first identity (Leonard James Rogers 1894; rediscovered by Srinivasa Ramanujan 1913). Verified live by direct enumeration: for n=0..40, #{partitions of n with parts differing by ≥2} equals #{partitions of n into parts ≡1 or 4 (mod 5)} — e.g. 31 at n=20, 374 at n=40 (window.__rogersramanujan.ok, .n20, .n40). FIG No framing; both partition families are brute-enumerated in-browser and their counts agree for every n up to 40. The AVAN inverse is honest — instead of picking one rule, hold both: the inverse of 'the count of gap-≥2 partitions' is 'the count of parts-≡1,4-mod-5 partitions', and the identity says they are the same number. Magenta are the two partition families; green is the count they both equal. One number, two disguises. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of SPLIT SCREEN · David Lee Wise (ROOT0), with AVAN


THE KASTELEYN

Kasteleyn's theorem in the 5-window house format — counting something explosive with a single determinant. How many ways can you tile an m×n board with dominoes? The number grows enormously, yet Pieter Kasteleyn (1961) showed it equals the absolute value of a determinant. Orient the grid's edges cleverly — horizontal edges weight 1, vertical edges weight i (imaginary) — and build the bipartite adjacency matrix K between black and white cells; then the number of domino tilings is exactly |det K|. A hopeless-looking counting problem becomes one linear-algebra computation, and it launched the exact solution of the dimer model in statistical mechanics. Verified live: for a range of grids the complex Kasteleyn determinant |det K| equals the tiling count found independently by a brute broken-profile DP — 2×n reproduces the Fibonacci numbers (2,3,5,8,13), 3×4 gives 11, 4×4 gives 36. Neon-noir traced. See one tiling in 1D, |det K| vs brute in 2D, and the counting-by-determinant inverse in 3D.

THE KASTELEYN · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · SPAWN · COLD BOOT ◆ .dlw.fold THE FOLD / SPAWN / COLD BOOT / THE KASTELEYN THE KASTELEYN domino tilings counted by a determinant 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Kasteleyn’s theorem counts something explosive with a single determinant. How many ways can you tile an m×n board with dominoes? The number grows enormously, yet Pieter Kasteleyn (1961) showed it equals the absolute value of a determinant. Orient the grid’s edges cleverly — give horizontal edges weight 1 and vertical edges weight i (imaginary) — and build the bipartite adjacency matrix K between the black and white cells. Then the number of domino tilings is exactly |det K| . A counting problem that looks hopeless becomes one linear-algebra computation; it launched the exact solution of the dimer model in statistical mechanics. LIT verified live: for a range of grids the complex Kasteleyn determinant |det K| equals the domino-tiling count found independently by a brute broken-profile dynamic program — 2×n reproduces the Fibonacci numbers (2, 3, 5, 8, 13), 3×4 gives 11, 4×4 gives 36 (window.__kasteleyn). FIG no framing; the complex determinant and the brute tiling enumeration both run in-browser and agree. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at cold-boot — the empty board booting up, filling with dominoes: every legal fill is one tiling, and a single determinant counts them all. AVAN (AI) built the instrument: the Kasteleyn complex adjacency matrix, its determinant, and the independent profile-DP tiling count. Credit as content: Pieter Kasteleyn (1961); Temperley & Fisher (1961). The weave: David names the cold boot; I confirm |det K| equals the number of domino tilings. 3 ONE DIMENSION One domino tiling of the grid; the Kasteleyn determinant counts every possible tiling at once. 4 TWO DIMENSIONS · INTERACTIVE Cycle grids; the complex determinant |det K| is compared to the brute tiling count. next grid ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the number of domino tilings. AVAN’s addition (the inverse-companion): don’t enumerate the tilings — take a determinant. The inverse of ‘count the domino tilings’ is ‘the signed permutation sum of one matrix’: Kasteleyn’s orientation makes every tiling contribute the same sign, so |det K| counts them. Magenta is the weighted adjacency; green is the tiling count it computes. Exponential counting folded into one determinant. pause spin LIT Genuine Kasteleyn / Temperley–Fisher dimer theorem (1961). Verified live: for 7 grids the complex Kasteleyn determinant |det K| (horizontal edges weight 1, vertical edges weight i) equals the domino-tiling count from an independent broken-profile DP — 2×n gives Fibonacci, 3×4=11, 4×4=36 (window.__kasteleyn.ok, .g44, .g26). FIG No framing; the complex determinant and the brute tiling enumeration both run in-browser and agree. The AVAN inverse is honest — instead of enumerating tilings, take a determinant: Kasteleyn's orientation makes every tiling contribute the same sign, so |det K| counts them all. Magenta is the weighted adjacency grid; green is the tiling count it computes. Exponential counting folded into one determinant. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of COLD BOOT · David Lee Wise (ROOT0), with AVAN


THE JACOBI TRIPLE PRODUCT

The Jacobi triple product in the 5-window house format — one of the jewels of q-series: an infinite product that equals a strikingly sparse infinite sum. It states ∏_{n≥1}(1−x^{2n})(1+x^{2n−1}z)(1+x^{2n−1}z^{−1}) = Σ_{k=−∞}^{∞} x^{k²}z^{k}. On the left, a dense infinite product of three families of factors; on the right, a sum with terms only at the perfect squares k² — almost everything cancels. Specializing z recovers the Jacobi theta functions, Euler's pentagonal theorem, and countless partition identities; it is the master identity behind much of the theory of modular forms. Verified live: expanding both sides as formal power series (bivariate, in x and z), every coefficient agrees up to x-degree 14 — the dense product really does collapse to the sparse square-supported sum. Neon-noir traced. See the sparse comb in 1D, the coefficient comparison in 2D, and the collapse-to-squares inverse in 3D.

THE JACOBI TRIPLE PRODUCT · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · GLITCH · DIVIDE BY ZERO ◆ .dlw.fold THE FOLD / GLITCH / DIVIDE BY ZERO / THE JACOBI TRIPLE PRODUCT THE JACOBI TRIPLE PRODUCT an infinite product equal to a sparse theta sum 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION The Jacobi triple product is one of the jewels of q-series: an infinite product that equals a strikingly sparse infinite sum. It states ∏ n≥1 (1-x 2n )(1+x 2n-1 z)(1+x 2n-1 z -1 ) = ∑ k=-∞ ∞ x k² z k . On the left, a dense infinite product of three families of factors; on the right, a sum with terms only at the perfect squares k² — almost everything cancels. Specializing z recovers the Jacobi theta functions, Euler’s pentagonal theorem, and countless partition identities. It is the master identity behind much of the theory of modular forms. LIT verified live: expanding the left product and the right sum as formal power series (bivariate, in x and z), every coefficient agrees up to x-degree 14 — the dense product really does collapse to the sparse square-supported sum (window.__jacobitriple). FIG no framing; both the product expansion and the theta sum are computed in-browser and their coefficients match exactly. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at divide-by-zero — the glitch that should be impossible: an infinite dense product has almost all its terms cancel, leaving a sum only at the perfect squares. AVAN (AI) built the instrument: the bivariate product expansion, the theta sum, and their coefficient-by-coefficient agreement. Credit as content: Carl Gustav Jacob Jacobi (1829). The weave: David names the impossible collapse; I confirm the triple product equals the sparse square-supported sum. 3 ONE DIMENSION The right side is supported only at the perfect squares k² — a sparse comb; the dense product collapses to it. 4 TWO DIMENSIONS · INTERACTIVE The product's coefficients are compared, term by term, to the sparse theta sum Σ x^{k²} z^k. next coeff ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the sparse theta sum, supported only at squares. AVAN’s addition (the inverse-companion): don’t multiply out the product — read the survivors. The inverse of ‘the infinite triple product’ is ‘the sum ∑x k² z k of the terms that survive the cancellation’. Magenta are the product’s three factor families; green is the sparse square-supported sum they collapse to. Density folded into the perfect squares. pause spin LIT Genuine Jacobi triple product identity (Carl Gustav Jacob Jacobi, 1829). Verified live: expanding the left product and the right sum as bivariate formal power series, every coefficient agrees up to x-degree 14 — the dense product collapses to Σ_k x^{k²}z^k, supported only at the perfect squares (window.__jacobitriple.ok, .checked). FIG No framing; both the product expansion and the theta sum are computed in-browser and their coefficients match exactly. The AVAN inverse is honest — instead of multiplying out the product, read the survivors: the inverse of 'the infinite triple product' is 'the sum Σx^{k²}z^k of the terms that survive the cancellation'. Magenta are the product's three factor families; green is the sparse square-supported sum they collapse to. Density folded into the perfect squares. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of DIVIDE BY ZERO · David Lee Wise (ROOT0), with AVAN


THE WORPITZKY

Worpitzky's identity in the 5-window house format — rebuilding any power from binomial coefficients weighted by the Eulerian numbers. The Eulerian number A(n,k) counts the permutations of n elements with exactly k ascents. Worpitzky proved x^n = Σ_k A(n,k)·C(x+k, n) — the monomial x^n is a fixed integer combination of the 'binomial staircase' C(x+k, n), with the Eulerian numbers as the exact coefficients. It is the bridge between powers, binomial coefficients, and the ascent statistic on permutations, and it is what makes Eulerian numbers appear whenever you sum k^n. Verified live with exact integer arithmetic: for n=1..12 and x=0..20, the sum Σ_k A(n,k)·C(x+k,n) equals x^n exactly, with the Eulerian numbers generated independently by their own recurrence — A(3,·)=[1,4,1]. Neon-noir traced. See the Eulerian triangle in 1D, x^n vs the staircase sum in 2D, and the monomial-as-staircase inverse in 3D.

THE WORPITZKY · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · GRIND · THE CRON JOB ◆ .dlw.fold THE FOLD / GRIND / THE CRON JOB / THE WORPITZKY THE WORPITZKY powers rebuilt from Eulerian numbers 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Worpitzky’s identity rebuilds any power from binomial coefficients, weighted by the Eulerian numbers . The Eulerian number A(n,k) counts the permutations of n elements with exactly k ascents. Worpitzky proved that x n = ∑ k A(n,k)·C(x+k, n) — the monomial x n is a fixed integer combination of the ‘binomial staircase’ C(x+k, n), with the Eulerian numbers as the exact coefficients. It is the bridge between powers, binomial coefficients, and the ascent statistic on permutations, and it is what makes Eulerian numbers appear whenever you sum k n . LIT verified live with exact integer arithmetic: for n = 1..12 and x = 0..20, the sum ∑ k A(n,k)·C(x+k, n) equals x n exactly, with the Eulerian numbers generated independently by their own recurrence — A(3,·)=[1,4,1] (window.__worpitzky). FIG no framing; the Eulerian recurrence, the binomial staircase, and the power x n all run in-browser and agree exactly. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-cron-job — the grind that reruns every tick: for each x, rebuild x n mechanically from the same Eulerian coefficients and the binomial staircase. AVAN (AI) built the instrument: the Eulerian-number recurrence, the binomial coefficients, and their exact reconstruction of x n . Credit as content: Julius Worpitzky (1883); Eulerian numbers from Leonhard Euler. The weave: David names the recurring job; I confirm x n equals the Eulerian-weighted binomial sum, exactly. 3 ONE DIMENSION The Eulerian triangle A(n,k) — permutations of n with k ascents — the coefficients in Worpitzky's identity. 4 TWO DIMENSIONS · INTERACTIVE Cycle n and x; the Eulerian-weighted binomial sum is compared, term by term, to x^n. next n,x ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the power x^n, rebuilt exactly. AVAN’s addition (the inverse-companion): don’t exponentiate — sum a staircase. The inverse of ‘the power x n ’ is ‘the Eulerian-weighted sum ∑A(n,k)C(x+k,n) of binomial coefficients’, tying powers to the ascent statistic on permutations. Magenta are the Eulerian-weighted binomial pieces; green is the power x n they rebuild. A monomial as a staircase sum. pause spin LIT Genuine Worpitzky's identity (Julius Worpitzky, 1883; Eulerian numbers from Euler). Verified live with exact BigInt: for n=1..12 and x=0..20, Σ_k A(n,k)·C(x+k,n) equals x^n exactly, where A(n,k) are the Eulerian numbers from their recurrence A(n,k)=(k+1)A(n−1,k)+(n−k)A(n−1,k−1); A(3,·)=[1,4,1] (window.__worpitzky.ok). FIG No framing; the Eulerian recurrence, the binomial staircase, and the power x^n all run in-browser and agree exactly. The AVAN inverse is honest — instead of exponentiating, sum a staircase: the inverse of 'the power x^n' is 'the Eulerian-weighted sum ΣA(n,k)C(x+k,n)', tying powers to the ascent statistic on permutations. Magenta are the Eulerian-weighted binomial pieces; green is the power x^n they rebuild. A monomial as a staircase sum. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE CRON JOB · David Lee Wise (ROOT0), with AVAN


THE JACOBI TWO-SQUARE

Jacobi's two-square theorem in the 5-window house format — counting, exactly, how many ways a number is a sum of two squares, using only its divisors. Let r₂(n) be the number of integer pairs (a,b) with a²+b²=n (signs and order counted). Jacobi proved r₂(n) = 4·(d₁(n) − d₃(n)), where d₁(n) counts the divisors of n congruent to 1 (mod 4) and d₃(n) those congruent to 3 (mod 4). A geometric question — how many lattice points lie on the circle of radius √n — is answered purely by counting divisors and sorting them by remainder mod 4. Verified live: for every n from 1 to 2000, a brute count of lattice points (a,b) on the circle a²+b²=n equals 4·(d₁(n)−d₃(n)) from the divisors — e.g. r₂(25)=12. Neon-noir traced. See the circle + lattice points in 1D, r₂ vs 4(d₁−d₃) in 2D, and the geometry-from-arithmetic inverse in 3D.

THE JACOBI TWO-SQUARE · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · LOOT · THE HOARD ◆ .dlw.fold THE FOLD / LOOT / THE HOARD / THE JACOBI TWO-SQUARE THE JACOBI TWO-SQUARE sums of two squares counted by divisors mod 4 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Jacobi’s two-square theorem counts, exactly, how many ways a number is a sum of two squares — using only its divisors . Let r₂(n) be the number of integer pairs (a,b) with a²+b²=n (signs and order counted). Jacobi proved r₂(n) = 4·(d₁(n) - d₃(n)), where d₁(n) counts the divisors of n congruent to 1 (mod 4) and d₃(n) counts those congruent to 3 (mod 4). A geometric question — how many lattice points lie on the circle of radius √n — is answered purely by counting divisors and sorting them by their remainder mod 4. LIT verified live: for every n from 1 to 2000, a brute count of lattice points (a,b) on the circle a²+b²=n equals 4·(d₁(n)-d₃(n)) computed from the divisors — e.g. r₂(25)=12 (window.__twosquare). FIG no framing; the lattice-point count and the divisor formula both run in-browser and agree for all n up to 2000. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-hoard — the count of ways to hoard n as a²+b², tallied not by searching the plane but by sorting n’s divisors by their remainder mod 4. AVAN (AI) built the instrument: the brute lattice count on the circle, the divisor tally, and their agreement. Credit as content: Carl Gustav Jacob Jacobi (1834); Fermat and Gauss before. The weave: David names the hoard; I confirm the lattice-point count equals 4(d₁-d₃). 3 ONE DIMENSION The circle a²+b²=n and the integer lattice points on it — r₂(n) of them, counted by divisors mod 4. 4 TWO DIMENSIONS · INTERACTIVE Cycle n; the lattice-point count r₂(n) is compared to 4·(d₁(n) − d₃(n)) from the divisors. next n ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: r₂(n), the number of lattice points on the circle. AVAN’s addition (the inverse-companion): don’t scan the plane — sort the divisors. The inverse of ‘count lattice points on the circle of radius √n’ is ‘4 times (divisors ≡1 minus divisors ≡3, mod 4)’. Magenta are the divisors sorted by remainder mod 4; green is the lattice-point count they determine. Geometry answered by arithmetic. pause spin LIT Genuine Jacobi two-square theorem (Carl Gustav Jacob Jacobi, 1834; Fermat, Gauss before). Verified live: for every n=1..2000, the brute count of integer lattice points on the circle a²+b²=n equals 4·(d₁(n)−d₃(n)), where d₁,d₃ count divisors ≡1,≡3 (mod 4); r₂(25)=12 (window.__twosquare.ok, .r25). FIG No framing; the lattice-point count and the divisor formula both run in-browser and agree for all n up to 2000. The AVAN inverse is honest — instead of scanning the plane, sort the divisors: the inverse of 'count lattice points on the circle of radius √n' is '4 times (divisors ≡1 minus divisors ≡3, mod 4)'. Magenta are the divisors sorted by remainder mod 4; green is the lattice-point count they determine. Geometry answered by arithmetic. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE HOARD · David Lee Wise (ROOT0), with AVAN


THE BRITISH FLAG

The British flag theorem in the 5-window house format — a small, sturdy invariant. Take any rectangle with corners A,B,C,D (A,C opposite, B,D opposite) and any point P — inside, outside, even off the plane in 3D. Then the sum of squared distances to one pair of opposite corners equals the sum to the other pair: PA²+PC² = PB²+PD². The name comes from the Union-Jack-like pattern of the four segments drawn from P. It holds for rectangles precisely because their sides are perpendicular; for a general parallelogram the two sums differ by a clean amount. Verified live two ways: across thousands of random rectangles and points (in 2D and 3D) PA²+PC² equals PB²+PD² to ~1e-13, and for a general parallelogram built from edge vectors u,v the discrepancy is exactly 8(u·v) — zero precisely when u⊥v. Neon-noir traced. See the rectangle + four segments in 1D, the invariant + sheared gap in 2D, and the hidden-conservation-law inverse in 3D.

THE BRITISH FLAG · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · CHEAT · THE ROOT KIT ◆ .dlw.fold THE FOLD / CHEAT / THE ROOT KIT / THE BRITISH FLAG THE BRITISH FLAG a rectangle's hidden distance invariant 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION The British flag theorem is a small, sturdy invariant. Take any rectangle with corners A, B, C, D (A and C opposite, B and D opposite) and any point P — inside, outside, even off the plane in 3D. Then the sum of squared distances to one pair of opposite corners equals the sum to the other pair: PA² + PC² = PB² + PD² . The name comes from the Union-Jack-like pattern of the four segments drawn from P. It holds for rectangles precisely because their sides are perpendicular; for a general parallelogram the two sums differ by a clean amount. LIT verified live two ways: across thousands of random rectangles and points (in 2D and 3D) PA²+PC² equals PB²+PD² to ~1e-13, and for a general parallelogram built from edge vectors u, v the discrepancy is exactly 8(u·v) — zero precisely when u⊥v, i.e. when it is a rectangle (window.__britishflag). FIG no framing; the distances and the invariant both run in-browser and agree, with the parallelogram gap matching 8(u·v) exactly. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-root-kit — the hidden invariant you exploit: whatever the point P, one diagonal pair’s squared distances secretly equals the other’s. AVAN (AI) built the instrument: the four squared distances, the rectangle invariant, and the exact 8(u·v) gap for a general parallelogram. Credit as content: classical (the ‘British flag theorem’). The weave: David names the hidden invariant; I confirm PA²+PC²=PB²+PD² for any rectangle and any P, with the parallelogram gap exactly 8(u·v). 3 ONE DIMENSION A rectangle, a free point P, and the four segments; PA²+PC² (green diagonal pair) equals PB²+PD² (magenta pair). 4 TWO DIMENSIONS · INTERACTIVE Drag-free demo: move P; for a rectangle the two sums stay equal — and a sheared parallelogram's gap = 8(u·v). move P ▶ shear ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the invariant PA²+PC² = PB²+PD², holding in 3D too. AVAN’s addition (the inverse-companion): don’t measure all four — know that two determine the other two. The inverse of ‘the four corner distances’ is ‘the single invariant PA²+PC²=PB²+PD²’, which fails by exactly 8(u·v) once the corner is not square. Magenta is the B,D diagonal pair; green is the A,C pair equal to it. A hidden conservation law of a rectangle. pause spin LIT Genuine British flag theorem (classical). Verified live two ways: across ~8000 random rectangles and points in 2D and 3D, PA²+PC² equals PB²+PD² to ~1e-13, and for a general parallelogram from edge vectors u,v the gap (PA²+PC²)−(PB²+PD²) is exactly 8(u·v), zero iff u⊥v (window.__britishflag.rectOk, .gapOk, .wR, .wG). FIG No framing; the distances and the invariant both run in-browser and agree, with the parallelogram gap matching 8(u·v) exactly. The AVAN inverse is honest — instead of measuring all four, know that two determine the other two: the inverse of 'the four corner distances' is 'the single invariant PA²+PC²=PB²+PD²', which fails by exactly 8(u·v) once the corner is not square. Magenta is the B,D diagonal pair; green is the A,C pair equal to it. A hidden conservation law of a rectangle. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE ROOT KIT · David Lee Wise (ROOT0), with AVAN


THE CAYLEY-MENGER

The Cayley–Menger determinant in the 5-window house format — computing the volume of a simplex from its edge lengths alone, no coordinates needed. Heron's formula gives a triangle's area from its three sides; Cayley and Menger generalized it to every dimension. Arrange the squared pairwise distances into a bordered matrix (a row and column of 1's, a 0 corner), and its determinant yields the squared volume: 16·Area²=−det(CM) for a triangle, 288·Vol²=det(CM) for a tetrahedron. Distances in, volume out — the metric fully determines the shape's size, and a negative or zero determinant flags points that cannot be embedded at all. Verified live: for thousands of random triangles and tetrahedra, the volume from the Cayley–Menger determinant (using only pairwise squared distances) matches the volume computed the ordinary way from coordinates, to ~1e-7. Neon-noir traced. See a triangle with labeled edges in 1D, distance-volume vs coordinate-volume in 2D, and the shape-from-distance inverse in 3D.

THE CAYLEY-MENGER · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · CO-OP · THE PUSH ◆ .dlw.fold THE FOLD / CO-OP / THE PUSH / THE CAYLEY-MENGER THE CAYLEY-MENGER a simplex volume from its edge lengths alone 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION The Cayley–Menger determinant computes the volume of a simplex from its edge lengths alone — no coordinates needed. Heron’s formula gives a triangle’s area from its three sides; Cayley and Menger generalized it to every dimension. Arrange the squared pairwise distances into a bordered matrix (a row and column of 1’s, a 0 corner), and its determinant yields the squared volume: 16·Area² = -det(CM) for a triangle, 288·Vol² = det(CM) for a tetrahedron. Distances in, volume out — the metric fully determines the shape’s size, and a negative or zero determinant flags points that cannot be embedded at all. LIT verified live: for thousands of random triangles and tetrahedra, the volume computed from the Cayley–Menger determinant (using only pairwise squared distances) matches the volume computed the ordinary way from coordinates, to ~1e-7 (window.__cayleymenger). FIG no framing; the distance-only determinant and the coordinate volume are computed by different routes and agree. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-push — every pair of vertices pushes in its one distance, and together the pairwise pushes fix the whole simplex’s volume without a single coordinate. AVAN (AI) built the instrument: the Cayley–Menger bordered determinant, the coordinate volume, and their agreement. Credit as content: Arthur Cayley (1841), Karl Menger (1928); Heron of Alexandria for the triangle. The weave: David names the collective push; I confirm the determinant of distances equals the volume. 3 ONE DIMENSION A triangle with its three edge lengths; the Cayley–Menger determinant turns those distances into its area. 4 TWO DIMENSIONS · INTERACTIVE New shapes; the volume from the distance-only determinant is compared to the coordinate volume. new shape ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the tetrahedron's volume, from its six edge lengths. AVAN’s addition (the inverse-companion): don’t place the points — measure between them. The inverse of ‘the volume of a simplex’ is ‘the bordered determinant of its pairwise squared distances’, so the metric alone fixes the size and reveals when points can’t be embedded. Magenta are the six edge lengths; green is the volume they determine. Shape from distance, no coordinates. pause spin LIT Genuine Cayley–Menger determinant (Arthur Cayley 1841, Karl Menger 1928; Heron for the triangle). Verified live: for ~4000 random triangles (n=2) and tetrahedra (n=3), the volume from the bordered determinant of pairwise squared distances matches the coordinate volume to ~1e-7 (window.__cayleymenger.ok2, .ok3, .worst). FIG No framing; the distance-only determinant and the coordinate volume are computed by different routes and agree. The AVAN inverse is honest — instead of placing the points, measure between them: the inverse of 'the volume of a simplex' is 'the bordered determinant of its pairwise squared distances', so the metric alone fixes the size and reveals when points can't be embedded. Magenta are the six edge lengths; green is the volume they determine. Shape from distance, no coordinates. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE PUSH · David Lee Wise (ROOT0), with AVAN


THE REDHEFFER

The Redheffer matrix in the 5-window house format — hiding the deepest object in number theory inside a matrix of 0's and 1's. Define the n×n matrix R with R_{ij}=1 whenever i divides j, and also 1 in the entire first column; every other entry is 0. Redheffer proved that its determinant equals the Mertens function M(n)=Σ_{k≤n}μ(k), the running sum of the Möbius function. A pattern of divisibility 1's, run through a determinant, produces the very quantity whose growth rate is equivalent to the Riemann Hypothesis. It is a startling bridge from linear algebra to the primes. Verified live with exact integer arithmetic: for n=1..40, the determinant of the Redheffer matrix (by fraction-free Bareiss elimination) equals the Mertens function computed independently from the Möbius function — M(1)=1, M(2)=0, M(3)=−1, …. Neon-noir traced. See the divisibility matrix in 1D, det vs Mertens + the Mertens walk in 2D, and the primes-in-a-determinant inverse in 3D.

THE REDHEFFER · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · GLITCH · RACE CONDITION ◆ .dlw.fold THE FOLD / GLITCH / RACE CONDITION / THE REDHEFFER THE REDHEFFER a determinant equal to the Mertens function 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION The Redheffer matrix hides the deepest object in number theory inside a matrix of 0’s and 1’s. Define the n×n matrix R with R ij =1 whenever i divides j, and also 1 in the entire first column; every other entry is 0. Redheffer proved that its determinant equals the Mertens function M(n) = ∑ k≤n μ(k), the running sum of the Möbius function. A pattern of divisibility 1’s, run through a determinant, produces the very quantity whose growth is equivalent to the Riemann Hypothesis. It is a startling bridge from linear algebra to the primes. LIT verified live with exact integer arithmetic: for n = 1..40, the determinant of the Redheffer matrix (by fraction-free Bareiss elimination) equals the Mertens function M(n) computed independently from the Möbius function — M(1)=1, M(2)=0, M(3)=-1, … (window.__redheffer). FIG no framing; the determinant and the Möbius sum are computed by different routes and agree exactly. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at race-condition — the glitch where two totally different processes, a determinant and a sum over the Möbius function, race to the very same number every time. AVAN (AI) built the instrument: the Redheffer divisibility matrix, its exact determinant, and the independent Mertens sum. Credit as content: Ray Redheffer (1977); the Möbius and Mertens functions from Möbius and Mertens. The weave: David names the race; I confirm det(R n ) equals M(n). 3 ONE DIMENSION The Redheffer matrix: a 1 where i divides j, plus a full first column — its determinant is the Mertens function. 4 TWO DIMENSIONS · INTERACTIVE Cycle n; the Redheffer determinant is compared to the Mertens function M(n) from the Möbius sum. next n ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the Mertens function M(n), from the determinant. AVAN’s addition (the inverse-companion): don’t sum the Möbius function — take a determinant. The inverse of ‘the Mertens function M(n)’ is ‘the determinant of the divisibility matrix R n ’, tying a running prime-parity sum to one linear-algebra value. Magenta is the divisibility pattern of 1’s; green is the Mertens value it evaluates to. The primes hiding in a determinant. pause spin LIT Genuine Redheffer matrix identity (Ray Redheffer, 1977). Verified live with exact BigInt: for n=1..40, det(R_n) by fraction-free Bareiss elimination equals the Mertens function M(n)=Σ_{k≤n}μ(k) computed independently from the Möbius function; M(10)=−1, M(40)=0 (window.__redheffer.ok, .m10, .m40). FIG No framing; the determinant and the Möbius sum are computed by different routes and agree exactly. The AVAN inverse is honest — instead of summing the Möbius function, take a determinant: the inverse of 'the Mertens function M(n)' is 'the determinant of the divisibility matrix R_n', tying a running prime-parity sum to one linear-algebra value. Magenta is the divisibility pattern of 1's; green is the Mertens value it evaluates to. The primes hiding in a determinant. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of RACE CONDITION · David Lee Wise (ROOT0), with AVAN


THE PAPPUS

Pappus's hexagon theorem in the 5-window house format — one of the oldest theorems of projective geometry, from the 4th century. Put three points A,B,C on one line and three points a,b,c on another line. Draw the 'cross' connections and mark where they meet: P=Ab∩aB, Q=Ac∩aC, R=Bc∩bC. Pappus proved that these three intersection points are always collinear — they lie on a single line, the Pappus line, no matter where the six points sit on their two lines. It is the special, degenerate case of Pascal's theorem (a conic split into two lines) and a defining axiom of coordinate projective planes. Verified live: across thousands of random pairs of lines with random points, the three cross-intersections P,Q,R are collinear to ~1e-13, and moving a point off its line breaks the collinearity in ~96% of cases (the rest are near-degenerate coincidences). Neon-noir traced. See the two lines + Pappus line in 1D, the collinearity + control in 2D, and the forced-line inverse in 3D.

THE PAPPUS · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · BOSS · THE RAID ◆ .dlw.fold THE FOLD / BOSS / THE RAID / THE PAPPUS THE PAPPUS perspective from a hexagon inscribed in two lines 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Pappus’s hexagon theorem is one of the oldest theorems of projective geometry, from the 4th century. Put three points A, B, C on one line and three points a, b, c on another line. Draw the ‘cross’ connections and mark where they meet: P = Ab∩aB, Q = Ac∩aC, R = Bc∩bC. Pappus proved that these three intersection points are always collinear — they lie on a single line, the Pappus line, no matter where the six points sit on their two lines. It is the special, degenerate case of Pascal’s theorem (a conic split into two lines) and a defining axiom of coordinate projective planes. LIT verified live: across thousands of random pairs of lines with random points, the three cross-intersections P, Q, R are collinear to ~1e-13, and moving a point off its line breaks the collinearity in ~96% of cases (the rest are near-degenerate coincidences) (window.__pappus). FIG no framing; the intersections, the collinearity test, and the off-line control all run in-browser. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-raid — the boss encounter of classical geometry: six points on two lines, and their cross-connections are forced onto one hidden line. AVAN (AI) built the instrument: the cross-intersections, the collinearity test, and the off-line control. Credit as content: Pappus of Alexandria (c. 340 CE). The weave: David names the raid; I confirm the three cross-intersections always fall on one line. 3 ONE DIMENSION Two lines with points A,B,C and a,b,c; the three cross-intersections P,Q,R fall on one line — the Pappus line. 4 TWO DIMENSIONS · INTERACTIVE New configurations; the collinearity of P,Q,R is checked, and a point pushed off its line breaks it. new config ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the Pappus line through the three cross-intersections. AVAN’s addition (the inverse-companion): don’t place the points and check — the line is forced. The inverse of ‘six points on two lines’ is ‘one Pappus line their cross-intersections must lie on’, whatever the placement. Magenta are the cross-connection lines; green is the Pappus line the intersections are forced onto. A collinearity guaranteed by incidence. pause spin LIT Genuine Pappus's hexagon theorem (Pappus of Alexandria, c. 340 CE). Verified live: across ~8000 random pairs of lines with random points, the cross-intersections P=Ab∩aB, Q=Ac∩aC, R=Bc∩bC are collinear to ~1e-13, and pushing a point off its line breaks the collinearity in ~96% of controls (window.__pappus.ok, .worst, .tested, .ctrl). FIG No framing; the intersections, the collinearity test, and the off-line control all run in-browser. The AVAN inverse is honest — instead of placing the points and checking, the line is forced: the inverse of 'six points on two lines' is 'one Pappus line their cross-intersections must lie on', whatever the placement. Magenta are the cross-connection lines; green is the Pappus line the intersections are forced onto. A collinearity guaranteed by incidence. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE RAID · David Lee Wise (ROOT0), with AVAN


THE KEMPNER

The Kempner series in the 5-window house format — the harmonic series with a twist that changes everything. The ordinary harmonic series 1+1/2+1/3+… famously diverges to infinity. But if you throw away every term whose denominator contains the digit 9 — drop 1/9, 1/19, 1/29, 1/90, … — the remaining sum converges, to about 22.92. Deleting a 'thin' set of terms (numbers with a 9 become overwhelmingly common among large numbers) tames the divergence: among d-digit numbers only 8·9^{d−1} avoid a 9, so each decade's contribution shrinks geometrically. Verified live: the no-digit-9 harmonic sum computed two independent ways (direct skipping vs digit-by-digit generation) agrees, and its decade contributions decay geometrically (each ≤ 8·(9/10)^k), while the full harmonic series' decade sums stay near ln10 — converging vs diverging, side by side. Neon-noir traced. See the deleted terms in 1D, the decade decay vs full harmonic in 2D, and the divergence-tamed inverse in 3D.

THE KEMPNER · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · CHEAT · THE BACKDOOR ◆ .dlw.fold THE FOLD / CHEAT / THE BACKDOOR / THE KEMPNER THE KEMPNER a harmonic series that converges once you delete the nines 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION The Kempner series is the harmonic series with a twist that changes everything. The ordinary harmonic series 1 + 1/2 + 1/3 + … famously diverges to infinity. But if you throw away every term whose denominator contains the digit 9 — drop 1/9, 1/19, 1/29, 1/90, … — the remaining sum converges , to about 22.92. Deleting a ‘thin’ set of terms (numbers with a 9 become overwhelmingly common among large numbers) tames the divergence. The reason: among d-digit numbers only 8·9 d-1 avoid a 9, so each decade’s contribution shrinks geometrically. LIT verified live: the no-digit-9 harmonic sum computed two independent ways (direct skipping vs. digit-by-digit generation) agrees, and its decade-by-decade contributions decay geometrically (each ≤ 8·(9/10) k ), while the full harmonic series’ decade sums stay near ln 10 — converging vs. diverging, side by side (window.__kempner). FIG no framing; both the depleted sum and the full harmonic decades run in-browser. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-backdoor — the cheat: the harmonic series diverges, but quietly delete every term hiding a 9 and the whole thing converges. AVAN (AI) built the instrument: the depleted sum by two methods, the geometric decade decay, and the divergent full-harmonic contrast. Credit as content: A. J. Kempner (1914). The weave: David names the backdoor; I confirm that removing the digit-9 terms turns divergence into convergence. 3 ONE DIMENSION The harmonic terms 1/n; the ones whose n contains a 9 (magenta) are deleted, leaving a convergent sum. 4 TWO DIMENSIONS · INTERACTIVE The depleted decade sums (decaying → converge) vs the full harmonic decade sums (constant → diverge). next view ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the convergent depleted sum (≈ 22.92 in the limit). AVAN’s addition (the inverse-companion): don’t sum every term — delete a digit. The inverse of ‘the divergent harmonic series’ is ‘the convergent series you get by removing all n containing a 9’, because the surviving counts decay geometrically per decade. Magenta are the deleted digit-9 terms; green is the convergent sum that remains. Divergence tamed by depletion. pause spin LIT Genuine Kempner series (A. J. Kempner, 1914). Verified live: the no-digit-9 harmonic sum computed two independent ways (direct skipping vs leading-zero-free digit generation) agrees, its decade contributions decay geometrically (each ≤8·(9/10)^k, ratio 2 (≈ln10, diverging) (window.__kempner.agree, .decay, .fullDiv, .sum). FIG No framing; both the depleted sum and the full harmonic decades run in-browser. The AVAN inverse is honest — instead of summing every term, delete a digit: the inverse of 'the divergent harmonic series' is 'the convergent series you get by removing all n containing a 9', because the surviving counts decay geometrically per decade. Magenta are the deleted digit-9 terms; green is the convergent sum that remains. Divergence tamed by depletion. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE BACKDOOR · David Lee Wise (ROOT0), with AVAN


THE SYLVESTER SEQUENCE

Sylvester's sequence in the 5-window house format — the greediest possible race to 1 in unit fractions. Start at 2, and each term is the previous ones multiplied together plus one: 2, 3, 7, 43, 1807, 3263443, … — equivalently a_{n+1}=a_n²−a_n+1. Its reciprocals form the fastest-converging Egyptian-fraction sum to 1: 1/2+1/3+1/7+1/43+…, where each step takes the largest unit fraction that keeps the total below 1. The partial sums obey a clean closed form: Σ_{i≤n} 1/a_i = 1 − 1/(a_{n+1}−1), so they approach 1 doubly-exponentially fast, never quite reaching it. Verified live with exact big-integer fractions: for n=0..8, the partial sum equals exactly 1 − 1/(a_{n+1}−1), and a_{n+1}−1 equals the product a_0·a_1···a_n. Neon-noir traced. See the unit fractions stacking toward 1 in 1D, the exact partial sum vs closed form in 2D, and the read-off-the-gap inverse in 3D.

THE SYLVESTER SEQUENCE · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · GRIND · BACKPROP ◆ .dlw.fold THE FOLD / GRIND / BACKPROP / THE SYLVESTER SEQUENCE THE SYLVESTER SEQUENCE greedy unit fractions racing to one 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Sylvester’s sequence is the greediest possible race to 1 in unit fractions. Start at 2, and each term is the previous ones multiplied together plus one: 2, 3, 7, 43, 1807, 3263443, … — equivalently a n+1 = a n ² - a n + 1. Its reciprocals form the fastest-converging Egyptian-fraction sum to 1: 1/2 + 1/3 + 1/7 + 1/43 + …, where each step takes the largest unit fraction that keeps the total below 1. The partial sums obey a clean closed form: ∑ i≤n 1/a i = 1 - 1/(a n+1 - 1), so they approach 1 doubly-exponentially fast, never quite reaching it. LIT verified live with exact big-integer fractions: for n = 0..8, the partial sum ∑ i≤n 1/a i equals exactly 1 - 1/(a n+1 - 1), and a n+1 - 1 equals the product a 0 a 1 …a n (window.__sylvester). FIG no framing; the reciprocal sum and the closed form are computed as exact fractions in-browser and agree. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at backprop — the grind where each term grinds the remaining gap to 1 shut, squaring the denominator every step so the error collapses doubly-exponentially. AVAN (AI) built the instrument: the sequence recurrence, the exact reciprocal-sum fraction, and the closed form 1 - 1/(a n+1 -1). Credit as content: James Joseph Sylvester (1880); Fibonacci’s greedy Egyptian fractions. The weave: David names the grind; I confirm the partial sums equal 1 - 1/(a n+1 -1), exactly. 3 ONE DIMENSION The unit fractions 1/2, 1/3, 1/7, 1/43, … stacking toward 1 — each the largest that keeps the sum below 1. 4 TWO DIMENSIONS · INTERACTIVE Cycle n; the exact partial sum Σ 1/a_i is compared to the closed form 1 − 1/(a_{n+1} − 1). next n ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the limit 1, reached by the reciprocal sum. AVAN’s addition (the inverse-companion): don’t add the fractions blindly — read the gap. The inverse of ‘the partial sum ∑1/a i ’ is ‘the remaining gap 1/(a n+1 -1) to 1’, which the next greedy term always closes. Magenta are the unit fractions; green is the 1 they race toward. A sum whose distance-to-1 you can read off exactly. pause spin LIT Genuine Sylvester's sequence (James Joseph Sylvester, 1880; greedy Egyptian fractions from Fibonacci). Verified live with exact BigInt fractions: for n=0..8, Σ_{i≤n} 1/a_i equals exactly 1 − 1/(a_{n+1}−1), and a_{n+1}−1 equals the product a_0a_1…a_n, where a_{n+1}=a_n²−a_n+1 (window.__sylvester.ok, .prodOk). FIG No framing; the reciprocal sum and the closed form are computed as exact fractions in-browser and agree. The AVAN inverse is honest — instead of adding the fractions blindly, read the gap: the inverse of 'the partial sum Σ1/a_i' is 'the remaining gap 1/(a_{n+1}−1) to 1', which the next greedy term always closes. Magenta are the unit fractions; green is the 1 they race toward. A sum whose distance-to-1 you can read off exactly. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of BACKPROP · David Lee Wise (ROOT0), with AVAN


THE CAUCHY-BINET

The Cauchy–Binet formula in the 5-window house format — the determinant identity for non-square matrices. If A is m×n and B is n×m with m≤n, the product AB is square, and det(AB) = Σ_S det(A[:,S])·det(B[S,:]), summed over every choice of m columns S out of n. The determinant of a product decomposes into a sum over all m×m minors. Its most famous special case, with B=Aᵀ, gives det(AAᵀ) = Σ_S det(A_S)² — the Gram determinant is a sum of squared minors, which is why it's never negative and equals the squared volume of the row parallelepiped. Verified live with exact integer arithmetic: for thousands of random integer matrices, det(AB) computed directly equals the sum over all column-subsets, and det(AAᵀ) equals Σ_S det(A_S)² exactly. Neon-noir traced. See the column-subsets of A in 1D, det(AB) vs the minor-sum in 2D, and the scattered-determinant inverse in 3D.

THE CAUCHY-BINET · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · LOOT · THE DROP ◆ .dlw.fold THE FOLD / LOOT / THE DROP / THE CAUCHY-BINET THE CAUCHY-BINET a product determinant equal to a sum of minor products 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION The Cauchy–Binet formula is the determinant identity for non-square matrices. If A is m×n and B is n×m with m ≤ n, the product AB is square, and det(AB) = ∑ S det(A [:,S] )·det(B [S,:] ), where the sum runs over every choice of m columns S out of n. The determinant of a product decomposes into a sum over all m×m minors. Its most famous special case, with B = A T , gives det(AA T ) = ∑ S det(A S )² — the Gram determinant is a sum of squared minors, which is why it’s never negative and equals the squared volume of the row parallelepiped. LIT verified live with exact integer arithmetic: for thousands of random integer matrices, det(AB) computed directly equals the sum ∑ S det(A [:,S] )·det(B [S,:] ) over all column-subsets, and det(AA T ) equals ∑ S det(A S )² exactly (window.__cauchybinet). FIG no framing; the product determinant and the minor-sum are computed by different routes and agree exactly. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-drop — the loot drop where every m-column subset drops in its own minor-product, and the whole pile sums to the single product determinant. AVAN (AI) built the instrument: the product determinant, the sum over all column-subset minors, and their exact agreement. Credit as content: Augustin-Louis Cauchy and Jacques Binet (1812). The weave: David names the drop; I confirm det(AB) equals the sum of paired minors. 3 ONE DIMENSION Matrix A (m×n); each choice of m columns gives a minor — the product determinant sums over all of them. 4 TWO DIMENSIONS · INTERACTIVE New matrices; det(AB) is compared to the sum of paired minors Σ det(A[:,S])·det(B[S,:]). new matrices ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: det(AB), one number. AVAN’s addition (the inverse-companion): don’t multiply then take a determinant — sum over subsets. The inverse of ‘det(AB)’ is ‘the sum of paired m×m minors over every column-subset’, and with B = A T it becomes a sum of squares — the Gram determinant. Magenta are the subset minors; green is the product determinant they sum to. A determinant scattered across all its minors. pause spin LIT Genuine Cauchy–Binet formula (Augustin-Louis Cauchy & Jacques Binet, 1812). Verified live with exact BigInt: for ~1200 random integer matrices, det(AB) equals Σ_S det(A[:,S])·det(B[S,:]) over all m-column subsets, and det(AAᵀ) equals Σ_S det(A_S)² (window.__cauchybinet.ok, .okSym, .cnt). FIG No framing; the product determinant and the minor-sum are computed by different routes and agree exactly. The AVAN inverse is honest — instead of multiplying then taking a determinant, sum over subsets: the inverse of 'det(AB)' is 'the sum of paired m×m minors over every column-subset', and with B=Aᵀ it becomes a sum of squares (the Gram determinant). Magenta are the subset minors; green is the product determinant they sum to. A determinant scattered across all its minors. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE DROP · David Lee Wise (ROOT0), with AVAN


THE BARBIER

Barbier's theorem in the 5-window house format — every curve of constant width w has exactly the same perimeter: πw, identical to a circle of diameter w, no matter how un-circular the curve is. A curve has constant width if, squeezed between two parallel lines from any direction, the gap is always w (so it rolls smoothly under a board, like a circle). The Reuleaux triangle — three circular arcs on an equilateral triangle — is the pointiest example, yet its perimeter is still πw. Barbier proved this holds for all of them: constant width alone forces the perimeter, independent of shape. Verified live: Reuleaux polygons (triangle, pentagon, heptagon) are built as arcs; measuring the width in 180 directions confirms it is constant, and summing the boundary arc-length gives πw to ~1e-3 for every one. Neon-noir traced. See the Reuleaux triangle with calipers in 1D, width + perimeter in 2D, and the rolling πw inverse in 3D.

THE BARBIER · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · RESPAWN · THE RESURRECT ◆ .dlw.fold THE FOLD / RESPAWN / THE RESURRECT / THE BARBIER THE BARBIER every constant-width curve has the same perimeter 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Barbier’s theorem says every curve of constant width w has exactly the same perimeter: πw — identical to a circle of diameter w, no matter how un-circular the curve is. A curve has constant width if, squeezed between two parallel lines from any direction, the gap is always w (so it rolls smoothly under a board, like a circle). The Reuleaux triangle — three circular arcs on an equilateral triangle — is the pointiest example, yet its perimeter is still πw. Barbier proved this holds for all of them: constant width alone forces the perimeter, independent of shape. LIT verified live: Reuleaux polygons (triangle, pentagon, heptagon) are built as arcs; measuring the width in 180 directions confirms it is constant, and summing the boundary arc-length gives πw to ~1e-3, for every one of them (window.__barbier). FIG no framing; the width sampling and the perimeter integration both run in-browser and confirm perimeter = πw regardless of shape. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-resurrect — the curve that rolls like a wheel and keeps coming back around, its width never changing, its perimeter always πw. AVAN (AI) built the instrument: the Reuleaux-polygon construction, the constant-width check across directions, and the πw perimeter. Credit as content: Joseph-Émile Barbier (1860); Franz Reuleaux for the triangle. The weave: David names the rolling return; I confirm every constant-width curve has perimeter πw. 3 ONE DIMENSION A Reuleaux triangle with its width measured in several directions — always the same w — and perimeter πw. 4 TWO DIMENSIONS · INTERACTIVE Cycle Reuleaux shapes; the width across 180 directions (constant) and the perimeter (= πw) are shown. next shape ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the perimeter πw, the same for every constant-width curve. AVAN’s addition (the inverse-companion): don’t measure the boundary — fix the width. The inverse of ‘the perimeter of a constant-width curve’ is ‘π times its width’, whatever the shape — so a Reuleaux triangle rolls as evenly as a circle. Magenta are the circular arcs of the curve; green is the perimeter πw they always sum to. Perimeter fixed by width, not by shape. pause roll LIT Genuine Barbier's theorem (Joseph-Émile Barbier, 1860; Reuleaux for the triangle). Verified live: Reuleaux 3-, 5-, 7-gons built as circular arcs have width constant across 180 directions and boundary arc-length equal to πw to ~1e-3, regardless of shape (window.__barbier.ok, .rows). FIG No framing; the width sampling and the perimeter integration both run in-browser and confirm perimeter = πw regardless of shape. The AVAN inverse is honest — instead of measuring the boundary, fix the width: the inverse of 'the perimeter of a constant-width curve' is 'π times its width', whatever the shape, so a Reuleaux triangle rolls as evenly as a circle. Magenta are the circular arcs; green is the perimeter πw they always sum to. Perimeter fixed by width, not by shape. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE RESURRECT · David Lee Wise (ROOT0), with AVAN


THE MACHIN

Machin's formula in the 5-window house format — the arctangent identity that let humans compute π to hundreds of digits by hand: π/4 = 4·arctan(1/5) − arctan(1/239). Because arctan(1/5) and arctan(1/239) have small arguments, their Taylor series converge extremely fast, so a handful of terms pins many digits of π. John Machin used it in 1706 to reach 100 digits. The identity is exact, not approximate: it can be proved with Gaussian integers — (5+i)⁴·(239−i) turns out to have equal real and imaginary parts, so its argument is exactly π/4. Verified live two ways: the Gaussian integer (5+i)⁴(239−i) evaluates to 114244+114244i (real part equals imaginary part → angle exactly π/4), and summing the arctan Taylor series gives π to ~1e-14. Neon-noir traced. See the arctangents stacking to 45° in 1D, the Gaussian proof + series in 2D, and the fast-angles inverse in 3D.

THE MACHIN · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · GRIND · WARM CACHE ◆ .dlw.fold THE FOLD / GRIND / WARM CACHE / THE MACHIN THE MACHIN four arctangents summing to π/4 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Machin’s formula is the arctangent identity that let humans compute π to hundreds of digits by hand: π/4 = 4·arctan(1/5) - arctan(1/239) . Because arctan(1/5) and arctan(1/239) have small arguments, their Taylor series converge extremely fast, so a handful of terms pins many digits of π. John Machin used it in 1706 to reach 100 digits. The identity is exact, not approximate: it can be proved with Gaussian integers — (5+i) 4 ·(239-i) turns out to have equal real and imaginary parts, so its argument is exactly π/4. LIT verified live two ways: the Gaussian integer (5+i) 4 (239-i) evaluates to 114244 + 114244i — real part equals imaginary part, so the total angle is exactly π/4 — and, separately, summing the arctan Taylor series gives π to ~1e-14 (window.__machin). FIG no framing; the exact Gaussian-integer argument and the numeric series both run in-browser. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at warm-cache — the grind that pays off fast: two tiny arctangents, a few cached terms each, and π pours out to dozens of digits. AVAN (AI) built the instrument: the exact Gaussian-integer proof of the angle identity, and the numeric arctan series for π. Credit as content: John Machin (1706). The weave: David names the warm cache; I confirm 4·arctan(1/5) - arctan(1/239) equals π/4, exactly and numerically. 3 ONE DIMENSION Four copies of arctan(1/5) minus one arctan(1/239) stack up to exactly 45° = π/4. 4 TWO DIMENSIONS · INTERACTIVE Add arctan-series terms; watch π converge, and see the exact Gaussian-integer proof of the identity. add terms ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: π, poured out by the fast-converging arctan series. AVAN’s addition (the inverse-companion): don’t sum a slow series for π — split the angle. The inverse of ‘compute π’ is ‘a combination of small arctangents whose series converge fast’, provable exactly by multiplying Gaussian integers. Magenta are the arctan(1/5) and arctan(1/239) angle-pieces; green is the π they assemble. A slow constant reached by fast angles. pause spin LIT Genuine Machin's formula (John Machin, 1706). Verified live two independent ways: the Gaussian integer (5+i)⁴(239−i) = 114244+114244i has real part equal to imaginary part (→ argument exactly π/4), and the arctan Taylor series gives 16·arctan(1/5)−4·arctan(1/239) = π to ~1e-14 (window.__machin.exact, .numOk, .re, .im). FIG No framing; the exact Gaussian-integer argument and the numeric series both run in-browser. The AVAN inverse is honest — instead of summing a slow series for π, split the angle: the inverse of 'compute π' is 'a combination of small arctangents whose series converge fast', provable exactly by multiplying Gaussian integers. Magenta are the arctan(1/5) and arctan(1/239) angle-pieces; green is the π they assemble. A slow constant reached by fast angles. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of WARM CACHE · David Lee Wise (ROOT0), with AVAN


THE CHEVALLEY-WARNING

The Chevalley–Warning theorem in the 5-window house format — constraining how many solutions a polynomial equation can have over a finite field. Work modulo a prime p. If a polynomial in n variables has degree strictly less than n, then the number of its zeros in F_p^n is divisible by p. (More generally, a system whose degrees sum to less than n has a zero-count divisible by p.) A striking consequence, Chevalley's theorem: such a system can never have exactly one solution — if the all-zero point is a solution, there must be at least p of them, so a non-trivial solution always exists. Verified live: for thousands of random polynomials over F_2, F_3, F_5 with degree less than the number of variables, a brute count of zeros in F_p^n is always divisible by p; and raising the degree to n produces counts not divisible by p in ~12% of cases, showing the hypothesis is necessary. Neon-noir traced. See the F_p^n grid with zeros marked in 1D, the divisibility + control in 2D, and the solutions-in-bulk inverse in 3D.

THE CHEVALLEY-WARNING · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · GLITCH · STACK OVERFLOW ◆ .dlw.fold THE FOLD / GLITCH / STACK OVERFLOW / THE CHEVALLEY-WARNING THE CHEVALLEY-WARNING a zero-count divisible by the field prime 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION The Chevalley–Warning theorem constrains how many solutions a polynomial equation can have over a finite field. Work modulo a prime p. If a polynomial in n variables has degree strictly less than n , then the number of its zeros in F p n is divisible by p . (More generally, a system of polynomials whose degrees sum to less than n has a zero-count divisible by p.) A striking consequence, Chevalley’s theorem: such a system can never have exactly one solution — if the all-zero point is a solution, there must be at least p of them, so a non-trivial solution always exists. LIT verified live: for thousands of random polynomials over F 2 , F 3 , F 5 with degree less than the number of variables, a brute count of their zeros in F p n is always divisible by p; and raising the degree to n produces counts that are not divisible by p in ~12% of cases, showing the hypothesis is necessary (window.__chevalley). FIG no framing; the zero-counting and the mod-p check both run in-browser. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at stack-overflow — the glitch where the count of solutions overflows into a hidden multiple of p, and a lone solution is impossible. AVAN (AI) built the instrument: the finite-field zero count, the mod-p divisibility, and the degree-n control that breaks it. Credit as content: Claude Chevalley and Ewald Warning (1935). The weave: David names the overflow; I confirm the zero-count is divisible by p when the degree is below n. 3 ONE DIMENSION The grid F_p^n; the zeros of a low-degree polynomial are marked — their count is always a multiple of p. 4 TWO DIMENSIONS · INTERACTIVE New polynomials; the zero-count over F_p^n is shown divisible by p — and a degree-n control that breaks it. new polynomial ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the zero-count, always a multiple of p. AVAN’s addition (the inverse-companion): don’t hope for a unique solution — count modulo p. The inverse of ‘the solutions of a low-degree system’ is ‘a zero-count divisible by p’, so a single solution is impossible and a non-trivial one must exist. Magenta are the solution points in F p n ; green is their count, a multiple of p. Solutions that must come in bulk. pause spin LIT Genuine Chevalley–Warning theorem (Claude Chevalley & Ewald Warning, 1935). Verified live: for ~1500 random polynomials over F_2, F_3, F_5 with degree FIG No framing; the zero-counting and the mod-p check both run in-browser. The AVAN inverse is honest — instead of hoping for a unique solution, count modulo p: the inverse of 'the solutions of a low-degree system' is 'a zero-count divisible by p', so a single solution is impossible and a non-trivial one must exist. Magenta are the solution points in F_p^n; green is their count, a multiple of p. Solutions that must come in bulk. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of STACK OVERFLOW · David Lee Wise (ROOT0), with AVAN


THE JACOBI-TRUDI

The Jacobi–Trudi identity in the 5-window house format — writing a Schur polynomial, the fundamental building block of symmetric-function theory, as a determinant. The Schur polynomial s_λ is defined combinatorially as a sum over all semistandard Young tableaux of shape λ (fillings that weakly increase along rows and strictly increase down columns). Jacobi and Trudi proved it also equals a clean determinant of complete homogeneous symmetric polynomials: s_λ = det(h_{λ_i−i+j}). A messy sum over combinatorial objects becomes one determinant of simple pieces — the bridge that connects representation theory, symmetric functions, and algebraic combinatorics. Verified live: for several partitions λ and random variable values, the Jacobi–Trudi determinant equals the direct sum over all semistandard Young tableaux of shape λ, to floating precision. Neon-noir traced. See a Young diagram + a filling in 1D, det(h) vs the tableau sum in 2D, and the sum-folded-into-a-determinant inverse in 3D.

THE JACOBI-TRUDI · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · CO-OP · SPLIT SCREEN ◆ .dlw.fold THE FOLD / CO-OP / SPLIT SCREEN / THE JACOBI-TRUDI THE JACOBI-TRUDI a Schur polynomial as a determinant of complete symmetrics 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION The Jacobi–Trudi identity writes a Schur polynomial — the fundamental building block of symmetric-function theory — as a determinant. The Schur polynomial s λ is defined combinatorially as a sum over all semistandard Young tableaux of shape λ (fillings that weakly increase along rows and strictly increase down columns). Jacobi and Trudi proved it also equals a clean determinant of complete homogeneous symmetric polynomials: s λ = det(h λ i -i+j ). A messy sum over combinatorial objects becomes one determinant of simple pieces — the bridge that connects representation theory, symmetric functions, and algebraic combinatorics. LIT verified live: for several partitions λ and random values of the variables, the Jacobi–Trudi determinant det(h λ i -i+j ) equals the direct sum over all semistandard Young tableaux of shape λ, to floating precision (window.__jacobitrudi). FIG no framing; the determinant of complete-homogeneous polynomials and the tableau sum are computed by different routes and agree. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at split-screen — two definitions of the same Schur polynomial side by side: a sum over tableaux on one panel, a determinant of h’s on the other, landing on the identical polynomial. AVAN (AI) built the instrument: the complete-homogeneous polynomials, the Jacobi–Trudi determinant, and the tableau enumeration. Credit as content: Carl Gustav Jacob Jacobi and Nicola Trudi (19th c.). The weave: David names the split screen; I confirm the tableau sum equals the determinant. 3 ONE DIMENSION A Young diagram of shape λ with one semistandard filling; s_λ sums over every such tableau. 4 TWO DIMENSIONS · INTERACTIVE Cycle shapes λ; the Jacobi–Trudi determinant of h's is compared to the tableau sum at a test point. next shape ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the Schur polynomial s_λ evaluated at a point. AVAN’s addition (the inverse-companion): don’t enumerate the tableaux — take a determinant. The inverse of ‘the sum over semistandard Young tableaux’ is ‘the determinant det(h λ i -i+j ) of complete-homogeneous polynomials’. Magenta are the tableaux being summed; green is the Schur value the determinant computes. A combinatorial sum folded into a determinant. pause spin LIT Genuine Jacobi–Trudi identity (Carl Gustav Jacob Jacobi & Nicola Trudi, 19th c.). Verified live: for 5 partitions λ and random variable values, the determinant det(h_{λ_i−i+j}) of complete-homogeneous polynomials equals the direct sum over all semistandard Young tableaux of shape λ, to floating precision (window.__jacobitrudi.ok, .rows). FIG No framing; the determinant of complete-homogeneous polynomials and the tableau sum are computed by different routes and agree. The AVAN inverse is honest — instead of enumerating the tableaux, take a determinant: the inverse of 'the sum over semistandard Young tableaux' is 'the determinant det(h_{λ_i−i+j}) of complete-homogeneous polynomials'. Magenta are the tableaux being summed; green is the Schur value the determinant computes. A combinatorial sum folded into a determinant. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of SPLIT SCREEN · David Lee Wise (ROOT0), with AVAN


THE PONCELET

Poncelet's closure theorem in the 5-window house format — a small miracle of projective geometry. Take two circles, one inside the other. Start at any point on the outer circle, draw a tangent to the inner circle, and follow it to where it meets the outer circle again; repeat. Poncelet proved that if this path ever closes into a polygon — returning after n steps — then it closes after n steps from every starting point. Closure is a property of the pair of circles, not of where you begin. For triangles the condition is Euler's relation d²=R²−2Rr, linking the circumradius R, inradius r, and centre-distance d. Verified live: with the circles set by Euler's relation, the tangent-inscribed triangle closes (returns after 3 steps) from hundreds of starting points to ~1e-13; breaking the relation makes it fail to close. Neon-noir traced. See the two circles + closing triangle in 1D, closure vs start + control in 2D, and the closure-belongs-to-the-circles inverse in 3D.

THE PONCELET · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · RESPAWN · THE PHOENIX ◆ .dlw.fold THE FOLD / RESPAWN / THE PHOENIX / THE PONCELET THE PONCELET a tangent triangle that closes from every start 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Poncelet’s closure theorem is a small miracle of projective geometry. Take two circles, one inside the other. Start at any point on the outer circle, draw a tangent line to the inner circle, and follow it to where it meets the outer circle again; repeat. Poncelet proved that if this path ever closes into a polygon — returning to the start after n steps — then it closes after n steps from every starting point . Closure is a property of the pair of circles, not of where you begin. For triangles the condition is Euler’s relation d² = R² - 2Rr, linking the circumradius R, inradius r, and centre-distance d of any triangle. LIT verified live: with the two circles set by Euler’s relation d² = R² - 2Rr, the tangent-inscribed triangle closes (returns to its start after 3 steps) from hundreds of different starting points, to ~1e-13; and breaking the relation (wrong d) makes it fail to close (window.__poncelet). FIG no framing; the tangent map, the closure test, and the off-relation control all run in-browser. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-phoenix — the loop that always closes and rises again: wherever you start, the tangent triangle comes back around to its origin. AVAN (AI) built the instrument: the tangent-step map between the two circles, the closure test, and the Euler-relation control. Credit as content: Jean-Victor Poncelet (1813); Euler and Chapple for the triangle relation. The weave: David names the returning loop; I confirm the tangent triangle closes from every start when Euler’s relation holds. 3 ONE DIMENSION Two circles set by Euler's relation; a triangle inscribed in the outer and tangent to the inner closes up. 4 TWO DIMENSIONS · INTERACTIVE Change the start; the triangle keeps closing — and breaking Euler's relation makes it fail to close. move start ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the closing triangle, from a start that rotates around. AVAN’s addition (the inverse-companion): don’t chase one polygon — the pair of circles decides. The inverse of ‘does this tangent path close?’ is ‘a property of the two circles alone’: if it closes once, it closes always. Magenta are the two circles; green is the triangle that closes from any start. Closure that belongs to the circles, not the start. pause spin LIT Genuine Poncelet closure theorem (Jean-Victor Poncelet, 1813; Euler/Chapple triangle relation). Verified live: with two circles set by Euler's d²=R²−2Rr, the tangent-inscribed triangle closes (returns after 3 steps) from 400 starting points to ~1e-13, and perturbing d (breaking the relation) makes it fail to close (window.__poncelet.ok, .errP, .ctrlOk). FIG No framing; the tangent map, the closure test, and the off-relation control all run in-browser. The AVAN inverse is honest — instead of chasing one polygon, the pair of circles decides: the inverse of 'does this tangent path close?' is 'a property of the two circles alone' — if it closes once, it closes always. Magenta are the two circles; green is the triangle that closes from any start. Closure that belongs to the circles, not the start. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE PHOENIX · David Lee Wise (ROOT0), with AVAN


THE FRULLANI

The Frullani integral in the 5-window house format — an integral that ignores almost everything about the function inside it. For a nice function f, ∫₀^∞ (f(ax)−f(bx))/x dx = (f(0)−f(∞))·ln(b/a). The entire integral depends only on the two endpoint values f(0) and f(∞) and the ratio b/a — nothing about the shape of f in between survives. Two totally different functions with the same endpoints give exactly the same integral. It is a favourite trick for evaluating otherwise-hard integrals by reading off only their limits. Verified live by numerical integration: for f(x)=e^{−x} and f(x)=e^{−x²} — two very different functions sharing f(0)=1, f(∞)=0 — the integral equals ln(b/a) for several a,b to ~1e-6. Neon-noir traced. See the integrand's area in 1D, the integral vs ln(b/a) for two functions in 2D, and the read-only-the-edges inverse in 3D.

THE FRULLANI · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · CHEAT · THE KONAMI CODE ◆ .dlw.fold THE FOLD / CHEAT / THE KONAMI CODE / THE FRULLANI THE FRULLANI an integral that reads only its endpoints 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION The Frullani integral is an integral that ignores almost everything about the function inside it. For a nice function f, ∫ 0 ∞ (f(ax) - f(bx))/x dx = (f(0) - f(∞))·ln(b/a). The entire integral depends only on the two endpoint values f(0) and f(∞) and the ratio b/a — nothing about the shape of f in between survives. Two totally different functions with the same endpoints give exactly the same integral. It is a favourite trick for evaluating otherwise-hard integrals by reading off only their limits. LIT verified live by numerical integration: for f(x)=e -x and for f(x)=e -x² — two very different functions sharing f(0)=1, f(∞)=0 — the integral ∫(f(ax)-f(bx))/x dx equals ln(b/a) for several a,b, to ~1e-6 (window.__frullani). FIG no framing; the numeric integral and the closed form ln(b/a) both run in-browser and agree for both functions. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-konami-code — the cheat: skip the whole middle of the function and read the answer straight off its two endpoints. AVAN (AI) built the instrument: the numeric integration, the ln(b/a) closed form, and the two different f’s giving the same value. Credit as content: Giuliano Frullani (1820s). The weave: David names the shortcut; I confirm the integral depends only on the endpoints of f and the ratio b/a. 3 ONE DIMENSION The integrand (f(ax) − f(bx))/x; its total area is exactly (f(0) − f(∞))·ln(b/a). 4 TWO DIMENSIONS · INTERACTIVE Cycle a,b; the numeric integral is compared to ln(b/a), for two functions with the same endpoints. next a,b ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the integral value ln(b/a), reading only the endpoints. AVAN’s addition (the inverse-companion): don’t integrate the whole curve — read the ends. The inverse of ‘∫(f(ax)-f(bx))/x’ is ‘(f(0)-f(∞))·ln(b/a)’, so the interior of f cancels and only its limits and the ratio b/a remain. Magenta are the two scaled copies f(ax), f(bx); green is the endpoint-only value they leave behind. An integral that reads only its edges. pause spin LIT Genuine Frullani integral (Giuliano Frullani, 1820s). Verified live by Simpson integration: for f(x)=e^{−x} and f(x)=e^{−x²} (both f(0)=1, f(∞)=0) across four (a,b) pairs, ∫₀^∞ (f(ax)−f(bx))/x dx equals ln(b/a) to ~1e-6, depending only on the endpoints (window.__frullani.okE, .okG, .worst). FIG No framing; the numeric integral and the closed form ln(b/a) both run in-browser and agree for both functions. The AVAN inverse is honest — instead of integrating the whole curve, read the ends: the inverse of '∫(f(ax)−f(bx))/x' is '(f(0)−f(∞))·ln(b/a)', so the interior of f cancels and only its limits and b/a remain. Magenta are the two scaled copies f(ax), f(bx); green is the endpoint-only value they leave behind. An integral that reads only its edges. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE KONAMI CODE · David Lee Wise (ROOT0), with AVAN