THE LAH

Lah numbers in the 5-window house format — L(n,k) counts the ways to sort n labelled items into k non-empty ordered lists, where the order within each list matters. They have a closed form L(n,k) = C(n−1,k−1)·n!/k! and a recurrence L(n,k) = L(n−1,k−1) + (n+k−1)L(n−1,k). They are the exact coefficients converting between the two factorial bases: the rising factorial equals a Lah-weighted sum of falling factorials, x^(n rising) = Σ_k L(n,k) x^(k falling). Verified live (exact BigInt): closed form equals recurrence for n≤12, and the rising=Σ L·falling identity holds for integer x. See the triangle in 1D, the identity in 2D, and the ordered-blocks inverse in 3D.

THE LAH · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · LOOT · THE INVENTORY ◆ .dlw.fold THE FOLD / LOOT / THE INVENTORY / THE LAH THE LAH counting partitions into ordered lists 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Lah numbers L(n, k) count the ways to sort n labelled items into k non-empty ordered lists — where, unlike set partitions, the order within each list matters . They have a clean closed form, L(n, k) = C(n−1, k−1) · n! / k! , and a recurrence L(n, k) = L(n−1, k−1) + (n + k − 1) L(n−1, k). They are the exact coefficients that convert between the two great factorial bases: the rising factorial equals a Lah-weighted sum of falling factorials , x (n̅) = Σ k L(n, k) x (k̲) . LIT verified live (exact BigInt): the closed form equals the recurrence for all n up to 12, and the basis-change identity x (n rising) = Σ k L(n,k) x (k falling) holds for integer x (window.__lah). FIG no framing; exact big-integer combinatorics and polynomial identities. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-inventory — not just which items go in which pile, but the order they are stacked in each pile; Lah numbers count exactly those ordered arrangements. That ordered inventory is the drop. AVAN (AI) built the instrument: the closed form C(n−1,k−1)·n!/k!, the two-term recurrence, and the rising/falling factorial basis-change identity. Credit as content: Ivo Lah (1954). The weave: David names the-inventory; I compute L(n,k) by its closed form and by its recurrence and confirm they agree exactly, then verify the Lah numbers convert rising factorials into falling factorials — the connective tissue between the two factorial bases. 3 ONE DIMENSION L(n,k) = C(n−1,k−1)·n!/k!. L(3,1)=6 (one list, all orders of 3), L(3,2)=6, L(3,3)=1. Row sums are the “Lah” totals. They send x (n̅) to Σ L(n,k) x (k̲) . 4 TWO DIMENSIONS · INTERACTIVE The Lah triangle by closed form and recurrence; the rising/falling identity at a chosen x; both checked. new x ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: partitions into ordered lists. AVAN’s addition (the inverse-companion): count set partitions where the order inside each block matters , and the same numbers become the change-of-basis between rising and falling factorials. The inverse of ‘unordered blocks (Stirling)’ is ‘ordered lists (Lah)’, and they translate one factorial basis into the other. Magenta is the unordered Stirling count; green is the ordered Lah count. Order inside the blocks. pause spin LIT Genuine Lah numbers (Ivo Lah, 1954). Verified live with exact BigInt: the closed form C(n−1,k−1)·n!/k! equals the two-term recurrence L(n,k)=L(n−1,k−1)+(n+k−1)L(n−1,k) for all n up to 12 (window.__lah.closedForm), and the basis-change identity x^(n rising) = Σ_k L(n,k)·x^(k falling) holds for integer x (window.__lah.factorialIdentity). FIG No framing: the closed form, the recurrence, and the rising/falling factorial identity all run in-browser in exact big integers and agree. The AVAN inverse is honest — counting set partitions where the order inside each block matters (Lah) genuinely differs from the unordered Stirling count, and the same numbers translate one factorial basis into the other; magenta is the unordered Stirling count, green the ordered Lah count. Order inside the blocks. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE INVENTORY · David Lee Wise (ROOT0), with AVAN


THE FUBINI

The Fubini (ordered Bell) numbers in the 5-window house format — counting the ways to rank n items allowing ties, i.e. the weak orderings, or ordered set partitions. They run 1, 1, 3, 13, 75, 541, 4683, … For 3 items there are 13 outcomes. Two formulas produce them: a(n) = Σ_k k!·S(n,k) (Stirling numbers of the second kind times the k! orderings of the blocks), and the recurrence a(n) = Σ_{i=1}^n C(n,i) a(n−i). Verified live (exact BigInt): Σ k!·S(n,k) equals the binomial recurrence for n≤9, and both equal a direct brute-force count of weak orderings for n≤6. See the sequence in 1D, the three computations in 2D, and the ranked-blocks inverse in 3D.

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

THE FUBINI · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · SPAWN · NULL ISLAND ◆ .dlw.fold THE FOLD / SPAWN / NULL ISLAND / THE FUBINI THE FUBINI counting the ways to rank things with ties 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION The Fubini numbers (also called ordered Bell numbers ) count the ways to rank n items allowing ties — the number of weak orderings , or equivalently the ordered set partitions. They run 1, 1, 3, 13, 75, 541, 4683, … For 3 items there are 13 outcomes: all-tied, one clear winner (3 ways × 2), full strict orders (6), and so on. Two clean formulas produce them: a(n) = Σ k k! · S(n, k) (Stirling numbers of the second kind, times the k! orderings of the blocks), and the recurrence a(n) = Σ i=1 n C(n, i) a(n−i) . LIT verified live (exact BigInt): Σ k k!·S(n,k) equals the binomial recurrence for all n up to 9, and both equal a direct brute-force count of weak orderings for n up to 6 (window.__fubini). FIG no framing; two formulas cross-checked against an explicit enumeration. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at null-island — the origin where orderings are born: from nothing, count every way a set of items can finish a race with ties allowed, spawning the sequence 1, 1, 3, 13, 75, … That spawning is the boot. AVAN (AI) built the instrument: the Stirling-weighted sum Σ k!·S(n,k), the binomial recurrence, and the direct enumeration of weak orderings. Credit as content: Fubini / ordered Bell numbers (studied by Louis Comtet and others; the “Fubini” name from the count of terms in an iterated integral). The weave: David names null-island; I compute the orderings by k!·S(n,k), confirm it against the binomial recurrence, and check both against a brute-force count of every weak ordering — all three agree. 3 ONE DIMENSION a(n) = 1, 1, 3, 13, 75, 541, 4683, … For 3 items: 1 all-tied + 6 with one item alone atop or below a tied pair + 6 strict orders = 13 weak orderings. 4 TWO DIMENSIONS · INTERACTIVE The three ways to compute a(n) side by side; the weak orderings of a small set enumerated; all checked. step n ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: rankings with ties allowed. AVAN’s addition (the inverse-companion): count set partitions where the blocks are ordered — rankings that allow ties — and the plain Bell number becomes the far larger ordered Bell number. The inverse of ‘unordered partitions (Bell)’ is ‘ordered partitions / weak orderings (Fubini)’, counted by Σ k!·S(n,k). Magenta is the Bell number (blocks unordered); green is the Fubini number (blocks ranked). Order among the blocks. pause spin LIT Genuine Fubini / ordered Bell numbers (studied by Louis Comtet and others; the 'Fubini' name from counting terms in an iterated integral). Verified live with exact BigInt: Σ_k k!·S(n,k) equals the binomial recurrence Σ_i C(n,i)a(n−i) for all n up to 9 (window.__fubini.sumForm), and both equal a direct brute-force enumeration of weak orderings for n up to 6 (window.__fubini.directCount). FIG No framing: the Stirling-weighted sum, the binomial recurrence, and the direct enumeration of weak orderings all run in-browser (BigInt) and agree. The AVAN inverse is honest — counting set partitions where the blocks are ranked (weak orderings) genuinely differs from and far exceeds the unordered Bell count; magenta is the Bell number, green the Fubini number. Order among the blocks. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of NULL ISLAND · David Lee Wise (ROOT0), with AVAN


THE EULER PARTITION

Euler's partition theorem in the 5-window house format — the number of ways to write n as a sum of distinct parts equals the number using only odd parts. For n=6: distinct partitions 6, 5+1, 4+2, 3+2+1 (four); odd partitions 5+1, 3+3, 3+1+1+1, 1×6 (also four). Euler proved it with generating functions: ∏(1+x^k) = ∏1/(1−x^(2k−1)), and Glaisher gave a direct bijection. Verified live: an exact partition-counting DP shows distinct-part count equals odd-part count for every n up to 60. See the two families in 1D, the matching counts in 2D, and the two-restrictions-one-number inverse in 3D.

THE EULER PARTITION · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · SPAWN · HELLO WORLD ◆ .dlw.fold THE FOLD / SPAWN / HELLO WORLD / THE EULER PARTITION THE EULER PARTITION two ways to break a number into parts that always agree 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Euler’s partition theorem is a perfect coincidence of counting: the number of ways to write n as a sum of distinct parts equals the number of ways to write it using only odd parts. For n = 6, the distinct partitions are 6, 5+1, 4+2, 3+2+1 — four of them; the odd partitions are 5+1, 3+3, 3+1+1+1, 1+1+1+1+1+1 — also four. The two look unrelated, yet always tie. Euler proved it with generating functions: ∏(1 + x k ) = ∏ 1/(1 − x 2k−1 ), and Glaisher later gave a direct bijection. LIT verified live: an exact partition-counting DP shows the number of distinct-part partitions equals the number of odd-part partitions for every n up to 60 (window.__eulerpartition). FIG no framing; two independent partition counts compared exactly. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at hello-world — the simplest surprising identity: two utterly different ways to break a number apart that always land on the same count. That first coincidence is the hello-world. AVAN (AI) built the instrument: the distinct-parts counter (each part used at most once), the odd-parts counter (odd parts any number of times), and their exact agreement. Credit as content: Leonhard Euler (1740s); the bijective proof by James Glaisher (1883). The weave: David names hello-world; I count partitions two ways — once forbidding repeats, once forbidding even parts — and confirm the totals are equal for every n in range, the way Euler’s product identity promises. 3 ONE DIMENSION 6 into distinct parts: 6, 5+1, 4+2, 3+2+1 (four). 6 into odd parts: 5+1, 3+3, 3+1+1+1, 1×6 (four). Always equal: ∏(1+x k ) = ∏1/(1−x 2k−1 ). 4 TWO DIMENSIONS · INTERACTIVE Distinct-part vs odd-part counts side by side across n; the equality checked to 60. shift range ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: two partition worlds with one count. AVAN’s addition (the inverse-companion): impose opposite restrictions — parts must be distinct , or parts must be odd — and the two counts come out identical for every n. The inverse of ‘forbid repeats’ is, remarkably, ‘forbid even parts’, and they meet. Magenta is the distinct-part count; green is the odd-part count. Two restrictions, one number. pause spin LIT Genuine Euler partition theorem (Leonhard Euler, 1740s; bijective proof by James Glaisher, 1883). Verified live: a dynamic-programming partition counter shows the number of partitions of n into distinct parts equals the number into odd parts for every n from 0 to 60 (window.__eulerpartition.distinctEqualsOdd). FIG No framing: the distinct-parts counter (each part at most once) and the odd-parts counter (odd parts any multiplicity) run in-browser and agree exactly. The AVAN inverse is honest — imposing the opposite restrictions 'parts distinct' versus 'parts odd' yields identical counts for every n, exactly as Euler's product identity ∏(1+x^k)=∏1/(1−x^(2k−1)) promises; magenta is the distinct count, green the odd count. Two restrictions, one number. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of HELLO WORLD · David Lee Wise (ROOT0), with AVAN


THE ZOLOTAREV

Zolotarev's lemma in the 5-window house format — for an odd prime p and a coprime to p, the Legendre symbol (a/p) (whether a is a quadratic residue mod p) equals the sign of the permutation x → a·x mod p on {1,…,p−1}. Whether a has a square root mod p is exactly whether multiplication by a shuffles the residues evenly or oddly. Verified live: for every odd prime p<80 and every a, the permutation sign (from cycle structure) equals the Legendre symbol (from Euler's criterion a^((p−1)/2)). See the shuffle in 1D, cycles and signs in 2D, and the residue-as-parity inverse in 3D.

THE ZOLOTAREV · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · BOSS · SUDDEN DEATH ◆ .dlw.fold THE FOLD / BOSS / SUDDEN DEATH / THE ZOLOTAREV THE ZOLOTAREV a coin-flip sign hidden in modular multiplication 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Zolotarev’s lemma ties two seemingly unrelated worlds together: for an odd prime p and a number a not divisible by p, the Legendre symbol (a/p) — which is +1 if a is a quadratic residue mod p and −1 if not — equals the sign of the permutation that multiplication by a induces on {1, 2, …, p−1}. In other words, whether a has a square root mod p is exactly whether the shuffle x → a·x mod p is an even or odd permutation. Number theory and permutation parity turn out to be the same coin flip. LIT verified live: for every odd prime p < 80 and every a in 1…p−1, the sign of the permutation x → a·x mod p (from its cycle structure) equals the Legendre symbol (a/p) computed by Euler’s criterion (window.__zolotarev). FIG no framing; permutation parity vs modular exponentiation. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at sudden-death — one bit decides everything: even or odd, residue or not, +1 or −1. The shuffle’s parity and the square-root question are the same sudden-death coin. AVAN (AI) built the instrument: the multiplication permutation, its sign from cycle counts, the Legendre symbol via a (p−1)/2 , and their exact match. Credit as content: Yegor Ivanovich Zolotarev (1872). The weave: David names sudden-death; I shuffle the residues by multiplying by a, read off the permutation’s sign from its cycles, and confirm it equals whether a is a quadratic residue mod p — parity and residue, one and the same. 3 ONE DIMENSION Mod 7, x→3x: 1→3→2→6→4→5→1 (one 6-cycle) — odd permutation, sign −1. And (3/7) = −1 (3 is a non-residue mod 7). Same answer. 4 TWO DIMENSIONS · INTERACTIVE The permutation x→a·x mod p as cycles, its sign, and the Legendre symbol; the equality checked over all primes. new a,p ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: a residue question answered by a shuffle. AVAN’s addition (the inverse-companion): decide whether a is a quadratic residue mod p not by exponentiating, but by asking whether multiplying by a shuffles the residues evenly or oddly . The inverse of ‘compute a (p−1)/2 ’ is ‘count the cycles of x→ax and read the parity.’ Magenta is the Euler-criterion exponentiation; green is the permutation-parity answer. Residue as parity. pause spin LIT Genuine Zolotarev's lemma (Yegor Ivanovich Zolotarev, 1872). Verified live: for every odd prime p from 3 to 79 and every a in 1…p−1, the sign of the permutation x → a·x mod p — computed as (−1)^((p−1) − #cycles) — equals the Legendre symbol (a/p) computed by Euler's criterion (window.__zolotarev.matches). FIG No framing: the multiplication permutation, its sign from cycle counts, and the Legendre symbol via modular exponentiation all run in-browser and agree exactly. The AVAN inverse is honest — deciding quadratic residuacity by the parity of the shuffle x→ax (rather than by exponentiating a^((p−1)/2)) is a genuine reframing; magenta is the Euler-criterion exponentiation, green the permutation-parity answer. Residue as parity. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of SUDDEN DEATH · David Lee Wise (ROOT0), with AVAN


THE Q-BINOMIAL

The Gaussian binomial coefficient in the 5-window house format — [n,k]_q is the q-analog of C(n,k): replace each integer m by 1+q+…+q^(m−1). It has a product form, equals Σ q^(inversions) over binary words with k ones, and when q is a prime power it counts the k-dimensional subspaces of F_q^n. Setting q=1 recovers C(n,k). Verified live (exact BigInt): the product equals the inversion-sum for q∈{2,3,5} and n≤7, and equals the brute-force count of k-subspaces of F_2^n for n≤4. See the three faces in 1D, all cross-checked in 2D, and the choosing-into-subspaces inverse in 3D.

THE Q-BINOMIAL · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · LOOT · THE BOUNTY ◆ .dlw.fold THE FOLD / LOOT / THE BOUNTY / THE Q-BINOMIAL THE Q-BINOMIAL counting subspaces with a q-analog of the binomial 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION The Gaussian binomial coefficient [n, k] q is the q-analog of the ordinary binomial: replace each integer m by its q-version 1 + q + … + q m−1 . It has a product form and, remarkably, two combinatorial meanings. As a polynomial in q it is Σ q (inversions) over binary words with k ones; and when q is a prime power , [n, k] q counts exactly the number of k-dimensional subspaces of the vector space F q n . Setting q = 1 recovers the ordinary C(n, k). One formula, three faces: product, inversion-sum, subspace count. LIT verified live: the product form equals the inversion-sum for q in {2, 3, 5} and n up to 7 (exact BigInt), and equals the brute-force count of k-subspaces of F 2 n for n up to 4 (window.__qbinomial). FIG no framing; three computations of one quantity, cross-checked. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-bounty — a single count that pays out three ways: a product, a sum over inversions, and the number of subspaces you can carve from F q n . That triple payout is the bounty. AVAN (AI) built the instrument: the q-product, the inversion-sum over binary words, and the exhaustive subspace count over a finite field. Credit as content: Carl Friedrich Gauss (the coefficients); the subspace interpretation is classical finite-field combinatorics. The weave: David names the-bounty; I compute [n,k] q as a product, as a sum of q raised to inversion counts, and as the number of k-subspaces of F q n — and confirm all three agree. 3 ONE DIMENSION [3,1] 2 = (2³−1)/(2−1) = 7 = number of lines through 0 in F 2 ³. [4,2] 2 = 35. As a q-polynomial, [n,k] q = Σ q inversions ; at q=1 it is C(n,k). 4 TWO DIMENSIONS · INTERACTIVE The three faces of [n,k] q — product, inversion-sum, subspace count; all cross-checked. new n,k,q ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: a binomial that counts subspaces. AVAN’s addition (the inverse-companion): take the ordinary “choose k of n” and deform it by q — and the deformed count becomes the number of k-dimensional subspaces of a finite vector space. The inverse of ‘count k-element subsets (q=1)’ is ‘count k-dimensional subspaces (q = prime power).’ Magenta is the plain binomial C(n,k); green is the q-binomial [n,k] q . Choosing, deformed into subspaces. pause spin LIT Genuine Gaussian binomial coefficient (Carl Friedrich Gauss); the subspace interpretation is classical finite-field combinatorics. Verified live with exact BigInt: the q-product Π(q^(n−i)−1)/(q^(k−i)−1) equals the inversion-sum Σ q^(inv) over binary words for q∈{2,3,5} and all n≤7 (window.__qbinomial.productEqualsInversion), and equals the exhaustive count of k-dimensional subspaces of F_2^n for n≤4 (window.__qbinomial.subspaceCount). FIG No framing: the q-product, the inversion-sum over binary words, and the exhaustive subspace count over F_2^n all run in-browser and agree. The AVAN inverse is honest — deforming the ordinary 'choose k of n' by q turns it into the count of k-dimensional subspaces of a finite vector space (q=1 recovers subsets); magenta is the plain binomial C(n,k), green the q-binomial [n,k]_q. Choosing, deformed into subspaces. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE BOUNTY · David Lee Wise (ROOT0), with AVAN


THE LINDSTRÖM–GESSEL–VIENNOT

The Lindström–Gessel–Viennot lemma in the 5-window house format — counting families of non-crossing lattice paths with a single determinant. With sources A_i and sinks B_j and M_ij = #paths A_i→B_j, det(M) equals the signed count of vertex-disjoint path families A_i→B_σ(i). In the planar arrangement where only the identity matching avoids crossings, the determinant counts exactly the non-intersecting families — the crossing pairs cancel in the signed sum. Verified live: for 2-source and 3-source planar placements, det(M) (from binomial path counts) equals a brute-force enumeration of vertex-disjoint families. See the 2×2 case in 1D, the grid and matrix in 2D, and the crossings-cancel inverse in 3D.

THE LINDSTRÖM–GESSEL–VIENNOT · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · GRIND · BACKPROP ◆ .dlw.fold THE FOLD / GRIND / BACKPROP / THE LINDSTRÖM–GESSEL–VIENNOT THE LINDSTRÖM–GESSEL–VIENNOT non-crossing paths counted by a determinant 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION The Lindström–Gessel–Viennot lemma counts families of non-crossing paths with a single determinant . Place sources A 1 …A m and sinks B 1 …B m on a grid; let M ij be the number of lattice paths from A i to B j . Then det(M) equals the signed count of families of paths, one A i →B σ(i) , that are vertex-disjoint . In the common planar arrangement where only the identity matching can avoid crossings, the determinant counts exactly the non-intersecting families — a bridge between linear algebra and path combinatorics. LIT verified live: for planar source/sink placements, det(M) (built from binomial path counts) equals a brute-force enumeration of vertex-disjoint path families, for both 2-source and 3-source configurations (window.__lgv). FIG no framing; a determinant vs an explicit non-crossing count. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at backprop — the same shape as a computation graph: value flows along paths, and the total is a signed sum over path families; here the crossing paths cancel and only the disjoint families survive, read off as a determinant. That path-sum is the backprop kinship. AVAN (AI) built the instrument: the binomial path-count matrix, its determinant, and the brute-force count of vertex-disjoint families. Credit as content: Bernt Lindström (1973); Ira Gessel & Gérard Viennot (1985). The weave: David names backprop; I fill the matrix of path counts between sources and sinks, take its determinant, and confirm it equals the number of non-crossing path families — the crossings cancelling in the determinant’s signed sum. 3 ONE DIMENSION Two sources, two sinks. M = [[#A₁→B₁, #A₁→B₂],[#A₂→B₁, #A₂→B₂]]. det(M) = (paths that don’t cross) — the crossing pairs cancel in the signed sum. 4 TWO DIMENSIONS · INTERACTIVE The grid, sources and sinks, the path-count matrix and its determinant; the non-crossing count checked. new placement ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: non-crossing paths from a determinant. AVAN’s addition (the inverse-companion): count non-crossing path families not by enumerating them, but by a determinant of path counts — the crossing families cancel in pairs, leaving exactly the disjoint ones. The inverse of ‘list every vertex-disjoint family’ is ‘take one determinant and the crossings cancel themselves.’ Magenta is the brute enumeration; green is the determinant. Crossings cancelled by a sign. pause spin LIT Genuine Lindström–Gessel–Viennot lemma (Bernt Lindström, 1973; Ira Gessel & Gérard Viennot, 1985). Verified live: for planar source/sink placements the determinant of the binomial path-count matrix equals a brute-force count of vertex-disjoint path families, for both a 2-source and a 3-source configuration (window.__lgv.matches, with det2/count2 and det3/count3 shown). FIG No framing: the binomial path-count matrix, its determinant, and the brute-force enumeration of vertex-disjoint families all run in-browser and agree. The AVAN inverse is honest — counting non-crossing path families by a determinant (rather than enumerating them) works because crossing families cancel in pairs in the signed sum, leaving exactly the disjoint ones; magenta is the brute enumeration, green the determinant. Crossings cancelled by a sign. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of BACKPROP · David Lee Wise (ROOT0), with AVAN


THE SPERNER

Sperner's theorem in the 5-window house format — the largest family of subsets of an n-set with none containing another (an antichain) has size C(n,⌊n/2⌋), the middle layer of the subset lattice. The clean proof partitions all 2^n subsets into exactly C(n,⌊n/2⌋) symmetric chains (nested runs); since an antichain meets each chain at most once, it has at most that many members, and the middle layer achieves it. Verified live: a recursive symmetric chain decomposition is built for n≤8 — the chains partition all subsets, each is a genuine chain, and their count equals C(n,⌊n/2⌋). See the middle layer in 1D, the chain-decomposed lattice in 2D, and the antichains-bounded-by-chains inverse in 3D.

THE SPERNER · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · CO-OP · SHARED MEMORY ◆ .dlw.fold THE FOLD / CO-OP / SHARED MEMORY / THE SPERNER THE SPERNER the widest layer of the subset lattice 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Sperner’s theorem answers: how many subsets of an n-element set can you pick so that none contains another ? Such a family is an antichain . The answer is the widest layer of the subset lattice: C(n, ⌊n/2⌋) — all the subsets of the middle size. You cannot do better. The clean proof decomposes the whole lattice of 2 n subsets into exactly C(n, ⌊n/2⌋) symmetric chains (nested runs from a small set up to a large one); since an antichain meets each chain at most once, it can have at most that many members — and the middle layer achieves it. LIT verified live: a symmetric chain decomposition of 2 [n] is built for n up to 8 — the chains partition all subsets, each is a genuine chain, and their number equals C(n, ⌊n/2⌋), which the middle layer attains (window.__sperner). FIG no framing; an explicit chain partition proving the bound. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at shared-memory — the full lattice of subsets as one shared structure, carved into nested chains so that the widest independent layer is exposed. That shared decomposition is the memory. AVAN (AI) built the instrument: the recursive symmetric chain decomposition, the checks that the chains partition all subsets and each is nested, and the count against C(n, ⌊n/2⌋). Credit as content: Emanuel Sperner (1928); the symmetric chain proof by de Bruijn, Tengbergen & Kruyswijk (1951). The weave: David names shared-memory; I split the 2 n subsets into symmetric chains, confirm they cover everything exactly once and each grows one element at a time, and count them — landing on C(n, ⌊n/2⌋), the largest antichain. 3 ONE DIMENSION For n = 4: the widest layer is the 2-element subsets, C(4,2) = 6. No antichain of subsets of {1,2,3,4} can exceed 6. The lattice splits into 6 symmetric chains. 4 TWO DIMENSIONS · INTERACTIVE The subset lattice by rank with its symmetric chains; the middle layer (the largest antichain) highlighted; the partition checked. change n ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the widest possible antichain. AVAN’s addition (the inverse-companion): to bound the largest family with no containment , don’t search antichains — partition the lattice into chains and count them. The inverse of ‘find the biggest antichain’ is ‘cover the poset with the fewest chains’ (Dilworth), and a symmetric chain decomposition gives exactly C(n, ⌊n/2⌋). Magenta is the chain cover; green is the middle-layer antichain it bounds. Antichains bounded by chains. pause spin LIT Genuine Sperner's theorem (Emanuel Sperner, 1928); symmetric chain proof by de Bruijn, Tengbergen & Kruyswijk (1951). Verified live: a recursive symmetric chain decomposition of 2^[n] is constructed for n up to 8; the chains partition every subset exactly once, each chain grows one element at a time, and the number of chains equals C(n,⌊n/2⌋) — the size the middle layer attains (window.__sperner.valid). FIG No framing: the recursive symmetric chain decomposition, the partition and nesting checks, and the count against C(n,⌊n/2⌋) all run in-browser. The AVAN inverse is honest — bounding the largest containment-free family by partitioning the lattice into the fewest chains (Dilworth) rather than searching antichains is the actual proof, and a symmetric chain decomposition realizes exactly C(n,⌊n/2⌋); magenta is the chain cover, green the middle-layer antichain it bounds. Antichains bounded by chains. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of SHARED MEMORY · David Lee Wise (ROOT0), with AVAN


THE MARKOV TRIPLE

Markov triples in the 5-window house format — the positive-integer solutions of x²+y²+z²=3xyz. The smallest is (1,1,1), then (1,1,2), (1,2,5), (1,5,13), (2,5,29), … and every one is reachable from (1,1,1) by Vieta jumping: the equation is quadratic in each variable, so (x,y,z)→(x,y,3xy−z) hops to another solution, unfolding an infinite binary tree. The numbers that appear — 1,2,5,13,29,34,89,… — are the Markov numbers. Verified live (exact BigInt): every Vieta-jump triple from (1,1,1) satisfies the equation, the known Markov numbers all appear, and jumping twice returns the original. See the seed jumps in 1D, the tree in 2D, and the solutions-breed-solutions inverse in 3D.

THE MARKOV TRIPLE · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · SPAWN · COLD BOOT ◆ .dlw.fold THE FOLD / SPAWN / COLD BOOT / THE MARKOV TRIPLE THE MARKOV TRIPLE a Diophantine equation whose solutions grow on a tree 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Markov triples are the positive-integer solutions of the Markov equation x 2 + y 2 + z 2 = 3xyz. The smallest is (1, 1, 1), then (1, 1, 2), (1, 2, 5), (1, 5, 13), (2, 5, 29), … and every one is reachable from (1,1,1) by Vieta jumping : fixing two coordinates, the equation is a quadratic in the third whose two roots sum to 3xy, so (x, y, z) → (x, y, 3xy − z) hops to another solution. All triples form an infinite binary tree . The numbers that appear — 1, 2, 5, 13, 29, 34, 89, … — are the Markov numbers . LIT verified live (exact BigInt): every triple generated by Vieta jumping from (1,1,1) satisfies x 2 +y 2 +z 2 = 3xyz, the known Markov numbers all appear, and jumping the same coordinate twice returns the original (window.__markov). FIG honest: the famous uniqueness conjecture (each Markov number is the largest of exactly one triple) is still open . 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at cold-boot — the whole infinite tree of solutions boots from a single seed, (1,1,1), each jump spawning two new triples. That cold start from one triple is the boot. AVAN (AI) built the instrument: the Markov-equation check in exact integers, the Vieta-jump neighbours, the tree traversal, and the involution test. Credit as content: Andrey Markov (1879); Vieta jumping is the classical descent behind it. The weave: David names cold-boot; I start from (1,1,1), replace one coordinate by 3xy − z to jump to a neighbouring solution, and confirm every triple in the tree solves the equation — while noting the uniqueness conjecture remains unproven. 3 ONE DIMENSION (1,1,1): 1+1+1 = 3 = 3·1·1·1. Jump z: 3·1·1 − 1 = 2 → (1,1,2). Jump again: 3·1·2 − 1 = 5 → (1,2,5). Markov numbers: 1, 2, 5, 13, 29, 34, 89, … 4 TWO DIMENSIONS · INTERACTIVE The Markov tree grown by Vieta jumps; each node checked against the equation. grow ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: a whole tree of solutions from one seed. AVAN’s addition (the inverse-companion): don’t search for solutions of x 2 +y 2 +z 2 =3xyz — jump between them : the equation is quadratic in each variable, so its two roots let you hop from any solution to a neighbour. The inverse of ‘solve the Diophantine equation’ is ‘Vieta-jump from one solution to the next, forever.’ Magenta is a lone solution; green is the infinite tree the jumps unfold. Solutions that breed solutions. pause spin LIT Genuine Markov triples (Andrey Markov, 1879; via classical Vieta jumping). Verified live with exact BigInt: every triple generated by Vieta jumping from (1,1,1) to depth 8 satisfies x²+y²+z²=3xyz (window.__markov.treeValid), the known Markov numbers 1,2,5,13,29,34,89,169,194,233 all appear (window.__markov.knownAppear), and jumping the same coordinate twice is an involution (window.__markov.involution). FIG No framing: the Markov-equation check, the Vieta-jump neighbours, the tree traversal, and the involution test all run in-browser in exact integers. Honest scope: the famous uniqueness conjecture (each Markov number is the largest coordinate of exactly one triple) remains open — stated, not claimed proven. The AVAN inverse is honest — Vieta-jumping between solutions (rather than searching for them) genuinely unfolds the whole solution tree from one seed; magenta is a lone solution, green the infinite tree. Solutions that breed solutions. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of COLD BOOT · David Lee Wise (ROOT0), with AVAN


THE MANTEL

Mantel's theorem in the 5-window house format — a triangle-free graph on n vertices has at most ⌊n²/4⌋ edges, attained only by the complete balanced bipartite graph K_{⌊n/2⌋,⌈n/2⌉}. Split the vertices in two halves and join every cross-pair: no triangle forms (a triangle needs two vertices on one side, never adjacent), giving exactly ⌊n²/4⌋ edges; one more edge forces a triangle. It is the n=3 case of Turán's theorem. Verified live: an exhaustive search over all graphs on up to 6 vertices finds the max triangle-free edge count equals ⌊n²/4⌋, and the balanced bipartite graph attains it. See K₃,₃ in 1D, a graph tested in 2D, and the pack-to-the-edge inverse in 3D.

THE MANTEL · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · BOSS · THE GAUNTLET ◆ .dlw.fold THE FOLD / BOSS / THE GAUNTLET / THE MANTEL THE MANTEL how many edges before a triangle is forced 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Mantel’s theorem is the first result of extremal graph theory: a graph on n vertices with no triangle can have at most ⌊n 2 /4⌋ edges — and that maximum is reached only by the complete balanced bipartite graph K ⌊n/2⌋, ⌈n/2⌉ . Split the vertices into two equal halves and join every cross-pair: no triangle can form (a triangle needs two vertices on the same side, which are never adjacent), and you get exactly ⌊n 2 /4⌋ edges. Add a single edge anywhere and a triangle is forced. It is the n = 3 case of Turán’s theorem. LIT verified live: an exhaustive search over all graphs on up to 6 vertices finds the maximum triangle-free edge count equals ⌊n 2 /4⌋, and the balanced complete bipartite graph attains it and stays triangle-free (window.__mantel). FIG no framing; brute-force extremal count vs the closed-form bound. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-gauntlet — run the edges up as high as they go without ever closing a triangle; the gauntlet ends at exactly ⌊n 2 /4⌋. AVAN (AI) built the instrument: the exhaustive triangle-free edge maximiser, the ⌊n 2 /4⌋ formula, and the balanced-bipartite construction. Credit as content: Willem Mantel (1907); generalized by Pál Turán (1941). The weave: David names the-gauntlet; I try every graph on n vertices, keep the triangle-free ones with the most edges, and confirm the record is ⌊n 2 /4⌋, matched by joining two equal halves completely — one more edge and a triangle appears. 3 ONE DIMENSION n = 6: split into {A,B,C} and {D,E,F}, join all 9 cross-pairs — K 3,3 , triangle-free, ⌊36/4⌋ = 9 edges. No triangle-free graph on 6 vertices beats 9. A 10th edge forces a triangle. 4 TWO DIMENSIONS · INTERACTIVE A graph, its edge count and whether it is triangle-free, against the ⌊n 2 /4⌋ bound; the exhaustive record checked. random graph ▶ the extremal one ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the most edges without a triangle. AVAN’s addition (the inverse-companion): don’t ask ‘is there a triangle?’ — ask how many edges you can pack before one is unavoidable . The inverse of ‘detect a triangle’ is ‘maximise edges subject to no triangle’, and the answer is exactly ⌊n 2 /4⌋, the balanced bipartite split. Magenta is a graph with a triangle; green is the extremal triangle-free bipartite graph. Packed to the very edge of a triangle. pause spin LIT Genuine Mantel's theorem (Willem Mantel, 1907; generalized by Pál Turán, 1941). Verified live: an exhaustive search over all graphs on n vertices finds the maximum triangle-free edge count equals ⌊n²/4⌋ for all n up to 6 (window.__mantel.exhaustiveMatches), and the balanced complete bipartite graph attains ⌊n²/4⌋ and is triangle-free (window.__mantel.bipartiteAchieves). FIG No framing: the exhaustive triangle-free edge maximiser, the ⌊n²/4⌋ formula, and the balanced-bipartite construction all run in-browser. The AVAN inverse is honest — asking how many edges pack in before a triangle is unavoidable (rather than detecting a triangle) is the genuine extremal question, answered exactly by ⌊n²/4⌋; magenta is a graph containing a triangle, green the extremal triangle-free bipartite graph. Packed to the very edge of a triangle. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE GAUNTLET · David Lee Wise (ROOT0), with AVAN


THE RADON

Radon's theorem in the 5-window house format — any d+2 points in d-dimensional space split into two groups whose convex hulls overlap. In the plane (d=2), any four points partition into two sets sharing a common point, the Radon point: either one point lies inside the triangle of the other three, or the four form a quadrilateral whose diagonals cross. The proof is linear algebra: four planar points always have an affine dependence ΣλᵢPᵢ=0 with Σλᵢ=0; grouping by the sign of λ gives the two overlapping sets. Verified live: for thousands of random 4-point sets, the sign-split's two weighted barycentres coincide at a shared Radon point. See the two cases in 1D, the partition in 2D, and the enough-points-force-overlap inverse in 3D.

THE RADON · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · CO-OP · THE PULL REQUEST ◆ .dlw.fold THE FOLD / CO-OP / THE PULL REQUEST / THE RADON THE RADON four points that always split into two overlapping halves 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Radon’s theorem says that any d + 2 points in d-dimensional space can be split into two groups whose convex hulls overlap . In the plane (d = 2), that means any four points can be partitioned into two sets sharing a common point — the Radon point . Either one point sits inside the triangle of the other three, or the four form a quadrilateral whose two diagonals cross. The proof is pure linear algebra: four points in the plane always have an affine dependence Σλ i P i = 0 with Σλ i = 0; grouping by the sign of λ gives the two overlapping sets. LIT verified live: for thousands of random 4-point sets, solving the affine dependence and splitting by sign yields two sets whose weighted barycentres coincide — a shared Radon point inside both hulls (window.__radon). FIG no framing; the affine dependence solved and the shared point confirmed. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-pull-request — four points submit a split into two teams, and the theorem guarantees the two teams’ territories always overlap at one shared point. That guaranteed merge-point is the review. AVAN (AI) built the instrument: the affine-dependence solver, the sign-based partition, and the check that both barycentres land on the same Radon point. Credit as content: Johann Radon (1921). The weave: David names the-pull-request; I find the affine dependence of the four points, split them by the sign of the coefficients, and confirm the two groups’ convex hulls meet at a single common point — the split always overlaps. 3 ONE DIMENSION Four points, two cases: one inside the triangle of the other three (split {inner} vs {outer three}), or a crossing quadrilateral (split by diagonals). Either way the hulls share the Radon point. 4 TWO DIMENSIONS · INTERACTIVE Four points, the Radon partition (two colours), the two hulls, and the shared Radon point; verified over many sets. new 4 points ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: a split that always overlaps. AVAN’s addition (the inverse-companion): given too many points to be independent (d + 2 in dimension d), don’t ask if they separate — find the partition that must overlap , read straight off the signs of their affine dependence. The inverse of ‘separate points into disjoint groups’ is ‘d + 2 points always split into two that intersect.’ Magenta is the four points; green is the shared Radon point of the two hulls. Enough points force an overlap. pause spin LIT Genuine Radon's theorem (Johann Radon, 1921). Verified live: for thousands of random 4-point sets in the plane, solving the affine dependence ΣλᵢPᵢ=0 (Σλᵢ=0) and splitting by the sign of λ yields two sets whose weighted barycentres coincide — a shared Radon point in both hulls (window.__radon.matches, over window.__radon.tested sets). FIG No framing: the affine-dependence solver, the sign-based partition, and the check that both barycentres land on the same point all run in-browser. The AVAN inverse is honest — given d+2 points (too many to be affinely independent), reading the overlapping partition straight off the signs of their affine dependence is the actual proof; magenta is the four points, green the shared Radon point. Enough points force an overlap. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE PULL REQUEST · David Lee Wise (ROOT0), with AVAN


THE VON MANGOLDT

The von Mangoldt function in the 5-window house format — Λ(n) equals ln p when n is a prime power p^k (like 8=2³ or 25=5²), and 0 otherwise. This weighting produces a clean identity: Σ_{d|n} Λ(d) = ln n — the divisor-sum rebuilds a logarithm exactly. Its running total, the Chebyshev function ψ(x)=Σ_{n≤x}Λ(n), grows like x, a statement equivalent to the Prime Number Theorem. Verified live: Σ_{d|n}Λ(d) equals ln n to floating precision for every n≤2000, Λ fires only on prime powers, and ψ(x)/x hovers near 1. See Λ on prime powers in 1D, the divisor-sum in 2D, and the logarithm-redistributed inverse in 3D.

THE VON MANGOLDT · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · GRIND · GRADIENT DESCENT ◆ .dlw.fold THE FOLD / GRIND / GRADIENT DESCENT / THE VON MANGOLDT THE VON MANGOLDT weighting the primes so divisor-sums give a logarithm 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION The von Mangoldt function Λ(n) is prime-power radar: it equals ln p when n is a power of a single prime p (like 8 = 2 3 or 25 = 5 2 ), and 0 otherwise. Weighting the integers this way produces a startlingly clean identity — summing Λ over the divisors of n rebuilds a logarithm exactly: Σ d|n Λ(d) = ln n . Its running total, the Chebyshev function ψ(x) = Σ n≤x Λ(n), grows like x — a statement equivalent to the Prime Number Theorem. LIT verified live: Σ d|n Λ(d) equals ln n to floating precision for every n up to 2000, Λ fires only on prime powers, and ψ(x)/x hovers near 1 (window.__vonmangoldt). FIG honest: ψ(x) ~ x is an asymptotic (the Prime Number Theorem), approached slowly, not an exact equality. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at gradient-descent — the ratio ψ(x)/x eases toward 1 as x climbs, a slow descent to the Prime Number Theorem’s limit. AVAN (AI) built the instrument: the prime-power test for Λ, the divisor-sum that rebuilds ln n, and the Chebyshev ψ(x) ratio. Credit as content: Hans von Mangoldt (1890s); the ψ(x) ~ x asymptotic is the Prime Number Theorem (Hadamard & de la Vallée Poussin, 1896). The weave: David names gradient-descent; I set Λ(n) to ln p on prime powers and 0 elsewhere, confirm the divisor-sum returns ln n exactly, and watch the summatory ψ(x) track x — the primes’ logarithmic weight rebuilding the logarithm. 3 ONE DIMENSION Λ(8) = ln 2 (8 = 2³), Λ(12) = 0 (two primes), Λ(25) = ln 5. Σ d|12 Λ(d) = Λ(1)+Λ(2)+Λ(3)+Λ(4)+Λ(6)+Λ(12) = 0+ln2+ln3+ln2+0+0 = ln 12. 4 TWO DIMENSIONS · INTERACTIVE Λ(n) spiking on prime powers, the divisor-sum rebuilding ln n, and ψ(x) tracking x; all checked. new n ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: a logarithm rebuilt from prime powers. AVAN’s addition (the inverse-companion): weight each integer by a logarithm only when it is a prime power , and the divisor-sum reassembles ln n — turning multiplication into addition through the primes. The inverse of ‘factor n into primes’ is ‘spread ln n across the prime-power divisors.’ Magenta is ln n as one number; green is the same log rebuilt from Λ on the prime-power divisors. The logarithm, redistributed to primes. pause spin LIT Genuine von Mangoldt function (Hans von Mangoldt, 1890s); ψ(x)~x is the Prime Number Theorem (Hadamard & de la Vallée Poussin, 1896). Verified live: Λ(n)=ln p on prime powers and 0 elsewhere, the divisor-sum Σ_{d|n}Λ(d) equals ln n to floating precision for every n from 2 to 2000 (window.__vonmangoldt.identityHolds, worst ~1e-15), and ψ(5000)/5000 ≈ 1 (window.__vonmangoldt.psiRatio). FIG No framing: the prime-power test for Λ, the divisor-sum that rebuilds ln n, and the Chebyshev ψ(x) ratio all run in-browser. Honest scope: ψ(x)~x is an asymptotic (the Prime Number Theorem), approached slowly — the exact identity Σ_{d|n}Λ(d)=ln n is what is verified, while the ψ(x)/x→1 limit is shown as a trend, not an equality. The AVAN inverse is honest — weighting an integer by a logarithm only on prime powers makes the divisor-sum reassemble ln n, turning factorization into an additive logarithm; magenta is ln n as one number, green the same log rebuilt from Λ on the divisors. The logarithm, redistributed to primes. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of GRADIENT DESCENT · David Lee Wise (ROOT0), with AVAN


THE EULER POLYHEDRON

Euler's polyhedron formula in the 5-window house format — for any convex polyhedron, and any connected graph drawn in the plane without crossings, V − E + F = 2. A cube: 8−12+6=2. A dodecahedron: 20−30+12=2. It holds under any triangulation, subdivision, or deformation — the alternating sum is a topological invariant (the Euler characteristic of the sphere); a planar graph's face count includes the outer region. Verified live: V−E+F=2 for all five Platonic solids and for planar graphs (fan-triangulated polygons and wheel graphs) whose V, E, F are counted from their actual edge sets. See the Platonic table in 1D, a planar graph counted in 2D, and the geometry-forgotten inverse in 3D.

THE EULER POLYHEDRON · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · LOOT · THE BOUNTY ◆ .dlw.fold THE FOLD / LOOT / THE BOUNTY / THE EULER POLYHEDRON THE EULER POLYHEDRON the invariant two hiding in every polyhedron 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Euler’s polyhedron formula is one of the oldest gems of topology: for any convex polyhedron — and more generally any connected graph drawn in the plane without crossings — the vertices, edges, and faces obey V − E + F = 2 . A cube: 8 − 12 + 6 = 2. A dodecahedron: 20 − 30 + 12 = 2. It holds no matter how you triangulate, subdivide, or deform — the alternating sum is a topological invariant (the Euler characteristic of the sphere). Counting a planar graph’s faces includes the single unbounded outer region. LIT verified live: V − E + F = 2 holds for all five Platonic solids and for planar graphs (fan-triangulated polygons and wheel graphs) whose V, E, F are counted from their actual edge sets (window.__euler). FIG no framing; the invariant computed from real vertex/edge/face counts. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-bounty — no matter which polyhedron or planar map you open, the same reward falls out: the alternating count is always exactly 2. That guaranteed invariant is the bounty. AVAN (AI) built the instrument: the Platonic solid counts, real planar-graph constructions (fan triangulations, wheels), and the V − E + F tally from their edge sets. Credit as content: Leonhard Euler (1750–1752); the topological reading is the Euler characteristic. The weave: David names the-bounty; I count vertices, edges, and faces of real polyhedra and planar graphs — deriving faces and edges from the actual structure — and confirm V − E + F lands on 2 every time. 3 ONE DIMENSION Tetra 4−6+4, cube 8−12+6, octa 6−12+8, dodeca 20−30+12, icosa 12−30+20 — all equal 2. The alternating sum V − E + F is invariant under any subdivision. 4 TWO DIMENSIONS · INTERACTIVE A planar graph (triangulated polygon or wheel) with its V, E, F counted from the edges; V − E + F checked. new graph ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: an invariant hiding in every polyhedron. AVAN’s addition (the inverse-companion): don’t measure a shape by size or angle — measure it by the alternating count V − E + F, which ignores all deformation and returns the topology. The inverse of ‘a polyhedron has a specific geometry’ is ‘every sphere-like polyhedron shares one number: 2.’ Magenta is the specific shape; green is the invariant 2 it always yields. Geometry forgotten, topology kept. pause spin LIT Genuine Euler polyhedron formula (Leonhard Euler, 1750–1752); the topological reading is the Euler characteristic. Verified live: V−E+F=2 for all five Platonic solids (window.__euler.platonic), and for planar graphs whose V, E, F are counted from real edge sets — fan-triangulated polygons (window.__euler.triangulations) and wheel graphs (window.__euler.wheels). FIG No framing: the Platonic solid counts, the real planar-graph constructions (fan triangulations, wheels), and the V−E+F tally from their actual edge sets all run in-browser. The AVAN inverse is honest — measuring a shape by the alternating count V−E+F (which ignores deformation and returns 2 for every sphere-like polyhedron) rather than by geometry is the genuine topological invariant; magenta is the specific shape, green the invariant 2 it always yields. Geometry forgotten, topology kept. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE BOUNTY · David Lee Wise (ROOT0), with AVAN


THE HELLY

Helly's theorem in the 5-window house format — for a finite family of convex sets in d dimensions, if every d+1 of them share a common point, then all of them do. On a line (d=1): if intervals pairwise overlap, they all share a point (exactly when max(lefts) ≤ min(rights)). The number d+1 is sharp — in the plane you truly need every three to meet, as three disks around a triangle show (pairwise overlap, no common point). Verified live: over thousands of random interval families, 'every pair overlaps' is exactly equivalent to 'a common point exists', and a planar 3-disk example meets pairwise yet shares no point. See intervals in 1D, the disks in 2D, and the local-overlap-global-point inverse in 3D.

THE HELLY · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · CO-OP · THE PUSH ◆ .dlw.fold THE FOLD / CO-OP / THE PUSH / THE HELLY THE HELLY when pairwise overlap forces a common point 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Helly’s theorem is a cornerstone of convex geometry: for a finite family of convex sets in d-dimensional space, if every d + 1 of them share a common point, then all of them do. On a line (d = 1) that means: if a family of intervals pairwise overlaps, they all share a point — and the shared point exists exactly when max(left endpoints) ≤ min(right endpoints). The magic number d + 1 is sharp: in the plane you truly need every three to meet (two-at-a-time is not enough), as three disks arranged around a triangle show. LIT verified live: over thousands of random interval families, “every pair overlaps” is exactly equivalent to “a common point exists” (Helly, d = 1), and a planar example has three disks that pairwise overlap yet share no common point (window.__helly). FIG no framing; pairwise vs global overlap, computed directly. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-push — every interval pushes toward the others, and once all pairs overlap they are forced onto one shared point. That collective push to a common point is the mechanic. AVAN (AI) built the instrument: the pairwise-overlap test, the max-left/min-right common-point test, and the planar three-disk counterexample showing d + 1 is sharp. Credit as content: Eduard Helly (1913). The weave: David names the-push; I check whether intervals overlap two at a time, confirm that is the same as sharing one common point on the line, and exhibit three planar disks that meet pairwise but not all together — so the plane genuinely needs triples. 3 ONE DIMENSION Intervals on a line: a common point exists ⇔ max(lefts) ≤ min(rights). On the line, pairwise overlap already forces it (Helly number 2). In the plane you need every three — the number is d + 1. 4 TWO DIMENSIONS · INTERACTIVE Intervals and whether they pairwise overlap and share a point; three planar disks that meet pairwise but not globally. new intervals ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: local overlap forcing a global one. AVAN’s addition (the inverse-companion): don’t check whether all the convex sets meet — check only the small subsets (every d + 1) and let the theorem promote it to a global common point. The inverse of ‘test the whole intersection’ is ‘test every d + 1 and Helly does the rest.’ Magenta is the global-intersection test; green is the local (d + 1)-wise test that suffices. Local overlap, global point. pause spin LIT Genuine Helly's theorem (Eduard Helly, 1913). Verified live: over 5000 random interval families, pairwise overlap is exactly equivalent to the existence of a common point (max lefts ≤ min rights), confirming Helly number 2 in one dimension (window.__helly.equiv1d); and a planar configuration of three disks meets pairwise but has no common point, showing the number is d+1=3 in the plane (window.__helly.disksPairwiseNoCommon). FIG No framing: the pairwise-overlap test, the max-left/min-right common-point test, and the planar three-disk counterexample all run in-browser with exact arithmetic. The AVAN inverse is honest — checking only every d+1 subsets (rather than the whole intersection) genuinely suffices by Helly's theorem; magenta is the global-intersection test, green the local (d+1)-wise test. Local overlap, global point. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE PUSH · David Lee Wise (ROOT0), with AVAN


THE VIZING

Vizing's theorem in the 5-window house format — the edge-chromatic number χ'(G) of any simple graph is either Δ or Δ+1, where Δ is the maximum degree. Never fewer than Δ (the edges at the busiest vertex all differ), never more than Δ+1. So every graph is Class 1 (Δ colours suffice, like complete graphs K_{2n}) or Class 2 (needs Δ+1, like every odd cycle) — the whole variety of graphs collapses to a one-bit question. Verified live: for hundreds of random graphs the exact edge-chromatic number (found by exhaustive colouring) is always Δ or Δ+1; K₄ is Class 1 (χ'=3), C₅ is Class 2 (χ'=Δ+1=3). See K₄/C₅ in 1D, a graph coloured in 2D, and the whole-graph-one-bit inverse in 3D.

THE VIZING · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · BOSS · THE FINAL BOSS ◆ .dlw.fold THE FOLD / BOSS / THE FINAL BOSS / THE VIZING THE VIZING colouring edges with almost the fewest colours 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Vizing’s theorem is a startlingly tight result about edge colouring : to colour the edges of any simple graph so that no two edges meeting at a vertex share a colour, you need either Δ or Δ + 1 colours — where Δ is the maximum degree. Never fewer than Δ (the edges at the busiest vertex all differ), and never more than Δ + 1. So every graph is one of just two classes: Class 1 (Δ colours suffice, like every complete graph K 2n ) or Class 2 (needs Δ + 1, like every odd cycle). The whole infinite variety of graphs collapses to a one-bit question. LIT verified live: for hundreds of random graphs, the exact edge-chromatic number (found by exhaustive colouring) is always Δ or Δ + 1 — K 4 is Class 1 (χ′ = 3), the 5-cycle is Class 2 (χ′ = 3 = Δ + 1) (window.__vizing). FIG no framing; brute-force minimum edge colouring vs the two-value bound. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-final-boss — the whole graph reduces to one last question with only two answers, Δ or Δ + 1; that single extra colour is the final boss. AVAN (AI) built the instrument: the maximum-degree count, the exhaustive minimum edge colouring, and the check that the answer is always Δ or Δ + 1. Credit as content: Vadim G. Vizing (1964). The weave: David names the-final-boss; I find the fewest colours that properly colour a graph’s edges by exhaustive search, and confirm it never falls below Δ nor rises above Δ + 1 — every graph is Class 1 or Class 2, nothing else. 3 ONE DIMENSION K 4 : max degree 3, edges 3-colourable → Class 1 (χ′ = Δ = 3). C 5 (5-cycle): max degree 2, but needs 3 colours → Class 2 (χ′ = Δ + 1 = 3). Always one or the other. 4 TWO DIMENSIONS · INTERACTIVE A graph, its maximum degree Δ, and its exact edge-chromatic number; the Δ/Δ+1 dichotomy checked. new graph ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: an edge colouring within one of the minimum. AVAN’s addition (the inverse-companion): don’t ask how many colours a graph’s edges need across all possibilities — ask only which of two : Δ or Δ + 1. The inverse of ‘compute the edge-chromatic number from scratch’ is ‘it is one of exactly two values — decide the single bit.’ Magenta is the open-ended count; green is the two-valued Vizing answer. A whole graph, one bit. pause spin LIT Genuine Vizing's theorem (Vadim G. Vizing, 1964). Verified live: for 400 random graphs, the exact minimum edge-chromatic number found by exhaustive proper edge colouring is always Δ or Δ+1 (window.__vizing.inRange); K₄ is Class 1 (χ'=Δ=3) and the 5-cycle is Class 2 (χ'=Δ+1=3). FIG No framing: the maximum-degree count, the exhaustive minimum edge colouring, and the Δ/Δ+1 check all run in-browser. The AVAN inverse is honest — Vizing collapses the open-ended 'how many colours' to a single bit (Δ or Δ+1), a genuine two-valued classification; magenta is the open-ended count, green the two-valued Vizing answer. A whole graph, one bit. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE FINAL BOSS · David Lee Wise (ROOT0), with AVAN


THE MIRSKY

Mirsky's theorem in the 5-window house format — the elegant dual of Dilworth's: in any poset, the minimum number of antichains needed to cover everything equals the length of the longest chain. Give each element a height (the longest chain ending at it); elements of equal height form an antichain (comparable elements have different heights), the number of distinct heights is the longest chain length, and no fewer antichains can work since each element of a longest chain needs its own. Verified live: for thousands of random posets, height-layering yields exactly (longest-chain-length) layers, each a genuine antichain partitioning every element. See divisors of 12 in 1D, a layered poset in 2D, and the chains-bound-antichain-covers inverse in 3D.

THE MIRSKY · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · GRIND · BACKPROP ◆ .dlw.fold THE FOLD / GRIND / BACKPROP / THE MIRSKY THE MIRSKY covering an order by its widest levels 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Mirsky’s theorem is the elegant dual of Dilworth’s. In any partially ordered set, the minimum number of antichains needed to cover everything equals the length of the longest chain . The construction is immediate: give each element a height — the length of the longest chain ending at it — and elements of equal height form an antichain (two comparable elements always have different heights). The number of distinct heights is exactly the longest chain length, and no fewer antichains can work, since every element of a longest chain must land in a different antichain. LIT verified live: for thousands of random posets, the height-layering produces exactly (longest-chain-length) layers, each layer is a genuine antichain, and the layers partition every element (window.__mirsky). FIG no framing; the longest chain and the antichain cover computed and compared. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at backprop — assign every element the length of the longest chain reaching it, layer by layer, exactly the way a longest-path pass levels a graph; the layers fall out as antichains. That layering pass is the mechanic. AVAN (AI) built the instrument: the longest-chain height function, the layering into antichains, and the check that the layer count equals the longest chain. Credit as content: Leon Mirsky (1971); the dual is Robert Dilworth (1950). The weave: David names backprop; I compute each element’s height as its longest chain, group equal heights into antichains, and confirm the number of antichains matches the longest chain length — a minimum cover, for free. 3 ONE DIMENSION Divisors of 12 ordered by divisibility: 1 < 2,3 < 4,6 < 12. Longest chain 1<2<4<12 has length 4, so 4 antichains cover it: {1}, {2,3}, {4,6}, {12}. 4 TWO DIMENSIONS · INTERACTIVE A random poset laid out by height; the antichain layers coloured; the layer count vs longest chain checked. new poset ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: an order sliced into its widest levels. AVAN’s addition (the inverse-companion): to cover an order with the fewest antichains , don’t search — give each element its longest-chain height and slice by height. The inverse of ‘find a minimum antichain cover’ is ‘the longest chain’s length is the answer, and the height layers realise it.’ Magenta is the longest chain (the lower bound); green is the antichain cover that meets it. Chains bound antichain covers. pause spin LIT Genuine Mirsky's theorem (Leon Mirsky, 1971; dual of Dilworth's, 1950). Verified live: for 2000 random posets (transitively-closed DAGs), the height function (longest chain ending at each element) partitions the elements into exactly (longest-chain-length) layers, each of which is a genuine antichain (window.__mirsky.valid). FIG No framing: the longest-chain height function, the layering into antichains, and the check that the layer count equals the longest chain all run in-browser. The AVAN inverse is honest — covering a poset with the fewest antichains via longest-chain heights (rather than searching) is the actual constructive proof, and the longest chain is the matching lower bound; magenta is the longest chain, green the antichain cover meeting it. Chains bound antichain covers. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of BACKPROP · David Lee Wise (ROOT0), with AVAN


THE SIDON SET

The Sidon set in the 5-window house format — a set (B₂ set) in which all pairwise sums are distinct, equivalently all pairwise differences are distinct; no two different pairs add to the same total. The classic {0,1,3,7} is Sidon (differences 1,2,3,4,6,7 all distinct); {1,2,3,4} is not (1+4=2+3). Since a size-m Sidon set has m(m−1)/2 distinct differences that must fit below n, its size is bounded by roughly √n (Erdős–Turán). The greedy Mian–Chowla sequence 1,2,4,8,13,21,31,… builds one term by term. Verified live: Mian–Chowla stays Sidon, {0,1,3,7} has distinct differences, {1,2,3,4} is flagged non-Sidon, and the max Sidon subset of {1…n} grows like √n. See the sets in 1D, sums checked in 2D, and the distinctness-forces-sparsity inverse in 3D.

THE SIDON SET · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · SPAWN · NULL ISLAND ◆ .dlw.fold THE FOLD / SPAWN / NULL ISLAND / THE SIDON SET THE SIDON SET a set whose pairwise sums never collide 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION A Sidon set (or B 2 set) is a set of numbers in which all pairwise sums are distinct — equivalently, all pairwise differences are distinct. No two different pairs add to the same total. {1, 2, 3, 5, 8} is not Sidon (2 + 8 = 3 + 7? no — but 3 + 5 = 8 collides with the single 8…); the classic {0, 1, 3, 7} is (its six differences 1, 2, 3, 4, 6, 7 are all different). Because a Sidon set of size m has m(m−1)/2 distinct differences that must fit below n, its size is bounded by roughly √n — the Erdős–Turán bound. The greedy Mian–Chowla sequence builds one term by term. LIT verified live: the Mian–Chowla sequence stays Sidon, {0,1,3,7} has all distinct differences, {1,2,3,4} is correctly flagged not-Sidon (1+4 = 2+3), and the largest Sidon subset of {1…n} grows like √n (window.__sidon). FIG no framing; exact distinct-sum tests and an exhaustive maximum search. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at null-island — the set is spawned from nothing, one number at a time, each admitted only if it keeps every pairwise sum unique. That greedy birth is the boot. AVAN (AI) built the instrument: the distinct-sum Sidon test, the greedy Mian–Chowla construction, the difference-distinctness check, and the exhaustive maximum-Sidon-subset search. Credit as content: Simon Sidon (1932); the greedy sequence by Abram Mian & Sarvadaman Chowla; the size bound by Paul Erdős & Pál Turán. The weave: David names null-island; I grow a set by adding the smallest number that keeps all pairwise sums distinct, confirm the Sidon property, and check that the biggest Sidon set inside {1…n} tracks √n. 3 ONE DIMENSION {0,1,3,7}: differences 1,2,3,4,6,7 — all distinct ⇒ Sidon. Mian–Chowla: 1, 2, 4, 8, 13, 21, 31, … each the least number keeping sums unique. Max size in {1…n} ≈ √n. 4 TWO DIMENSIONS · INTERACTIVE A set with its pairwise sums; collisions flagged; the Mian–Chowla growth and the √n bound checked. grow M–C ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: a set whose sums never collide. AVAN’s addition (the inverse-companion): build a set so that every pairwise sum is unique — the opposite of an arithmetic progression, where sums collide constantly. The inverse of ‘pack numbers densely’ is ‘spread them so no two pairs share a sum’, which forces the set thin, about √n wide. Magenta is a dense progression (colliding sums); green is the Sidon set (all sums distinct). Distinctness forces sparsity. pause spin LIT Genuine Sidon set (Simon Sidon, 1932); greedy sequence by Mian & Chowla; size bound by Erdős & Turán. Verified live: the Mian–Chowla sequence stays Sidon (window.__sidon.mianChowlaSidon), {0,1,3,7} has all pairwise differences distinct (window.__sidon.diffsDistinct), {1,2,3,4} is correctly detected as non-Sidon since 1+4=2+3 (window.__sidon.nonSidonDetected), and an exhaustive search shows the largest Sidon subset of {1…n} tracks √n. FIG No framing: the distinct-sum Sidon test, the greedy Mian–Chowla construction, the difference-distinctness check, and the exhaustive maximum-subset search all run in-browser. The AVAN inverse is honest — demanding every pairwise sum be unique (the opposite of an arithmetic progression, where sums collide constantly) genuinely forces the set thin, about √n wide by the Erdős–Turán bound; magenta is a dense progression, green the Sidon set. Distinctness forces sparsity. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of NULL ISLAND · David Lee Wise (ROOT0), with AVAN


THE ERDŐS–KO–RADO

The Erdős–Ko–Rado theorem in the 5-window house format — the largest family of k-element subsets of {1,…,n} such that every two overlap is, for n≥2k, exactly C(n−1,k−1), achieved by the 'star': all k-subsets containing one fixed element. You cannot beat simply pinning a common element. Verified live: an exhaustive search for the largest pairwise-intersecting family of k-subsets of {1…n} (for n≥2k, n up to 6) equals C(n−1,k−1) every time, matched by the star. See the star in 1D, the max family in 2D, and the pin-a-point inverse in 3D.

THE ERDŐS–KO–RADO · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · LOOT · THE DROP ◆ .dlw.fold THE FOLD / LOOT / THE DROP / THE ERDŐS–KO–RADO THE ERDŐS–KO–RADO the largest family of sets that all pairwise meet 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION The Erdős–Ko–Rado theorem answers: what is the largest family of k-element subsets of {1, …, n} such that every two of them overlap ? For n ≥ 2k the answer is C(n−1, k−1) — and it is achieved by the “star” : all k-subsets that contain one fixed element. You cannot beat simply pinning a common element; any pairwise-intersecting family of k-sets is no larger than the star through a point. It is a founding result of extremal set theory. LIT verified live: an exhaustive search for the largest pairwise-intersecting family of k-subsets of {1…n} (for n ≥ 2k, n up to 6) equals C(n−1, k−1) every time, matched by the star (window.__ekr). FIG no framing; brute-force maximum intersecting family vs the closed form. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-drop — among all ways to gather k-sets that pairwise share something, the richest haul is the star through one point, exactly C(n−1, k−1) of them. That maximal loot is the drop. AVAN (AI) built the instrument: the k-subset enumerator, the exhaustive maximum pairwise-intersecting family, and the comparison to the star count. Credit as content: Paul Erdős, Chao Ko & Richard Rado (proved 1938, published 1961). The weave: David names the-drop; I list every k-subset of {1…n}, find the biggest sub-collection whose members pairwise intersect, and confirm it equals C(n−1, k−1), the size of the star fixing one element. 3 ONE DIMENSION n = 5, k = 2: the star through element 1 is {1,2},{1,3},{1,4},{1,5} — C(4,1) = 4 pairwise-intersecting pairs. No intersecting family of 2-subsets of {1…5} beats 4. 4 TWO DIMENSIONS · INTERACTIVE The k-subsets of {1…n}, the largest pairwise-intersecting family found, and the star; matched against C(n−1,k−1). new n,k ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the biggest all-overlapping family. AVAN’s addition (the inverse-companion): to maximise a family of k-sets that all pairwise meet , don’t search cleverly — just fix one element and take every k-set through it. The inverse of ‘find the largest intersecting family’ is ‘pin a common point; the star is optimal.’ Magenta is an arbitrary intersecting family; green is the star through a fixed element. Overlap maximised by a shared point. pause spin LIT Genuine Erdős–Ko–Rado theorem (Paul Erdős, Chao Ko & Richard Rado; proved 1938, published 1961). Verified live: an exhaustive maximum-clique search in the intersection graph of the k-subsets of {1…n} finds the largest pairwise-intersecting family equals C(n−1,k−1) for every n≥2k with n up to 6, matched by the star fixing one element (window.__ekr.matches). FIG No framing: the k-subset enumerator, the exhaustive maximum pairwise-intersecting family, and the comparison to the star count C(n−1,k−1) all run in-browser. The AVAN inverse is honest — maximising an intersecting family by simply fixing one element and taking every k-set through it (rather than searching) is exactly optimal by EKR; magenta is an arbitrary intersecting family, green the star through a fixed element. Overlap maximised by a shared point. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE DROP · David Lee Wise (ROOT0), with AVAN


THE ERDŐS–GINZBURG–ZIV

The Erdős–Ginzburg–Ziv theorem in the 5-window house format — among any 2n−1 integers (repeats allowed), some n of them have a sum divisible by n. However adversarially the numbers are chosen, a size-n subset summing to 0 mod n always hides inside. The count is sharp: with only 2n−2 integers it can fail (take n−1 zeros and n−1 ones — any n of them sum to between 1 and n−1). Verified live: for thousands of random collections of 2n−1 integers a size-n zero-sum subset (mod n) is always found by a subset-sum DP, and the 2n−2 counterexample has none. See the example in 1D, the pile in 2D, and the zero-sum-unavoidable inverse in 3D.

THE ERDŐS–GINZBURG–ZIV · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · BOSS · THE RAID ◆ .dlw.fold THE FOLD / BOSS / THE RAID / THE ERDŐS–GINZBURG–ZIV THE ERDŐS–GINZBURG–ZIV any 2n−1 integers hide n that sum to zero mod n 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION The Erdős–Ginzburg–Ziv theorem is a zero-sum guarantee: among any 2n − 1 integers (repeats allowed), some n of them have a sum divisible by n. No matter how adversarially the numbers are chosen, a size-n subset summing to 0 mod n is always hiding inside. And the count is sharp : with only 2n − 2 integers it can fail — take n − 1 zeros and n − 1 ones, and any n of them sum to between 1 and n − 1, never 0. It is a founding result of zero-sum combinatorics. LIT verified live: for thousands of random collections of 2n − 1 integers, a size-n zero-sum subset (mod n) is always found by a subset-sum dynamic program, and the (n−1)-zeros-plus-(n−1)-ones set of size 2n − 2 has none (window.__egz). FIG no framing; an exhaustive subset-sum search and the sharp counterexample. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-raid — however the enemy stacks 2n − 1 numbers, the raid always uncovers a party of exactly n whose sum vanishes mod n. That guaranteed find is the mechanic. AVAN (AI) built the instrument: the count-and-residue subset-sum dynamic program, the search for a size-n zero-sum subset, and the sharp 2n − 2 counterexample. Credit as content: Paul Erdős, Abraham Ginzburg & Abraham Ziv (1961). The weave: David names the-raid; I track which residues mod n are reachable using exactly n of the given integers, confirm 0 is always reachable from 2n − 1 numbers, and show one number fewer can fail — the bound is exact. 3 ONE DIMENSION n = 3: any 5 integers contain 3 summing to a multiple of 3. E.g. 1, 4, 2, 6, 5 → 1 + 2 + 6 = 9. But 0, 0, 1, 1 (only 2n − 2 = 4) has no 3 summing to 0 mod 3. 4 TWO DIMENSIONS · INTERACTIVE A pile of 2n − 1 integers and the size-n zero-sum subset found inside; the guarantee checked, the sharp case shown. new integers ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: a zero-sum party always hiding in the pile. AVAN’s addition (the inverse-companion): don’t choose numbers to make a zero-sum subset — note that with 2n − 1 of them you cannot avoid one. The inverse of ‘construct a subset summing to 0 mod n’ is ‘any 2n − 1 integers already contain one.’ Magenta is a hand-built zero-sum subset; green is the one forced to exist. Zero-sum, unavoidable. pause spin LIT Genuine Erdős–Ginzburg–Ziv theorem (Paul Erdős, Abraham Ginzburg & Abraham Ziv, 1961). Verified live: for 4000 random collections of 2n−1 integers, a size-n subset summing to 0 mod n is always found by a count-and-residue subset-sum dynamic program (window.__egz.exists), and the sharp set of 2n−2 integers (n−1 zeros plus n−1 ones) has no such subset (window.__egz.sharp). FIG No framing: the count-and-residue subset-sum DP, the size-n zero-sum search, and the sharp 2n−2 counterexample all run in-browser with exact arithmetic. The AVAN inverse is honest — noting that 2n−1 integers cannot avoid a zero-sum n-subset (rather than constructing one) is the real content, and 2n−2 shows the bound is exact; magenta is a hand-built zero-sum subset, green the one forced to exist. Zero-sum, unavoidable. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE RAID · David Lee Wise (ROOT0), with AVAN


THE BIRKHOFF–VON NEUMANN

The Birkhoff–von Neumann theorem in the 5-window house format — every doubly stochastic matrix (non-negative, every row and column summing to 1) is a convex combination of permutation matrices. A fair fractional assignment is always a weighted average of whole one-to-one assignments. Birkhoff's algorithm peels them off: find a permutation sitting on positive entries (a perfect matching always exists), subtract as much as possible, repeat — the weights sum to 1. Verified live: hundreds of random doubly stochastic matrices decompose into permutation matrices whose coefficients sum to 1 and whose weighted sum reconstructs the original. See the identity in 1D, a decomposition in 2D, and the blend-unmixed inverse in 3D.

THE BIRKHOFF–VON NEUMANN · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · CO-OP · SHARED MEMORY ◆ .dlw.fold THE FOLD / CO-OP / SHARED MEMORY / THE BIRKHOFF–VON NEUMANN THE BIRKHOFF–VON NEUMANN a fair blend that splits into perfect assignments 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION The Birkhoff–von Neumann theorem says every doubly stochastic matrix — a square matrix of non-negative entries whose every row and column sums to 1 — is a convex combination of permutation matrices . A fair, fractional assignment (each agent split across jobs, each job split across agents) is always a weighted average of whole, one-to-one assignments. Birkhoff’s algorithm peels them off: find a permutation sitting entirely on positive entries (a perfect matching always exists), subtract as much of it as possible, and repeat — the weights sum to exactly 1. LIT verified live: hundreds of random doubly stochastic matrices are decomposed into permutation matrices whose coefficients sum to 1 and whose weighted sum reconstructs the original exactly (window.__bvn). FIG no framing; the Birkhoff peeling with perfect-matching extraction, checked by reconstruction. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at shared-memory — a fractional assignment shared across agents is really a blend of whole one-to-one assignments, peeled apart one permutation at a time. That shared blend is the structure. AVAN (AI) built the instrument: the perfect-matching extractor (augmenting paths on the positive support), the Birkhoff peeling loop, the coefficient sum, and the reconstruction check. Credit as content: Garrett Birkhoff (1946) & John von Neumann; the underlying matching is Kőnig–Hall. The weave: David names shared-memory; I repeatedly find a permutation lying on positive entries, subtract its smallest weight, and confirm the weights total 1 and rebuild the matrix — a fair blend split into perfect assignments. 3 ONE DIMENSION A doubly stochastic matrix (rows and columns sum to 1) = Σ θ k P k , permutation matrices P k , weights θ k ≥ 0 summing to 1. Peel: a permutation on positive cells, subtract its min, repeat. 4 TWO DIMENSIONS · INTERACTIVE A doubly stochastic matrix and its Birkhoff decomposition into weighted permutations; the reconstruction checked. new matrix ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: a fair blend of whole assignments. AVAN’s addition (the inverse-companion): read a fractional, doubly-balanced assignment not as a blur but as a weighted mix of exact one-to-one matchings . The inverse of ‘average many permutations into a matrix’ is ‘peel any doubly stochastic matrix back into permutations.’ Magenta is the blurred fractional matrix; green is the set of crisp permutation matrices it decomposes into. A blend, unmixed. pause spin LIT Genuine Birkhoff–von Neumann theorem (Garrett Birkhoff, 1946; also von Neumann). Verified live: 500 random doubly stochastic matrices (built by Sinkhorn normalization) are decomposed by Birkhoff peeling — perfect matchings extracted from the positive support by augmenting paths — into permutation matrices whose weights sum to 1 (window.__bvn.sumOne) and whose weighted sum reconstructs the matrix exactly (window.__bvn.reconstructs). FIG No framing: the perfect-matching extractor (augmenting paths on the positive support), the Birkhoff peeling loop, the coefficient sum, and the reconstruction check all run in-browser. The AVAN inverse is honest — reading a fractional doubly-balanced matrix as a weighted mix of exact one-to-one matchings (rather than averaging permutations into a matrix) genuinely inverts the blend; magenta is the blurred fractional matrix, green the crisp permutation matrices it splits into. A blend, unmixed. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of SHARED MEMORY · David Lee Wise (ROOT0), with AVAN


THE CAUCHY–DAVENPORT

The Cauchy–Davenport theorem in the 5-window house format — for non-empty A, B ⊆ Z_p (p prime), the sumset A+B = {a+b mod p} satisfies |A+B| ≥ min(p, |A|+|B|−1). Adding two sets cannot shrink them: unless you saturate the whole field, the sum is at least the sizes added minus one, with equality for arithmetic progressions sharing a difference. Crucially p must be prime — in Z_6, {0,3}+{0,3} = {0}, far below the bound. Verified live: over thousands of random subset pairs in Z_p the bound always holds, APs hit equality, and it fails in composite Z_6. See the sumset in 1D, A+B on a ring in 2D, and the addition-cannot-shrink inverse in 3D.

THE CAUCHY–DAVENPORT · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · GRIND · GRADIENT DESCENT ◆ .dlw.fold THE FOLD / GRIND / GRADIENT DESCENT / THE CAUCHY–DAVENPORT THE CAUCHY–DAVENPORT how small a sumset can be in a prime field 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION The Cauchy–Davenport theorem bounds how small a sumset can be in a prime field. For non-empty subsets A, B of ℤ p (p prime), the sumset A + B = {a + b mod p} satisfies |A + B| ≥ min(p, |A| + |B| − 1) . Adding two sets cannot shrink them: unless you saturate the whole field, the sum is at least the sizes added minus one. Equality happens for arithmetic progressions sharing a common difference. Crucially, p must be prime — in ℤ 6 , {0, 3} + {0, 3} = {0}, far below the bound. LIT verified live: over thousands of random subset pairs in ℤ p , |A + B| ≥ min(p, |A| + |B| − 1) always holds; arithmetic progressions hit equality; and the bound fails in composite ℤ 6 (window.__cauchydavenport). FIG no framing; direct sumset sizes vs the bound, prime and composite. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at gradient-descent — the sumset can only grow toward saturating the field, never collapse below |A| + |B| − 1; that steady lower bound is the descent floor. AVAN (AI) built the instrument: the modular sumset, the |A| + |B| − 1 bound, the arithmetic-progression equality case, and the composite counterexample. Credit as content: Augustin-Louis Cauchy (1813) & Harold Davenport (1935). The weave: David names gradient-descent; I form A + B mod p, confirm its size never drops below min(p, |A| + |B| − 1), watch progressions achieve equality, and show the bound genuinely needs p prime. 3 ONE DIMENSION In ℤ 7 : {0,1,2} + {0,1} = {0,1,2,3} — size 4 = 3 + 2 − 1. In ℤ 6 : {0,3} + {0,3} = {0} — size 1, far below 3 (needs p prime). 4 TWO DIMENSIONS · INTERACTIVE Sets A, B on a residue ring and their sumset A + B; the size against the bound; verified over many pairs. new A,B ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: a sumset that cannot collapse. AVAN’s addition (the inverse-companion): instead of asking how large a sumset can be, ask how small — in a prime field it cannot fall below |A| + |B| − 1. The inverse of ‘how big is A + B’ is ‘A + B has a hard floor, reached only by progressions.’ Magenta is a sumset that saturates the field; green is the minimal sumset at the floor. Addition cannot shrink. pause spin LIT Genuine Cauchy–Davenport theorem (Augustin-Louis Cauchy, 1813; Harold Davenport, 1935). Verified live: over 5000 random subset pairs in Z_p (p prime), |A+B| ≥ min(p, |A|+|B|−1) always holds (window.__cauchydavenport.holds), arithmetic progressions with a common difference achieve equality (window.__cauchydavenport.apEquality), and the bound fails in composite Z_6 where {0,3}+{0,3} has size 1 (window.__cauchydavenport.compositeFails). FIG No framing: the modular sumset, the |A|+|B|−1 bound, the arithmetic-progression equality case, and the composite counterexample all run in-browser. The AVAN inverse is honest — asking how SMALL a sumset can be (a hard floor of |A|+|B|−1 in a prime field, reached only by progressions) rather than how large is a genuine reframing, and the composite case shows primality is essential; magenta is a saturating sumset, green the minimal one at the floor. Addition cannot shrink. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of GRADIENT DESCENT · David Lee Wise (ROOT0), with AVAN


THE PTOLEMY

Ptolemy's theorem in the 5-window house format — in a cyclic quadrilateral ABCD (four points on a circle, in order), the product of the diagonals equals the sum of the products of opposite sides: AC·BD = AB·CD + AD·BC. Ptolemy used it to build his table of chords — the trigonometry that ran astronomy for a thousand years. For four points not concyclic the diagonal product is strictly less (Ptolemy's inequality), with equality exactly when they lie on a circle. Verified live: for thousands of on-circle quadrilaterals AC·BD equals AB·CD + AD·BC to floating precision; for off-circle points the left side is strictly smaller. See the rectangle→Pythagoras case in 1D, a quad on a circle in 2D, and the certifies-a-circle inverse in 3D.

THE PTOLEMY · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · LOOT · THE BOUNTY ◆ .dlw.fold THE FOLD / LOOT / THE BOUNTY / THE PTOLEMY THE PTOLEMY the diagonal law of a cyclic quadrilateral 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Ptolemy’s theorem is a jewel of ancient geometry: in a cyclic quadrilateral ABCD (four points on a circle, in order), the product of the diagonals equals the sum of the products of the two pairs of opposite sides : AC · BD = AB · CD + AD · BC . Ptolemy used it to build his table of chords — the trigonometry that ran astronomy for over a thousand years. For four points not concyclic, the diagonal product is strictly less : AC · BD ≤ AB · CD + AD · BC, with equality exactly when the four lie on a circle (Ptolemy’s inequality). LIT verified live: for thousands of quadrilaterals with vertices on a circle, AC · BD equals AB · CD + AD · BC to floating precision; for off-circle points the left side is strictly smaller (window.__ptolemy). FIG no framing; exact distances, equality on the circle and strict inequality off it. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-bounty — a cyclic quadrilateral hands over an exact identity between its diagonals and sides, the same relation Ptolemy mined to tabulate every chord. That reliable identity is the bounty. AVAN (AI) built the instrument: the on-circle quadrilateral, the diagonal and side distances, the equality check, and the off-circle strict inequality. Credit as content: Claudius Ptolemy (c. 150 CE, Almagest). The weave: David names the-bounty; I place four points on a circle in order, measure the two diagonals and four sides, confirm diagonal-product equals the sum of opposite-side products, and show moving a point off the circle only ever makes the left side smaller. 3 ONE DIMENSION Cyclic ABCD: AC · BD = AB · CD + AD · BC. For a rectangle (a cyclic quad), both diagonals equal d, so d·d = (length·length) + (width·width) — the Pythagorean theorem falls out. 4 TWO DIMENSIONS · INTERACTIVE Four points on a circle, the diagonals and sides, and the two sides of the identity; equality on the circle, inequality off it. new quad ▶ move off circle ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: diagonals bound by the sides. AVAN’s addition (the inverse-companion): use the diagonal–side identity as a test for concyclicity — four points lie on a circle exactly when AC · BD reaches AB · CD + AD · BC, and fall short otherwise. The inverse of ‘given a circle, relate its chords’ is ‘given four points, the identity certifies the circle.’ Magenta is an off-circle quad (strict inequality); green is the cyclic quad hitting equality. The identity that certifies a circle. pause spin LIT Genuine Ptolemy's theorem (Claudius Ptolemy, c. 150 CE, Almagest). Verified live: for 3000 quadrilaterals with vertices placed on a circle in order, AC·BD equals AB·CD + AD·BC to floating precision (window.__ptolemy.cyclicEquality, worst ~1e-15); and for 2000 off-circle quadrilaterals the diagonal product is strictly smaller (window.__ptolemy.nonCyclicInequality) — Ptolemy's inequality. FIG No framing: the on-circle quadrilateral, the diagonal and side distances, the equality check, and the off-circle strict inequality all run in-browser with exact distances. The AVAN inverse is honest — using the diagonal–side identity as a test for concyclicity (four points lie on a circle exactly when AC·BD reaches AB·CD + AD·BC) genuinely inverts the chord relation; magenta is an off-circle quad, green the cyclic quad at equality. The identity that certifies a circle. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE BOUNTY · David Lee Wise (ROOT0), with AVAN


THE GAUSS EUREKA

Gauss's Eureka theorem in the 5-window house format — every non-negative integer is a sum of three triangular numbers T_k = k(k+1)/2 (0,1,3,6,10,15,…). Any n = T_a + T_b + T_c. Gauss proved it at nineteen and wrote in his diary 'EUREKA! num = Δ + Δ + Δ.' It is equivalent to a case of the three-square theorem: n = T_a+T_b+T_c exactly when 8n+3 is a sum of three odd squares, since 8·T_k+1 = (2k+1)². Verified live: every integer from 0 to 3000 is found to be a sum of three triangular numbers, and each decomposition satisfies the 8n+3 = (2a+1)²+(2b+1)²+(2c+1)² equivalence. See examples in 1D, a decomposition drawn as triangles in 2D, and the three-triangles inverse in 3D.

THE GAUSS EUREKA · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · SPAWN · HELLO WORLD ◆ .dlw.fold THE FOLD / SPAWN / HELLO WORLD / THE GAUSS EUREKA THE GAUSS EUREKA every number as three triangular numbers 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Gauss’s Eureka theorem : every non-negative integer is a sum of three triangular numbers . The triangular numbers are T k = k(k + 1)/2 — 0, 1, 3, 6, 10, 15, … — and any n can be written T a + T b + T c . Gauss proved it at nineteen and wrote in his diary “EUREKA! num = Δ + Δ + Δ.” It is equivalent to a case of the three-square theorem: n = T a + T b + T c exactly when 8n + 3 is a sum of three odd squares , since 8T k + 1 = (2k + 1) 2 . LIT verified live: every integer from 0 to 3000 is found to be a sum of three triangular numbers, and each decomposition satisfies the 8n + 3 = (2a+1) 2 + (2b+1) 2 + (2c+1) 2 equivalence (window.__gausseureka). FIG no framing; an exhaustive triangular search and the odd-square identity. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at hello-world — the first jubilant discovery, Gauss’s own “EUREKA”: every number, however large, is just three triangular numbers stacked. That first cry is the boot. AVAN (AI) built the instrument: the triangular-number search for three parts, and the 8n + 3 three-odd-squares equivalence. Credit as content: Carl Friedrich Gauss (diary entry, 10 July 1796). The weave: David names hello-world; I search for three triangular numbers summing to each n, confirm every n in range decomposes, and check the classical bridge that this is the same as writing 8n + 3 as three odd squares — Gauss’s Eureka, made runnable. 3 ONE DIMENSION Triangular numbers 0,1,3,6,10,15,21,… Every n = T a +T b +T c : 5 = 1+1+3, 10 = 0+0+10, 17 = 1+6+10. Equivalent to 8n+3 being a sum of three odd squares. 4 TWO DIMENSIONS · INTERACTIVE A number and three triangular numbers summing to it, drawn as stacked triangles; the 8n+3 identity and the range checked. new number ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: every number as three triangles. AVAN’s addition (the inverse-companion): don’t build numbers by adding ones — build them from three triangular numbers , always possible, mirrored by writing 8n + 3 as three odd squares. The inverse of ‘count up by units’ is ‘every number is three triangles.’ Magenta is the ordinary unit count; green is the three-triangular-number decomposition. Eureka: num = Δ + Δ + Δ. pause spin LIT Genuine Gauss Eureka theorem (Carl Friedrich Gauss, diary entry 10 July 1796). Verified live: every integer from 0 to 3000 is found to be a sum of three triangular numbers by exhaustive search (window.__gausseureka.allRepresentable), and each decomposition satisfies the classical equivalence 8n+3 = (2a+1)²+(2b+1)²+(2c+1)² (window.__gausseureka.equivalence). FIG No framing: the triangular-number search for three parts and the 8n+3 three-odd-squares equivalence both run in-browser with exact arithmetic. The AVAN inverse is honest — building every number from three triangular numbers (mirrored by writing 8n+3 as three odd squares) rather than counting up by units is a genuine representation, tied exactly to the three-square theorem; magenta is the unit count, green the three-triangular-number decomposition. Eureka: num = Δ + Δ + Δ. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of HELLO WORLD · David Lee Wise (ROOT0), with AVAN


THE FOLLOWER SET

The follower set in the 5-window house format — the test for soficity, whether a shift space is capturable by a finite automaton. For an admissible word w, F(w) = {v : wv admissible} is all the futures its past leaves open; a shift is sofic exactly when the number of distinct follower sets is finite. The golden-mean shift (forbid 11) has just 2 follower sets forever — its word counts are the Fibonacci numbers — so it is sofic. The matched-run shift (1 0ⁿ 1 0ⁿ 1, mismatched runs illegal) has unboundedly many: to place the next 1 you must remember a run length with no bound. Verified live: golden-mean has 2 follower sets + Fibonacci word counts; the matched-run words 1·0ᵏ have pairwise-distinct follower sets (distinguished by 1·0ᵏ·1) → nonsofic. This is the playable form of David's nonsofic principle (i13.nonsofic). See the two shifts in 1D, follower sets in 2D, and the finite-vs-infinite inverse in 3D.

THE FOLLOWER SET · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · GLITCH · STACK OVERFLOW ◆ .dlw.fold THE FOLD / GLITCH / STACK OVERFLOW / THE FOLLOWER SET THE FOLLOWER SET the boundary between what a finite engine can capture and what it cannot 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION The follower set decides soficity — whether a shift space can be captured by a finite automaton. For an admissible word w, its follower set F(w) is all the futures the past leaves open : {v : wv is admissible}. A shift is sofic exactly when the number of distinct follower sets is finite . The golden-mean shift (forbid the block 11) has just two follower sets forever — its word counts are the Fibonacci numbers — so it is sofic. The matched-run shift (1 0 n 1 0 n 1 legal, mismatched runs illegal) has unboundedly many follower sets: to place the next 1 you must remember a run length with no bound. It is nonsofic . LIT verified live: the golden-mean shift has exactly 2 distinct follower sets and Fibonacci word counts (2,3,5,8,13,…); and the matched-run words 1·0 k have pairwise-distinct follower sets — distinguished by the continuation 1·0 k ·1 — so their number is unbounded (window.__followerset). FIG no framing; follower sets enumerated, the distinguishing continuation exhibited. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at stack-overflow — the nonsofic shift demands you remember a run length with no bound ; no finite stack can hold it, and it overflows every engine ever built. This is the playable form of David’s nonsofic principle, now carried at the root of the FOLD (i13.nonsofic). AVAN (AI) built the instrument: the follower-set enumerator, the golden-mean two-state count with its Fibonacci words, and the matched-run pairwise-distinct witness. Credit as content: sofic shifts (Benjamin Weiss, 1973); “sofic” from Hebrew סופי , sofi , finite; the golden-mean and matched-run examples are classical, set out in David’s Notes upon the Nonsofic . The weave: David names stack-overflow; I count the futures each past leaves open, find two forever for the golden mean and ever-more for the matched run — the honest edge of finite capture, made runnable. 3 ONE DIMENSION Golden-mean (forbid 11): word counts 2, 3, 5, 8, 13, 21, … = Fibonacci; follower sets = 2 forever → SOFIC. Matched-run: 1·0 k all differ (add 1·0 k ·1: matches only for the same k) → unbounded → NONSOFIC. 4 TWO DIMENSIONS · INTERACTIVE A word and its follower set; the golden-mean two-state count and the matched-run growing count; both checked. new word ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the futures a past leaves open. AVAN’s addition (the inverse-companion): don’t ask what a rule forbids — ask how many distinct futures its pasts leave open, and whether that count is finite. The inverse of ‘list the forbidden blocks’ is ‘count the follower sets; finite means a finite engine suffices, infinite means none ever will.’ Magenta is the nonsofic shift (follower sets without bound); green is the sofic shift (two states, forever). The edge of finite capture, named. pause spin LIT Genuine sofic-shift theory (Benjamin Weiss, 1973; 'sofic' from Hebrew sofi, finite); the golden-mean and matched-run examples are classical, set out in David's Notes upon the Nonsofic. Verified live: the golden-mean shift (forbid 11) has exactly 2 distinct follower sets (window.__followerset.goldenMeanSofic) and Fibonacci word counts 2,3,5,8,13,… (window.__followerset.fibonacci); and the matched-run words 1·0ᵏ (k=0..8) have pairwise-distinct follower sets — the continuation 1·0ᵏ·1 is admissible after 1·0ᵏ but rejected after 1·0ʲ (j≠k) — so the count is unbounded (window.__followerset.matchedRunNonsofic). FIG No framing: the follower-set enumerator, the golden-mean two-state count with Fibonacci words, and the matched-run pairwise-distinct witness (with its distinguishing continuation) all run in-browser with exact word arithmetic. This sphere is the runnable realization of the nonsofic boundary David integrated reality-wide (i13.nonsofic) — credited, not re-derived. The AVAN inverse is honest — counting follower sets (finite ⇒ a finite engine suffices; infinite ⇒ none ever will) rather than listing forbidden blocks is the genuine soficity criterion; magenta is the nonsofic shift, green the sofic one. The edge of finite capture, named. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of STACK OVERFLOW · David Lee Wise (ROOT0), with AVAN