THE SINKHORN

Sinkhorn's algorithm in the 5-window house format — take any matrix of positive numbers and, by the simplest loop (divide every row by its sum, then every column by its sum, and repeat), drive it to a doubly stochastic matrix where every row and column sums to exactly 1. Sinkhorn's theorem guarantees convergence, and that the result is the unique D₁·A·D₂ rescaling of the original by positive diagonal matrices. This little iteration is the computational heart of modern optimal transport (entropic regularization) and of matching problems across machine learning. Verified live: over 3000 random positive matrices, alternating row/column normalization drives every row and column sum to 1 (~1e-16), and the result is exactly diag(u)·A·diag(v) — the ratio to the original is rank-one. Neon-noir traced. See the sums converging in 1D, the step-by-step balancing in 2D, and the scaling-factor inverse in 3D.

THE SINKHORN · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · RESPAWN · SECOND WIND ◆ .dlw.fold THE FOLD / RESPAWN / SECOND WIND / THE SINKHORN THE SINKHORN alternate row and column normalizing to perfect balance 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Sinkhorn’s algorithm takes any matrix of positive numbers and, by the simplest imaginable loop — divide every row by its sum, then divide every column by its sum, and repeat — drives it to a doubly stochastic matrix, where every row and every column sums to exactly 1. Sinkhorn’s theorem guarantees this converges, and that the result is the unique D₁·A·D₂ rescaling of the original by positive diagonal matrices. This little iteration is the computational heart of modern optimal transport (entropic regularization) and of matching problems across machine learning. LIT verified live: over 3000 random positive matrices, alternating row/column normalization drives every row sum and column sum to 1 (deviation ~1e-16), and the result is exactly diag(u)·A·diag(v) — the ratio to the original is rank-one (window.__sinkhorn). FIG no framing; the alternating normalization, the row/column sum checks, and the diagonal-scaling structure all run in-browser. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at second-wind — a system knocked out of balance keeps rebalancing, row then column then row, each pass a fresh breath, until it settles into perfect equilibrium. AVAN (AI) built the instrument: the alternating row/column normalization, the doubly-stochastic convergence check, and the diagonal-scaling structure verification. Credit as content: Richard Sinkhorn (1964); central to entropic optimal transport (Cuturi, 2013). The weave: David names second wind; I confirm the row-then-column breathing settles to a doubly stochastic matrix. 3 ONE DIMENSION A positive matrix as a grid of intensities; the row-sum and column-sum bars converge toward 1 as the loop runs. 4 TWO DIMENSIONS · INTERACTIVE Step the normalization (row then column) and watch the worst row/column deviation from 1 collapse toward zero. step ▶ run ▶ new matrix ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the doubly stochastic matrix — every row and column summing to 1. AVAN’s addition (the inverse-companion): don’t solve for the scaling — alternate. The inverse of ‘find diagonal D₁, D₂ making D₁AD₂ balanced’ is ‘just normalize rows, then columns, forever’ — the fixed point is exactly that scaling. Magenta is the row/column scaling factors; green is the balanced matrix they produce. Balance by breathing, not by solving. pause spin LIT Genuine Sinkhorn-Knopp iterative scaling (Richard Sinkhorn, 1964; central to entropic optimal transport, Cuturi 2013). Verified live: over 3000 random positive matrices alternating row/column normalization drives every row and column sum to 1 (deviation ~1e-16), and the result equals diag(u)·A·diag(v) (the ratio B/A is rank-one) (window.__sinkhorn.doublyStochastic, .structOk). FIG No framing; the alternating normalization, the row/column sum checks, and the diagonal-scaling structure all run in-browser. The AVAN inverse is honest — instead of solving for the diagonal scaling that balances A, just normalize rows then columns forever; the fixed point is exactly that scaling. Magenta is the row/column scaling factors; green is the balanced matrix they produce. Balance by breathing, not by solving. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of SECOND WIND · David Lee Wise (ROOT0), with AVAN


THE VIETA JUMPING

Vieta jumping in the 5-window house format — a proof technique built on the fact that a quadratic has two roots summing to a rational you read off the coefficients (Vieta's formulas). Its most famous victory is IMO 1988 Problem 6: if a and b are positive integers such that (a²+b²)/(ab+1) is an integer k, then k must be a perfect square. The proof: fix k, and from any solution jump to another by replacing a with the quadratic's other root a′ = k·b − a; this produces a smaller solution, and infinite descent drives b to 0, where k = a² is manifestly a square. Verified live: over all 0≤b≤a≤200, every integer value of (a²+b²)/(ab+1) is a perfect square (0,1,4,9,16,25,36,49…), and the Vieta jump always yields another valid solution that is strictly smaller. Neon-noir traced. See the solution ladder in 1D, the descent step in 2D, and the reflect-across-the-quadratic inverse in 3D.

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

THE VIETA JUMPING · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · CHEAT · THE SPEEDRUN ◆ .dlw.fold THE FOLD / CHEAT / THE SPEEDRUN / THE VIETA JUMPING THE VIETA JUMPING an integer ratio that can only be a perfect square 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Vieta jumping is a proof technique built on the fact that a quadratic has two roots summing to a rational you can read off the coefficients (Vieta’s formulas). Its most famous victory is IMO 1988 Problem 6 : if a and b are positive integers such that (a 2 + b 2 )/(ab + 1) is an integer k, then k must be a perfect square . The proof: fix k, and from any solution ‘jump’ to another by replacing a with the quadratic’s other root a′ = k·b - a; this produces a smaller solution, and infinite descent drives b to 0, where k = a 2 is manifestly a square. LIT verified live: over all 0 ≤ b ≤ a ≤ 200, every integer value of (a 2 +b 2 )/(ab+1) is a perfect square (the values seen are 0,1,4,9,16,25,36,49…), and the Vieta jump a′ = k·b - a always yields another valid solution that is strictly smaller (window.__vieta). FIG no framing; the exhaustive integer search and the descent step both run in-browser. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-speedrun — the jump is a shortcut past a brute search: instead of grinding, hop to the quadratic’s other root and slide straight down the ladder of solutions to the base case. AVAN (AI) built the instrument: the exhaustive integer search, the perfect-square test, and the Vieta descent a′ = k·b - a. Credit as content: Vieta’s formulas (François Viète, 1590s); the technique crystallized by IMO 1988 Problem 6. The weave: David names the speedrun; I confirm every integer ratio is a perfect square and the jump descends to the base case. 3 ONE DIMENSION The (a,b) solution lattice for k = g²: solutions climb a ladder, each the Vieta jump of the last, down to (g, 0). 4 TWO DIMENSIONS · INTERACTIVE Pick a perfect square k = g² and jump down the ladder: each step a′ = k·b − a lands on a smaller valid solution. next k ▶ jump down ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: a solution (a, b) with (a²+b²)/(ab+1) = k. AVAN’s addition (the inverse-companion): don’t search for solutions — jump between them. The inverse of ‘a is a root of x² - k·b·x + (b² - k)’ is ‘its other root a′ = k·b - a’, a reflection that descends the ladder to (g, 0) where k = g². Magenta is the jumped partner; green is the current solution. Descent by reflecting across the quadratic. pause spin LIT Genuine Vieta jumping (Vieta's formulas, François Viète 1590s; technique crystallized by IMO 1988 Problem 6). Verified live: over all 0≤b≤a≤200, every integer (a²+b²)/(ab+1) is a perfect square (values seen 0,1,4,9,16,25,36,49), and the Vieta jump a′=k·b−a always yields a valid solution strictly smaller than a (window.__vieta.allSquare, .jumpDescends). FIG No framing; the exhaustive integer search, the perfect-square test, and the Vieta descent all run in-browser. The AVAN inverse is honest — instead of searching for solutions, jump between them: the other root of x²−k·b·x+(b²−k) is a′=k·b−a, a reflection that descends the ladder to (g,0) where k=g². Magenta is the jumped partner; green is the current solution. Descent by reflecting across the quadratic. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE SPEEDRUN · David Lee Wise (ROOT0), with AVAN


THE FROBENIUS COIN

The Frobenius coin problem (the Chicken McNugget theorem) in the 5-window house format — with only coins of two coprime denominations a and b, what is the largest amount you cannot make from non-negative whole numbers of each? The answer is startlingly clean: the Frobenius number is a·b − a − b. Everything above it is payable; below it, exactly (a−1)(b−1)/2 amounts are impossible. With 3s and 5s the biggest unmakeable total is 7; with the famous 6,9,20 nuggets the largest impossible order is 43. Verified live: over hundreds of coprime pairs the largest non-representable integer is exactly ab−a−b, the count of gaps is exactly (a−1)(b−1)/2, and every integer beyond the Frobenius number is representable. Neon-noir traced. See the payable/impossible number line in 1D, the two formulas vs brute in 2D, and the finite-gaps-vs-infinite-reach inverse in 3D.

THE FROBENIUS COIN · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · LOOT · THE MINT ◆ .dlw.fold THE FOLD / LOOT / THE MINT / THE FROBENIUS COIN THE FROBENIUS COIN the largest amount two coins cannot make 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION The Frobenius coin problem (the ‘Chicken McNugget theorem’) asks: with only coins of two coprime denominations a and b, what is the largest amount you cannot make from non-negative whole numbers of each? The answer is startlingly clean: the Frobenius number is a·b - a - b . Everything above it is payable; below it, exactly (a-1)(b-1)/2 amounts are impossible. With 3s and 5s the biggest unmakeable total is 7; with the famous 6, 9, 20 nuggets the largest impossible order is 43. Two coprime numbers carve the integers into a finite island of gaps and an endless mainland of the reachable. LIT verified live: over hundreds of coprime pairs (a,b) the largest non-representable integer is exactly a·b - a - b, the count of non-representable integers is exactly (a-1)(b-1)/2, and every integer beyond the Frobenius number is representable (window.__frobenius). FIG no framing; the representability search, the Frobenius-number formula, and the gap-count formula all run in-browser. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-mint — coins are struck in fixed denominations, and this is the exact boundary of what those coins can and cannot pay: a last impossible sum, then the mint’s reach is total. AVAN (AI) built the instrument: the representability test, the Frobenius-number check, and the gap-count formula. Credit as content: Ferdinand Frobenius (the problem bears his name); James Sylvester proved the two-coin formulas (1884). The weave: David names the mint; I confirm ab-a-b is the last unpayable amount and (a-1)(b-1)/2 the number of gaps. 3 ONE DIMENSION The integers: green = payable with coins a and b, red = impossible. The last red is the Frobenius number ab-a-b. 4 TWO DIMENSIONS · INTERACTIVE Cycle coprime denominations; the Frobenius number and gap count are computed by brute search and by formula, and compared. next coins ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the endless mainland of representable amounts. AVAN’s addition (the inverse-companion): don’t list what you can make — bound what you cannot. The inverse of ‘the reachable amounts’ is ‘the finite island of gaps below ab-a-b, exactly (a-1)(b-1)/2 of them.’ Magenta is that finite gap-set; green is the infinite reachable ray past the Frobenius number. A last impossibility, then total reach. pause spin LIT Genuine Frobenius (Chicken McNugget) two-coin theorem (problem named for Frobenius; formulas proved by J. J. Sylvester, 1884). Verified live: over ~580 coprime pairs (a,b) the largest non-representable integer equals ab−a−b, the number of non-representable integers equals (a−1)(b−1)/2, and every integer above the Frobenius number is representable (window.__frobenius.frobOk, .countOk, .allAbove). FIG No framing; the representability search, the Frobenius-number formula, and the gap-count formula all run in-browser. The AVAN inverse is honest — instead of listing what you can make, bound what you cannot: the finite island of gaps below ab−a−b, exactly (a−1)(b−1)/2 of them. Magenta is that finite gap-set; green is the infinite reachable ray past the Frobenius number. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE MINT · David Lee Wise (ROOT0), with AVAN


THE PERRON-FROBENIUS

The Perron-Frobenius theorem in the 5-window house format — the reason PageRank, Markov chains, and population models all converge. A matrix of strictly positive entries has a single dominant eigenvalue that is real, positive, and strictly larger in magnitude than every other, with an all-positive eigenvector. Repeatedly multiplying any positive vector by the matrix and renormalizing drives it straight to that Perron eigenvector, and the eigenvalue is pinned between the smallest and largest row sums. Verified live: over 4000 random positive matrices, power iteration converges to A·v=λv with residual below 1e-6, λ is positive and the eigenvector all one sign, and λ always lies between the minimum and maximum row sums. Neon-noir traced. See the iterate rotating to the Perron vector in 1D, λ entering the row-sum band in 2D, and the iterate-don't-solve inverse in 3D.

THE PERRON-FROBENIUS · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · GRIND · THE MAINFRAME ◆ .dlw.fold THE FOLD / GRIND / THE MAINFRAME / THE PERRON-FROBENIUS THE PERRON-FROBENIUS a positive matrix's one dominant real eigenvalue 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION The Perron–Frobenius theorem is the reason PageRank, Markov chains, and population models all converge. It says a matrix of strictly positive entries has a single dominant eigenvalue that is real, positive, and strictly larger in magnitude than every other eigenvalue — and its eigenvector can be chosen with all-positive entries. Repeatedly multiplying any positive starting vector by the matrix and renormalizing drives it straight to that Perron eigenvector, and the eigenvalue is pinned between the smallest and largest row sums. It is the mathematics of ‘the long-run steady state exists and is unique’. LIT verified live: over 4000 random positive matrices, power iteration converges to an eigenpair A·v = λv with residual below 1e-6, the eigenvalue is positive and the eigenvector is all one sign, and λ always lies between the minimum and maximum row sums (window.__perron). FIG no framing; the power iteration, the residual check, the sign check, and the row-sum bound all run in-browser. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-mainframe — the heavy repeated matrix–vector grind a mainframe runs to find a steady state, converging on the one dominant direction. AVAN (AI) built the instrument: the power iteration, the Rayleigh eigenvalue, the positivity check, and the row-sum bounds that bracket the Perron root. Credit as content: Oskar Perron (1907) & Georg Frobenius (1912). The weave: David names the mainframe; I confirm the iteration lands on a positive dominant eigenpair with λ between the row sums. 3 ONE DIMENSION A positive matrix; the iterate vector (green) rotates toward the all-positive Perron eigenvector as the loop runs. 4 TWO DIMENSIONS · INTERACTIVE Step the power iteration; λ climbs into the band between the smallest and largest row sums and the residual collapses. step ▶ run ▶ new matrix ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the all-positive Perron eigenvector, the long-run steady direction. AVAN’s addition (the inverse-companion): don’t solve the characteristic polynomial — iterate. The inverse of ‘find the dominant eigenvalue’ is ‘multiply any positive vector by A over and over; every other direction decays and only the Perron eigenvector survives.’ Magenta is the row-sum band bracketing λ; green is the surviving eigenvector. The steady state, reached by repetition. pause spin LIT Genuine Perron-Frobenius theorem (Oskar Perron 1907; Georg Frobenius 1912). Verified live: over 4000 random positive matrices power iteration converges to an eigenpair A·v=λv with residual FIG No framing; the power iteration, the residual check, the sign check, and the row-sum bound all run in-browser. The AVAN inverse is honest — instead of solving the characteristic polynomial, multiply any positive vector by A repeatedly; every other direction decays and only the Perron eigenvector survives. Magenta is the row-sum band bracketing λ; green is the surviving eigenvector. The steady state, reached by repetition. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE MAINFRAME · David Lee Wise (ROOT0), with AVAN


THE PADÉ

The Padé approximant in the 5-window house format — replacing a power series with a ratio of two polynomials P(x)/Q(x) chosen so its own Taylor expansion agrees with the original to the highest possible order m+n. Because it has a denominator, it captures poles: where a Taylor series diverges the instant you pass its radius of convergence, the Padé approximant sails on, its denominator's roots sitting right where the true function blows up. It underlies function libraries, control theory, and the resummation of divergent series. Verified live: the [3/3] Padé of eˣ reproduces the Taylor coefficients through order 6 exactly, and at x=1 its error (~3e-5) is an order of magnitude smaller than the degree-6 Taylor polynomial's (~2e-4). Neon-noir traced. See f, Padé, and Taylor through a pole in 1D, the coefficient match + error in 2D, and the divide-past-the-radius inverse in 3D.

THE PADÉ · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · GLITCH · DIVIDE BY ZERO ◆ .dlw.fold THE FOLD / GLITCH / DIVIDE BY ZERO / THE PADÉ THE PADÉ a rational that captures the poles a polynomial cannot 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION A Padé approximant replaces a power series with a ratio of two polynomials P(x)/Q(x) chosen so its own Taylor expansion agrees with the original series to the highest possible order, m+n, for a numerator of degree m and denominator of degree n. Because it has a denominator, it can do something a Taylor polynomial never can: capture poles . Where a Taylor series diverges the instant you pass its radius of convergence, the Padé approximant sails on — its denominator’s roots sit right where the true function blows up. It is the workhorse behind function libraries, control theory, and resummation of divergent series. LIT verified live: the [3/3] Padé approximant of e x reproduces the Taylor coefficients through order 6 exactly, and at x=1 its error (≈3e-5) is an order of magnitude smaller than the degree-6 Taylor polynomial’s (≈2e-4) (window.__pade). FIG no framing; the Padé linear solve, the series re-expansion, and the error comparison all run in-browser. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at divide-by-zero — the very thing a polynomial fears, the Padé embraces: its denominator is allowed to hit zero, and it places those zeros exactly at the function’s poles. AVAN (AI) built the instrument: the Padé coefficient solve, the re-expansion match, and the accuracy comparison against Taylor. Credit as content: Henri Padé (1892); anticipated by Frobenius and Jacobi. The weave: David names the divide-by-zero; I confirm the rational matches the series to order m+n and beats the Taylor polynomial. 3 ONE DIMENSION f (cyan), its Padé approximant (green), and the Taylor polynomial (magenta): near a pole the Taylor diverges while Padé holds. 4 TWO DIMENSIONS · INTERACTIVE Cycle the target function; the Padé coefficients match the series to order m+n and its max error beats the Taylor polynomial. next f ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the Padé rational, tracking f through the pole. AVAN’s addition (the inverse-companion): don’t just add powers — divide by them. The inverse of ‘a polynomial that can only grow’ is ‘a denominator whose zeros land on the function’s poles, so the ratio survives where the sum explodes.’ Magenta is the Taylor polynomial diverging; green is the Padé holding through the singularity. Reach past the radius by dividing. pause spin LIT Genuine Padé approximant (Henri Padé, 1892; anticipated by Frobenius and Jacobi). Verified live: the [3/3] Padé of eˣ re-expands to the Taylor coefficients through order 6 (worst ~1e-17) and at x=1 has error ~3e-5, smaller than the degree-6 Taylor polynomial's ~2e-4; the ln(1+x) match is also exact (window.__pade.matchesSeries, .beatsTaylor). FIG No framing; the Padé linear solve, the series re-expansion, and the error comparison all run in-browser. The AVAN inverse is honest — instead of only adding powers, divide by them: a denominator whose zeros land on the function's poles, so the ratio survives where the sum explodes. Magenta is the Taylor polynomial diverging; green is the Padé holding through the singularity. Reach past the radius by dividing. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of DIVIDE BY ZERO · David Lee Wise (ROOT0), with AVAN


THE BARKER CODE

The Barker code in the 5-window house format — a finite sequence of +1s and −1s with an almost magical property: its aperiodic autocorrelation (slide a copy against itself and sum the products) has a tall central peak equal to the code length, and every off-centre value is at most 1 in magnitude. A receiver correlating an incoming signal against a Barker code sees a single sharp spike at alignment and almost nothing elsewhere — which is why they mark the start of radar pulses and Wi-Fi and GPS frames. Barker codes are known only for lengths 2,3,4,5,7,11,13, and it is conjectured none longer exist. Verified live: for each known Barker code the zero-shift autocorrelation equals its length, and every non-zero shift gives a value in {−1,0,+1}. Neon-noir traced. See the code and its autocorrelation in 1D, every shift listed in 2D, and the correlation-peak inverse in 3D.

THE BARKER CODE · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · CO-OP · THE SYNC ◆ .dlw.fold THE FOLD / CO-OP / THE SYNC / THE BARKER CODE THE BARKER CODE a ±1 code whose echoes never rise above one 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION A Barker code is a finite sequence of +1s and -1s with an almost magical property: its aperiodic autocorrelation — slide a copy of the code against itself and sum the products — has a tall central peak equal to the code length, and every off-centre value is at most 1 in magnitude . That means a receiver correlating an incoming signal against a Barker code sees a single sharp spike exactly at alignment and almost nothing elsewhere, which is why they are used for radar pulse compression and to mark the start of Wi-Fi and GPS frames. Remarkably, Barker codes are known only for lengths 2, 3, 4, 5, 7, 11, and 13 — and it is conjectured none longer exist. LIT verified live: for each known Barker code the zero-shift autocorrelation equals its length, and every non-zero shift gives a value in {-1, 0, +1} (window.__barker). FIG no framing; the autocorrelation at every shift runs in-browser. That no Barker code longer than 13 exists is a famous conjecture , not shown here. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-sync — a Barker code is the marker that says ‘the frame starts here ’: its correlation spikes at perfect alignment and stays flat everywhere else, so two ends synchronize on the instant. AVAN (AI) built the instrument: the aperiodic autocorrelation at every shift, the peak check, and the sidelobe bound. Credit as content: Ronald Hugh Barker (1953). The weave: David names the sync; I confirm each known Barker code’s sidelobes never exceed 1. 3 ONE DIMENSION The ±1 code (top) and its autocorrelation (bottom): a tall spike at zero shift, sidelobes never above 1. 4 TWO DIMENSIONS · INTERACTIVE Cycle the known Barker lengths; the autocorrelation at every shift is listed, and the sidelobe bound is checked. next length ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the sharp correlation peak at perfect alignment. AVAN’s addition (the inverse-companion): don’t read the code — correlate against it. The inverse of ‘a string of ±1s’ is ‘its autocorrelation’, and a Barker code is exactly the string whose autocorrelation is a lone spike with flat sidelobes. Magenta are the suppressed sidelobes (never above 1); green is the peak equal to the length. A code defined by its own echo. pause spin LIT Genuine Barker codes (Ronald Hugh Barker, 1953). Verified live: for every known Barker code (lengths 2,3,4,5,7,11,13) the zero-shift aperiodic autocorrelation equals the code length and every non-zero shift gives a value in {−1,0,+1} (window.__barker.sidelobesBounded, .lengths). FIG No framing; the autocorrelation at every shift runs in-browser. Honest scope: that no Barker code longer than 13 exists is a famous conjecture, not shown here. The AVAN inverse is honest — instead of reading the code, correlate against it: a Barker code is exactly the ±1 string whose autocorrelation is a lone spike with flat sidelobes. Magenta are the suppressed sidelobes; green is the peak equal to the length. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE SYNC · David Lee Wise (ROOT0), with AVAN


THE DIFFERENCE SET

The cyclic difference set in the 5-window house format — a small set of residues D in Z_v so perfectly arranged that every non-zero residue arises as a difference dᵢ−dⱼ (mod v) the same number of times, λ. A (v,k,λ)-difference set generates a symmetric block design: rotate D through all v shifts and you get v blocks where every pair of points meets in exactly λ blocks. The set {0,1,3} mod 7 is the smallest example — its six differences are exactly 1,2,3,4,5,6 each once — and it is nothing less than the Fano plane in disguise. Verified live: for several classical difference sets — (7,3,1), (13,4,1), (21,5,1), and the (11,5,2) Paley set — every non-zero residue appears exactly λ times among the differences, and a non-example is correctly rejected. Neon-noir traced. See the residue circle and its differences in 1D, the uniform histogram in 2D, and the differences-not-points inverse in 3D.

THE DIFFERENCE SET · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · CHEAT · THE KONAMI CODE ◆ .dlw.fold THE FOLD / CHEAT / THE KONAMI CODE / THE DIFFERENCE SET THE DIFFERENCE SET a set whose differences hit every target the same number of times 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION A cyclic difference set is a small set of residues D in Z v so perfectly arranged that every non-zero residue arises as a difference d i - d j (mod v) the same number of times , λ. A (v, k, λ)-difference set of k elements generates a symmetric block design: rotate D through all v shifts and you get v blocks where every pair of points meets in exactly λ blocks. The set {0, 1, 3} mod 7 is the smallest example — its six differences are exactly 1, 2, 3, 4, 5, 6, each once — and it is nothing less than the Fano plane in disguise. LIT verified live: for several classical difference sets — (7,3,1), (13,4,1), (21,5,1), and the (11,5,2) Paley set — every non-zero residue appears exactly λ times among the differences, and a non-example is correctly rejected (window.__diffset). FIG no framing; the full difference multiset and its uniformity check run in-browser. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-konami-code — a short secret sequence with perfect structure: a handful of residues whose differences unlock every target uniformly, a cheat pattern hiding a whole design. AVAN (AI) built the instrument: the difference multiset, the exactly-λ uniformity check, and a non-example rejection. Credit as content: the theory of difference sets (Singer, 1938; Paley); the (7,3,1) set is the Fano plane. The weave: David names the konami code; I confirm each set’s differences cover every residue exactly λ times. 3 ONE DIMENSION The residues of Z_v on a circle; the difference-set points are lit, and every arc-difference is drawn — each residue hit λ times. 4 TWO DIMENSIONS · INTERACTIVE Cycle the classical difference sets; the histogram of differences is shown — every non-zero residue exactly λ. next set ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the k residues of the difference set on the cycle. AVAN’s addition (the inverse-companion): don’t list the points — list their differences. The inverse of ‘a set of k residues’ is ‘the multiset of its pairwise differences’, and a difference set is exactly the set whose differences are perfectly uniform. Magenta are the difference vectors covering the circle; green are the chosen residues. A design hidden in the gaps between points. pause spin LIT Genuine cyclic (v,k,λ)-difference sets (Singer, 1938; Paley construction; the (7,3,1) set is the Fano plane). Verified live: for (7,3,1), (13,4,1), (21,5,1) and the (11,5,2) Paley set every non-zero residue appears exactly λ times among the pairwise differences, and the non-example {0,1,2} mod 7 is correctly rejected (window.__diffset.allValid, .nonExampleRejected). FIG No framing; the full difference multiset and its uniformity check run in-browser. The AVAN inverse is honest — instead of listing the k residues, list the multiset of their pairwise differences: a difference set is exactly the set whose differences are perfectly uniform. Magenta are the difference vectors covering the circle; green are the chosen residues. A design hidden in the gaps between points. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE KONAMI CODE · David Lee Wise (ROOT0), with AVAN


THE WYNN

Wynn's epsilon algorithm in the 5-window house format — a machine for accelerating convergence. Given the crawling partial sums of a slowly-converging series, it fills a triangular table by one simple rule — ε(n)_{k+1} = ε(n+1)_{k-1} + 1/(ε(n+1)_k − ε(n)_k) — and its even columns leap toward the limit far faster than the sums themselves. It is equivalent to Padé approximation applied to the series, and can wring a dozen correct digits from a series that summed directly would need billions of terms. The Leibniz series for π is the classic victim: agonizingly slow raw, nearly instant accelerated. Verified live: from just 16 terms of the Leibniz series the raw partial sum is off by ~0.06 while Wynn's accelerated estimate is off by ~3×10⁻¹² — over nine orders of magnitude better. Neon-noir traced. See the partial sums vs the accelerated snap in 1D, the plunging error in 2D, and the reach-sideways inverse in 3D.

THE WYNN · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · CHEAT · THE BACKDOOR ◆ .dlw.fold THE FOLD / CHEAT / THE BACKDOOR / THE WYNN THE WYNN an accelerator that squeezes π from a crawling series 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Wynn’s epsilon algorithm is a machine for accelerating convergence . Given the crawling partial sums of a slowly-converging series, it fills a triangular table by one deceptively simple rule — ε (n) k+1 = ε (n+1) k-1 + 1/(ε (n+1) k - ε (n) k ) — and its even columns leap toward the limit far faster than the sums themselves. It is equivalent to Padé approximation applied to the series, and it can wring a dozen correct digits out of a series that, summed directly, would need billions of terms. The Leibniz series for π is the classic victim: agonizingly slow raw, nearly instant accelerated. LIT verified live: from just 16 terms of the Leibniz series the raw partial sum is off by ~0.06, while Wynn’s accelerated estimate is off by ~3×10 -12 — more than nine orders of magnitude better (window.__wynn). FIG no framing; the epsilon table and the error comparison run in-browser. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-backdoor — instead of grinding through billions of terms, slip through a side entrance: a table that reaches the limit from a mere handful of partial sums. AVAN (AI) built the instrument: the epsilon-table recurrence, the even-column extraction, and the error comparison against the raw partial sum. Credit as content: Peter Wynn (1956), accelerating Shanks’ transformation. The weave: David names the backdoor; I confirm the table reaches π to twelve digits from sixteen terms. 3 ONE DIMENSION The partial sums (magenta) oscillate slowly toward π; the accelerated even-column estimate (green) snaps to it. 4 TWO DIMENSIONS · INTERACTIVE Add terms one at a time; the raw partial-sum error barely shrinks while the Wynn-accelerated error plunges. add 2 terms ▶ reset ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the accelerated estimate, pinned to the true limit π. AVAN’s addition (the inverse-companion): don’t sum more terms — transform the sums you have. The inverse of ‘add another term’ is ‘feed the partial sums through the epsilon table; its even columns already hold the limit.’ Magenta is the crawling sequence of partial sums; green is the accelerated value. Reach the limit sideways. pause spin LIT Genuine Wynn epsilon algorithm (Peter Wynn, 1956; accelerating Shanks' transformation). Verified live: from 16 terms of the Leibniz series for π the raw partial sum has error ~6×10⁻² while the Wynn even-column accelerated estimate has error ~3×10⁻¹² (est 3.141592653586), more than nine orders of magnitude better (window.__wynn.accelerates, .partialErr, .accErr). FIG No framing; the epsilon table and the error comparison run in-browser. The AVAN inverse is honest — instead of summing more terms, transform the sums you have: the epsilon table's even columns already hold the limit. Magenta is the crawling sequence of partial sums; green is the accelerated value. Reach the limit sideways. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE BACKDOOR · David Lee Wise (ROOT0), with AVAN


THE LAGUERRE

Laguerre's method in the 5-window house format — a root-finder of unreasonable robustness. To locate a root of a degree-n polynomial it uses both derivatives to build a step that assumes all the other roots are bunched at one distant point — a pessimistic guess that nonetheless lands on a root with cubic convergence and converges from almost any starting point, even to complex roots from a real start. Find one root, divide it out by deflation, and repeat until every root — real and complex — is captured. It is a mainstay of polynomial solvers precisely because it so rarely fails. Verified live: for polynomials built from known roots (mixing real values and complex-conjugate pairs), Laguerre with deflation recovers all roots to about 1e-8. Neon-noir traced. See the roots on the complex plane in 1D, the recovered-vs-true match in 2D, and the deflation inverse in 3D.

THE LAGUERRE · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · BOSS · SUDDEN DEATH ◆ .dlw.fold THE FOLD / BOSS / SUDDEN DEATH / THE LAGUERRE THE LAGUERRE a solver that hunts every root, real and complex 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Laguerre’s method is a root-finder of almost unreasonable robustness. To locate a root of a degree-n polynomial it uses both the first and second derivatives to build a step that assumes all the other roots are bunched at one distant point — a wildly pessimistic guess that nonetheless lands the iterate on a root with cubic convergence and, remarkably, converges from almost any starting point , even to complex roots from a real start. Find one root, divide it out (deflation), and repeat until every root — real and complex — is captured. It is a mainstay of polynomial solvers precisely because it so rarely fails. LIT verified live: for polynomials built from known roots (mixing real values and complex-conjugate pairs), Laguerre with deflation recovers all roots to about 1e-8 (window.__laguerre). FIG no framing; the complex arithmetic, the Laguerre step, the deflation, and the root-matching all run in-browser. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at sudden-death — one root felled per round, then deflated away, and on to the next until the whole polynomial is defeated: no root survives the sweep. AVAN (AI) built the instrument: the complex-number kernel, the Laguerre iteration, the synthetic-division deflation, and the recovered-vs-true root matching. Credit as content: Edmond Laguerre (1880). The weave: David names sudden-death; I confirm every root — real and complex — is found and matched to the true set. 3 ONE DIMENSION The complex plane: the true roots (gold rings) and the roots Laguerre recovers (green dots) coincide. 4 TWO DIMENSIONS · INTERACTIVE Cycle polynomials (real and complex roots); Laguerre + deflation recovers the full root set and matches the truth. next polynomial ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: all the polynomial’s roots, plucked from the complex plane. AVAN’s addition (the inverse-companion): don’t just find a root — remove it. The inverse of ‘solve p(x)=0’ is ‘deflate: divide out (x - root) to shrink the problem, and the next root falls the same way.’ Magenta is the deflated factor being peeled off; green are the roots as they fall. Defeat the polynomial one root at a time. pause spin LIT Genuine Laguerre's method (Edmond Laguerre, 1880). Verified live with a complex-arithmetic kernel: for polynomials built from known roots (including complex-conjugate pairs like 1±i, ±i, 0.5±0.5i) Laguerre with synthetic-division deflation recovers all roots and matches the true set to ~1e-8 worst-case (window.__laguerre.recoversAll, .worst). FIG No framing; the complex arithmetic, the Laguerre step, the deflation, and the root-matching all run in-browser. The AVAN inverse is honest — instead of just finding a root, remove it: deflate by dividing out (x−root) to shrink the problem, and the next root falls the same way. Magenta is the deflated factor being peeled off; green are the roots as they fall. Defeat the polynomial one root at a time. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of SUDDEN DEATH · David Lee Wise (ROOT0), with AVAN


THE GOLD CODE

Gold codes in the 5-window house format — the sequences that let dozens of GPS satellites and phones talk over the same frequency at the same time. They start from maximum-length LFSR sequences (m-sequences), whose cyclic autocorrelation is a single tall spike of value N at zero shift and a flat −1 everywhere else. Taking a special preferred pair of m-sequences and XOR-ing their shifts produces a family of codes whose cross-correlation takes only three small values, so any two users' signals look nearly orthogonal — the mathematical basis of code-division multiple access. Verified live: for n=5 (period 31) the m-sequence's autocorrelation is 31 at shift 0 and exactly −1 at all other shifts, and the preferred-pair cross-correlation takes only the three values {−1,−9,7} (t(5)=9), as Gold's theorem predicts. Neon-noir traced. See the m-sequence and its autocorrelation in 1D, the three-valued cross-correlation in 2D, and the many-voices-one-channel inverse in 3D.

THE GOLD CODE · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · CO-OP · SPLIT SCREEN ◆ .dlw.fold THE FOLD / CO-OP / SPLIT SCREEN / THE GOLD CODE THE GOLD CODE near-orthogonal codes that share one channel 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Gold codes are the sequences that let dozens of GPS satellites and phones talk over the same frequency at the same time . They start from maximum-length LFSR sequences (‘m-sequences’), whose cyclic autocorrelation is a single tall spike of value N at zero shift and a flat -1 everywhere else. Taking a special ‘preferred pair’ of m-sequences and XOR-ing their shifts produces a whole family of codes whose cross-correlation takes only three small values , so any two users’ signals look nearly orthogonal — the mathematical basis of code-division multiple access. LIT verified live: for n=5 (period 31) the m-sequence’s autocorrelation is 31 at shift 0 and exactly -1 at all other shifts, and the preferred-pair cross-correlation takes only the three values {-1, -9, 7} (t(5)=9), as Gold’s theorem predicts (window.__gold). FIG no framing; the LFSR generation, the autocorrelation, and the cross-correlation all run in-browser. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at split-screen — many players sharing one screen without interfering, exactly as many signals share one frequency band because their codes barely correlate. AVAN (AI) built the instrument: the LFSR m-sequence generator, the two-valued autocorrelation check, and the three-valued preferred-pair cross-correlation. Credit as content: Robert Gold (1967); m-sequences from primitive polynomials. The weave: David names split-screen; I confirm the autocorrelation is two-valued and the cross-correlation only three-valued. 3 ONE DIMENSION An m-sequence (±1) and its cyclic autocorrelation: a spike of 31 at zero shift, a flat -1 at every other shift. 4 TWO DIMENSIONS · INTERACTIVE The cross-correlation of the preferred pair over all shifts — a histogram landing on only three values {-1, -9, 7}. auto ↔ cross ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the m-sequence’s sharp autocorrelation spike. AVAN’s addition (the inverse-companion): don’t send one code — overlap many. The inverse of ‘a signal keyed to its own code’ is ‘every other user’s code correlates near zero’, so their signals coexist on one band. Magenta is the low three-valued cross-correlation between users; green is each user’s own tall autocorrelation peak. Many voices, one channel. pause spin LIT Genuine Gold codes (Robert Gold, 1967; m-sequences from primitive polynomials). Verified live: for n=5 (period 31) the m-sequence built from taps [5,2] has cyclic autocorrelation 31 at shift 0 and exactly −1 at all 30 other shifts, and the preferred-pair cross-correlation with taps [5,4,3,2] takes only the three values {−1,−9,7} (t(5)=9) (window.__gold.twoValuedAuto, .threeValuedCross, .values). FIG No framing; the LFSR generation, the autocorrelation, and the cross-correlation all run in-browser. The AVAN inverse is honest — instead of sending one code, overlap many: every other user's code correlates near zero, so their signals coexist on one band. Magenta is the low three-valued cross-correlation between users; green is each user's own tall autocorrelation peak. Many voices, one channel. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of SPLIT SCREEN · David Lee Wise (ROOT0), with AVAN


THE OSTROWSKI

Ostrowski numeration in the 5-window house format — a whole number system built out of the continued fraction of an irrational α. Instead of powers of ten, the place values are the denominators q_k of α's convergents, and every non-negative integer has a unique representation as a digit-weighted sum of them, with digits bounded by the continued-fraction terms and a rule forbidding a maxed digit from sitting on a non-zero one. For the golden ratio this is exactly Zeckendorf's Fibonacci representation; for √2 the place values are the Pell numbers 1,2,5,12,29,70… It is the deep reason the Fibonacci and Pell numbers form clean bases. Verified live: using √2 (place values 1,2,5,12,29,70,169), the greedy Ostrowski digits reconstruct every integer in [0,169) exactly, all digits obey the bounds and the no-adjacent-max rule, and every representation is unique. Neon-noir traced. See the place-values and a digit sum in 1D, per-N digits in 2D, and the exotic-address inverse in 3D.

THE OSTROWSKI · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · GRIND · THE MAINFRAME ◆ .dlw.fold THE FOLD / GRIND / THE MAINFRAME / THE OSTROWSKI THE OSTROWSKI a numeral system carved from a continued fraction 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Ostrowski numeration builds a whole number system out of the continued fraction of an irrational α. Instead of powers of ten, the ‘place values’ are the denominators q k of α’s convergents , and every non-negative integer has a unique representation as a digit-weighted sum of them, with digits bounded by the continued-fraction terms and a rule forbidding a maxed digit from sitting on a non-zero one. For the golden ratio this is exactly Zeckendorf’s Fibonacci representation ; for √2 the place values are the Pell numbers 1, 2, 5, 12, 29, 70… It is the deep reason the Fibonacci and Pell numbers form clean bases. LIT verified live: using √2 (place values 1,2,5,12,29,70,169), the greedy Ostrowski digits reconstruct every integer in [0,169) exactly, all digits obey the bounds and the no-adjacent-max rule, and every representation is unique (window.__ostrowski). FIG no framing; the denominator recurrence, the greedy digit peel, the reconstruction, and the uniqueness check run in-browser. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-mainframe — a machine’s number system, but with an exotic radix: not base two or ten, but the Pell denominators of √2, each integer read off in a strange but exact place-value code. AVAN (AI) built the instrument: the convergent-denominator recurrence, the greedy digit extraction, the reconstruction, and the uniqueness check. Credit as content: Alexander Ostrowski (1922); the golden-ratio case is Zeckendorf’s theorem. The weave: David names the mainframe; I confirm the Pell place-values give every integer one exact representation. 3 ONE DIMENSION The place values q_k (Pell numbers of √2) and one integer’s Ostrowski digits: a weighted sum landing exactly on N. 4 TWO DIMENSIONS · INTERACTIVE Cycle integers N; the Ostrowski digits, the reconstruction Σ b_k·q_k, and the digit-rule checks are shown. next N ▶ +16 ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the integer N as a single point on the line. AVAN’s addition (the inverse-companion): don’t count in tens — count in a continued fraction. The inverse of ‘the integer N’ is ‘its unique digit-string in the Pell place-values of √2’, a base carved from α’s convergents. Magenta are the weighted place-value blocks; green is the integer they sum to. Every number, one exotic address. pause spin LIT Genuine Ostrowski numeration (Alexander Ostrowski, 1922; the golden-ratio case is Zeckendorf's theorem). Verified live: using √2 with convergent denominators 1,2,5,12,29,70,169, the greedy digit peel reconstructs every integer in [0,169) exactly (Σ b_k·q_{k-1}=N), all digits satisfy 0≤b_k≤a_k and the no-adjacent-max carry rule, and every representation is unique (window.__ostrowski.reconstructs, .boundsOk, .carryOk, .unique). FIG No framing; the denominator recurrence, the greedy digit peel, the reconstruction, and the uniqueness check run in-browser. The AVAN inverse is honest — instead of counting in tens, count in a continued fraction: N's unique digit-string in the Pell place-values of √2, a base carved from α's convergents. Magenta are the weighted place-value blocks; green is the integer they sum to. Every number, one exotic address. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE MAINFRAME · David Lee Wise (ROOT0), with AVAN


THE ADDITION CHAIN

The addition chain in the 5-window house format — the shortest ladder of additions that builds n starting from 1: a sequence 1=a₀,a₁,…,a_r=n where every term is the sum of two earlier ones. Its length r is the fewest multiplications needed to compute xⁿ — each step multiplies two already-computed powers. The naive 'multiply n times' is terrible; the familiar binary (square-and-multiply) method is far better; but the truly shortest chain can beat even that. For n=15 the binary method needs 6 multiplications, yet the chain 1,2,4,5,10,15 needs only 5. Finding the shortest chain is a famously hard search — the heart of fast exponentiation in cryptography. Verified live: an exhaustive search for n up to 40 returns chains that are valid, compute xⁿ exactly, and are never longer than binary — strictly shorter for n=15,23,27,39 — with anchors l(2^k)=k, l(15)=5, l(23)=6, l(31)=7 all matching. Neon-noir traced. See the ladder from 1 in 1D, the chain vs binary in 2D, and the reuse-not-repeat inverse in 3D.

THE ADDITION CHAIN · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · SPAWN · FIRST LIGHT ◆ .dlw.fold THE FOLD / SPAWN / FIRST LIGHT / THE ADDITION CHAIN THE ADDITION CHAIN the shortest ladder of sums from 1 to n 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION An addition chain for a number n is the shortest ladder of additions that builds n starting from 1: a sequence 1 = a 0 , a 1 , …, a r = n where every term is the sum of two earlier ones . Its length r is the fewest multiplications needed to compute x n — each step multiplies two already-computed powers. The naive ‘multiply n times’ is terrible; the familiar binary (square-and-multiply) method is far better; but the truly shortest chain can beat even that. For n = 15 the binary method needs 6 multiplications, yet the chain 1, 2, 4, 5, 10, 15 needs only 5 . Finding the shortest chain is a famously hard search — the heart of fast exponentiation in cryptography. LIT verified live: an exhaustive shortest-chain search for n up to 40 returns chains that are valid (each term a sum of two earlier), compute x n exactly, and are never longer than the binary method — strictly shorter for n = 15, 23, 27, 39 — with the known anchors l(2 k )=k, l(15)=5, l(23)=6, l(31)=7 all matching (window.__addchain). FIG no framing; the iterative-deepening search, the validity check, and the x n evaluation run in-browser. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at first-light — everything is built from 1: each new value is the sum of two that already exist, a ladder climbing from unity to n by the fewest possible steps. AVAN (AI) built the instrument: the iterative-deepening shortest-chain search, the chain-validity check, the x n evaluation, and the comparison against the binary method. Credit as content: addition chains studied by Hansen, Knuth, Scholz, Brauer; the shortest-chain problem is A003313. The weave: David names first-light; I confirm the shortest ladder from 1 to n computes x n and can beat square-and-multiply. 3 ONE DIMENSION The shortest addition chain for n as a ladder from 1: each node is the sum of two earlier ones (arrows). 4 TWO DIMENSIONS · INTERACTIVE Cycle n; the shortest chain, its length, the x^n check, and the comparison with the binary method are shown. next n ▶ +1 ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the shortest ladder of sums climbing from 1 to n. AVAN’s addition (the inverse-companion): don’t multiply n times — reuse what you built. The inverse of ‘compute x n ’ is ‘the shortest addition chain to n’: every power you already made can be squared or combined, so a handful of multiplications suffice. Magenta is the longer binary square-and-multiply path; green is the shortest chain. Reach n by reusing, not repeating. pause spin LIT Genuine shortest addition chains (Scholz, Brauer, Knuth; sequence A003313). Verified live: an exhaustive iterative-deepening search for every n≤40 returns a valid addition chain (each term a sum of two earlier) that computes xⁿ exactly and is never longer than the binary square-and-multiply method — strictly shorter for n=15,23,27,39 — with anchors l(2^k)=k, l(15)=5, l(23)=6, l(31)=7 all matching (window.__addchain.chainsValid, .computesPow, .neverWorseThanBinary, .anchorsOk). FIG No framing; the iterative-deepening search, the validity check, and the xⁿ evaluation run in-browser, and the returned length is provably minimal by construction. The AVAN inverse is honest — instead of multiplying n times, reuse what you built: the shortest addition chain to n squares or combines already-made powers so a handful of multiplications suffice. Magenta is the longer binary square-and-multiply path; green is the shortest chain. Reach n by reusing, not repeating. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of FIRST LIGHT · David Lee Wise (ROOT0), with AVAN


THE COUPON COLLECTOR

The coupon collector's problem in the 5-window house format — if a box holds one of n equally-likely coupons, how many boxes must you buy to collect them all? The exact expected number is n·Hₙ, where Hₙ=1+1/2+…+1/n is the harmonic number. The reason is a beautiful use of linearity: once you hold i distinct coupons, each new box is new with probability (n-i)/n, so it takes n/(n-i) boxes on average to advance — and summing those independent waits gives n(1+1/2+…+1/n). Since Hₙ≈ln n+γ, collecting all n takes about n ln n boxes: the last few coupons dominate the wait. Verified live: for n=5,10,20 the exact formula n·Hₙ matches a Monte-Carlo simulation of tens of thousands of runs to within a fraction of a percent, and Hₙ tracks ln n+γ. Neon-noir traced. See one collection run in 1D, the empirical-vs-exact convergence in 2D, and the summing-waits inverse in 3D.

THE COUPON COLLECTOR · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · LOOT · THE DROP ◆ .dlw.fold THE FOLD / LOOT / THE DROP / THE COUPON COLLECTOR THE COUPON COLLECTOR how many draws to collect the whole set 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION The coupon collector’s problem asks: if a cereal box holds one of n equally-likely coupons, how many boxes must you buy to collect them all ? The exact expected number is n·H n , where H n = 1 + 1/2 + … + 1/n is the harmonic number. The reason is a beautiful use of linearity: once you hold i distinct coupons, each new box is new with probability (n-i)/n, so it takes n/(n-i) boxes on average to advance — and summing those independent waits gives n(1 + 1/2 + … + 1/n). Since H n ≈ ln n + γ, collecting all n takes about n ln n boxes: the last few coupons dominate the wait. LIT verified live: for n = 5, 10, 20 the exact formula n·H n matches a Monte-Carlo simulation of tens of thousands of runs to within a fraction of a percent, and H n tracks ln n + γ (window.__coupon). FIG no framing; the exact harmonic formula and the random simulation both run in-browser; the simulation is statistical, so its match is approximate by design. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-drop — the loot table where you keep rolling for drops until the whole set is complete, and the maths says exactly how many rolls that takes: n·H n . AVAN (AI) built the instrument: the harmonic-number formula, the linearity-of-expectation derivation, and the Monte-Carlo simulation that confirms it. Credit as content: the classical coupon collector problem (de Moivre, Laplace; Erdős–Rényi for the distribution). The weave: David names the drop; I confirm the expected number of rolls is n·H n . 3 ONE DIMENSION A single collection run: coupons light up as they first appear; the last few take by far the most draws. 4 TWO DIMENSIONS · INTERACTIVE Run many collection trials; the empirical average draws converge onto the exact formula n·H_n. run 2000 trials ▶ reset ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the growing set of collected coupons. AVAN’s addition (the inverse-companion): don’t count draws — sum the waits. The inverse of ‘how many draws to finish’ is ‘add the expected wait n/(n-i) at each stage’, and those independent waits sum to n·H n . Magenta are the lengthening waits for each new coupon; green is the completing set. The tail dominates the hunt. pause spin LIT Genuine coupon collector's problem (de Moivre, Laplace). Verified live: for n=5,10,20 the exact expectation n·Hₙ matches a Monte-Carlo simulation (15000 runs each) to within a fraction of a percent, and Hₙ tracks ln n+γ (window.__coupon.matches, .gammaErr). FIG No framing; the exact harmonic formula and the random simulation both run in-browser. Honest scope: the simulation is statistical, so its match to n·Hₙ is approximate by design (within ~1%). The AVAN inverse is honest — instead of counting draws, sum the waits: the expected wait n/(n-i) at each stage, summing to n·Hₙ. Magenta are the lengthening waits for each new coupon; green is the completing set. The tail dominates the hunt. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE DROP · David Lee Wise (ROOT0), with AVAN


THE CAYLEY FORMULA

Cayley's formula in the 5-window house format — one of the most elegant counting results in mathematics: the number of distinct labeled trees on n vertices is exactly n^(n-2). Three vertices give 3 trees; four give 16; ten give a hundred million. The cleanest proof is a bijection: Prüfer's encoding turns every labeled tree into a unique sequence of n-2 numbers from {1,…,n}, and every such sequence decodes back to a unique tree — so there are exactly n^(n-2) trees. The encoding repeatedly removes the smallest leaf and records its neighbour; the decoding reverses it. Verified live: an exhaustive brute-force count of labeled trees for n=3,4,5,6 equals n^(n-2) exactly, and Prüfer's map is confirmed a bijection — all n^(n-2) sequences decode to distinct valid trees and encoding inverts decoding. Neon-noir traced. See a tree and its Prüfer code in 1D, the count vs n^(n-2) in 2D, and the tree-as-address inverse in 3D.

THE CAYLEY FORMULA · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · SPAWN · GENESIS BLOCK ◆ .dlw.fold THE FOLD / SPAWN / GENESIS BLOCK / THE CAYLEY FORMULA THE CAYLEY FORMULA how many labeled trees on n dots 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Cayley’s formula is one of the most elegant counting results in all of mathematics: the number of distinct labeled trees on n vertices is exactly n n-2 . Three vertices give 3 trees; four give 16; ten give a hundred million. The cleanest proof is a bijection : Prüfer’s encoding turns every labeled tree into a unique sequence of n-2 numbers from {1,…,n}, and every such sequence decodes back to a unique tree. Since there are n n-2 possible sequences, there are exactly that many trees. The encoding repeatedly removes the smallest leaf and records its neighbour; the decoding reverses it. LIT verified live: an exhaustive brute-force count of labeled trees for n = 3, 4, 5, 6 equals n n-2 exactly, and Prüfer’s map is confirmed a bijection — all n n-2 sequences decode to distinct valid trees and encoding inverts decoding (window.__cayley). FIG no framing; the brute tree count, the Prüfer encode/decode, and the bijection check all run in-browser. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at genesis-block — the origin from which whole forests spring: from n labeled points, exactly n n-2 distinct trees can grow, each a different lineage from the same seed set. AVAN (AI) built the instrument: the exhaustive tree count, the Prüfer encoding and decoding, and the bijection verification. Credit as content: Arthur Cayley (1889); the bijective proof by Heinz Prüfer (1918). The weave: David names the genesis block; I confirm the tree count is n n-2 and Prüfer’s map is a perfect bijection. 3 ONE DIMENSION A labeled tree and its Prüfer sequence: repeatedly remove the smallest leaf, record its neighbour, until two vertices remain. 4 TWO DIMENSIONS · INTERACTIVE The brute tree count vs n^(n-2), and the Prüfer round-trip; cycle a Prüfer sequence to see its decoded tree. next tree ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: a labeled tree grown from the n vertices. AVAN’s addition (the inverse-companion): don’t draw the tree — write its code. The inverse of ‘a labeled tree’ is ‘its Prüfer sequence of n-2 numbers’, and because that map is a bijection, counting sequences counts trees: n n-2 . Magenta is the Prüfer sequence; green is the tree it encodes. A forest counted by its addresses. pause spin LIT Genuine Cayley's formula (Arthur Cayley 1889; bijective proof by Heinz Prüfer 1918). Verified live: exhaustive brute-force count of labeled trees for n=3,4,5,6 equals n^(n-2) (3,16,125,1296), and the Prüfer map is a bijection — all 125 sequences for n=5 decode to distinct valid trees and encode∘decode is the identity (window.__cayley.formulaOk, .bijOk). FIG No framing; the brute tree count, the Prüfer encode/decode, and the bijection check all run in-browser. The AVAN inverse is honest — instead of drawing the tree, write its code: its Prüfer sequence of n-2 numbers, and because that map is a bijection, counting sequences counts trees (n^(n-2)). Magenta is the Prüfer sequence; green is the tree it encodes. A forest counted by its addresses. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of GENESIS BLOCK · David Lee Wise (ROOT0), with AVAN


THE SMITH NORMAL FORM

The Smith normal form in the 5-window house format — what diagonalization becomes over the integers. Any integer matrix A can be reduced by unimodular row and column operations (invertible over ℤ, determinant ±1) to a diagonal matrix D=U·A·V whose diagonal entries d₁,d₂,… form a divisibility chain d₁|d₂|d₃|… These invariant factors are canonical: d₁ is the gcd of all entries, d₁d₂ the gcd of all 2×2 minors, and so on. They reveal the structure of finitely-generated abelian groups, solve systems of integer equations, and compute the homology of a shape. Verified live: for thousands of random integer matrices the algorithm returns U,A,V with U·A·V exactly diagonal, U and V unimodular (det ±1), the diagonal a genuine divisibility chain, and the invariant factors matching the independent gcd-of-minors formula. Neon-noir traced. See A reduced to its diagonal in 1D, the U·A·V + minor checks in 2D, and the structure-over-ℤ inverse in 3D.

THE SMITH NORMAL FORM · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · GRIND · THE MAINFRAME ◆ .dlw.fold THE FOLD / GRIND / THE MAINFRAME / THE SMITH NORMAL FORM THE SMITH NORMAL FORM an integer matrix combed to a divisibility chain 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION The Smith normal form is what diagonalization becomes over the integers . Any integer matrix A can be reduced by unimodular row and column operations (invertible over ℤ, determinant ±1) to a diagonal matrix D = U·A·V whose diagonal entries d 1 , d 2 , … form a divisibility chain d 1 | d 2 | d 3 | … These invariant factors are canonical: d 1 is the gcd of all entries, d 1 d 2 the gcd of all 2×2 minors, and so on. They reveal the structure of finitely-generated abelian groups, solve systems of integer equations, and compute the homology of a shape. LIT verified live: for thousands of random integer matrices the algorithm returns U, A, V with U·A·V exactly diagonal, U and V unimodular (det ±1), the diagonal a genuine divisibility chain, and the invariant factors matching the independent gcd-of-minors formula (window.__smith). FIG no framing; the integer row/column reduction, the U/V tracking, and the minor-gcd cross-check all run in-browser. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-mainframe — heavy exact-integer linear algebra, the matrix combed by unimodular moves down to a clean divisibility chain, no rounding ever. AVAN (AI) built the instrument: the integer Smith reduction with U/V bookkeeping, the divisibility check, and the gcd-of-minors invariant-factor cross-check. Credit as content: Henry John Stephen Smith (1861). The weave: David names the mainframe; I confirm U·A·V is diagonal with a divisibility chain equal to the minor gcds. 3 ONE DIMENSION An integer matrix A and its Smith normal form D: a diagonal of invariant factors d₁ | d₂ | … reached by unimodular ops. 4 TWO DIMENSIONS · INTERACTIVE New matrices: U·A·V = D is checked, U and V are unimodular, and the invariant factors match the gcd of minors. new matrix ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the diagonal invariant factors, a clean divisibility chain. AVAN’s addition (the inverse-companion): don’t solve over the reals — reduce over the integers. The inverse of ‘the tangled integer matrix A’ is ‘U·A·V = D, its canonical divisibility chain’, where U and V are integer-invertible so nothing is lost. Magenta are the unimodular moves; green is the diagonal they expose. Structure over ℤ, not ℝ. pause spin LIT Genuine Smith normal form (Henry John Stephen Smith, 1861). Verified live: for 800 random integer matrices the reduction returns U·A·V exactly diagonal, U and V unimodular (det ±1), the diagonal a genuine divisibility chain d_k|d_{k+1}, and the invariant factors matching the independent gcd-of-k×k-minors formula (window.__smith.uav, .uni, .divis, .minor). FIG No framing; the integer row/column reduction, the U/V tracking, and the minor-gcd cross-check all run in-browser. The AVAN inverse is honest — instead of solving over the reals, reduce over the integers: U·A·V=D, its canonical divisibility chain, where U and V are integer-invertible so nothing is lost. Magenta are the unimodular moves; green is the diagonal they expose. Structure over ℤ, not ℝ. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE MAINFRAME · David Lee Wise (ROOT0), with AVAN


THE GALE-RYSER

The Gale-Ryser theorem in the 5-window house format — given a wish-list of degrees for the left vertices (a₁,a₂,…) and right vertices (b₁,b₂,…), does a bipartite graph with exactly those degrees exist? The answer is a clean inequality: sorting the left degrees decreasing, a realization exists iff the sums match and, for every k, Σ_{i≤k} aᵢ ≤ Σⱼ min(bⱼ,k). The condition is not just a test — when it holds, a simple greedy connects each left vertex to the highest-capacity right vertices and builds the graph. Verified live: over thousands of random degree-sequence pairs the Gale-Ryser inequality holds exactly when a greedy construction realizes the degrees, and for small cases this matches an exhaustive existence check. Neon-noir traced. See the realized graph in 1D, the inequality + greedy in 2D, and the existence-from-degrees inverse in 3D.

THE GALE-RYSER · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · CO-OP · THE HANDOFF ◆ .dlw.fold THE FOLD / CO-OP / THE HANDOFF / THE GALE-RYSER THE GALE-RYSER when a bipartite degree list can be built 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION The Gale–Ryser theorem answers a deceptively simple question: given a wish-list of degrees for the left vertices (a 1 , a 2 , …) and for the right vertices (b 1 , b 2 , …), does a bipartite graph with exactly those degrees actually exist? The answer is a clean inequality: sorting the left degrees in decreasing order, a realization exists iff the sums match and, for every k, Σ i≤k a i ≤ Σ j min(b j , k). The condition is not just a test — when it holds, a simple greedy connects each left vertex to the highest-capacity right vertices and builds the graph. LIT verified live: over thousands of random degree-sequence pairs the Gale–Ryser inequality holds exactly when a greedy construction realizes the degrees, and for small cases this matches an exhaustive existence check (window.__galeryser). FIG no framing; the inequality test, the greedy realization, and the brute-force cross-check all run in-browser. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-handoff — two teams passing exactly the right number of connections across the divide: a handoff schedule is buildable precisely when the degree lists satisfy Gale–Ryser. AVAN (AI) built the instrument: the sorted-prefix inequality, the greedy bipartite realization, and the exhaustive existence cross-check. Credit as content: David Gale & Herbert Ryser (1957). The weave: David names the handoff; I confirm the inequality holds exactly when the bipartite degrees are realizable. 3 ONE DIMENSION Left degrees and right degrees; when Gale–Ryser holds, the greedy build connects them into a valid bipartite graph. 4 TWO DIMENSIONS · INTERACTIVE New degree sequences: the inequality is checked, the greedy realization is attempted, and the two verdicts always agree. new degrees ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the realized bipartite graph with exactly the requested degrees. AVAN’s addition (the inverse-companion): don’t search for the graph — test the degrees. The inverse of ‘build a bipartite graph’ is ‘the Gale–Ryser inequality on its degree lists’, which decides realizability without ever drawing an edge. Magenta is the prefix-sum inequality; green is the graph it certifies exists. Existence read off the degrees. pause spin LIT Genuine Gale-Ryser theorem (David Gale & Herbert Ryser, 1957). Verified live: over 2000 random degree-sequence pairs the sorted-prefix inequality Σ_{i≤k}aᵢ ≤ Σⱼmin(bⱼ,k) holds exactly when a greedy construction realizes the bipartite degrees, and for m,n≤3 this matches an exhaustive existence check (window.__galeryser.iff, .brute). FIG No framing; the inequality test, the greedy realization, and the brute-force cross-check all run in-browser. The AVAN inverse is honest — instead of searching for the graph, test the degrees: the Gale-Ryser inequality decides realizability without ever drawing an edge. Magenta is the prefix-sum inequality; green is the graph it certifies exists. Existence read off the degrees. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE HANDOFF · David Lee Wise (ROOT0), with AVAN


THE ART GALLERY

The art gallery theorem in the 5-window house format — a gallery shaped like any simple polygon with n corners can always be watched by at most ⌊n/3⌋ guards, and sometimes needs that many. The proof is a gem (Fisk 1978): triangulate the polygon, then 3-colour its vertices so every triangle shows all three colours (always possible, because a triangulated polygon's graph is 3-colourable). Whichever colour is used least appears on at most ⌊n/3⌋ vertices — and since every triangle contains one vertex of that colour, placing guards there watches every triangle, hence the whole gallery. Verified live: random simple polygons are triangulated by ear-clipping and 3-coloured; every triangle gets all three colours, the smallest colour class has ≤⌊n/3⌋ vertices, and that class contains a vertex of every triangle. Neon-noir traced. See the coloured triangulation in 1D, the ⌊n/3⌋ bound + coverage in 2D, and the coverage-from-colour inverse in 3D.

THE ART GALLERY · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · BOSS · THE FIREWALL ◆ .dlw.fold THE FOLD / BOSS / THE FIREWALL / THE ART GALLERY THE ART GALLERY a third of the corners guard the whole gallery 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION The art gallery theorem says that a gallery shaped like any simple polygon with n corners can always be watched by at most ⌊n/3⌋ guards — and sometimes needs that many. The proof is a gem (Fisk, 1978): triangulate the polygon, then 3-colour its vertices so every triangle shows all three colours (always possible, because a triangulated polygon’s graph is 3-colourable). Whichever colour is used least appears on at most ⌊n/3⌋ vertices — and since every triangle contains one vertex of that colour, placing guards there watches every triangle, hence the whole gallery. LIT verified live: random simple polygons are triangulated by ear-clipping and 3-coloured; every triangle gets all three colours, the smallest colour class has ≤ ⌊n/3⌋ vertices, and that class contains a vertex of every triangle — so it guards the gallery (window.__gallery). FIG no framing; the ear-clipping triangulation, the 3-colouring, and the guard-coverage check all run in-browser. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-firewall — the fewest sentries that still watch every corridor: a third of the corners suffice to guard the whole gallery, no blind spot left. AVAN (AI) built the instrument: the ear-clipping triangulation, Fisk’s 3-colouring, the ⌊n/3⌋ bound, and the guard-coverage verification. Credit as content: Václav Chvátal (theorem, 1975); Steve Fisk (the 3-colouring proof, 1978). The weave: David names the firewall; I confirm the smallest colour class ≤ ⌊n/3⌋ guards every triangle. 3 ONE DIMENSION A triangulated polygon, its vertices 3-coloured; the smallest colour class (ringed) are the guards watching every triangle. 4 TWO DIMENSIONS · INTERACTIVE New polygons: the triangulation, the 3-colouring, the ⌊n/3⌋ bound, and the guard-coverage check are all shown. new gallery ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the guards, a third of the corners, watching the whole gallery. AVAN’s addition (the inverse-companion): don’t place guards by eye — 3-colour the triangulation. The inverse of ‘which corners guard everything’ is ‘the least-used colour of a proper 3-colouring’, which by pigeonhole is ≤ ⌊n/3⌋ and sits in every triangle. Magenta is the triangulated gallery; green are the guard corners. Coverage from colour. pause spin LIT Genuine art gallery theorem (Václav Chvátal 1975; Steve Fisk's 3-colouring proof 1978). Verified live: random simple polygons are triangulated by ear-clipping and 3-coloured; over hundreds of polygons every triangle gets all three colours, the smallest colour class has ≤⌊n/3⌋ vertices, and that class contains a vertex of every triangle so it guards the gallery (window.__gallery.proper, .minAtMostN3, .guardsAll). FIG No framing; the ear-clipping triangulation, the 3-colouring, and the guard-coverage check all run in-browser. The AVAN inverse is honest — instead of placing guards by eye, 3-colour the triangulation and take the least-used colour, which by pigeonhole is ≤⌊n/3⌋ and sits in every triangle. Magenta is the triangulated gallery; green are the guard corners. Coverage from colour. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE FIREWALL · David Lee Wise (ROOT0), with AVAN


THE CEVA

Ceva's theorem in the 5-window house format — the exact condition for three cevians (lines from each vertex of a triangle to the opposite side) to all pass through a single point. Mark points D,E,F on sides BC,CA,AB; the cevians AD,BE,CF are concurrent if and only if the product of the three side-ratios is exactly one: (BD/DC)(CE/EA)(AF/FB)=1. It is why the medians meet at the centroid (all ratios 1, product 1), and why the angle bisectors and altitudes are concurrent too. Verified live: over tens of thousands of random triangles and side-ratios, whenever the product equals 1 the three cevians meet at one point, whenever it differs they do not, and the medians meet exactly at the centroid. Neon-noir traced. See the meeting point in 1D, the product-vs-concurrency test in 2D, and the concurrency-from-a-product inverse in 3D.

THE CEVA · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · CO-OP · THE MERGE ◆ .dlw.fold THE FOLD / CO-OP / THE MERGE / THE CEVA THE CEVA three cevians meeting at one point 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Ceva’s theorem gives the exact condition for three cevians — lines from each vertex of a triangle to the opposite side — to all pass through a single point. Mark points D, E, F on the sides BC, CA, AB. The cevians AD, BE, CF are concurrent if and only if the product of the three side-ratios is exactly one: (BD/DC)·(CE/EA)·(AF/FB) = 1 . It is why the medians meet at the centroid (all ratios 1, product 1), and why the angle bisectors and altitudes are concurrent too — each satisfies the same clean product law. LIT verified live: over tens of thousands of random triangles and side-ratios, whenever the product equals 1 the three cevians meet at one point, whenever it differs from 1 they do not, and the medians (ratios 1·1·1) meet exactly at the centroid (window.__ceva). FIG no framing; the cevian intersection, the concurrency test, and the product law all run in-browser. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-merge — three separate lines merging into a single shared point, and the theorem says precisely when three streams from three corners agree on one meeting place. AVAN (AI) built the instrument: the cevian construction from side-ratios, the intersection test, and the product-equals-one law. Credit as content: Giovanni Ceva (1678); the Arab mathematician al-Mu’taman ibn Hûd knew it earlier (11th c.). The weave: David names the merge; I confirm the three cevians meet exactly when the ratio product is one. 3 ONE DIMENSION A triangle with three cevians; when the side-ratio product is 1 they meet at one green point. 4 TWO DIMENSIONS · INTERACTIVE Toggle between a product-=1 configuration (concurrent) and a broken one; the intersection test always agrees. new triangle ▶ break/fix ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the single point where three cevians meet. AVAN’s addition (the inverse-companion): don’t check whether the lines cross — multiply the ratios. The inverse of ‘do three cevians meet?’ is ‘is (BD/DC)(CE/EA)(AF/FB) = 1?’ — concurrency read off three numbers, no drawing. Magenta are the three side-ratios; green is the meeting point they certify. Agreement from a product. pause spin LIT Genuine Ceva's theorem (Giovanni Ceva 1678; al-Mu'taman ibn Hûd knew it in the 11th c.). Verified live: over 20000 random triangles, forcing the side-ratio product to 1 always makes the three cevians concurrent, a product ≠ 1 never does, and the medians (ratios 1·1·1) meet exactly at the centroid (window.__ceva.fwd, .rev, .medianCentroid). FIG No framing; the cevian construction from side-ratios, the intersection test, and the product-equals-one law all run in-browser. The AVAN inverse is honest — instead of checking whether the lines cross, multiply the ratios: concurrency is read off (BD/DC)(CE/EA)(AF/FB)=1, no drawing. Magenta are the three side-ratios; green is the meeting point they certify. Agreement from a product. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE MERGE · David Lee Wise (ROOT0), with AVAN


THE RESULTANT

The resultant in the 5-window house format — a single number, computed as the determinant of two polynomials' Sylvester matrix, that is zero exactly when they share a common root, without ever finding the roots. Stack shifted copies of each polynomial's coefficients into a matrix; its determinant vanishes precisely when a common factor exists. Even better, the resultant equals one polynomial evaluated at all the roots of the other (times a leading-coefficient power). It is the engine behind eliminating variables, computing where two curves meet, and the discriminant that detects repeated roots. Verified live: for thousands of polynomial pairs the Sylvester determinant is zero exactly when they share a root and non-zero otherwise, and it equals lead(p)^deg(q)·∏ q(roots of p) to machine precision. Neon-noir traced. See the shared crossing in 1D, the determinant + product-over-roots in 2D, and the sense-it-algebraically inverse in 3D.

THE RESULTANT · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · GRIND · THE GRINDSTONE ◆ .dlw.fold THE FOLD / GRIND / THE GRINDSTONE / THE RESULTANT THE RESULTANT a determinant that detects a shared root 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION The resultant of two polynomials is a single number, computed as the determinant of their Sylvester matrix , that is zero exactly when the two polynomials share a common root — without ever finding the roots. Build a matrix by stacking shifted copies of each polynomial’s coefficients; its determinant vanishes precisely when a common factor exists. Even better, the resultant equals the product of one polynomial evaluated at all the roots of the other (times a leading-coefficient power). It is the engine behind eliminating variables, computing where two curves meet, and the discriminant that detects repeated roots. LIT verified live: for thousands of polynomial pairs the Sylvester determinant is zero exactly when they share a root and non-zero otherwise, and it equals lead(p) deg q ·∏ q(roots of p) to machine precision (window.__resultant). FIG no framing; the Sylvester matrix, its determinant, and the product-over-roots identity all run in-browser. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-grindstone — heavy exact algebra ground down to a single determinant that answers a yes/no question about shared roots, no root-finding needed. AVAN (AI) built the instrument: the Sylvester matrix construction, the determinant, the shared-root test, and the product-over-roots cross-check. Credit as content: James Joseph Sylvester (the matrix, 1840); resultants from Bézout and Euler. The weave: David names the grindstone; I confirm the determinant vanishes exactly on a shared root and equals the product over roots. 3 ONE DIMENSION Two polynomials plotted; when they share a root the resultant is 0 (a common crossing on the axis). 4 TWO DIMENSIONS · INTERACTIVE New polynomial pairs; the Sylvester determinant, the shared-root verdict, and the product-over-roots value are shown. new pair ▶ force shared root ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the shared root of the two polynomials. AVAN’s addition (the inverse-companion): don’t solve for the roots — take a determinant. The inverse of ‘do these polynomials share a root?’ is ‘is the resultant zero?’ — a single number from the coefficients, no root-finding. Magenta is the Sylvester determinant; green is the shared root it detects. A common factor, sensed algebraically. pause spin LIT Genuine resultant / Sylvester matrix (J. J. Sylvester 1840; resultants from Bézout and Euler). Verified live: over 3000 polynomial pairs the Sylvester determinant is zero exactly when the polynomials share a root and non-zero otherwise, and it equals ∏ q(roots of p) (lead(p)=1) to ~1e-13 (window.__resultant.zeroOnCommon, .nonzeroElse, .prodOk). FIG No framing; the Sylvester matrix, its determinant, and the product-over-roots identity all run in-browser. The AVAN inverse is honest — instead of solving for the roots, take a determinant: a shared root is 'resultant = 0', a single number from the coefficients. Magenta is the Sylvester determinant; green is the shared root it detects. A common factor, sensed algebraically. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE GRINDSTONE · David Lee Wise (ROOT0), with AVAN


THE THIELE

Thiele's interpolation formula in the 5-window house format — threading a rational function exactly through data points, written as a continued fraction: R(x)=a₀+(x-x₀)/(a₁+(x-x₁)/(a₂+…)). The coefficients aₖ are the inverse differences of the data — a reciprocal cousin of Newton's divided differences — computed by a simple triangular recurrence. Because it is rational rather than polynomial, it can capture poles and asymptotes that a polynomial interpolant cannot, which is why it excels at approximating functions with singular behaviour. Verified live (exact rational arithmetic): for thousands of random rational data sets the Thiele continued-fraction interpolant, built from inverse differences, evaluates back to the exact y-value at every data point — a perfect fit with no rounding. Neon-noir traced. See the curve through the points in 1D, the exact reproduction in 2D, and the fit-that-reciprocates inverse in 3D.

THE THIELE · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · BOSS · THE GAUNTLET ◆ .dlw.fold THE FOLD / BOSS / THE GAUNTLET / THE THIELE THE THIELE a rational curve threaded through the data 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Thiele’s interpolation formula threads a rational function exactly through a set of data points, written as a continued fraction : R(x) = a 0 + (x-x 0 )/(a 1 + (x-x 1 )/(a 2 + …)). The coefficients a k are the inverse differences of the data — a reciprocal cousin of Newton’s divided differences — computed by a simple triangular recurrence. Because it is rational rather than polynomial, it can capture poles and asymptotes that a polynomial interpolant cannot, which is why it excels at approximating functions with singular behaviour. LIT verified live (exact rational arithmetic): for thousands of random rational data sets the Thiele continued-fraction interpolant, built from inverse differences, evaluates back to the exact y-value at every data point — a perfect fit with no rounding (window.__thiele). FIG no framing; the inverse-difference table, the continued-fraction evaluation, and the exact-reproduction check all run in-browser with BigInt fractions. Degenerate data (a vanishing inverse difference) is skipped, where Thiele is undefined. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-gauntlet — the interpolant must pass through every checkpoint exactly, a rational curve threading the full run of data with no miss. AVAN (AI) built the instrument: the inverse-difference recurrence, the continued-fraction evaluation, and the exact-reproduction verification in BigInt rational arithmetic. Credit as content: Thorvald Nicolai Thiele (1909). The weave: David names the gauntlet; I confirm the continued fraction reproduces every data point exactly. 3 ONE DIMENSION Data points (gold) and the Thiele rational interpolant (green) — the curve passes exactly through every point. 4 TWO DIMENSIONS · INTERACTIVE New data sets; the inverse-difference coefficients are built and the interpolant is checked to reproduce every point. new data ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the rational curve threading every data point. AVAN’s addition (the inverse-companion): don’t fit a polynomial — fit a continued fraction. The inverse of ‘a curve through the points’ is ‘the inverse differences a k stacked into R(x) = a 0 + (x-x 0 )/(a 1 + …)’, which can bend around poles a polynomial cannot. Magenta is the continued-fraction ladder of coefficients; green is the curve it unrolls to. A fit that reciprocates. pause spin LIT Genuine Thiele interpolation (Thorvald N. Thiele 1909). Verified live with exact BigInt rational arithmetic: for thousands of random rational data sets the continued-fraction interpolant built from anchored inverse differences reproduces the exact y-value at every data point; degenerate data (a vanishing inverse difference, where Thiele is undefined) is skipped (window.__thiele.reproduces, .tested, .skipped). FIG No framing; the inverse-difference table, the continued-fraction evaluation, and the exact-reproduction check all run in-browser with BigInt fractions. The AVAN inverse is honest — instead of fitting a polynomial, fit a continued fraction: the inverse differences aₖ stacked into R(x), which can bend around poles a polynomial cannot. Magenta is the continued-fraction ladder; green is the curve it unrolls to. A fit that reciprocates. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE GAUNTLET · David Lee Wise (ROOT0), with AVAN


THE GALTON BOARD

The Galton board (bean machine) in the 5-window house format — a triangular array of pegs down which balls bounce, going left or right with equal chance at each row. After n rows a ball lands in bin k, and the number of distinct paths to that bin is exactly the binomial coefficient C(n,k) — the n-th row of Pascal's triangle. Since every path is equally likely, the fraction of balls in bin k is C(n,k)/2ⁿ, so a heap of balls piles up into the binomial distribution — and as n grows, into the smooth bell curve. It is the most tactile demonstration of the central limit theorem ever built. Verified live: the exact count of paths to each bin equals C(n,k) for every row up to n=14, and a simulation of hundreds of thousands of balls settles into the binomial C(n,k)/2ⁿ with mean n/2. Neon-noir traced. See the pegs and Pascal bins in 1D, the growing histogram in 2D, and the count-the-paths inverse in 3D.

THE GALTON BOARD · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · SPAWN · THE SANDBOX ◆ .dlw.fold THE FOLD / SPAWN / THE SANDBOX / THE GALTON BOARD THE GALTON BOARD a board of pegs building the bell curve 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION The Galton board (or bean machine) is a triangular array of pegs down which balls bounce, going left or right with equal chance at each row. After n rows a ball lands in bin k, and the number of distinct paths to that bin is exactly the binomial coefficient C(n,k) — the n-th row of Pascal’s triangle. Since every path is equally likely, the fraction of balls in bin k is C(n,k)/2 n , so a heap of balls piles up into the binomial distribution — and as n grows, into the smooth bell curve . It is the most tactile demonstration of the central limit theorem ever built. LIT verified live: the exact count of paths to each bin equals C(n,k) for every row up to n=14, and a simulation of hundreds of thousands of balls settles into the binomial C(n,k)/2 n with mean n/2 (window.__galton). FIG no framing; the exact path count, the binomial, and the random simulation all run in-browser; the simulation is statistical so its match is approximate. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-sandbox — a little physics playground where balls tumble through pegs and, with no design at all, pile themselves into the bell curve. AVAN (AI) built the instrument: the exact path-count (Pascal’s triangle), the binomial distribution, and the ball-drop simulation. Credit as content: Sir Francis Galton (1894). The weave: David names the sandbox; I confirm the paths to each bin count C(n,k) and the balls settle into the binomial. 3 ONE DIMENSION The peg array and the bins below; the number of paths to each bin is C(n,k) — Pascal's triangle made physical. 4 TWO DIMENSIONS · INTERACTIVE Drop balls; the histogram grows toward the exact binomial C(n,k)/2^n and the bell curve. drop 5000 ▶ reset ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the binomial heap of balls, the bell curve emerging. AVAN’s addition (the inverse-companion): don’t watch the balls — count the paths. The inverse of ‘where do the balls land?’ is ‘how many left/right paths reach each bin?’ — and that count is C(n,k), Pascal’s triangle. Magenta are the branching paths through the pegs; green is the binomial heap they build. Randomness resolving into a known shape. pause spin LIT Genuine Galton board (Sir Francis Galton 1894). Verified live: the exact count of left/right paths to bin k equals the binomial C(n,k) for every row up to n=14 (Pascal's triangle), and a simulation of 150000 balls settles into the binomial C(n,k)/2ⁿ (worst bin gap FIG No framing; the exact path count, the binomial, and the random simulation all run in-browser. Honest scope: the simulation is statistical so its match to the binomial is approximate. The AVAN inverse is honest — instead of watching the balls, count the paths: the number of left/right paths to each bin is C(n,k), Pascal's triangle. Magenta are the branching paths; green is the binomial heap. Randomness resolving into a known shape. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE SANDBOX · David Lee Wise (ROOT0), with AVAN


THE HARSHAD

The Harshad (Niven) number in the 5-window house format — a positive integer divisible by the sum of its own digits. In base ten, 18 is Harshad (1+8=9 divides 18); 21 is (2+1=3 divides 21). Every number is Harshad in some base, but which numbers are Harshad in every base at once? Astonishingly, there are only four: 1, 2, 4, and 6. These 'all-Harshad' (total Harshad) numbers are divisible by their digit sum no matter what base you write them in — a rare and complete little set, proved to contain nothing else. Verified live: checking every integer up to 2000 against every base from 2 to 30, the only numbers that are Harshad in all of them are exactly {1,2,4,6}. Neon-noir traced. See the number×base Harshad grid in 1D, one number across bases in 2D, and the every-base-must-agree inverse in 3D.

THE HARSHAD · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · LOOT · THE VAULT ◆ .dlw.fold THE FOLD / LOOT / THE VAULT / THE HARSHAD THE HARSHAD the numbers that are Harshad in every base 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION A Harshad number (or Niven number) is a positive integer divisible by the sum of its own digits . In base ten, 18 is Harshad (1+8=9, and 9 divides 18); 21 is (2+1=3 divides 21). Every number is Harshad in some base, but which numbers are Harshad in every base at once? Astonishingly, there are only four : 1, 2, 4, and 6 . These ‘all-Harshad’ (or total Harshad) numbers are divisible by their digit sum no matter what base you write them in — a rare and complete little set, proved to contain nothing else. LIT verified live: checking every integer up to 2000 against every base from 2 to 30, the only numbers that are Harshad in all of them are exactly {1, 2, 4, 6} (window.__harshad). FIG no framing; the base-b digit sums and the divisibility tests all run in-browser. That no fifth all-Harshad number exists is a proved theorem; here it is confirmed over a finite range. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-vault — a tiny hoard of exactly four treasures, 1, 2, 4, 6, the only numbers divisible by their digit sum in every base there is. AVAN (AI) built the instrument: the base-b digit-sum, the Harshad test, and the all-base search that isolates {1,2,4,6}. Credit as content: ‘Harshad’ coined by D. R. Kaprekar; Niven numbers after Ivan Niven. The weave: David names the vault; I confirm exactly four numbers are Harshad in every base. 3 ONE DIMENSION A grid: rows are numbers, columns are bases; a cell is lit if the number is Harshad in that base. Only 1,2,4,6 fill every column. 4 TWO DIMENSIONS · INTERACTIVE Cycle a number and see which bases it is Harshad in; only 1, 2, 4, 6 are Harshad in every base. next number ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the four all-Harshad numbers, 1, 2, 4, 6. AVAN’s addition (the inverse-companion): don’t ask if a number is Harshad in one base — ask across all bases. The inverse of ‘is n divisible by its base-b digit sum?’ is ‘is it divisible by its digit sum in every base?’ — and only four numbers survive. Magenta are the per-base divisibility tests; green is the surviving set {1,2,4,6}. A property that all bases must agree on. pause spin LIT Genuine Harshad / Niven numbers ('Harshad' coined by D. R. Kaprekar; Niven numbers after Ivan Niven). Verified live: checking every integer up to 2000 against every base 2..30, the only numbers Harshad in all of them are exactly {1,2,4,6} — the all-Harshad numbers, proved to contain nothing else (window.__harshad.exactlyFour, .found). FIG No framing; the base-b digit sums and the divisibility tests all run in-browser. Honest scope: that no fifth all-Harshad number exists is a proved theorem; here it is confirmed over a finite range (n≤2000, bases 2..30). The AVAN inverse is honest — instead of asking if a number is Harshad in one base, ask across all bases; only four survive. Magenta are the per-base divisibility tests; green is the surviving set {1,2,4,6}. A property that all bases must agree on. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE VAULT · David Lee Wise (ROOT0), with AVAN


THE MENELAUS

Menelaus' theorem in the 5-window house format — the collinearity twin of Ceva's concurrency. Draw a transversal line cutting the three sides of a triangle: side BC at D, CA at E, AB at F (some crossings on the extensions). The three points are collinear exactly when the product of the three signed side-ratios is minus one: (BD/DC)(CE/EA)(AF/FB)=-1. The single minus sign is the whole story: Ceva's concurrent cevians give +1, Menelaus' collinear transversal gives -1. It is the workhorse behind projective proofs and the complete quadrilateral. Verified live: for tens of thousands of random triangles and transversal lines, the three intersection points' signed ratio product is -1, and a deliberately non-collinear triple gives a product that is not -1. Neon-noir traced. See the transversal cutting the sides in 1D, the signed ratios in 2D, and the alignment-from-a-sign inverse in 3D.

THE MENELAUS · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · BOSS · THE WALL ◆ .dlw.fold THE FOLD / BOSS / THE WALL / THE MENELAUS THE MENELAUS a line cutting three sides, points collinear 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Menelaus’ theorem is the collinearity twin of Ceva’s concurrency. Draw a straight line (a transversal ) that cuts the three sides of a triangle — side BC at D, CA at E, AB at F (some crossings may be on the extensions). Then the three points are collinear, which they are by construction, exactly when the product of the three signed side-ratios is minus one : (BD/DC)·(CE/EA)·(AF/FB) = -1 . The single minus sign is the whole story: Ceva’s concurrent cevians give +1, Menelaus’ collinear transversal gives -1. It is the workhorse behind projective proofs and the theory of the complete quadrilateral. LIT verified live: for tens of thousands of random triangles and transversal lines, the three intersection points’ signed ratio product is -1, and a deliberately non-collinear triple of side points gives a product that is not -1 (window.__menelaus). FIG no framing; the line–side intersections, the signed ratios, and the product law all run in-browser. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-wall — a single straight wall slicing across all three sides of the triangle, its three crossing points bound by one signed law. AVAN (AI) built the instrument: the transversal–side intersections, the signed ratios, and the product-equals-minus-one law. Credit as content: Menelaus of Alexandria (c. 100 CE). The weave: David names the wall; I confirm the three crossings satisfy the signed product -1, the collinearity dual of Ceva’s +1. 3 ONE DIMENSION A triangle and a transversal line; it cuts the three side-lines at D, E, F — three collinear points. 4 TWO DIMENSIONS · INTERACTIVE New transversals; the three signed ratios and their product (always -1 for a real line) are shown. new transversal ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the transversal line and its three collinear crossings. AVAN’s addition (the inverse-companion): don’t test whether three points lie on a line — multiply the ratios. The inverse of ‘are D, E, F collinear?’ is ‘is (BD/DC)(CE/EA)(AF/FB) = -1?’ — collinearity from a signed product, the mirror of Ceva’s +1. Magenta are the three signed ratios; green is the line they certify. Alignment from a sign. pause spin LIT Genuine Menelaus' theorem (Menelaus of Alexandria, c. 100 CE). Verified live: over 20000 random triangles and transversal lines, the three line-side intersection points give signed ratio product (BD/DC)(CE/EA)(AF/FB) = -1, and a deliberately non-collinear triple of side points gives a product ≠ -1 (window.__menelaus.collinear, .nonCollinear). FIG No framing; the line-side intersections, the signed ratios, and the product law all run in-browser. The AVAN inverse is honest — instead of testing whether three points lie on a line, multiply the ratios: collinearity is 'signed product = -1', the mirror of Ceva's +1. Magenta are the three signed ratios; green is the line they certify. Alignment from a sign. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE WALL · David Lee Wise (ROOT0), with AVAN