THE WEYL EQUIDISTRIBUTION

Weyl's equidistribution theorem in the 5-window house format — the fractional parts of the multiples of an irrational number spread out perfectly evenly. Take any irrational α and look at the sequence {α}, {2α}, {3α}, … (fractional parts, mod 1). Weyl proved these points become equidistributed in [0,1): the fraction landing in any subinterval [a,b) converges to its length b−a. The sequence never settles into a pattern — it fills the interval as uniformly as possible. For a rational α=p/q, by contrast, the fractional parts cycle through only q values and are never equidistributed. Verified live: for α=√2, φ, π, e, the star discrepancy of {nα} shrinks toward zero as N grows — below 1e-3 by N=20000 — while for rational α=1/3 the discrepancy stays large. Neon-noir traced. See the points filling [0,1) in 1D, the discrepancy → 0 + control in 2D, and the order-into-uniformity inverse in 3D.

THE WEYL EQUIDISTRIBUTION · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · SPAWN · COLD BOOT ◆ .dlw.fold THE FOLD / SPAWN / COLD BOOT / THE WEYL EQUIDISTRIBUTION THE WEYL EQUIDISTRIBUTION irrational multiples filling the interval evenly 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Weyl’s equidistribution theorem says the fractional parts of the multiples of an irrational number spread out perfectly evenly. Take any irrational α and look at the sequence {α}, {2α}, {3α}, … (fractional parts, mod 1). Weyl proved these points become equidistributed in [0,1): the fraction landing in any subinterval [a,b) converges to its length b-a. The sequence never settles into a pattern — it fills the interval as uniformly as possible. For a rational α = p/q, by contrast, the fractional parts cycle through only q values and are never equidistributed. LIT verified live: for α = √2, φ, π, e, the star discrepancy of {nα} (the maximum gap between the empirical and uniform distribution) shrinks toward zero as N grows — below 1e-3 by N = 20000 — while for a rational α = 1/3 the discrepancy stays large (window.__weyl). FIG no framing; the fractional-part sequence and the discrepancy measure both run in-browser. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at cold-boot — the spawn: each new multiple of an irrational drops a point into the interval, and cold-booting up from nothing they fill it perfectly evenly. AVAN (AI) built the instrument: the fractional-part sequence, the star-discrepancy measure, and the rational control. Credit as content: Hermann Weyl (1916). The weave: David names the even fill; I confirm {nα} equidistributes for irrational α and not for rational. 3 ONE DIMENSION The points {nα} accumulating in [0,1) for an irrational α — filling the interval with no gaps or clumps. 4 TWO DIMENSIONS · INTERACTIVE Cycle α; the discrepancy (deviation from uniform) shrinks for irrationals, stays large for rationals. next α ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the uniformly-filled interval from an irrational's multiples. AVAN’s addition (the inverse-companion): don’t track each point — know the density. The inverse of ‘the sequence {nα}’ is ‘the uniform distribution on [0,1)’, which it converges to exactly when α is irrational. Magenta are the sequence points; green is the flat uniform density they fill out. Order dissolving into uniformity. pause spin LIT Genuine Weyl equidistribution theorem (Hermann Weyl, 1916). Verified live: for α=√2, φ, π, e the star discrepancy of {nα} falls below 0.01 by N=20000 (equidistributed), while a rational α=1/3 keeps discrepancy ≈0.33 (window.__weyl.ok, .ctrl, .rows). FIG No framing; the fractional-part sequence and the discrepancy measure both run in-browser. The AVAN inverse is honest — instead of tracking each point, know the density: the inverse of 'the sequence {nα}' is 'the uniform distribution on [0,1)', which it converges to exactly when α is irrational. Magenta are the sequence points; green is the flat uniform density they fill out. Order dissolving into uniformity. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of COLD BOOT · David Lee Wise (ROOT0), with AVAN


THE GAMMA REFLECTION

Euler's reflection formula in the 5-window house format — tying the gamma function to the sine in one clean stroke: Γ(x)·Γ(1−x) = π/sin(πx). The gamma function Γ extends the factorial to all real (and complex) numbers, and it looks nothing like a trig function — yet multiply its value at x by its value at the mirror point 1−x, and the messy transcendental factorials collapse into a simple cosecant. Setting x=½ gives Γ(½)²=π, so Γ(½)=√π — the gateway to the Gaussian integral. The poles of the gamma function at 0,−1,−2,… line up exactly with the zeros of sine. Verified live: computing Γ by the Lanczos approximation, the product Γ(x)·Γ(1−x) equals π/sin(πx) to relative error ~1e-14 for thousands of x in (0,1), and Γ(½)²=π. Neon-noir traced. See the mirrored gamma curves in 1D, product vs cosecant in 2D, and the factorials-reflected inverse in 3D.

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

THE GAMMA REFLECTION · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · CHEAT · THE BACKDOOR ◆ .dlw.fold THE FOLD / CHEAT / THE BACKDOOR / THE GAMMA REFLECTION THE GAMMA REFLECTION a gamma product equal to a cosecant 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Euler’s reflection formula ties the gamma function to the sine in one clean stroke: Γ(x)·Γ(1-x) = π/sin(πx). The gamma function Γ extends the factorial to all real (and complex) numbers, and it looks nothing like a trig function — yet multiply its value at x by its value at the mirror point 1-x, and the messy transcendental factorials collapse into a simple cosecant. Setting x = ½ gives Γ(½)² = π, so Γ(½) = √π — the gateway to the Gaussian integral. The poles of the gamma function at 0, -1, -2, … line up exactly with the zeros of sine. LIT verified live: computing Γ by the Lanczos approximation, the product Γ(x)·Γ(1-x) equals π/sin(πx) to a relative error ~1e-14 for thousands of x in (0,1), and Γ(½)² = π (window.__gammareflection). FIG no framing; the gamma product and the cosecant are computed by different routes in-browser and agree. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-backdoor — the cheat: instead of evaluating a hard factorial, reflect across x = ½ and read the answer off a sine. AVAN (AI) built the instrument: the Lanczos gamma, the reflection product, and the π/sin(πx) cross-check. Credit as content: Leonhard Euler (reflection formula). The weave: David names the backdoor; I confirm Γ(x)Γ(1-x) equals π/sin(πx). 3 ONE DIMENSION Γ(x) and its mirror Γ(1−x) on (0,1); their product traces exactly the curve π/sin(πx). 4 TWO DIMENSIONS · INTERACTIVE Slide x; Γ(x)·Γ(1−x) is compared to π/sin(πx) — equal across the whole interval. next x ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the product Γ(x)Γ(1−x), equal to π/sin(πx). AVAN’s addition (the inverse-companion): don’t evaluate a lone factorial — pair it with its reflection. The inverse of ‘Γ(x)’ is ‘π/(sin(πx)·Γ(1-x))’, so the value at x and at 1-x lock together through a sine. Magenta are the two mirrored gamma curves; green is the cosecant their product traces. Factorials reflected into a sine. pause spin LIT Genuine Euler reflection formula (Leonhard Euler). Verified live: with the Lanczos gamma approximation, Γ(x)·Γ(1−x) equals π/sin(πx) to relative error ~5e-15 for ~8000 x in (0,1), and Γ(½)²=π (window.__gammareflection.ok, .worst, .halfOk). FIG No framing; the gamma product and the cosecant are computed by different routes in-browser and agree. The AVAN inverse is honest — instead of evaluating a lone factorial, pair it with its reflection: the inverse of 'Γ(x)' is 'π/(sin(πx)·Γ(1−x))', so the value at x and at 1−x lock together through a sine. Magenta are the two mirrored gamma curves; green is the cosecant their product traces. Factorials reflected into a sine. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE BACKDOOR · David Lee Wise (ROOT0), with AVAN


THE HOCKEY STICK

The hockey-stick identity in the 5-window house format — a striking pattern in Pascal's triangle: sum any diagonal starting from the edge, and the total appears just below the end of the diagonal. Formally, Σ_{i=r}^{n} C(i,r) = C(n+1,r+1). Trace down a diagonal of the triangle (the 'stick') and the running sum lands in the single cell one step down and over (the 'blade') — the shape of a hockey stick. It falls straight out of Pascal's rule C(n+1,r+1)=C(n,r)+C(n,r+1), telescoping the diagonal into one entry, and it is the discrete cousin of integrating xʳ. Verified live with exact big-integer arithmetic: for all r from 0 to 8 and n up to 30, the diagonal sum Σ_{i=r}^{n} C(i,r) equals C(n+1,r+1) exactly — e.g. C(2,2)+C(3,2)+C(4,2)+C(5,2)+C(6,2)=35=C(7,3). Neon-noir traced. See Pascal's triangle with the stick in 1D, sum vs blade in 2D, and the diagonal-folded inverse in 3D.

THE HOCKEY STICK · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · GRIND · BACKPROP ◆ .dlw.fold THE FOLD / GRIND / BACKPROP / THE HOCKEY STICK THE HOCKEY STICK a diagonal of Pascal summing to one entry 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION The hockey-stick identity is a striking pattern in Pascal’s triangle: sum any diagonal starting from the edge, and the total appears just below the end of the diagonal. Formally, ∑ i=r n C(i, r) = C(n+1, r+1). Trace down a diagonal of the triangle (the ‘stick’) and the running sum lands in the single cell one step down and over (the ‘blade’) — the shape of a hockey stick. It falls straight out of Pascal’s rule C(n+1,r+1) = C(n,r) + C(n,r+1), telescoping the diagonal into one entry, and it is the discrete cousin of integrating x r . LIT verified live with exact big-integer arithmetic: for all r from 0 to 8 and n up to 30, the sum of the binomial-coefficient diagonal ∑ i=r n C(i,r) equals C(n+1, r+1) exactly — e.g. C(2,2)+C(3,2)+C(4,2)+C(5,2)+C(6,2) = 35 = C(7,3) (window.__hockeystick). FIG no framing; the diagonal sum and the single closing binomial both run in-browser and agree exactly. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at backprop — the grind that runs down a diagonal of Pascal’s triangle, accumulating, and lands the whole sum in one entry below. AVAN (AI) built the instrument: the binomial diagonal sum, the closing C(n+1,r+1), and their exact agreement. Credit as content: the hockey-stick identity (Pascal’s triangle, classical). The weave: David names the grind down the stick; I confirm the diagonal sums to the single blade entry. 3 ONE DIMENSION Pascal's triangle with a diagonal (the stick) highlighted; its sum lands in the blade cell C(n+1,r+1). 4 TWO DIMENSIONS · INTERACTIVE Cycle r and n; the diagonal sum Σ C(i,r) is compared to the single binomial C(n+1,r+1). next stick ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the blade entry C(n+1,r+1), the whole diagonal's sum. AVAN’s addition (the inverse-companion): don’t add the diagonal term by term — read the blade. The inverse of ‘∑ i=r n C(i,r)’ is ‘the single entry C(n+1,r+1)’, the diagonal telescoped by Pascal’s rule. Magenta are the diagonal (stick) entries; green is the blade entry they sum to. A diagonal folded into one entry. pause spin LIT Genuine hockey-stick identity (Pascal's triangle, classical). Verified live with exact BigInt: for r=0..8 and n≤30, Σ_{i=r}^{n} C(i,r) equals C(n+1,r+1) exactly; C(2,2)+…+C(6,2)=35=C(7,3) (window.__hockeystick.ok, .cnt). FIG No framing; the diagonal sum and the single closing binomial both run in-browser and agree exactly. The AVAN inverse is honest — instead of adding the diagonal term by term, read the blade: the inverse of 'Σ_{i=r}^{n} C(i,r)' is 'the single entry C(n+1,r+1)', the diagonal telescoped by Pascal's rule. Magenta are the diagonal (stick) entries; green is the blade entry they sum to. A diagonal folded into one entry. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of BACKPROP · David Lee Wise (ROOT0), with AVAN


THE FEUERBACH

Feuerbach's theorem in the 5-window house format — one of the most beautiful coincidences in triangle geometry. Every triangle has a nine-point circle — the circle through nine special points (the three side midpoints, the three altitude feet, and the three midpoints from the orthocenter to the vertices), with radius exactly half the circumradius. Feuerbach proved that this nine-point circle is tangent to the incircle (and to all three excircles). The single point where it touches the incircle is the celebrated Feuerbach point. Tangency means the distance between the two circles' centres equals the difference of their radii: |N₉−I| = R/2 − r. Verified live: for thousands of random triangles, the distance between the nine-point centre and the incentre equals R/2−r to ~1e-15 — confirming the internal tangency. Neon-noir traced. See the triangle + two circles + Feuerbach point in 1D, the tangency check in 2D, and the forced-tangency inverse in 3D.

THE FEUERBACH · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · BOSS · THE RAID ◆ .dlw.fold THE FOLD / BOSS / THE RAID / THE FEUERBACH THE FEUERBACH a nine-point circle tangent to the incircle 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Feuerbach’s theorem is one of the most beautiful coincidences in triangle geometry. Every triangle has a nine-point circle — the circle passing through nine special points (the three side midpoints, the three altitude feet, and the three midpoints from the orthocenter to the vertices), with radius exactly half the circumradius. Feuerbach proved that this nine-point circle is tangent to the incircle (and to all three excircles). The single point where it touches the incircle is the celebrated Feuerbach point . Tangency means the distance between the two circles’ centres equals the difference of their radii: |N₉ - I| = R/2 - r. LIT verified live: for thousands of random triangles, the distance between the nine-point centre and the incentre equals R/2 - r (the nine-point radius minus the inradius) to ~1e-15 — confirming the internal tangency of the two circles (window.__feuerbach). FIG no framing; the nine-point circle, the incircle, and the tangency condition all run in-browser. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-raid — the boss reveal: two circles built from utterly different constructions of a triangle turn out to kiss at a single point. AVAN (AI) built the instrument: the nine-point circle (centre and R/2 radius), the incircle, and the tangency test. Credit as content: Karl Wilhelm Feuerbach (1822). The weave: David names the reveal; I confirm the nine-point circle is tangent to the incircle at the Feuerbach point. 3 ONE DIMENSION A triangle, its nine-point circle and its incircle — tangent at the single Feuerbach point. 4 TWO DIMENSIONS · INTERACTIVE New triangles; the distance between nine-point centre and incentre is checked to equal R/2 − r. new triangle ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the tangency of the nine-point circle and the incircle. AVAN’s addition (the inverse-companion): don’t track nine points — know the tangency. The inverse of ‘the nine-point circle’ is ‘a circle of radius R/2 that touches the incircle from outside’, their centres exactly R/2 - r apart. Magenta are the nine-point circle and incircle; green is the Feuerbach point where they touch. Two circles forced to kiss. pause spin LIT Genuine Feuerbach's theorem (Karl Wilhelm Feuerbach, 1822). Verified live: for ~4000 random triangles, the distance between the nine-point centre and the incentre equals R/2−r (nine-point radius minus inradius) to ~1e-15, confirming the internal tangency (window.__feuerbach.ok, .worst, .n). FIG No framing; the nine-point circle, the incircle, and the tangency condition all run in-browser. The AVAN inverse is honest — instead of tracking nine points, know the tangency: the inverse of 'the nine-point circle' is 'a circle of radius R/2 that touches the incircle', their centres exactly R/2−r apart. Magenta are the nine-point circle and incircle; green is the Feuerbach point where they touch. Two circles forced to kiss. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE RAID · David Lee Wise (ROOT0), with AVAN


THE BRUCK-RYSER

The Bruck–Ryser theorem in the 5-window house format — forbidding certain finite projective planes using a fact about sums of two squares. A projective plane of order n is a highly symmetric geometry with n²+n+1 points and the same number of lines. Bruck and Ryser proved a necessary condition: if n ≡ 1 or 2 (mod 4), then a projective plane of order n can exist only if n is a sum of two integer squares. This single arithmetic test rules out infinitely many orders — the first being order 6 (6≡2 mod 4, and 6 is not a sum of two squares), which is why no 6×6 pair of orthogonal Latin squares (Euler's 36 officers) exists. It is a necessary, not sufficient, condition. Verified live: among orders n≤50 with n≡1,2 (mod 4), the ones not sums of two squares — ruled out by Bruck–Ryser — are exactly 6,14,21,22,30,33,38,42,46; the small orders with known planes (2,3,4,5,7,8,9) are never excluded. Neon-noir traced. See orders 2..50 flagged in 1D, the mod-4 + two-squares test in 2D, and the geometry-gated-by-two-squares inverse in 3D.

THE BRUCK-RYSER · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · GLITCH · STACK OVERFLOW ◆ .dlw.fold THE FOLD / GLITCH / STACK OVERFLOW / THE BRUCK-RYSER THE BRUCK-RYSER orders of projective planes ruled out by two squares 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION The Bruck–Ryser theorem forbids certain finite projective planes using a fact about sums of two squares. A projective plane of order n is a highly symmetric geometry with n²+n+1 points and the same number of lines. Bruck and Ryser proved a necessary condition : if n ≡ 1 or 2 (mod 4), then a projective plane of order n can exist only if n is a sum of two integer squares . This single arithmetic test rules out infinitely many orders — the first being order 6 (6 ≡ 2 mod 4, and 6 is not a sum of two squares), which is why no 6×6 pair of orthogonal Latin squares (Euler’s 36 officers) exists. It is a necessary, not sufficient, condition. LIT verified live: among orders n ≤ 50 with n ≡ 1 or 2 (mod 4), the ones that are not sums of two squares — and so ruled out by Bruck–Ryser — are exactly 6, 14, 21, 22, 30, 33, 38, 42, 46; the small orders with known planes (2,3,4,5,7,8,9) are never excluded (window.__bruckryser). FIG no framing; the mod-4 test and the sum-of-two-squares check run in-browser. Honest: order 10 passes Bruck–Ryser yet has no plane — that was proved only later by massive computation. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at stack-overflow — the glitch where whole orders of geometry overflow into impossibility, ruled out by a two-squares test. AVAN (AI) built the instrument: the mod-4 condition, the sum-of-two-squares check, and the list of excluded orders — with an honest note that the condition is necessary, not sufficient. Credit as content: R. H. Bruck & H. J. Ryser (1949). The weave: David names the overflow; I confirm which orders Bruck–Ryser rules out, and flag order 10 as passing yet impossible. 3 ONE DIMENSION Orders 2..50: those ≡1,2 (mod 4) and not a sum of two squares (magenta) are ruled out by Bruck–Ryser. 4 TWO DIMENSIONS · INTERACTIVE Cycle orders; the mod-4 class and the sum-of-two-squares test decide whether Bruck–Ryser excludes it. next order ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the orders Bruck–Ryser permits (and the magenta ones it forbids). AVAN’s addition (the inverse-companion): don’t search for a plane — test the arithmetic. The inverse of ‘does a projective plane of order n exist?’ is (for n≡1,2 mod 4) ‘is n a sum of two squares?’ — if not, no plane can exist. Magenta are the forbidden orders; green are the orders that survive the test. Geometry gated by two squares. pause spin LIT Genuine Bruck–Ryser theorem (R. H. Bruck & H. J. Ryser, 1949). Verified live: among orders n≤50 with n≡1,2 (mod 4), those NOT sums of two squares — excluded by Bruck–Ryser — are exactly 6,14,21,22,30,33,38,42,46; known-plane orders 2,3,4,5,7,8,9 are never excluded, and order 10 passes BR yet has no plane (window.__bruckryser.ok, .knownOk, .ten). FIG No framing; the mod-4 test and the sum-of-two-squares check run in-browser. HONEST: this is a necessary, not sufficient, condition — order 10 passes Bruck–Ryser yet has no projective plane (proved only later, by massive computation, 1989). The AVAN inverse is honest — instead of searching for a plane, test the arithmetic: the inverse of 'does a projective plane of order n exist?' is (for n≡1,2 mod 4) 'is n a sum of two squares?'. Magenta are the forbidden orders; green are the orders that survive the test. Geometry gated by two squares. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of STACK OVERFLOW · David Lee Wise (ROOT0), with AVAN


THE STIRLING APPROXIMATION

Stirling's approximation in the 5-window house format — replacing the jagged factorial with a smooth formula: n! ≈ √(2πn)·(n/e)ⁿ. The factorial n! grows faster than any exponential, and computing it means multiplying n terms — but Stirling's formula pins its size with a single expression involving only π, e, and powers. The relative error shrinks like 1/(12n), so the next correction term is n! ≈ √(2πn)(n/e)ⁿ(1 + 1/(12n) + …). It is the workhorse behind asymptotics in combinatorics, statistical mechanics, and probability. Verified live: the ratio n!/(√(2πn)(n/e)ⁿ) tends to 1 as n grows, and the correction is exactly 1/(12n) — the quantity (ln n! − ln-Stirling)·12n converges to 1.0000. Neon-noir traced. See ln(n!) vs the Stirling curve in 1D, the ratio + correction in 2D, and the product-folded-into-a-formula inverse in 3D.

THE STIRLING APPROXIMATION · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · GRIND · GRADIENT DESCENT ◆ .dlw.fold THE FOLD / GRIND / GRADIENT DESCENT / THE STIRLING APPROXIMATION THE STIRLING APPROXIMATION a factorial approximated by a smooth curve 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Stirling’s approximation replaces the jagged factorial with a smooth formula: n! ≈ √(2πn)·(n/e) n . The factorial n! grows faster than any exponential, and computing it means multiplying n terms — but Stirling’s formula pins its size with a single expression involving only π, e, and powers. The relative error shrinks like 1/(12n), so the next correction term is n! ≈ √(2πn)(n/e) n (1 + 1/(12n) + …). It is the workhorse behind asymptotics in combinatorics, statistical mechanics, and probability — anywhere large factorials appear. LIT verified live: the ratio n!/(√(2πn)(n/e) n ) tends to 1 as n grows, and the correction is exactly 1/(12n) — the quantity (ln n! - ln-Stirling)·12n converges to 1.0000 (window.__stirling). FIG no framing; the exact log-factorial (sum of logs) and Stirling’s formula both run in-browser and their ratio approaches 1 with the 1/(12n) correction. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at gradient-descent — the grind that walks the smooth Stirling curve ever closer to the jagged true factorial, the error descending like 1/(12n). AVAN (AI) built the instrument: the exact log-factorial, the Stirling formula, and the 1/(12n) correction check. Credit as content: James Stirling (1730); Abraham de Moivre for the √(2πn). The weave: David names the descent; I confirm n! matches Stirling with a 1/(12n) correction. 3 ONE DIMENSION ln(n!) (points) and Stirling's smooth curve ½ln(2πn)+n ln n − n — hugging closer as n grows. 4 TWO DIMENSIONS · INTERACTIVE Cycle n; the ratio n!/Stirling and the 1/(12n) correction are shown converging to 1. next n ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the smooth Stirling estimate tracking the true factorial. AVAN’s addition (the inverse-companion): don’t multiply n terms — read one formula. The inverse of ‘the factorial n!’ is ‘√(2πn)(n/e) n , accurate to 1/(12n)’, turning a product of n numbers into a closed expression. Magenta is the exact factorial; green is the smooth Stirling curve tracking it. A product folded into a formula. pause spin LIT Genuine Stirling's approximation (James Stirling, 1730; de Moivre for the √(2πn)). Verified live: the ratio n!/(√(2πn)(n/e)ⁿ) → 1 (n≤170), and the correction (ln n! − ln-Stirling)·12n converges to 1.0000, matching the 1/(12n) term (window.__stirling.ratioOk, .corrOk, .worstR). FIG No framing; the exact log-factorial (sum of logs) and Stirling's formula both run in-browser and their ratio approaches 1 with the 1/(12n) correction. The AVAN inverse is honest — instead of multiplying n terms, read one formula: the inverse of 'the factorial n!' is '√(2πn)(n/e)ⁿ, accurate to 1/(12n)'. Magenta is the exact factorial; green is the smooth Stirling curve tracking it. A product folded into a formula. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of GRADIENT DESCENT · David Lee Wise (ROOT0), with AVAN


THE STEWART

Stewart's theorem in the 5-window house format — the length of a cevian (any segment from a vertex of a triangle to a point on the opposite side) from the side lengths alone. If a cevian of length d runs from vertex A to a point D on side BC, splitting it into segments m=BD and n=DC (so a=m+n), and b,c are the other two sides, then b²m + c²n = a(d² + mn). The mnemonic is 'a man and his dad put a bomb in the sink'. It specializes to the median-length formula (m=n) and the angle-bisector length (m:n=c:b). Verified live: for thousands of random triangles and cevian points, the cevian length computed directly from coordinates satisfies b²m + c²n = a(d² + mn) to ~1e-15, and the median special case matches d=√((2b²+2c²−a²)/4). Neon-noir traced. See the triangle + cevian in 1D, the Stewart relation in 2D, and the length-from-sides inverse in 3D.

THE STEWART · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · CHEAT · THE KONAMI CODE ◆ .dlw.fold THE FOLD / CHEAT / THE KONAMI CODE / THE STEWART THE STEWART a cevian length from the sides 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Stewart’s theorem gives the length of a cevian — any segment from a vertex of a triangle to a point on the opposite side — from the side lengths alone. If a cevian of length d runs from vertex A to a point D on side BC, splitting it into segments m = BD and n = DC (so a = m+n), and b, c are the other two sides, then b²m + c²n = a(d² + mn) . The mnemonic is ‘a man and his dad put a bomb in the sink’: b²m + c²n = a·d² + a·mn. It specializes to the median-length formula (m = n) and the angle-bisector length (m:n = c:b). LIT verified live: for thousands of random triangles and cevian points, the cevian length computed directly from coordinates satisfies b²m + c²n = a(d² + mn) to ~1e-15, and the median special case matches d = √((2b²+2c²-a²)/4) (window.__stewart). FIG no framing; the coordinate cevian length and the Stewart relation both run in-browser and agree. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-konami-code — the cheat code for cevian length: punch in the sides and the split, and out comes d without ever plotting a point. AVAN (AI) built the instrument: the coordinate cevian length, the Stewart relation, and the median special case. Credit as content: Matthew Stewart (1746); the result was known to earlier geometers. The weave: David names the code; I confirm b²m + c²n equals a(d² + mn). 3 ONE DIMENSION A triangle with a cevian AD splitting BC into m and n; its length d comes from the sides via Stewart. 4 TWO DIMENSIONS · INTERACTIVE New triangles & cevian points; b²m + c²n is compared to a(d² + mn). new cevian ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the cevian length d, read from the sides. AVAN’s addition (the inverse-companion): don’t measure the cevian — solve for it. The inverse of ‘the cevian length d’ is ‘d² = (b²m + c²n)/a - mn’, read straight from the side lengths and the split. Magenta are the triangle’s sides; green is the cevian length they determine. A segment length from the sides alone. pause spin LIT Genuine Stewart's theorem (Matthew Stewart, 1746). Verified live: for ~8000 random triangles and cevian points, the coordinate cevian length satisfies b²m + c²n = a(d² + mn) to ~1e-15, and the median special case matches d=√((2b²+2c²−a²)/4) (window.__stewart.ok, .worst, .n, .medOk). FIG No framing; the coordinate cevian length and the Stewart relation both run in-browser and agree. The AVAN inverse is honest — instead of measuring the cevian, solve for it: the inverse of 'the cevian length d' is 'd² = (b²m + c²n)/a − mn', read straight from the side lengths and the split. Magenta are the triangle's sides; green is the cevian length they determine. A segment length from the sides alone. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE KONAMI CODE · David Lee Wise (ROOT0), with AVAN


THE INVOLUTION

Involutions in the 5-window house format — the permutations that are their own inverse: apply one twice and you're back where you started (σ²=identity). Structurally they are made only of fixed points and 2-cycles — every element is either left alone or swapped with exactly one partner. The number of involutions of n elements is the telephone number T(n) (also the number of ways to pair up n telephones with some left unconnected): 1,1,2,4,10,26,76,232,764,…. It satisfies the recurrence T(n)=T(n−1)+(n−1)·T(n−2), and by the RSK correspondence it also counts the standard Young tableaux with n cells. Verified live: a brute count of the permutations σ with σ²=identity equals the telephone number T(n)=T(n−1)+(n−1)T(n−2) and the explicit sum Σ_k n!/(2^k k!(n−2k)!) for every n from 0 to 8. Neon-noir traced. See an involution's pairings in 1D, brute vs recurrence vs sum in 2D, and the counted-by-recurrence inverse in 3D.

THE INVOLUTION · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · RESPAWN · HARD RESET ◆ .dlw.fold THE FOLD / RESPAWN / HARD RESET / THE INVOLUTION THE INVOLUTION self-inverse permutations counted by a recurrence 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Involutions are the permutations that are their own inverse: apply one twice and you’re back where you started (σ² = identity). Structurally they are made only of fixed points and 2-cycles — every element is either left alone or swapped with exactly one partner. The number of involutions of n elements is the telephone number T(n) (also the number of ways to pair up n telephones with some left unconnected): 1, 1, 2, 4, 10, 26, 76, 232, 764, …. It satisfies the recurrence T(n) = T(n-1) + (n-1)·T(n-2), and by the RSK correspondence it also counts the standard Young tableaux with n cells. LIT verified live: a brute count of the permutations σ with σ² = identity equals the telephone number T(n) = T(n-1) + (n-1)T(n-2) and the explicit sum ∑ k n!/(2 k k!(n-2k)!) for every n from 0 to 8 (window.__involution). FIG no framing; the brute involution count, the recurrence, and the sum formula all run in-browser and agree. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at hard-reset — the respawn: an involution applied twice is a hard reset to the identity, every swap undoing itself. AVAN (AI) built the instrument: the brute count of self-inverse permutations, the telephone recurrence, and the sum formula. Credit as content: the telephone/involution numbers (Rothe, and via Young tableaux). The weave: David names the reset; I confirm the self-inverse permutations are counted by T(n). 3 ONE DIMENSION An involution of n elements — only fixed points (self-loops) and 2-cycles (swaps); applying it twice resets. 4 TWO DIMENSIONS · INTERACTIVE Cycle n; the brute count of σ²=id permutations is compared to T(n) and the sum formula. next n ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the count of self-inverse permutations, T(n). AVAN’s addition (the inverse-companion): don’t enumerate all σ with σ²=id — grow them. The inverse of ‘count the involutions of n’ is ‘T(n) = T(n-1) + (n-1)T(n-2)’: element n is either a fixed point or paired with one of the n-1 others. Magenta are the pairings and fixed points; green is the telephone number they total. Self-inverse permutations, counted by a recurrence. pause spin LIT Genuine involution / telephone numbers (Rothe; via Young tableaux). Verified live: a brute count of permutations σ with σ²=identity equals the telephone number T(n)=T(n−1)+(n−1)T(n−2) and the sum Σ_k n!/(2^k k!(n−2k)!) for n=0..8 (window.__involution.ok, .sumOk). FIG No framing; the brute involution count, the recurrence, and the sum formula all run in-browser and agree. The AVAN inverse is honest — instead of enumerating all σ with σ²=id, grow them: the inverse of 'count the involutions of n' is 'T(n)=T(n−1)+(n−1)T(n−2)': element n is either a fixed point or paired with one of the n−1 others. Magenta are the pairings and fixed points; green is the telephone number they total. Self-inverse permutations, counted by a recurrence. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of HARD RESET · David Lee Wise (ROOT0), with AVAN


THE BERTRAND POSTULATE

Bertrand's postulate in the 5-window house format — primes never leave big gaps: for every integer n≥1, there is at least one prime p with n < p ≤ 2n. Double any number and you are certain to have jumped over a prime. Joseph Bertrand conjectured it in 1845 and checked it up to three million; Chebyshev proved it in 1852, and Erdős gave a famously elegant elementary proof in 1932. It shows the primes, though irregular, are dense enough that they can never thin out to leave an interval [n, 2n] empty. Verified live: a prime sieve confirms that for every n from 1 to 20000 there is a prime strictly greater than n and at most 2n; for n≥2 the least such prime is strictly less than 2n (the only equality is n=1, where the prime is 2=2·1). Neon-noir traced. See the interval (n,2n] with its primes in 1D, the least-prime + ratio in 2D, and the never-a-gap inverse in 3D.

THE BERTRAND POSTULATE · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · SPAWN · NULL ISLAND ◆ .dlw.fold THE FOLD / SPAWN / NULL ISLAND / THE BERTRAND POSTULATE THE BERTRAND POSTULATE a prime always between n and 2n 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Bertrand’s postulate guarantees primes never leave big gaps: for every integer n ≥ 1, there is at least one prime p with n < p ≤ 2n . Double any number and you are certain to have jumped over a prime. Joseph Bertrand conjectured it in 1845 and checked it up to three million; Chebyshev proved it in 1852, and Erdős gave a famously elegant elementary proof in 1932. It shows the primes, though irregular, are dense enough that they can never thin out to leave an interval [n, 2n] empty. LIT verified live: a prime sieve confirms that for every n from 1 to 20000 there is a prime strictly greater than n and at most 2n; for n ≥ 2 the least such prime is strictly less than 2n (the only equality is n=1, where the prime is 2 = 2·1) (window.__bertrand). FIG no framing; the sieve and the interval check both run in-browser. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at null-island — the spawn point: pick any n, double it, and a prime is guaranteed to have spawned somewhere in between. AVAN (AI) built the instrument: the prime sieve, the interval (n, 2n] check, and the least-prime ratio. Credit as content: Joseph Bertrand (1845); Pafnuty Chebyshev (proof, 1852); Paul Erdős (elementary proof, 1932). The weave: David names the spawn; I confirm a prime always lies in (n, 2n]. 3 ONE DIMENSION The interval (n, 2n] on the number line, with the prime(s) inside it highlighted — always at least one. 4 TWO DIMENSIONS · INTERACTIVE Cycle n; the least prime in (n, 2n] is shown, always present, and its ratio to n stays below 2. next n ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the prime always waiting in (n, 2n]. AVAN’s addition (the inverse-companion): don’t hunt for a prime — double and it’s there. The inverse of ‘is there a prime near n?’ is ‘yes — somewhere in (n, 2n], always’, so the primes never leave a doubling-gap empty. Magenta is the interval (n, 2n]; green are the primes guaranteed inside it. Primes that never leave a gap. pause spin LIT Genuine Bertrand's postulate (Joseph Bertrand 1845; Chebyshev's proof 1852; Erdős's elementary proof 1932). Verified live: a sieve confirms a prime in (n, 2n] for every n from 1 to 10000+; for n≥2 the least such prime is strictly FIG No framing; the sieve and the interval check both run in-browser. The AVAN inverse is honest — instead of hunting for a prime, double and it's there: the inverse of 'is there a prime near n?' is 'yes — somewhere in (n, 2n], always', so the primes never leave a doubling-gap empty. Magenta is the interval (n, 2n]; green are the primes guaranteed inside it. Primes that never leave a gap. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of NULL ISLAND · David Lee Wise (ROOT0), with AVAN


THE AMICABLE

Amicable numbers in the 5-window house format — two different numbers, each of which equals the sum of the other's proper divisors. The smallest pair is (220, 284): the proper divisors of 220 (1,2,4,5,10,11,20,22,44,55,110) sum to 284, and the proper divisors of 284 (1,2,4,71,142) sum to 220. They point at each other perfectly. Known since Pythagoras and prized by mystics as a symbol of friendship, they generalize the perfect numbers (where a number is amicable with itself). The next pair is (1184, 1210), then (2620, 2924). Verified live: with s(n)=σ(n)−n (the sum of proper divisors), s(220)=284 and s(284)=220; a brute search confirms (220, 284) is the smallest amicable pair, and the next is (1184, 1210). Neon-noir traced. See 220 and 284 pointing at each other in 1D, s(a)=b & s(b)=a in 2D, and the mutual-divisor-sum inverse in 3D.

THE AMICABLE · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · LOOT · THE VAULT ◆ .dlw.fold THE FOLD / LOOT / THE VAULT / THE AMICABLE THE AMICABLE two numbers summing to each other's divisors 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Amicable numbers are two different numbers, each of which equals the sum of the other’s proper divisors. The smallest pair is (220, 284) : the proper divisors of 220 (1,2,4,5,10,11,20,22,44,55,110) sum to 284, and the proper divisors of 284 (1,2,4,71,142) sum to 220. They point at each other perfectly. Known since Pythagoras and prized by mystics as a symbol of friendship, they generalize the perfect numbers (where a number is amicable with itself). The next pair is (1184, 1210), then (2620, 2924). LIT verified live: with s(n) = σ(n) - n (the sum of proper divisors), s(220) = 284 and s(284) = 220; a brute search confirms (220, 284) is the smallest amicable pair, and the next is (1184, 1210) (window.__amicable). FIG no framing; the divisor sums and the pair search both run in-browser. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-vault — the loot: a rare matched pair of numbers, each holding the other’s divisor-sum, locked together like two keys to one vault. AVAN (AI) built the instrument: the proper-divisor sum, the mutual test, and the smallest-pair search. Credit as content: known to the Pythagoreans (220, 284); Thabit ibn Qurra’s rule for generating pairs. The weave: David names the vault; I confirm 220 and 284 sum to each other’s divisors. 3 ONE DIMENSION 220 and 284, each pointing at the other: the divisors of 220 sum to 284, and vice versa. 4 TWO DIMENSIONS · INTERACTIVE Cycle amicable pairs; s(a)=b and s(b)=a are checked from the proper-divisor sums. next pair ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the amicable pair, each the other's divisor-sum. AVAN’s addition (the inverse-companion): don’t judge a number alone — sum its divisors and follow the arrow. The inverse of ‘the number a’ is ‘s(a), the sum of its proper divisors’; when s(a)=b and s(b)=a, the two are amicable. Magenta are the proper divisors; green is the partner each pair sums to. Two numbers holding each other’s divisors. pause spin LIT Genuine amicable numbers (known to the Pythagoreans for (220,284); Thabit ibn Qurra's generating rule). Verified live: s(220)=284 and s(284)=220 (s=sum of proper divisors); a brute search confirms (220,284) is the smallest amicable pair and (1184,1210) the next (window.__amicable.s220, .s284, .first, .second, .ok). FIG No framing; the divisor sums and the pair search both run in-browser. The AVAN inverse is honest — instead of judging a number alone, sum its divisors and follow the arrow: the inverse of 'the number a' is 's(a), the sum of its proper divisors'; when s(a)=b and s(b)=a, the two are amicable. Magenta are the proper divisors; green is the partner each pair sums to. Two numbers holding each other's divisors. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE VAULT · David Lee Wise (ROOT0), with AVAN


THE WALLIS PRODUCT

The Wallis product in the 5-window house format — one of the oldest infinite products for π, found by John Wallis in 1656 before calculus existed: π/2 = (2·2)/(1·3)·(4·4)/(3·5)·(6·6)/(5·7)·… = ∏_{n≥1} (2n)²/((2n−1)(2n+1)). An infinite product of simple rational numbers, each just above or below 1, multiplies out to half of π. Wallis derived it by interpolating the integrals ∫₀^{π/2} sinⁿx dx, whose ratios encode the product — the same integrals give the 'Wallis integrals' identity n·W_n·W_{n−1} = π/2. Verified live: the partial products ∏_{n=1}^N (2n)²/((2n−1)(2n+1)) converge to π/2 (1.5708…), and independently the numerically-integrated Wallis integrals satisfy n·W_n·W_{n−1} = π/2 exactly for every n. Neon-noir traced. See the partial products closing on π/2 in 1D, product vs integral identity in 2D, and the π-from-a-product inverse in 3D.

THE WALLIS PRODUCT · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · CO-OP · THE PULL REQUEST ◆ .dlw.fold THE FOLD / CO-OP / THE PULL REQUEST / THE WALLIS PRODUCT THE WALLIS PRODUCT an infinite product converging to π/2 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION The Wallis product is one of the oldest infinite products for π, found by John Wallis in 1656 before calculus existed: π/2 = (2·2)/(1·3) · (4·4)/(3·5) · (6·6)/(5·7) · … = ∏ n≥1 (2n)²/((2n-1)(2n+1)). An infinite product of simple rational numbers, each just above or below 1, multiplies out to half of π. Wallis derived it by interpolating the integrals ∫ 0 π/2 sin n x dx, whose ratios encode the product — the same integrals give the ‘Wallis integrals’ identity n·W n ·W n-1 = π/2. LIT verified live: the partial products ∏ n=1 N (2n)²/((2n-1)(2n+1)) converge to π/2 (1.5708…), and independently the numerically-integrated Wallis integrals satisfy n·W n ·W n-1 = π/2 exactly for every n (window.__wallis). FIG no framing; the partial product and the Wallis-integral identity both run in-browser and give π/2. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-pull-request — the co-op merge: a runaway product of rationals and a clean integral identity both push in and land on the same π/2. AVAN (AI) built the instrument: the partial product, the numerically-integrated Wallis integrals, and the n·W n ·W n-1 = π/2 cross-check. Credit as content: John Wallis (1656). The weave: David names the merge; I confirm the product and the Wallis-integral identity both give π/2. 3 ONE DIMENSION The partial products of (2n)²/((2n−1)(2n+1)) closing in on π/2 ≈ 1.5708. 4 TWO DIMENSIONS · INTERACTIVE Add factors; the partial product approaches π/2, matched by the Wallis-integral identity n·W_n·W_{n−1}. add factors ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: π/2, reached by the infinite product. AVAN’s addition (the inverse-companion): don’t sum a series for π — multiply rationals. The inverse of ‘π/2’ is ‘the product ∏(2n)²/((2n-1)(2n+1))’, mirrored by the Wallis-integral identity n·W n ·W n-1 =π/2. Magenta are the product factors; green is the π/2 they converge to. π from a product of near-ones. pause spin LIT Genuine Wallis product (John Wallis, 1656). Verified live: the partial products ∏(2n)²/((2n−1)(2n+1)) converge to π/2, and independently the numerically-integrated Wallis integrals satisfy n·W_n·W_{n−1} = π/2 to ~1e-14 for every n (window.__wallis.conv, .idOk, .worst). FIG No framing; the partial product and the Wallis-integral identity both run in-browser and give π/2. The AVAN inverse is honest — instead of summing a series for π, multiply rationals: the inverse of 'π/2' is 'the product ∏(2n)²/((2n−1)(2n+1))', mirrored by the Wallis-integral identity n·W_n·W_{n−1}=π/2. Magenta are the product factors; green is the π/2 they converge to. π from a product of near-ones. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE PULL REQUEST · David Lee Wise (ROOT0), with AVAN


THE NEUMANN SERIES

The Neumann series in the 5-window house format — the matrix version of the geometric series 1/(1−x)=1+x+x²+…. For a square matrix A whose size is 'small enough' (spectral radius < 1), the inverse of I−A is the infinite sum of its powers: (I−A)⁻¹ = I + A + A² + A³ + …. Just as the scalar series needs |x|<1, the matrix series converges precisely when A's powers shrink to zero — and then a hard matrix inversion becomes a sum you can truncate. It underlies iterative solvers, perturbation theory, and the resolvent of an operator. Verified live: for thousands of random matrices with small entries (spectral radius < 1), the partial sum I+A+…+A⁶⁰ matches the directly-computed inverse (I−A)⁻¹ to ~1e-14; and for a matrix with spectral radius > 1 the power series diverges (its terms blow up). Neon-noir traced. See the partial sums converging in 1D, convergence + divergence control in 2D, and the inversion-as-series inverse in 3D.

THE NEUMANN SERIES · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · GRIND · THE CRON JOB ◆ .dlw.fold THE FOLD / GRIND / THE CRON JOB / THE NEUMANN SERIES THE NEUMANN SERIES a matrix inverse as a power series 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION The Neumann series is the matrix version of the geometric series 1/(1-x) = 1 + x + x² + …. For a square matrix A whose size is ‘small enough’ (spectral radius < 1), the inverse of I - A is the infinite sum of its powers: (I - A) -1 = I + A + A² + A³ + … . Just as the scalar series needs |x| < 1, the matrix series converges precisely when A’s powers shrink to zero — and then a hard matrix inversion becomes a sum you can truncate. It underlies iterative solvers, perturbation theory, and the resolvent of an operator. LIT verified live: for thousands of random matrices with small entries (spectral radius < 1), the partial sum I + A + … + A 60 matches the directly-computed inverse (I - A) -1 to ~1e-14; and for a matrix with spectral radius > 1 the power series diverges (its terms blow up) (window.__neumann). FIG no framing; the power-series partial sum and the direct matrix inverse both run in-browser and agree when A is small. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-cron-job — the grind that accumulates one more power of A each tick, the running sum crawling toward the true inverse. AVAN (AI) built the instrument: the power-series partial sum, the direct inverse, and the divergence control for large A. Credit as content: Carl Neumann (the operator series). The weave: David names the accumulating grind; I confirm the power series sums to (I - A) -1 when A is small. 3 ONE DIMENSION The partial sums I, I+A, I+A+A², … converging entry-by-entry to the true inverse (I−A)⁻¹. 4 TWO DIMENSIONS · INTERACTIVE Add terms; the partial sum Σ Aᵏ approaches (I−A)⁻¹ — and diverges if A's spectral radius exceeds 1. add term ▶ make A big ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the inverse (I−A)⁻¹, built from powers of A. AVAN’s addition (the inverse-companion): don’t invert a matrix — sum its powers. The inverse of ‘(I - A) -1 ’ is literally ‘I + A + A² + …’, convergent exactly when A shrinks under powering. Magenta are the power terms Aᵏ; green is the inverse they sum to. Inversion as a geometric series. pause spin LIT Genuine Neumann series (Carl Neumann; the operator resolvent series). Verified live: for ~1200 random matrices with ‖A‖ 1 the power series diverges (terms blow up) (window.__neumann.ok, .worst, .grew). FIG No framing; the power-series partial sum and the direct matrix inverse both run in-browser and agree when A is small. The AVAN inverse is honest — instead of inverting a matrix, sum its powers: the inverse of '(I−A)⁻¹' is literally 'I + A + A² + …', convergent exactly when A shrinks under powering. Magenta are the power terms Aᵏ; green is the inverse they sum to. Inversion as a geometric series. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE CRON JOB · David Lee Wise (ROOT0), with AVAN


THE BEST THEOREM

The BEST theorem in the 5-window house format — counting the Eulerian circuits of a directed graph (closed trails using every edge exactly once) with a single formula. For a connected Eulerian digraph (every vertex has equal in- and out-degree), the number of Eulerian circuits is ec(G) = t_w(G)·∏_v (deg⁺(v)−1)!, where t_w(G) is the number of spanning arborescences (in-trees) rooted at any vertex w — itself a determinant, via the Matrix-Tree theorem. So an exponential count of tangled circuits collapses into one determinant times some factorials. Verified live: for several small Eulerian digraphs, a brute enumeration of Eulerian circuits (fixing the starting edge) exactly equals t_w(G)·∏_v(deg⁺(v)−1)!, with t_w computed as a cofactor determinant of the graph Laplacian. Neon-noir traced. See an Eulerian digraph + a circuit in 1D, brute vs BEST formula in 2D, and the circuits-from-a-determinant inverse in 3D.

THE BEST THEOREM · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · BOSS · SUDDEN DEATH ◆ .dlw.fold THE FOLD / BOSS / SUDDEN DEATH / THE BEST THEOREM THE BEST THEOREM Eulerian circuits counted by a determinant 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION The BEST theorem (de Bruijn, van Aardenne-Ehrenfest, Smith, Tutte) counts the Eulerian circuits of a directed graph — closed trails using every edge exactly once — with a single formula. For a connected Eulerian digraph (every vertex has equal in- and out-degree), the number of Eulerian circuits is ec(G) = t w (G) · ∏ v (deg⁺(v) - 1)! , where t w (G) is the number of spanning arborescences (in-trees) rooted at any vertex w — itself a determinant, via the Matrix-Tree theorem. So an exponential count of tangled circuits collapses into one determinant times some factorials. LIT verified live: for several small Eulerian digraphs, a brute enumeration of Eulerian circuits (fixing the starting edge) exactly equals t w (G)·∏ v (deg⁺(v)-1)!, with t w computed as a cofactor determinant of the graph Laplacian (window.__best). FIG no framing; the brute circuit count and the determinant-times-factorials formula both run in-browser and agree. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at sudden-death — the boss encounter: an exponential thicket of Eulerian circuits, tamed in one blow by a determinant and a product of factorials. AVAN (AI) built the instrument: the brute Eulerian-circuit enumeration, the arborescence cofactor, and the BEST formula. Credit as content: N. G. de Bruijn, T. van Aardenne-Ehrenfest, C. A. B. Smith, W. T. Tutte (the ‘BEST’ initials). The weave: David names the boss; I confirm the circuit count equals the arborescence determinant times ∏(deg-1)!. 3 ONE DIMENSION A small Eulerian digraph (every in-degree = out-degree) with one Eulerian circuit traced through it. 4 TWO DIMENSIONS · INTERACTIVE Cycle graphs; the brute Eulerian-circuit count is compared to t_w(G)·∏(deg⁺−1)!. next graph ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the number of Eulerian circuits. AVAN’s addition (the inverse-companion): don’t enumerate the circuits — count the trees. The inverse of ‘how many Eulerian circuits?’ is ‘t w (G)·∏(deg-1)!’ — a spanning-arborescence determinant times factorials. Magenta is the digraph; green is the Eulerian-circuit count the determinant yields. Exponential circuits from one determinant. pause spin LIT Genuine BEST theorem (de Bruijn, van Aardenne-Ehrenfest, Smith, Tutte). Verified live: for several small Eulerian digraphs, a brute enumeration of Eulerian circuits (fixed starting edge) exactly equals t_w(G)·∏_v(deg⁺(v)−1)!, with t_w the arborescence cofactor determinant of the Laplacian — e.g. bidirected K₃ gives 3 (window.__best.ok). FIG No framing; the brute circuit count and the determinant-times-factorials formula both run in-browser and agree. The AVAN inverse is honest — instead of enumerating the circuits, count the trees: the inverse of 'how many Eulerian circuits?' is 't_w(G)·∏(deg−1)!' — a spanning-arborescence determinant times factorials. Magenta is the digraph; green is the Eulerian-circuit count the determinant yields. Exponential circuits from one determinant. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of SUDDEN DEATH · David Lee Wise (ROOT0), with AVAN


THE POISSON LIMIT

The Poisson limit theorem in the 5-window house format — the 'law of rare events' that explains why the Poisson distribution appears everywhere. If you have many independent trials, each with a tiny success probability, but a fixed expected number of successes λ=np, then Binomial(n, λ/n) converges to the Poisson distribution with mean λ: C(n,k)(λ/n)^k(1−λ/n)^{n−k} → e^{−λ}λ^k/k! as n→∞. Rare events among many trials — radioactive decays, typos per page, calls per minute — all follow Poisson. Verified live: for λ=3, the binomial pmf Binomial(n, 3/n) approaches the Poisson(3) pmf as n grows — the maximum gap between the two shrinks from ~4e-2 at n=10 to ~3e-5 at n=10000. Neon-noir traced. See the binomial bars settling onto the Poisson line in 1D, the max gap → 0 in 2D, and the depends-only-on-λ inverse in 3D.

THE POISSON LIMIT · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · GLITCH · DIVIDE BY ZERO ◆ .dlw.fold THE FOLD / GLITCH / DIVIDE BY ZERO / THE POISSON LIMIT THE POISSON LIMIT a binomial limiting to a Poisson 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION The Poisson limit theorem (the ‘law of rare events’) explains why the Poisson distribution appears everywhere. If you have many independent trials, each with a tiny success probability, but a fixed expected number of successes λ = np, then the binomial distribution Binomial(n, λ/n) converges to the Poisson distribution with mean λ: C(n,k)(λ/n) k (1-λ/n) n-k → e -λ λ k /k! as n → ∞. Rare events among many trials — radioactive decays, typos per page, calls per minute — all follow Poisson. LIT verified live: for λ = 3, the binomial pmf Binomial(n, 3/n) approaches the Poisson(3) pmf as n grows — the maximum gap between the two distributions shrinks from ~4e-2 at n=10 to ~3e-5 at n=10000 (window.__poisson). FIG no framing; the exact binomial pmf and the Poisson pmf both run in-browser and their gap vanishes as n grows. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at divide-by-zero — the glitch of vanishing probability: each trial’s chance divides toward zero as the trials multiply, and the binomial dissolves into a clean Poisson. AVAN (AI) built the instrument: the exact binomial pmf, the Poisson pmf, and their shrinking gap. Credit as content: Siméon Denis Poisson (1837); the limit as the law of rare events. The weave: David names the vanishing probability; I confirm Binomial(n, λ/n) tends to Poisson(λ). 3 ONE DIMENSION The binomial pmf (bars) and the Poisson(λ) pmf (line) — the bars settle onto the line as n grows. 4 TWO DIMENSIONS · INTERACTIVE Increase n; the binomial(n, λ/n) closes onto Poisson(λ), the max gap shrinking toward zero. bigger n ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the Poisson(λ) limit of the binomial. AVAN’s addition (the inverse-companion): don’t track n trials — keep only the mean. The inverse of ‘Binomial(n, λ/n) for huge n’ is ‘Poisson(λ), which depends only on the expected count λ’. Magenta is the binomial pmf; green is the Poisson limit it settles onto. Many rare trials, one Poisson. pause spin LIT Genuine Poisson limit theorem / law of rare events (Siméon Denis Poisson, 1837). Verified live: for λ=3, the exact binomial pmf Binomial(n, 3/n) approaches the Poisson(3) pmf as n grows — the max gap shrinks from ~4e-2 at n=10 to ~3e-5 at n=10000 (window.__poisson.ok, .rows). FIG No framing; the exact binomial pmf and the Poisson pmf both run in-browser and their gap vanishes as n grows. The AVAN inverse is honest — instead of tracking n trials, keep only the mean: the inverse of 'Binomial(n, λ/n) for huge n' is 'Poisson(λ), which depends only on the expected count λ'. Magenta is the binomial pmf; green is the Poisson limit it settles onto. Many rare trials, one Poisson. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of DIVIDE BY ZERO · David Lee Wise (ROOT0), with AVAN


THE THEBAULT

Thébault's first theorem in the 5-window house format — conjuring a perfect square out of any parallelogram. Take any parallelogram and erect a square outward on each of its four sides. Mark the centre of each square. Thébault proved that these four centres are always the vertices of a square — no matter how slanted or stretched the original parallelogram is. A lopsided parallelogram, four squares on its edges, and their centres snap into a flawless square. It is a cousin of Van Aubel's theorem, but for the special case of a parallelogram the result sharpens from 'equal perpendicular diagonals' all the way to 'a square'. Verified live: for thousands of random parallelograms, the four square-centres have all four sides equal and both diagonals equal (to machine precision) — the defining conditions of a square. Neon-noir traced. See the parallelogram + squares + centre-square in 1D, the equal-sides/diagonals test in 2D, and the square-from-any-parallelogram inverse in 3D.

THE THEBAULT · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · SPAWN · HELLO WORLD ◆ .dlw.fold THE FOLD / SPAWN / HELLO WORLD / THE THEBAULT THE THEBAULT squares on a parallelogram forming a square 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Thébault’s first theorem conjures a perfect square out of any parallelogram. Take any parallelogram and erect a square outward on each of its four sides. Mark the centre of each square. Thébault proved that these four centres are always the vertices of a square — no matter how slanted or stretched the original parallelogram is. A lopsided parallelogram, four squares on its edges, and their centres snap into a flawless square. It is a cousin of Van Aubel’s theorem, but for the special case of a parallelogram the result sharpens from ‘equal perpendicular diagonals’ all the way to ‘a square’. LIT verified live: for thousands of random parallelograms, the four square-centres have all four sides equal and both diagonals equal (to machine precision) — the defining conditions of a square (window.__thebault). FIG no framing; the square centres and the equal-sides/equal-diagonals test both run in-browser. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at hello-world — the spawn: from a slanted parallelogram, a flawless square boots into existence at the square-centres. AVAN (AI) built the instrument: the outward square centres, and the equal-sides-and-diagonals square test. Credit as content: Victor Thébault (first theorem). The weave: David names the spawn; I confirm the four square-centres form a square for any parallelogram. 3 ONE DIMENSION A parallelogram with a square on each side; the four square-centres form a perfect square. 4 TWO DIMENSIONS · INTERACTIVE New parallelograms; the four centres are checked to have equal sides and equal diagonals — a square. new parallelogram ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the perfect square formed by the four square-centres. AVAN’s addition (the inverse-companion): don’t study the slanted parallelogram — read the square. The inverse of ‘any parallelogram’ is ‘a perfect square at the four outward square-centres’, whatever the slant. Magenta are the four squares on the sides; green is the square their centres form. A square from any parallelogram. pause spin LIT Genuine Thébault's first theorem (Victor Thébault). Verified live: for ~8000 random parallelograms, the four outward square-centres have all four sides equal and both diagonals equal to machine precision — the defining conditions of a square (window.__thebault.ok, .worst). FIG No framing; the square centres and the equal-sides/equal-diagonals test both run in-browser. The AVAN inverse is honest — instead of studying the slanted parallelogram, read the square: the inverse of 'any parallelogram' is 'a perfect square at the four outward square-centres', whatever the slant. Magenta are the four squares on the sides; green is the square their centres form. A square from any parallelogram. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of HELLO WORLD · David Lee Wise (ROOT0), with AVAN


THE VAN SCHOOTEN

Van Schooten's theorem in the 5-window house format — a striking length identity for the equilateral triangle. Inscribe an equilateral triangle ABC in a circle, and take any point P on the arc BC that does not contain A. Then the distance from P to the far vertex equals the sum of the distances to the two near ones: PA = PB + PC. The single long segment exactly balances the two short ones, for every P on that arc. It is a cousin of Ptolemy's theorem specialized to the equilateral case, where the equal sides make three of Ptolemy's four terms collapse into this clean sum. Verified live: for an equilateral triangle on a circle and thousands of points P on the arc BC, PA equals PB + PC to ~1e-15; and on that arc the 'wrong' identity PB = PA + PC does not hold. Neon-noir traced. See the triangle + P + distances in 1D, the PA=PB+PC identity in 2D, and the one-length-as-sum-of-two inverse in 3D.

THE VAN SCHOOTEN · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · CO-OP · SHARED MEMORY ◆ .dlw.fold THE FOLD / CO-OP / SHARED MEMORY / THE VAN SCHOOTEN THE VAN SCHOOTEN the far distance equal to the sum of the two near ones 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Van Schooten’s theorem is a striking length identity for the equilateral triangle. Inscribe an equilateral triangle ABC in a circle, and take any point P on the arc BC that does not contain A. Then the distance from P to the far vertex equals the sum of the distances to the two near ones: PA = PB + PC . The single long segment exactly balances the two short ones, for every P on that arc. It is a cousin of Ptolemy’s theorem specialized to the equilateral case, where the equal sides make three of Ptolemy’s four terms collapse into this clean sum. LIT verified live: for an equilateral triangle on a circle and thousands of points P on the arc BC, the distance PA equals PB + PC to ~1e-15; and on that arc the ‘wrong’ identity PB = PA + PC does not hold (window.__vanschooten). FIG no framing; the three distances and the PA = PB + PC identity both run in-browser. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at shared-memory — the co-op cell where the two near distances write into one shared total that is exactly the far distance: PB + PC = PA. AVAN (AI) built the instrument: the equilateral-on-a-circle construction, the three distances, and the PA = PB + PC identity with its control. Credit as content: Frans van Schooten (17th c.); a special case of Ptolemy. The weave: David names the shared total; I confirm PA equals PB + PC for P on the far arc. 3 ONE DIMENSION An equilateral triangle on a circle, P on arc BC, and the three distances — PA equals PB + PC. 4 TWO DIMENSIONS · INTERACTIVE Move P along arc BC; PA is checked to equal PB + PC (and off the arc the identity breaks). move P ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: PA, equal to the sum of the two near distances. AVAN’s addition (the inverse-companion): don’t measure the long segment — add the two short ones. The inverse of ‘the distance PA’ is ‘PB + PC’, whenever P sits on the arc opposite A. Magenta are the two near distances PB and PC; green is the far distance PA they sum to. One length as the sum of two. pause spin LIT Genuine Van Schooten's theorem (Frans van Schooten, 17th c.; a special case of Ptolemy). Verified live: for an equilateral triangle on a circle and ~10000 points P on arc BC, PA equals PB + PC to ~1e-15, and the control identity PB = PA + PC does not hold on that arc (window.__vanschooten.ok, .ctrl, .worst). FIG No framing; the three distances and the PA = PB + PC identity both run in-browser. The AVAN inverse is honest — instead of measuring the long segment, add the two short ones: the inverse of 'the distance PA' is 'PB + PC', whenever P sits on the arc opposite A. Magenta are the two near distances PB and PC; green is the far distance PA they sum to. One length as the sum of two. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of SHARED MEMORY · David Lee Wise (ROOT0), with AVAN


THE DOTTIE

The Dottie number in the 5-window house format — the unique real solution of cos(x)=x, approximately 0.7390851332. Punch any number into a calculator and press cosine over and over — cos, cos, cos, … — and the display always drifts to the same value, 0.739085…, no matter where you start. That value is the Dottie number, named after a professor who noticed the phenomenon. It works because the map x→cos(x) is a contraction near its fixed point: the slope there is −sin(D), whose size ~0.674 is less than 1, so every start is drawn in. Verified live: iterating cosine from five different starting points all converge to the same D=0.7390851332, Newton's method on cos(x)−x reaches the same value, cos(D)=D holds, and the multiplier |cos′(D)|=|−sin(D)|≈0.674<1 confirms it is an attracting fixed point. Neon-noir traced. See the cobweb iteration into D in 1D, iteration + Newton in 2D, and the root-by-repetition inverse in 3D.

THE DOTTIE · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · RESPAWN · EVENT HORIZON ◆ .dlw.fold THE FOLD / RESPAWN / EVENT HORIZON / THE DOTTIE THE DOTTIE the fixed point of cosine 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION The Dottie number is the unique real solution of cos(x) = x, approximately 0.7390851332. Punch any number into a calculator and press cosine over and over — cos, cos, cos, … — and the display always drifts to the same value, 0.739085…, no matter where you start. That value is the Dottie number, named after a professor who noticed the phenomenon. It works because the map x → cos(x) is a contraction near its fixed point: the slope there is -sin(D), whose size ~0.674 is less than 1, so every start is drawn in. LIT verified live: iterating cosine from five different starting points all converge to the same D = 0.7390851332, Newton’s method on cos(x)-x reaches the same value, cos(D) = D holds, and the multiplier |cos′(D)| = |-sin(D)| ≈ 0.674 < 1 confirms it is an attracting fixed point (window.__dottie). FIG no framing; the cosine iteration, Newton’s method, and the contraction check all run in-browser. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at event-horizon — the respawn: whatever start you fall in from, the cosine map pulls you across the same horizon to the one fixed point 0.739. AVAN (AI) built the instrument: the cosine fixed-point iteration, Newton’s method, and the contraction-multiplier check. Credit as content: the ‘Dottie number’ (folklore name; the cosine fixed point). The weave: David names the horizon; I confirm every start iterates to cos’s unique fixed point. 3 ONE DIMENSION The curves y = cos(x) and y = x cross once, at the Dottie number; the cobweb iteration spirals into it. 4 TWO DIMENSIONS · INTERACTIVE Pick a start; watch the cosine iterates converge to D — and compare to Newton's method. new start ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the Dottie number D, cos's unique fixed point. AVAN’s addition (the inverse-companion): don’t solve cos(x) = x — just iterate. The inverse of ‘the equation cos(x) = x’ is ‘the attracting fixed point of the map x → cos(x)’, reached from any start because the map contracts. Magenta are the successive cosine iterates; green is the Dottie number they spiral into. A root found by repetition. pause spin LIT Genuine Dottie number (folklore name; the cosine fixed point). Verified live: cosine iteration from five different starts all converge to D=0.7390851332, Newton's method on cos(x)−x reaches the same value, cos(D)=D, and the multiplier |−sin(D)|≈0.674 FIG No framing; the cosine iteration, Newton's method, and the contraction check all run in-browser. The AVAN inverse is honest — instead of solving cos(x)=x, just iterate: the inverse of 'the equation cos(x)=x' is 'the attracting fixed point of the map x→cos(x)', reached from any start because the map contracts. Magenta are the successive cosine iterates; green is the Dottie number they spiral into. A root found by repetition. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of EVENT HORIZON · David Lee Wise (ROOT0), with AVAN


THE WEITZENBOCK

Weitzenböck's inequality in the 5-window house format — bounding a triangle's squared side lengths below by its area: for any triangle with sides a,b,c and area T, a²+b²+c² ≥ 4√3·T. The constant 4√3≈6.928 is the best possible, and equality holds exactly for the equilateral triangle. In other words, for a fixed area, the equilateral triangle has the smallest sum of squared sides — the most 'compact' shape. It is a favourite olympiad inequality and a special case of the sharper Hadwiger–Finsler inequality. Verified live: for tens of thousands of random triangles, a²+b²+c² is always at least 4√3·T — the ratio (a²+b²+c²)/(4√3·T) never drops below 1, and reaches exactly 1 for the equilateral triangle. Neon-noir traced. See the triangle above its floor in 1D, the ratio ≥1 + equilateral equality in 2D, and the squared-sides-floored inverse in 3D.

THE WEITZENBOCK · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · BOSS · THE CHOKE POINT ◆ .dlw.fold THE FOLD / BOSS / THE CHOKE POINT / THE WEITZENBOCK THE WEITZENBOCK a triangle's squared sides bounded below by its area 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Weitzenböck’s inequality bounds a triangle’s squared side lengths below by its area: for any triangle with sides a, b, c and area T, a² + b² + c² ≥ 4√3·T . The constant 4√3 ≈ 6.928 is the best possible, and equality holds exactly for the equilateral triangle . In other words, for a fixed area, the equilateral triangle has the smallest sum of squared sides — the most ‘compact’ shape. It is a favourite olympiad inequality and a special case of the sharper Hadwiger–Finsler inequality. LIT verified live: for tens of thousands of random triangles, a² + b² + c² is always at least 4√3·T — the ratio (a²+b²+c²)/(4√3·T) never drops below 1, and reaches exactly 1 for the equilateral triangle (window.__weitzenbock). FIG no framing; the side lengths, the area, and the inequality all run in-browser. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-choke-point — the boss floor a triangle can never sink below: whatever its shape, its squared sides sum to at least 4√3 times its area, with the equilateral pinned to the floor. AVAN (AI) built the instrument: the side lengths, the area, the inequality ratio, and the equilateral equality case. Credit as content: Roland Weitzenböck (1919). The weave: David names the floor; I confirm a²+b²+c² ≥ 4√3·T, tight at the equilateral. 3 ONE DIMENSION A triangle with its squared sides and its area; a²+b²+c² sits above the floor 4√3·T. 4 TWO DIMENSIONS · INTERACTIVE New triangles; the ratio (a²+b²+c²)/(4√3·T) is shown ≥ 1, reaching 1 only when equilateral. new triangle ▶ make equilateral ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the floor 4√3·T that the squared sides sit above. AVAN’s addition (the inverse-companion): don’t just add the squared sides — know their floor. The inverse of ‘a²+b²+c²’ is ‘at least 4√3 times the area, with equality only for the equilateral triangle’. Magenta is the triangle; green is the 4√3·T floor its squared sides can never cross. Squared sides floored by area. pause spin LIT Genuine Weitzenböck's inequality (Roland Weitzenböck, 1919). Verified live: for ~40000 random triangles, a²+b²+c² ≥ 4√3·T always — the ratio (a²+b²+c²)/(4√3·T) never drops below 1 (min ~1.00001) and equals 1 exactly for the equilateral triangle (window.__weitzenbock.ok, .minR). FIG No framing; the side lengths, the area, and the inequality all run in-browser. The AVAN inverse is honest — instead of just adding the squared sides, know their floor: the inverse of 'a²+b²+c²' is 'at least 4√3 times the area, with equality only for the equilateral triangle'. Magenta is the triangle; green is the 4√3·T floor its squared sides can never cross. Squared sides floored by area. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE CHOKE POINT · David Lee Wise (ROOT0), with AVAN


THE LIOUVILLE NUMBER

Liouville's number in the 5-window house format — L = Σ_{k≥1} 10^{−k!} = 0.110001000000000000000001… (a 1 at every factorial position, 0 elsewhere) was the first number ever proven transcendental (Liouville, 1844). The trick: its digits leave enormous runs of zeros, so the truncations p_n/q_n approximate L absurdly well — |L−p_n/q_n| < 1/q_n^n for every n. But Liouville proved an algebraic number of degree d can never be approximated better than c/q^d. Since L can be approximated to any power, it is not algebraic of any degree — it is transcendental. Verified live with exact big-integer arithmetic: for the truncations of L, the error |L−p_n/q_n| is strictly less than 1/q_n^n for n=1..5, and the approximation exponent (n+1) grows without bound — beating any fixed algebraic degree. Neon-noir traced. See the factorial-position digits in 1D, the error below the Liouville bound in 2D, and the transcendence-from-approximation inverse in 3D.

THE LIOUVILLE NUMBER · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · LOOT · THE STASH ◆ .dlw.fold THE FOLD / LOOT / THE STASH / THE LIOUVILLE NUMBER THE LIOUVILLE NUMBER a number approximated absurdly well by rationals 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Liouville’s number L = ∑ k≥1 10 -k! = 0.110001000000000000000001… (a 1 at every factorial position, 0 elsewhere) was the first number ever proven transcendental (Liouville, 1844). The trick: its digits leave enormous runs of zeros, so the truncations p n /q n approximate L absurdly well — |L - p n /q n | < 1/q n n for every n. But Liouville proved an algebraic number of degree d can never be approximated better than c/q d . Since L can be approximated to any power, it is not algebraic of any degree — it is transcendental. LIT verified live with exact big-integer arithmetic: for the truncations of L = ∑10 -k! , the approximation error |L - p n /q n | is strictly less than 1/q n n for n = 1..5, and the approximation exponent (n+1) grows without bound — beating any fixed algebraic degree (window.__liouvillenumber). FIG no framing; the exact BigInt inequality runs in-browser and confirms the super-fast approximation that forces transcendence. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-stash — the loot: a rare transcendental number, hand-built to be approximated so well by rationals that no polynomial can ever pin it down. AVAN (AI) built the instrument: the factorial-position digits, the truncation errors, and the exact-BigInt Liouville inequality. Credit as content: Joseph Liouville (1844). The weave: David names the stash; I confirm |L - p n /q n | < 1/q n n , the mark of a transcendental. 3 ONE DIMENSION The digits of L: a 1 at positions 1, 2, 6, 24, 120, … (the factorials), long runs of 0 between them. 4 TWO DIMENSIONS · INTERACTIVE Cycle n; the truncation error |L − p_n/q_n| is shown below the Liouville bound 1/q_n^n. next n ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: L, a transcendental pinned by super-good rational approximations. AVAN’s addition (the inverse-companion): don’t ask if L is a root — measure how well rationals catch it. The inverse of ‘is L algebraic?’ is ‘how large can its approximation exponent be?’ — unbounded here, so no polynomial can have L as a root. Magenta are the rational truncations racing toward L; green is the transcendental L they can never quite reach algebraically. Transcendence read from approximation speed. pause spin LIT Genuine Liouville number / Liouville's theorem (Joseph Liouville, 1844). Verified live with exact BigInt: for the truncations of L=Σ10^{−k!}, the error |L−p_n/q_n| is strictly less than 1/q_n^n for n=1..5, and the approximation exponent (n+1) grows without bound, beating any fixed algebraic degree (window.__liouvillenumber.ok, .exps). FIG No framing; the exact BigInt inequality runs in-browser and confirms the super-fast approximation that forces transcendence. The AVAN inverse is honest — instead of asking if L is a root, measure how well rationals catch it: the inverse of 'is L algebraic?' is 'how large can its approximation exponent be?' — unbounded here, so no polynomial can have L as a root. Magenta are the rational truncations racing toward L; green is the transcendental L they can never quite reach algebraically. Transcendence read from approximation speed. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE STASH · David Lee Wise (ROOT0), with AVAN


THE CHU-VANDERMONDE

The Chu–Vandermonde identity in the 5-window house format — collapsing a whole convolution of binomial coefficients into a single one: Σ_k C(m,k)·C(n,r−k) = C(m+n,r). Choosing r objects from a combined pile of m+n is the same as splitting the choice — k from the first pile, r−k from the second — and summing over all splits. Its most famous special case, with m=n=r, gives Σ_k C(n,k)² = C(2n,n): the sum of squared binomial coefficients across a row of Pascal's triangle is the central coefficient two rows down. Verified live with exact big-integer arithmetic: for all m,n up to 15 and every r, the convolution sum Σ_k C(m,k)C(n,r−k) equals C(m+n,r) exactly; and the special case Σ_k C(n,k)²=C(2n,n) holds for n up to 12. Neon-noir traced. See the two Pascal rows convolving in 1D, sum vs C(m+n,r) in 2D, and the convolution-folded inverse in 3D.

THE CHU-VANDERMONDE · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · CHEAT · GOD MODE ◆ .dlw.fold THE FOLD / CHEAT / GOD MODE / THE CHU-VANDERMONDE THE CHU-VANDERMONDE a binomial convolution collapsing to one entry 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION The Chu–Vandermonde identity collapses a whole convolution of binomial coefficients into a single one: ∑ k C(m,k)·C(n,r-k) = C(m+n,r). Choosing r objects from a combined pile of m + n is the same as splitting the choice — k from the first pile, r-k from the second — and summing over all splits. Its most famous special case, with m = n = r, gives ∑ k C(n,k)² = C(2n,n): the sum of squared binomial coefficients across a row of Pascal’s triangle is the central coefficient two rows down. LIT verified live with exact big-integer arithmetic: for all m, n up to 15 and every r, the convolution sum ∑ k C(m,k)C(n,r-k) equals C(m+n,r) exactly; and the special case ∑ k C(n,k)² = C(2n,n) holds for n up to 12 (window.__chuvandermonde). FIG no framing; the binomial convolution and the single closing coefficient both run in-browser and agree exactly. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at god-mode — the cheat: a whole convolution of binomials clipped instantly into one coefficient C(m+n,r), no summing required. AVAN (AI) built the instrument: the binomial convolution, the closing C(m+n,r), and the ∑C(n,k)² special case. Credit as content: Zhu Shijie (Chu, 1303) and Alexandre-Théophile Vandermonde. The weave: David names the cheat; I confirm the convolution equals a single binomial. 3 ONE DIMENSION Two Pascal rows C(m,·) and C(n,·); their convolution at position r equals the single entry C(m+n,r). 4 TWO DIMENSIONS · INTERACTIVE Cycle m, n, r; the convolution sum Σ C(m,k)C(n,r−k) is compared to C(m+n,r). next m,n,r ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the single coefficient C(m+n,r). AVAN’s addition (the inverse-companion): don’t compute a convolution — read one coefficient. The inverse of ‘∑ k C(m,k)C(n,r-k)’ is ‘C(m+n,r)’: choosing r from a combined pile, however you split it. Magenta are the convolution terms C(m,k)C(n,r-k); green is the single binomial they sum to. A convolution folded into one entry. pause spin LIT Genuine Chu–Vandermonde identity (Zhu Shijie 1303; Alexandre-Théophile Vandermonde). Verified live with exact BigInt: for all m,n≤15 and every r, Σ_k C(m,k)C(n,r−k) equals C(m+n,r) exactly, and the special case Σ_k C(n,k)²=C(2n,n) holds for n≤12 (window.__chuvandermonde.ok, .cnt, .sq). FIG No framing; the binomial convolution and the single closing coefficient both run in-browser and agree exactly. The AVAN inverse is honest — instead of computing a convolution, read one coefficient: the inverse of 'Σ_k C(m,k)C(n,r−k)' is 'C(m+n,r)': choosing r from a combined pile, however you split it. Magenta are the convolution terms C(m,k)C(n,r−k); green is the single binomial they sum to. A convolution folded into one entry. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of GOD MODE · David Lee Wise (ROOT0), with AVAN


THE EULER TOTIENT THEOREM

Euler's totient theorem in the 5-window house format — generalizing Fermat's little theorem to any modulus. For any integer a coprime to n, a^φ(n) ≡ 1 (mod n), where φ(n) is Euler's totient — the count of integers from 1 to n coprime to n. Raise a coprime residue to the φ(n)-th power and it snaps back to 1. When n is prime, φ(n)=n−1 and this is exactly Fermat's little theorem. The multiplicative order of a (the smallest k with a^k≡1) always divides φ(n) — a consequence of Lagrange's theorem in the group of units. It is the engine behind RSA. Verified live: for every modulus n up to 200 and every a coprime to n, a^φ(n)≡1 (mod n) by modular exponentiation, and the order of a divides φ(n) — e.g. φ(10)=4 and 3⁴=81≡1 (mod 10). Neon-noir traced. See the power cycle returning to 1 in 1D, a^φ(n)≡1 + order-divides-φ in 2D, and the cycle-length inverse in 3D.

THE EULER TOTIENT THEOREM · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · GRIND · BACKPROP ◆ .dlw.fold THE FOLD / GRIND / BACKPROP / THE EULER TOTIENT THEOREM THE EULER TOTIENT THEOREM a power cycling back to one modulo n 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Euler’s totient theorem generalizes Fermat’s little theorem to any modulus. For any integer a coprime to n, a φ(n) ≡ 1 (mod n) , where φ(n) is Euler’s totient — the count of integers from 1 to n that are coprime to n. Raise a coprime residue to the φ(n)-th power and it snaps back to 1. When n is prime, φ(n) = n-1 and this is exactly Fermat’s little theorem. The multiplicative order of a (the smallest k with a k ≡ 1) always divides φ(n) — a consequence of Lagrange’s theorem in the group of units. It is the engine behind RSA and modular arithmetic. LIT verified live: for every modulus n up to 200 and every a coprime to n, a φ(n) ≡ 1 (mod n) by modular exponentiation, and the order of a divides φ(n) — e.g. φ(10) = 4 and 3 4 = 81 ≡ 1 (mod 10) (window.__eulertotient). FIG no framing; the modular power and the totient are computed independently in-browser and agree. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at backprop — the grind that keeps powering a modulo n, and after φ(n) steps the whole chain resets exactly to 1. AVAN (AI) built the instrument: the totient, the modular exponentiation, and the order-divides-φ(n) check. Credit as content: Leonhard Euler (1763); Fermat for the prime case. The weave: David names the grind; I confirm a φ(n) returns to 1 and the order divides φ(n). 3 ONE DIMENSION The powers a, a², a³, … mod n cycle around and land back on 1 after ord(a) steps (which divides φ(n)). 4 TWO DIMENSIONS · INTERACTIVE Cycle n and a; a^φ(n) mod n is shown equal to 1, and the order of a divides φ(n). next a,n ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the return to 1 after φ(n) powers. AVAN’s addition (the inverse-companion): don’t iterate powers blindly — count the coprimes. The inverse of ‘when does a k return to 1?’ is ‘at k = φ(n) (and its divisors)’ — the totient sets the period. Magenta are the powers of a stepping around mod n; green is the 1 they return to after φ(n) steps. A cycle whose length divides φ(n). pause spin LIT Genuine Euler's totient theorem (Leonhard Euler, 1763; Fermat for the prime case). Verified live: for every modulus n up to 200 and every a coprime to n, a^φ(n)≡1 (mod n) by modular exponentiation, and the multiplicative order of a divides φ(n) (window.__eulertotient.ok, .ordOk). FIG No framing; the modular power and the totient are computed independently in-browser and agree. The AVAN inverse is honest — instead of iterating powers blindly, count the coprimes: the inverse of 'when does a^k return to 1?' is 'at k=φ(n) (and its divisors)' — the totient sets the period. Magenta are the powers of a stepping around mod n; green is the 1 they return to after φ(n) steps. A cycle whose length divides φ(n). ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of BACKPROP · David Lee Wise (ROOT0), with AVAN


THE ERDOS-MORDELL

The Erdős–Mordell inequality in the 5-window house format — relating a point's distances to a triangle's corners and to its sides. For any point P inside triangle ABC, the sum of distances to the three vertices is at least twice the sum of the perpendicular distances to the three sides: PA+PB+PC ≥ 2(dₐ+d_b+d_c). Erdős posed it in 1935; Mordell and Barrow proved it. Equality holds precisely when the triangle is equilateral and P is its centre. The far distances always dominate the near ones by at least a factor of two. Verified live: for tens of thousands of random triangles and interior points P, PA+PB+PC is always at least 2(dₐ+d_b+d_c) — the ratio never drops below 1, approaching 1 only for the equilateral triangle with P at its centre. Neon-noir traced. See the vertex + side distances in 1D, the ≥2 ratio in 2D, and the far-floored-by-near inverse in 3D.

THE ERDOS-MORDELL · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · BOSS · THE FIREWALL ◆ .dlw.fold THE FOLD / BOSS / THE FIREWALL / THE ERDOS-MORDELL THE ERDOS-MORDELL a point's vertex distances bounded below by its side distances 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION The Erdős–Mordell inequality relates a point’s distances to a triangle’s corners and to its sides. For any point P inside triangle ABC, the sum of distances to the three vertices is at least twice the sum of the (perpendicular) distances to the three sides : PA + PB + PC ≥ 2(d a + d b + d c ). Erdős posed it in 1935; Mordell and Barrow proved it. Equality holds precisely when the triangle is equilateral and P is its centre. The far distances always dominate the near ones by at least a factor of two. LIT verified live: for tens of thousands of random triangles and interior points P, PA + PB + PC is always at least 2(d a + d b + d c ) — the ratio never drops below 1, approaching 1 only for the equilateral triangle with P at its centre (window.__erdosmordell). FIG no framing; the vertex distances, the perpendicular side distances, and the inequality all run in-browser. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-firewall — the boss barrier: the sum of a point’s distances to the vertices can never fall below twice its distances to the walls it sits between. AVAN (AI) built the instrument: the vertex distances, the perpendicular side distances, and the ≥2 ratio. Credit as content: Paul Erdős (1935); Louis Mordell and David Barrow (proof). The weave: David names the barrier; I confirm PA+PB+PC ≥ 2(d a +d b +d c ). 3 ONE DIMENSION A triangle with interior P: the three distances to the vertices, and the three perpendiculars to the sides. 4 TWO DIMENSIONS · INTERACTIVE Move P; PA+PB+PC is checked to be ≥ 2(dₐ+d_b+d_c), the ratio ≥ 1 always. move P ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the vertex-distance sum, at least twice the side-distance sum. AVAN’s addition (the inverse-companion): don’t just add the vertex distances — bound them by the side distances. The inverse of ‘PA+PB+PC’ is ‘at least 2(d a +d b +d c )’, tight only for the equilateral centre. Magenta are the perpendicular side distances; green is the vertex-distance sum, floored at twice their total. Far distances floored by near ones. pause spin LIT Genuine Erdős–Mordell inequality (Paul Erdős 1935; Mordell & Barrow, proof). Verified live: for ~40000 random triangles and interior points P, PA+PB+PC ≥ 2(dₐ+d_b+d_c) always — the ratio never drops below 1 (min ~1.002), tight only for the equilateral triangle with P at its centre (window.__erdosmordell.ok, .minR). FIG No framing; the vertex distances, the perpendicular side distances, and the inequality all run in-browser. The AVAN inverse is honest — instead of just adding the vertex distances, bound them by the side distances: the inverse of 'PA+PB+PC' is 'at least 2(dₐ+d_b+d_c)', tight only for the equilateral centre. Magenta are the perpendicular side distances; green is the vertex-distance sum, floored at twice their total. Far distances floored by near ones. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE FIREWALL · David Lee Wise (ROOT0), with AVAN


THE ALTERNATING PERMUTATIONS

Alternating permutations in the 5-window house format — arrangements that zig-zag: a₁<a₂>a₃<a₄>…, going up, down, up, down. The number of them on n elements is the zigzag number (or Euler number) — 1,1,1,2,5,16,61,272,1385,… — and Désiré André proved in 1879 that they are packaged by a beautiful exponential generating function: Σ_n Z(n)xⁿ/n! = sec(x)+tan(x). The even-indexed terms come from the secant (the 'secant numbers'), the odd from the tangent (the 'tangent numbers') — two everyday trig functions counting a purely combinatorial object. Verified live: a brute count of the up-down alternating permutations of n elements equals the coefficient of xⁿ/n! in the Taylor series of sec(x)+tan(x), for every n from 0 to 8 — giving 1,1,1,2,5,16,61,272,1385. Neon-noir traced. See a zigzag permutation in 1D, brute vs sec+tan in 2D, and the counted-by-trigonometry inverse in 3D.

THE ALTERNATING PERMUTATIONS · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · LOOT · THE DROP ◆ .dlw.fold THE FOLD / LOOT / THE DROP / THE ALTERNATING PERMUTATIONS THE ALTERNATING PERMUTATIONS zigzag permutations counted by secant plus tangent 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Alternating permutations are arrangements that zig-zag: a 1 < a 2 > a 3 < a 4 > …, going up, down, up, down. The number of them on n elements is the zigzag number (or Euler number) — 1, 1, 1, 2, 5, 16, 61, 272, 1385, … — and Désiré André proved in 1879 that they are packaged by a beautiful exponential generating function : ∑ n Z(n) x n /n! = sec(x) + tan(x) . The even-indexed terms come from the secant (the ‘secant numbers’), the odd from the tangent (the ‘tangent numbers’) — two everyday trig functions counting a purely combinatorial object. LIT verified live: a brute count of the up-down alternating permutations of n elements equals the coefficient of x n /n! in the Taylor series of sec(x) + tan(x), for every n from 0 to 8 — giving 1, 1, 1, 2, 5, 16, 61, 272, 1385 (window.__alternating). FIG no framing; the brute permutation count and the sec+tan series coefficients both run in-browser and agree. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-drop — the loot: the zigzag count dropping out of two trig functions, sec and tan, as if by magic. AVAN (AI) built the instrument: the brute alternating-permutation count and the sec+tan Taylor coefficients. Credit as content: Désiré André (1879); the Euler zigzag numbers. The weave: David names the drop; I confirm the zigzag count equals the sec+tan series coefficient. 3 ONE DIMENSION An up-down alternating permutation drawn as a zigzag: up, down, up, down through the values. 4 TWO DIMENSIONS · INTERACTIVE Cycle n; the brute count of up-down permutations is compared to the sec+tan series coefficient. next n ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the zigzag number Z(n), the count of alternating permutations. AVAN’s addition (the inverse-companion): don’t list the zigzags — read a trig series. The inverse of ‘count the up-down permutations of n’ is ‘the coefficient of x n /n! in sec(x) + tan(x)’ — secant for even n, tangent for odd. Magenta are the zigzag permutations; green is the Z(n) that sec+tan delivers. Combinatorics counted by trigonometry. pause spin LIT Genuine alternating-permutation / André's theorem (Désiré André, 1879; the Euler zigzag numbers). Verified live: a brute count of up-down alternating permutations of [n] equals the coefficient of xⁿ/n! in the Taylor series of sec(x)+tan(x) for n=0..8, giving 1,1,1,2,5,16,61,272,1385 (window.__alternating.ok). FIG No framing; the brute permutation count and the sec+tan series coefficients both run in-browser and agree. The AVAN inverse is honest — instead of listing the zigzags, read a trig series: the inverse of 'count the up-down permutations of n' is 'the coefficient of xⁿ/n! in sec(x)+tan(x)' — secant for even n, tangent for odd. Magenta are the zigzag permutations; green is the Z(n) that sec+tan delivers. Combinatorics counted by trigonometry. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE DROP · David Lee Wise (ROOT0), with AVAN


THE GREGORY-LEIBNIZ

The Gregory–Leibniz series in the 5-window house format — the most famous, and most beautifully slow, series for π: π/4 = 1 − 1/3 + 1/5 − 1/7 + 1/9 − … = Σ_{k≥0} (−1)^k/(2k+1). Every odd reciprocal, alternating in sign, sums to a quarter of π. It comes straight from the arctangent series arctan(x)=x−x³/3+x⁵/5−… evaluated at x=1, since arctan(1)=π/4. It is exact but converges agonizingly slowly — the error after N terms is only about 1/(2N), so you need hundreds of terms for two decimals. Verified live: 4·Σ(−1)^k/(2k+1) approaches π, and independently the numerical integral 4·∫₀¹ 1/(1+x²) dx (which is 4·arctan(1)) equals π to ~1e-9 — the two routes agree. Neon-noir traced. See the partial sums bracketing π in 1D, series vs integral in 2D, and the π-from-odd-reciprocals inverse in 3D.

THE GREGORY-LEIBNIZ · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · CO-OP · THE PUSH ◆ .dlw.fold THE FOLD / CO-OP / THE PUSH / THE GREGORY-LEIBNIZ THE GREGORY-LEIBNIZ a slow alternating series for π 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION The Gregory–Leibniz series is the most famous — and most beautifully slow — series for π: π/4 = 1 - 1/3 + 1/5 - 1/7 + 1/9 - … = ∑ k≥0 (-1) k /(2k+1). Every odd reciprocal, alternating in sign, sums to a quarter of π. It comes straight from the arctangent series arctan(x) = x - x³/3 + x⁵/5 - … evaluated at x = 1, since arctan(1) = π/4. It is exact but converges agonizingly slowly — the error after N terms is only about 1/(2N), so you need hundreds of terms for two decimals. LIT verified live: 4·∑(-1) k /(2k+1) approaches π, and independently the numerical integral 4·∫ 0 1 1/(1+x²) dx (which is 4·arctan(1)) equals π to ~1e-9 — the two routes agree (window.__gregoryleibniz). FIG no framing; the alternating series and the arctangent integral both run in-browser and give π. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-push — the co-op merge: a crawling alternating series and a clean arctangent integral push in from two directions and meet at π. AVAN (AI) built the instrument: the alternating series partial sums, the arctangent integral, and their agreement on π. Credit as content: James Gregory (1671) and Gottfried Leibniz (1673); Madhava of Sangamagrama earlier. The weave: David names the merge; I confirm the series and the integral both give π. 3 ONE DIMENSION The partial sums of 4(1 − 1/3 + 1/5 − …) oscillating slowly toward π, bracketing it from both sides. 4 TWO DIMENSIONS · INTERACTIVE Add terms; the alternating series crawls toward π, matched by 4·∫₀¹ 1/(1+x²) dx = 4·arctan(1). add terms ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: π, reached by the alternating odd-reciprocal series. AVAN’s addition (the inverse-companion): don’t trust the slow sum alone — cross it with an integral. The inverse of ‘the series ∑(-1) k /(2k+1)’ is ‘the integral ∫ 0 1 1/(1+x²) dx = arctan(1) = π/4’. Magenta are the alternating series terms; green is the π they and the integral both reach. π from the odd reciprocals. pause spin LIT Genuine Gregory–Leibniz series (James Gregory 1671, Gottfried Leibniz 1673; Madhava earlier). Verified live: 4·Σ(−1)^k/(2k+1) approaches π, and independently 4·∫₀¹ 1/(1+x²) dx (= 4·arctan 1) equals π to ~1e-9 — the two routes agree (window.__gregoryleibniz.integOk, .agree). FIG No framing; the alternating series and the arctangent integral both run in-browser and give π. The AVAN inverse is honest — instead of trusting the slow sum alone, cross it with an integral: the inverse of 'the series Σ(−1)^k/(2k+1)' is 'the integral ∫₀¹ 1/(1+x²) dx = arctan(1) = π/4'. Magenta are the alternating series terms; green is the π they and the integral both reach. π from the odd reciprocals. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE PUSH · David Lee Wise (ROOT0), with AVAN