THE PIZZA THEOREM

The pizza theorem in the 5-window house format — a slice of surprising fairness. Take a circular pizza and pick any point P inside it — not necessarily the centre. Make cuts through P at equal angles, and if you make eight slices (four cuts, 45° apart), then two people taking alternate slices always get exactly equal total area — no matter where P was or how the knife was rotated. The off-centre gains of the big slices are exactly cancelled by the losses of the small ones. It works for any number of slices that is a multiple of four and at least eight; curiously, for four slices it fails (whoever gets the slices containing the centre wins). Verified live: for eight slices from a random interior point, the two alternating groups have equal area (to ~1e-4 by fine integration), the total equals πR², and the four-slice control is confirmed unequal. Neon-noir traced. See the coloured alternate slices in 1D, the equal sums in 2D, and the fairness-from-alternation inverse in 3D.

THE PIZZA THEOREM · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · CO-OP · THE SYNC ◆ .dlw.fold THE FOLD / CO-OP / THE SYNC / THE PIZZA THEOREM THE PIZZA THEOREM a pizza split fairly from any interior cut-point 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION The pizza theorem is a slice of surprising fairness. Take a circular pizza and pick any point P inside it — not necessarily the centre. Make cuts through P at equal angles, and if you make eight slices (four cuts, 45° apart), then two people taking alternate slices always get exactly equal total area — no matter where P was or how the knife was rotated. The off-centre gains of the big slices are exactly cancelled by the losses of the small ones. It works for any number of slices that is a multiple of four and at least eight; curiously, for four slices it fails (whoever gets the slices containing the centre wins). LIT verified live: for eight slices from a random interior point, the two alternating groups have equal area (to ~1e-4 by fine integration), the total equals πR², and the control case of four slices is confirmed unequal (window.__pizza). FIG no framing; the sector areas are integrated from the interior point independently in-browser. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-sync — the co-op: two players take turns and, against all intuition, come out exactly even from any off-centre cut. AVAN (AI) built the instrument: the sector-area integration from an interior point, the alternate-sum comparison, and the four-slice control. Credit as content: the pizza theorem (Upton, 1968; Goldberg; and others). The weave: David names the fair split; I confirm eight alternate slices tie while four do not. 3 ONE DIMENSION A pizza cut into 8 slices through an off-centre point; the two alternating colours have equal total area. 4 TWO DIMENSIONS · INTERACTIVE Move the cut-point and rotate; the two alternating 8-slice sums stay equal (4 slices would not). move point ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: one player’s four alternate slices, equal to the other’s. AVAN’s addition (the inverse-companion): don’t aim for the centre — trust the alternation. The inverse of ‘an unfair off-centre cut’ is ‘alternate eighths that always tie, wherever the point is’. Magenta are the other player’s slices; green are yours — equal totals from any interior cut-point. Fairness hidden in the alternation. pause spin LIT Genuine pizza theorem (Upton, 1968; Goldberg and others). Verified live: for eight slices from a random interior point, the two alternating groups have equal area (to ~1e-4 by fine integration), the total equals πR², and the control case of four slices is confirmed unequal (window.__pizza.eq8, .neq4). FIG No framing; the sector areas are integrated from the interior point independently in-browser. The AVAN inverse is honest — instead of aiming for the centre, trust the alternation: the inverse of 'an unfair off-centre cut' is 'alternate eighths that always tie, wherever the point is'. Magenta are the other player's slices; green are yours — equal totals from any interior cut-point. Fairness hidden in the alternation. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE SYNC · David Lee Wise (ROOT0), with AVAN


THE BRIANCHON

Brianchon's theorem in the 5-window house format — the exact mirror-image of Pascal's. Where Pascal takes six points on a conic and finds a line, Brianchon takes six lines tangent to a conic — a hexagon circumscribed about it — and finds a point: the three main diagonals (joining opposite vertices) all pass through one common point. This point-line swap is the deepest idea in projective geometry, duality: every theorem about points on a conic has a twin about tangent lines, trading 'point' for 'line', 'lies on' for 'passes through', 'collinear' for 'concurrent'. Charles-Julien Brianchon proved it in 1810. Verified live: for tens of thousands of random hexagons circumscribed about an ellipse, the three main diagonals are concurrent — the third diagonal passes through the intersection of the first two, normalized residual below 1e-6. Neon-noir traced. See the circumscribed hexagon and its Brianchon point in 1D, the concurrency check in 2D, and the dual-of-Pascal inverse in 3D.

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

THE BRIANCHON · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · LOOT · THE BOUNTY ◆ .dlw.fold THE FOLD / LOOT / THE BOUNTY / THE BRIANCHON THE BRIANCHON six tangents to a conic whose diagonals meet at a point 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Brianchon’s theorem is the exact mirror-image of Pascal’s. Where Pascal takes six points on a conic and finds a line, Brianchon takes six lines tangent to a conic — a hexagon circumscribed about it — and finds a point : the three main diagonals (joining opposite vertices) all pass through one common point . This point-line swap is the deepest idea in projective geometry, duality : every theorem about points on a conic has a twin about tangent lines, obtained by trading ‘point’ for ‘line’, ‘lies on’ for ‘passes through’, ‘collinear’ for ‘concurrent’. Charles-Julien Brianchon proved it in 1810. LIT verified live: for tens of thousands of random hexagons circumscribed about an ellipse (six tangent lines), the three main diagonals are concurrent — the third diagonal passes through the intersection of the first two, normalized residual below 1e-6 (window.__brianchon). FIG no framing; the tangent lines, the vertices, and the diagonal concurrency are computed independently in-browser. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-bounty — the loot: six tangent lines, drawn out to a hexagon, hand over a single meeting-point of all three diagonals. AVAN (AI) built the instrument: the tangent lines, their vertex intersections, and the diagonal concurrency check. Credit as content: Charles-Julien Brianchon (1810); the projective dual of Pascal’s theorem. The weave: David names the collected point; I confirm the three diagonals of a circumscribed hexagon concur. 3 ONE DIMENSION A hexagon of six tangent lines around an ellipse; its three main diagonals meet at the Brianchon point. 4 TWO DIMENSIONS · INTERACTIVE Cycle circumscribed hexagons; the three diagonals are checked to concur at one point. next hexagon ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the Brianchon point where all three diagonals meet. AVAN’s addition (the inverse-companion): don’t only put points on the conic — wrap tangent lines around it. The inverse of ‘Pascal’s line from six points’ is ‘Brianchon’s point from six tangents’ — the projective dual, points ↔ lines. Magenta are the three main diagonals; green is the single point they all pass through. Pascal’s theorem, dualized into a point. pause spin LIT Genuine Brianchon's theorem (Charles-Julien Brianchon, 1810; the projective dual of Pascal's theorem). Verified live: for tens of thousands of random hexagons circumscribed about an ellipse (six tangent lines), the three main diagonals are concurrent — the third diagonal passes through the intersection of the first two, normalized residual below 1e-6 (window.__brianchon.ok, .worst). FIG No framing; the tangent lines, the vertices, and the diagonal concurrency are computed independently in-browser. The AVAN inverse is honest — instead of only putting points on the conic, wrap tangent lines around it: the inverse of 'Pascal's line from six points' is 'Brianchon's point from six tangents' — the projective dual, points ↔ lines. Magenta are the three main diagonals; green is the single point they all pass through. Pascal's theorem, dualized into a point. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE BOUNTY · David Lee Wise (ROOT0), with AVAN


THE BERNOULLI NUMBERS

The Bernoulli numbers in the 5-window house format — a sequence of rationals that surface all over mathematics: 1, −½, 1/6, 0, −1/30, 0, 1/42, 0, −1/30, … They are defined by the recurrence Σ C(n+1,k) B_k = 0, and every odd-indexed one past B₁ is exactly zero. They give the coefficients in Faulhaber's formulas for sums of powers, the Taylor series of tan and coth — and, most beautifully, Euler's closed form for the even zeta values: ζ(2n) = (−1)^{n+1} B_{2n} (2π)^{2n} / (2·(2n)!). Setting n = 1 recovers ζ(2) = π²/6 from B₂ = 1/6. Verified live: the recurrence yields B₂ = 1/6, B₄ = −1/30, B₆ = 1/42, all odd B (past B₁) zero; and Euler's formula gives ζ(2) = π²/6 and ζ(4) = π⁴/90 to ~1e-10. Neon-noir traced. See the sequence with vanishing odd terms in 1D, the ζ(2n) rebuild in 2D, and the sums-pinned-to-a-sequence inverse in 3D.

THE BERNOULLI NUMBERS · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · GRIND · WARM CACHE ◆ .dlw.fold THE FOLD / GRIND / WARM CACHE / THE BERNOULLI NUMBERS THE BERNOULLI NUMBERS a rational sequence hiding inside power sums and the zeta values 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION The Bernoulli numbers B 0 , B 1 , B 2 , … are a sequence of rationals that surface all over mathematics: 1, -½, 1/6, 0, -1/30, 0, 1/42, 0, -1/30, … They are defined by the recurrence ∑ k=0 n C(n+1,k) B k = 0, and every odd-indexed one past B 1 is exactly zero. They give the coefficients in Faulhaber’s formulas for sums of powers, the Taylor series of tan and coth — and, most beautifully, Euler’s closed form for the even zeta values: ζ(2n) = (-1) n+1 B 2n (2π) 2n / (2·(2n)!). Setting n = 1 recovers ζ(2) = π²/6 from B 2 = 1/6. LIT verified live: the recurrence yields B 2 = 1/6, B 4 = -1/30, B 6 = 1/42, all odd B (past B 1 ) zero; and Euler’s formula gives ζ(2) = π²/6 and ζ(4) = π⁴/90 to ~1e-10 (window.__bernoulli). FIG no framing; the recurrence and the zeta formula are computed independently in-browser. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at warm-cache — the grind: one rational recurrence, ground out term by term, that the power-sums and the zeta values all quietly read from. AVAN (AI) built the instrument: the Bernoulli recurrence, the vanishing odd terms, and Euler’s zeta-even formula. Credit as content: Jacob Bernoulli (Ars Conjectandi, 1713); Leonhard Euler (the zeta connection). The weave: David names the shared cache; I confirm the recurrence and ζ(2n) = (-1) n+1 B 2n (2π) 2n /(2(2n)!). 3 ONE DIMENSION The Bernoulli numbers as a sequence — the odd ones (past B₁) all vanish exactly. 4 TWO DIMENSIONS · INTERACTIVE Step the recurrence; each Bₙ is computed, and ζ(2n) is rebuilt from B₂ₙ (ζ(2)=π²/6, ζ(4)=π⁴/90). next n ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: ζ(2) = π²/6, read off from the Bernoulli number B₂. AVAN’s addition (the inverse-companion): don’t sum ζ(2n) directly — read it from a rational. The inverse of ‘the infinite sum ζ(2n)’ is ‘the Bernoulli number B 2n times (2π) 2n /(2(2n)!)’. Magenta are the Bernoulli numbers; green is the ζ(2) = π²/6 that B 2 delivers. Infinite sums pinned to a rational sequence. pause spin LIT Genuine Bernoulli numbers (Jacob Bernoulli, Ars Conjectandi 1713; Euler's zeta connection). Verified live: the recurrence Σ C(n+1,k)B_k = 0 yields B₂ = 1/6, B₄ = −1/30, B₆ = 1/42, all odd B past B₁ zero; and Euler's formula gives ζ(2) = π²/6 and ζ(4) = π⁴/90 to ~1e-10 (window.__bernoulli.ok). FIG No framing; the recurrence and the zeta formula are computed independently in-browser. The AVAN inverse is honest — instead of summing ζ(2n) directly, read it from a rational: the inverse of 'the infinite sum ζ(2n)' is 'the Bernoulli number B_{2n} times (2π)^{2n}/(2(2n)!)'. Magenta are the Bernoulli numbers; green is the ζ(2) = π²/6 that B₂ delivers. Infinite sums pinned to a rational sequence. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of WARM CACHE · David Lee Wise (ROOT0), with AVAN


THE LUCKY EULER

Euler's lucky numbers in the 5-window house format — a startling coincidence Euler found in 1772: the polynomial n²+n+41 produces a prime for every n from 0 to 39 — forty primes in an unbroken run: 41, 43, 47, 53, 61, 71, …, 1601. The streak finally breaks at n = 40, where 40²+40+41 = 1681 = 41². Even beyond that it stays astonishingly prime-rich (about 58% of values up to n = 1000 are prime). The magic isn't luck: 41 is the largest of the six 'lucky numbers of Euler', tied to the fact that the imaginary quadratic field of discriminant −163 = 1 − 4·41 has class number one — unique factorization, the deepest reason the primes line up. Verified live: n²+n+41 is prime for all n = 0 to 39, composite at n = 40 (= 41²), and about 58% of values up to n = 1000 are prime. Neon-noir traced. See the 40-prime run breaking at 41² in 1D, the per-n factorization in 2D, and the class-number-one inverse in 3D.

THE LUCKY EULER · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · CHEAT · THE ROOT KIT ◆ .dlw.fold THE FOLD / CHEAT / THE ROOT KIT / THE LUCKY EULER THE LUCKY EULER a polynomial that spits primes forty times in a row 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Euler’s lucky numbers come from a startling coincidence he found in 1772: the polynomial n² + n + 41 produces a prime for every n from 0 to 39 — forty primes in an unbroken run: 41, 43, 47, 53, 61, 71, …, 1601. The streak finally breaks at n = 40, where 40²+40+41 = 1681 = 41². Even beyond that it stays astonishingly prime-rich (about 58% of values up to n = 1000 are prime). The magic isn’t luck: 41 is the largest of the six ‘lucky numbers of Euler’, tied to the fact that the imaginary quadratic field of discriminant -163 = 1 - 4·41 has class number one — unique factorization, the deepest reason the primes line up. LIT verified live: n²+n+41 is confirmed prime for all n = 0 to 39, composite at n = 40 (equal to 41²), and about 58% of values up to n = 1000 are prime (window.__luckyeuler). FIG no framing; the polynomial values and their primality are computed independently in-browser. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-root-kit — the cheat: a single quadratic that injects forty primes in a row, no sieve required. AVAN (AI) built the instrument: the polynomial n²+n+41, its forty-prime streak, and the break at 41². Credit as content: Leonhard Euler (1772); the connection to discriminant -163 and class number one. The weave: David names the prime-cheat; I confirm the forty-long streak and its exact break. 3 ONE DIMENSION n²+n+41 for n = 0…44 — an unbroken run of green primes, breaking to red at n = 40 (=41²). 4 TWO DIMENSIONS · INTERACTIVE Step n; n²+n+41 is factored and tested — prime through n=39, then 41² at n=40. next n ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the forty-long streak of primes from one quadratic. AVAN’s addition (the inverse-companion): don’t marvel at the streak — ask why it holds. The inverse of ‘forty primes in a row’ is ‘discriminant -163 has class number one — unique factorization forces it’. Magenta are the polynomial values; green is the unbroken run of primes they form. A coincidence that is really a deep theorem. pause spin LIT Genuine Euler's lucky numbers / prime-generating polynomial (Leonhard Euler, 1772; discriminant −163, class number one). Verified live: n²+n+41 is prime for all n = 0 to 39, composite at n = 40 (= 41²), and about 58% of values up to n = 1000 are prime (window.__luckyeuler.allPrime, .fail40, .cnt). FIG No framing; the polynomial values and their primality are computed independently in-browser. The AVAN inverse is honest — instead of marvelling at the streak, ask why it holds: the inverse of 'forty primes in a row' is 'discriminant −163 has class number one — unique factorization forces it'. Magenta are the polynomial values; green is the unbroken run of primes they form. A coincidence that is really a deep theorem. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE ROOT KIT · David Lee Wise (ROOT0), with AVAN


THE NINE-POINT CIRCLE

The nine-point circle in the 5-window house format — one of the most elegant facts about a triangle: nine special points all lie on a single circle. They are the three midpoints of the sides, the three feet of the altitudes, and the three midpoints of the segments from each vertex to the orthocentre. No matter how the triangle is shaped, these nine points are perfectly concyclic. The circle's centre N is the midpoint between the circumcentre O and the orthocentre H (so N sits on the Euler line), and its radius is exactly half the circumradius, R/2. It touches the incircle and the three excircles (Feuerbach's theorem). Verified live: for tens of thousands of random triangles, all nine points are equidistant from N = midpoint(O, H), at distance exactly R/2, to ~1e-14. Neon-noir traced. See the nine points on one circle in 1D, the nine equal radii in 2D, and the three-constructions-one-circle inverse in 3D.

THE NINE-POINT CIRCLE · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · SPAWN · THE TOOLCHAIN ◆ .dlw.fold THE FOLD / SPAWN / THE TOOLCHAIN / THE NINE-POINT CIRCLE THE NINE-POINT CIRCLE nine special triangle points on one circle 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION The nine-point circle is one of the most elegant facts about a triangle: nine special points all lie on a single circle. They are the three midpoints of the sides , the three feet of the altitudes , and the three midpoints of the segments from each vertex to the orthocentre. No matter how the triangle is shaped, these nine points are perfectly concyclic. The circle’s centre N is the midpoint between the circumcentre O and the orthocentre H (so N sits on the Euler line), and its radius is exactly half the circumradius , R/2. It touches the incircle and the three excircles (Feuerbach’s theorem). LIT verified live: for tens of thousands of random triangles, all nine points are equidistant from N = midpoint(O, H), at distance exactly R/2, to ~1e-14 (window.__ninepoint). FIG no framing; the nine points, the centre, and the radius are computed independently in-browser. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-toolchain — the spawn: nine points from three different constructions compile, every time, onto one circle of radius R/2. AVAN (AI) built the instrument: the side-midpoints, altitude-feet, Euler-point midpoints, and the shared circle. Credit as content: Poncelet and Brianchon (the nine-point circle); Karl Feuerbach (the tangency). The weave: David names the compile; I confirm all nine points sit on the circle of radius R/2 about the midpoint of O and H. 3 ONE DIMENSION A triangle with its nine points — side midpoints, altitude feet, Euler-point midpoints — all on one circle. 4 TWO DIMENSIONS · INTERACTIVE Cycle triangles; all nine distances to the centre N are checked equal to R/2. next triangle ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the nine-point circle, radius R/2 about the midpoint of O and H. AVAN’s addition (the inverse-companion): don’t track nine points from three constructions — read the one circle they share. The inverse of ‘nine special points’ is ‘a single circle of radius R/2 centred midway between circumcentre and orthocentre’. Magenta are the nine points; green is the circle carrying them all. Three constructions, one circle. pause spin LIT Genuine nine-point circle (Poncelet and Brianchon; Feuerbach for the tangency). Verified live: for ~40000 random triangles, all nine points (three side-midpoints, three altitude-feet, three Euler-point midpoints) are equidistant from N = midpoint(O, H) at distance exactly R/2, to ~1e-14 (window.__ninepoint.ok, .worst). FIG No framing; the nine points, the centre, and the radius are computed independently in-browser. The AVAN inverse is honest — instead of tracking nine points from three constructions, read the one circle they share: the inverse of 'nine special points' is 'a single circle of radius R/2 centred midway between circumcentre and orthocentre'. Magenta/gold/cyan are the nine points; green is the circle carrying them all. Three constructions, one circle. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE TOOLCHAIN · David Lee Wise (ROOT0), with AVAN


THE GAUSSIAN INTEGRAL

The Gaussian integral in the 5-window house format — the beautiful fact that the area under the bell curve is the square root of π: ∫_{−∞}^{∞} e^{−x²} dx = √π. There is no elementary antiderivative for e^{−x²} — you cannot integrate it term by term — yet the total area is exactly √π ≈ 1.7724539. The classic trick squares the integral and switches to polar coordinates, turning an impossible one-dimensional integral into an easy two-dimensional one. Rescaled, it gives the normalization of the normal distribution: ∫ e^{−x²/2} dx = √(2π), which is why the bell curve of statistics divides by √(2π). Verified live: numerical integration of e^{−x²} over the real line gives 1.7724539… = √π to ~1e-7, and e^{−x²/2} integrates to √(2π). Neon-noir traced. See the shaded bell curve area in 1D, the quadrature converging to √π in 2D, and the square-it-to-solve-it inverse in 3D.

THE GAUSSIAN INTEGRAL · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · BOSS · THE RAID ◆ .dlw.fold THE FOLD / BOSS / THE RAID / THE GAUSSIAN INTEGRAL THE GAUSSIAN INTEGRAL a bell curve whose area is the square root of pi 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION The Gaussian integral is the beautiful fact that the area under the bell curve is the square root of π: ∫ -∞ ∞ e -x² dx = √π . There is no elementary antiderivative for e -x² — you cannot integrate it term by term — yet the total area is exactly √π ≈ 1.7724539. The classic trick squares the integral and switches to polar coordinates, turning an impossible one-dimensional integral into an easy two-dimensional one. Rescaled, it gives the normalization of the normal distribution: ∫ e -x²/2 dx = √(2π), which is why the bell curve of statistics divides by √(2π). LIT verified live: numerical integration of e -x² over the real line gives 1.7724539… = √π to ~1e-7, and e -x²/2 integrates to √(2π) (window.__gaussianintegral). FIG no framing; the integral is computed by fine numerical quadrature independently in-browser. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-raid — the boss with no elementary antiderivative: the bell curve resists term-by-term integration, yet yields its whole area √π to the polar trick. AVAN (AI) built the instrument: the quadrature of e -x² and its match to √π (and √(2π) for the normal). Credit as content: Carl Friedrich Gauss and Pierre-Simon Laplace (the integral and the normal distribution). The weave: David names the un-antidifferentiable boss; I confirm the area equals √π. 3 ONE DIMENSION The bell curve e^(−x²); the shaded area under the whole curve equals √π ≈ 1.77245. 4 TWO DIMENSIONS · INTERACTIVE Refine the quadrature; the numerical area converges to √π (and e^(−x²/2) to √(2π)). refine ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the area √π under the one-dimensional bell curve. AVAN’s addition (the inverse-companion): don’t fight the missing antiderivative — go up a dimension. The inverse of ‘the 1-D integral of e -x² ’ is ‘its square as a 2-D polar integral, which equals π — so the original is √π’. Magenta is the bell curve’s area strip; green is the √π it totals. An impossible integral solved by squaring it. pause spin LIT Genuine Gaussian integral (Carl Friedrich Gauss and Pierre-Simon Laplace). Verified live: numerical quadrature of e^{−x²} over the real line gives 1.7724539… = √π to ~1e-7, and e^{−x²/2} integrates to √(2π) (window.__gaussianintegral.Iok, .I2ok). FIG No framing; the integral is computed by fine numerical quadrature independently in-browser. The AVAN inverse is honest — instead of fighting the missing antiderivative, go up a dimension: the inverse of 'the 1-D integral of e^{−x²}' is 'its square as a 2-D polar integral, which equals π — so the original is √π'. Magenta is the bell curve's area strip; green is the √π it totals. An impossible integral solved by squaring it. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE RAID · David Lee Wise (ROOT0), with AVAN


THE MONTY HALL

The Monty Hall problem in the 5-window house format — the most famous counter-intuitive result in probability. You pick one of three doors; behind one is a car, behind the others goats. The host — who knows where the car is — opens a different door revealing a goat, then offers you the chance to switch. Should you? Yes: switching wins 2/3 of the time, staying only 1/3. Your first pick is right 1/3 of the time, so the other door hides the car the remaining 2/3 — and the host's reveal concentrates all of that onto the single unopened door. It scales: with N doors and one goat revealed, switching to a random remaining door wins (N−1)/(N(N−2)). Verified live: a Monte-Carlo simulation gives switch ≈ 2/3 and stay ≈ 1/3 for three doors, and matches (N−1)/(N(N−2)) for four and five doors. Neon-noir traced. See the three doors and outcomes in 1D, the win-rate settling on 2/3 in 2D, and the where-did-the-2/3-go inverse in 3D.

THE MONTY HALL · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · GLITCH · UNDEFINED BEHAVIOR ◆ .dlw.fold THE FOLD / GLITCH / UNDEFINED BEHAVIOR / THE MONTY HALL THE MONTY HALL a game show where switching doubles your odds 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION The Monty Hall problem is the most famous counter-intuitive result in probability. You pick one of three doors; behind one is a car, behind the others goats. The host — who knows where the car is — opens a different door revealing a goat, then offers you the chance to switch . Should you? Yes: switching wins 2/3 of the time, staying only 1/3. Your first pick is right 1/3 of the time, so the other door hides the car the remaining 2/3 — and the host’s reveal concentrates all of that onto the single unopened door. It scales: with N doors and one goat revealed, switching to a random remaining door wins (N-1)/(N(N-2)). LIT verified live: a Monte-Carlo simulation gives switch ≈ 2/3 and stay ≈ 1/3 for three doors, and matches (N-1)/(N(N-2)) for four and five doors (window.__montyhall). FIG no framing; the game is simulated with a fair random generator independently in-browser. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at undefined-behavior — the glitch in intuition: the odds seem 50/50 after a door opens, yet switching quietly wins twice as often. AVAN (AI) built the instrument: the three-door simulation, the 2/3-vs-1/3 split, and the N-door generalization. Credit as content: the Monty Hall problem (Steve Selvin, 1975; popularized by Marilyn vos Savant, 1990). The weave: David names the intuition-glitch; I confirm switching wins 2/3 and the N-door formula. 3 ONE DIMENSION The three doors: your pick, the host's reveal, and the two outcomes — switch (2/3) vs stay (1/3). 4 TWO DIMENSIONS · INTERACTIVE Run more trials; the switch-win rate settles on 2/3 (and the N-door rate on its formula). doors: 3 ▶ +trials ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the 2/3 win rate from switching. AVAN’s addition (the inverse-companion): don’t re-price the two closed doors as 50/50 — track where the 2/3 went. The inverse of ‘your 1/3 first pick’ is ‘the other 2/3, swept by the host onto the one unopened door’. Magenta is the stay probability (1/3); green is the switch probability (2/3) it complements. The host’s reveal hands you the better two-thirds. pause spin LIT Genuine Monty Hall problem (Steve Selvin, 1975; popularized by Marilyn vos Savant, 1990). Verified live: a Monte-Carlo simulation gives switch ≈ 2/3 and stay ≈ 1/3 for three doors, and matches (N−1)/(N(N−2)) for four and five doors (window.__montyhall.ok). FIG No framing; the game is simulated with a fair random generator independently in-browser. The AVAN inverse is honest — instead of re-pricing the two closed doors as 50/50, track where the 2/3 went: the inverse of 'your 1/3 first pick' is 'the other 2/3, swept by the host onto the one unopened door'. Magenta is the stay probability (1/3); green is the switch probability (2/3) it complements. The host's reveal hands you the better two-thirds. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of UNDEFINED BEHAVIOR · David Lee Wise (ROOT0), with AVAN


THE EULER-MASCHERONI

The Euler–Mascheroni constant in the 5-window house format — γ ≈ 0.5772156649, the mysterious gap between two things that both grow without bound: the harmonic series H_n = 1 + 1/2 + … + 1/n, and the natural logarithm ln(n). Both march to infinity, but their difference settles onto a single fixed number: γ = lim(H_n − ln n). It appears in the gamma function, the prime-counting function, and the zeta function — yet after 250 years no one knows whether γ is even irrational. The plain limit crawls (error ~1/2n), but a corrected form H_n − ln n − 1/(2n) + 1/(12n²) sprints to γ. Verified live: H_n − ln n approaches 0.5772156649… (to ~1e-6 at n = 2×10⁶), and the corrected form reaches γ to ~1e-12. Neon-noir traced. See the gap settling onto γ in 1D, plain-vs-corrected in 2D, and the two-infinities-one-remainder inverse in 3D.

THE EULER-MASCHERONI · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · GRIND · THE EPOCH ◆ .dlw.fold THE FOLD / GRIND / THE EPOCH / THE EULER-MASCHERONI THE EULER-MASCHERONI the constant left over between the harmonic series and the logarithm 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION The Euler–Mascheroni constant γ ≈ 0.5772156649 is the mysterious gap between two things that both grow without bound: the harmonic series H n = 1 + 1/2 + 1/3 + … + 1/n, and the natural logarithm ln(n). Both march off to infinity, but their difference settles down to a single fixed number: γ = lim n→∞ (H n - ln n). It appears everywhere — in the gamma function, the prime-counting function, and the zeta function — yet after 250 years no one knows whether γ is even irrational . The plain limit crawls (error ~1/2n), but a corrected form H n - ln n - 1/(2n) + 1/(12n²) sprints to γ. LIT verified live: H n - ln n approaches 0.5772156649… (to ~1e-6 at n = 2×10⁶), and the corrected form reaches γ to ~1e-12 (window.__eulermascheroni). FIG no framing; the harmonic sum, the logarithm, and their difference are computed independently in-browser. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-epoch — the grind: the harmonic sum ground out term by term, always staying exactly γ ahead of the logarithm. AVAN (AI) built the instrument: the harmonic partial sum, the logarithm, their difference, and the fast-converging correction. Credit as content: Leonhard Euler and Lorenzo Mascheroni. The weave: David names the grind; I confirm H n - ln n → γ = 0.5772156649… 3 ONE DIMENSION The gap H_n − ln(n) settling onto γ ≈ 0.57722 as n grows. 4 TWO DIMENSIONS · INTERACTIVE Grow n; the plain gap crawls toward γ while the corrected form sprints there. grow n ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: γ, the fixed gap between the harmonic series and the logarithm. AVAN’s addition (the inverse-companion): don’t chase two diverging quantities — read the constant they leave behind. The inverse of ‘H n and ln n both → ∞’ is ‘their difference → the fixed number γ’. Magenta is the harmonic staircase above the logarithm; green is the constant gap γ between them. Two infinities, one finite remainder. pause spin LIT Genuine Euler–Mascheroni constant γ (Leonhard Euler and Lorenzo Mascheroni). Verified live: H_n − ln n approaches 0.5772156649… (to ~1e-6 at n = 2×10⁶), and the corrected form H_n − ln n − 1/(2n) + 1/(12n²) reaches γ to ~1e-12 (window.__eulermascheroni.ok). FIG No framing; the harmonic sum, the logarithm, and their difference are computed independently in-browser. The AVAN inverse is honest — instead of chasing two diverging quantities, read the constant they leave behind: the inverse of 'H_n and ln n both → ∞' is 'their difference → the fixed number γ'. Magenta is the harmonic staircase above the logarithm; green is the constant gap γ between them. Two infinities, one finite remainder. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE EPOCH · David Lee Wise (ROOT0), with AVAN


THE BIRTHDAY PARADOX

The birthday paradox in the 5-window house format — the shock that in a room of just 23 people, it is more likely than not that two share a birthday. It feels wrong — 365 days, surely you'd need ~180 people? But you are not matching one fixed birthday; you are checking all pairs, and 23 people make 253 pairs. The probability of at least one shared birthday is 1 − (365/365)(364/365)…((365−n+1)/365); at n = 23 it crosses 0.507, past a half. By 57 people it is over 99%. The counter-intuition comes from confusing 'a match with me' (linear) with 'a match among anyone' (quadratic in the number of people). Verified live: the exact formula gives P = 0.5073 at 23 people (> 1/2) and 0.9901 at 57; a Monte-Carlo simulation matches across several group sizes. Neon-noir traced. See the probability crossing ½ at 23 in 1D, exact-vs-simulation in 2D, and the count-pairs-not-people inverse in 3D.

THE BIRTHDAY PARADOX · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · GLITCH · UNDEFINED BEHAVIOR ◆ .dlw.fold THE FOLD / GLITCH / UNDEFINED BEHAVIOR / THE BIRTHDAY PARADOX THE BIRTHDAY PARADOX twenty-three people enough to share a birthday 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION The birthday paradox is the shock that in a room of just 23 people, it is more likely than not that two share a birthday. It feels wrong — there are 365 days, so surely you’d need ~180 people? But you are not matching one fixed birthday; you are checking all pairs , and 23 people make 253 pairs. The probability of at least one shared birthday is 1 - (365/365)(364/365)(363/365)…((365-n+1)/365); at n = 23 it crosses 0.507 , past a half. By 57 people it is over 99%. The counter-intuition comes from confusing ‘a match with me’ (linear) with ‘a match among anyone’ (quadratic in the number of people). LIT verified live: the exact formula gives P = 0.5073 at 23 people (> 1/2) and 0.9901 at 57; a Monte-Carlo simulation matches the exact probabilities across several group sizes (window.__birthday). FIG no framing; the exact product and the random simulation are computed independently in-browser. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at undefined-behavior — the glitch in intuition: 23 people feel far too few, yet the pair-count quietly tips the odds past a half. AVAN (AI) built the instrument: the exact collision formula and the Monte-Carlo confirmation. Credit as content: the birthday problem (Richard von Mises and others). The weave: David names the intuition-glitch; I confirm 23 people cross 1/2 and simulation agrees. 3 ONE DIMENSION P(shared birthday) rising with group size — crossing 1/2 at exactly 23 people. 4 TWO DIMENSIONS · INTERACTIVE Change the group size; the exact probability is checked against a random simulation. group size ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the >50% collision chance at 23 people. AVAN’s addition (the inverse-companion): don’t count people — count pairs. The inverse of ‘23 people’ is ‘253 pairs, each a chance to collide’ — the quadratic that beats intuition. Magenta are the pairwise comparisons; green is the >1/2 chance they add up to. Not you-versus-one, but everyone-versus-everyone. pause spin LIT Genuine birthday problem (Richard von Mises and others). Verified live: the exact formula gives P = 0.5073 at 23 people (> 1/2) and 0.9901 at 57; a Monte-Carlo simulation matches the exact probabilities across several group sizes (window.__birthday.ok, .simOk). FIG No framing; the exact product and the random simulation are computed independently in-browser. The AVAN inverse is honest — instead of counting people, count pairs: the inverse of '23 people' is '253 pairs, each a chance to collide' — the quadratic that beats intuition. Magenta are the pairwise comparisons; green is the >1/2 chance they add up to. Not you-versus-one, but everyone-versus-everyone. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of UNDEFINED BEHAVIOR · David Lee Wise (ROOT0), with AVAN


THE FIBONACCI-GCD

The Fibonacci–GCD identity in the 5-window house format — a jewel of divisibility: the greatest common divisor of two Fibonacci numbers is itself a Fibonacci number, and exactly the one whose index is the gcd of the indices. In symbols, gcd(F_m, F_n) = F_{gcd(m,n)}. For example gcd(F₁₂, F₁₈) = gcd(144, 2584) = 8 = F₆, and gcd(12,18) = 6. The Fibonacci sequence carries the whole divisibility structure of the integers on its back. A clean corollary: for m ≥ 3, F_m divides F_n if and only if m divides n — every third Fibonacci is even (divisible by F₃ = 2), every fourth divisible by F₄ = 3, and so on. Verified live with exact BigInt: gcd(F_m, F_n) = F_{gcd(m,n)} for all m, n up to 40, and F_m | F_n ⟺ m | n for m ≥ 3. Neon-noir traced. See the gcd land back on F_{gcd(m,n)} in 1D, the identity check in 2D, and the sequence-commutes-with-gcd inverse in 3D.

THE FIBONACCI-GCD · 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 FIBONACCI-GCD THE FIBONACCI-GCD a greatest common divisor that stays inside the Fibonacci sequence 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION The Fibonacci–GCD identity is a jewel of divisibility: the greatest common divisor of two Fibonacci numbers is itself a Fibonacci number — and exactly the one whose index is the gcd of the indices. In symbols, gcd(F m , F n ) = F gcd(m,n) . For example gcd(F 12 , F 18 ) = gcd(144, 2584) = 8 = F 6 , and gcd(12,18) = 6. The Fibonacci sequence carries the whole divisibility structure of the integers on its back. A clean corollary follows: for m ≥ 3, F m divides F n if and only if m divides n — every third Fibonacci is even (divisible by F 3 = 2), every fourth is divisible by F 4 = 3, and so on. LIT verified live with exact BigInt: gcd(F m , F n ) = F gcd(m,n) for all m, n up to 40, and F m | F n ⇔ m | n for m ≥ 3 (window.__fibonaccigcd). FIG no framing; the Fibonacci numbers and both gcd sides are computed independently in-browser. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-pull-request — the co-op: two Fibonacci numbers merge to a common divisor that is itself a Fibonacci, indexed by the merge of their indices. AVAN (AI) built the instrument: the exact Fibonacci sequence, the two gcd sides, and the divisibility corollary. Credit as content: the Fibonacci divisibility sequence (a strong divisibility sequence). The weave: David names the merge; I confirm gcd(F m , F n ) = F gcd(m,n) . 3 ONE DIMENSION gcd(F_m, F_n) landing back on F_{gcd(m,n)} — the gcd of indices, read into the sequence. 4 TWO DIMENSIONS · INTERACTIVE Cycle indices m, n; gcd(F_m, F_n) is checked equal to F_{gcd(m,n)}. next m,n ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: F_{gcd(m,n)}, the common divisor that stays Fibonacci. AVAN’s addition (the inverse-companion): don’t take the gcd of the big numbers — take it of the indices first. The inverse of ‘gcd(F m , F n )’ is ‘F evaluated at gcd(m, n)’ — the sequence commutes with gcd. Magenta are the two Fibonacci numbers; green is the Fibonacci common divisor F gcd(m,n) . Divisibility carried inside the sequence. pause spin LIT Genuine Fibonacci divisibility sequence / GCD identity (a strong divisibility sequence). Verified live with exact BigInt: gcd(F_m, F_n) = F_{gcd(m,n)} for all m, n up to 40, and F_m | F_n ⟺ m | n for m ≥ 3 (window.__fibonaccigcd.ok, .divOk). FIG No framing; the Fibonacci numbers and both gcd sides are computed independently in-browser. The AVAN inverse is honest — instead of taking the gcd of the big numbers, take it of the indices first: the inverse of 'gcd(F_m, F_n)' is 'F evaluated at gcd(m, n)' — the sequence commutes with gcd. Magenta are the two Fibonacci numbers; green is the Fibonacci common divisor F_{gcd(m,n)}. Divisibility carried inside the sequence. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE PULL REQUEST · David Lee Wise (ROOT0), with AVAN


THE GERGONNE

The Gergonne point in the 5-window house format — a hidden meeting-point every triangle carries. Inscribe the incircle — the circle tangent to all three sides. It touches the sides at three contact points. Now draw a line (a cevian) from each vertex to the contact point on the opposite side. Astonishingly, all three of these lines meet at a single point: the Gergonne point. It works for every triangle, guaranteed by Ceva's theorem, because the contact point on side a sits at distance s−b from one end and s−c from the other (s the semiperimeter), and the three ratios multiply to exactly 1. Named for Joseph Diez Gergonne. Verified live: for tens of thousands of random triangles, the three cevians from the vertices to the incircle's contact points are concurrent — the third passes through the intersection of the first two, normalized residual below 1e-6. Neon-noir traced. See the incircle, contact points, and meeting cevians in 1D, the concurrency check in 2D, and the three-lines-one-meeting inverse in 3D.

THE GERGONNE · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · BOSS · THE FINAL BOSS ◆ .dlw.fold THE FOLD / BOSS / THE FINAL BOSS / THE GERGONNE THE GERGONNE triangle cevians to the incircle meeting at one point 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION The Gergonne point is a hidden meeting-point every triangle carries. Inscribe the incircle — the circle tangent to all three sides. It touches the sides at three contact points . Now draw a line (a cevian) from each vertex to the contact point on the opposite side. Astonishingly, all three of these lines meet at a single point: the Gergonne point. It works for every triangle, guaranteed by Ceva’s theorem, because the contact point on side a sits at distance s-b from one end and s-c from the other (s the semiperimeter), and the three ratios multiply to exactly 1. Named for Joseph Diez Gergonne. LIT verified live: for tens of thousands of random triangles, the three cevians from the vertices to the incircle’s contact points are concurrent — the third passes through the intersection of the first two, normalized residual below 1e-6 (window.__gergonne). FIG no framing; the incircle contact points and the cevian concurrency are computed independently in-browser. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-final-boss — the boss gate: three cevians drawn to the incircle’s touch-points always converge on one point, no exceptions. AVAN (AI) built the instrument: the incircle contact points, the three cevians, and the concurrency check. Credit as content: Joseph Diez Gergonne; Ceva’s theorem. The weave: David names the gate; I confirm the three contact-point cevians meet at the Gergonne point. 3 ONE DIMENSION A triangle, its incircle, the three contact points, and the cevians meeting at the Gergonne point. 4 TWO DIMENSIONS · INTERACTIVE Cycle triangles; the three contact-point cevians are checked to concur at one point. next triangle ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the Gergonne point where all three cevians meet. AVAN’s addition (the inverse-companion): don’t track three separate cevians — read the single point they force. The inverse of ‘three contact-point cevians’ is ‘one Gergonne point, guaranteed by Ceva’s ratio product = 1’. Magenta are the three cevians; green is the point they all pass through. Three lines, one forced meeting. pause spin LIT Genuine Gergonne point (Joseph Diez Gergonne; Ceva's theorem). Verified live: for tens of thousands of random triangles, the three cevians from the vertices to the incircle's contact points are concurrent — the third passes through the intersection of the first two, normalized residual below 1e-6 (window.__gergonne.ok, .worst). FIG No framing; the incircle contact points and the cevian concurrency are computed independently in-browser. The AVAN inverse is honest — instead of tracking three separate cevians, read the single point they force: the inverse of 'three contact-point cevians' is 'one Gergonne point, guaranteed by Ceva's ratio product = 1'. Magenta are the three cevians; green is the point they all pass through. Three lines, one forced meeting. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE FINAL BOSS · David Lee Wise (ROOT0), with AVAN


THE VANDERMONDE DETERMINANT

The Vandermonde determinant in the 5-window house format — a stunningly clean answer to a messy-looking question. Build a matrix whose rows are the powers of some numbers x₀, x₁, …, x_{n−1} — row i is (1, x_i, x_i², …, x_i^{n−1}). Its determinant, which looks like it should be a horrible polynomial, factors perfectly into a product of all pairwise differences: det = ∏_{i<j} (x_j − x_i). So the determinant is zero exactly when two of the numbers coincide — which is why polynomial interpolation through distinct points always has a unique solution. It underlies interpolation, coding theory (Reed–Solomon), and the theory of symmetric functions. Verified live: for tens of thousands of random node sets (n = 3 to 6), the determinant computed by Gaussian elimination equals ∏_{i<j}(x_j − x_i) to ~1e-9. Neon-noir traced. See the power matrix and its product-of-gaps determinant in 1D, det-vs-product in 2D, and the determinant-is-just-the-gaps inverse in 3D.

THE VANDERMONDE DETERMINANT · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · LOOT · THE MINT ◆ .dlw.fold THE FOLD / LOOT / THE MINT / THE VANDERMONDE DETERMINANT THE VANDERMONDE DETERMINANT a determinant that factors into pairwise differences 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION The Vandermonde determinant gives a stunningly clean answer to a messy-looking question. Build a matrix whose rows are the powers of some numbers x 0 , x 1 , …, x n-1 — row i is (1, x i , x i ², …, x i n-1 ). Its determinant, which looks like it should be a horrible polynomial, factors perfectly into a product of all pairwise differences: det = ∏ i<j (x j - x i ). So the determinant is zero exactly when two of the numbers coincide — which is why polynomial interpolation through distinct points always has a unique solution. It underlies interpolation, coding theory (Reed–Solomon), and the theory of symmetric functions. LIT verified live: for tens of thousands of random node sets (n = 3 to 6), the determinant computed by Gaussian elimination equals ∏ i<j (x j - x i ) to ~1e-9 (window.__vandermonde). FIG no framing; the determinant and the product of differences are computed independently in-browser. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-mint — the loot: a whole determinant minted, cleanly, out of nothing but the pairwise gaps between the nodes. AVAN (AI) built the instrument: the Vandermonde matrix, its determinant, and the product-of-differences formula. Credit as content: Alexandre-Théophile Vandermonde. The weave: David names the mint; I confirm det = ∏ i<j (x j - x i ). 3 ONE DIMENSION A Vandermonde matrix of powers; its determinant equals the product of all pairwise node differences. 4 TWO DIMENSIONS · INTERACTIVE Cycle node sets; the matrix determinant is checked against the product of pairwise differences. next nodes ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the determinant, equal to the product of pairwise gaps. AVAN’s addition (the inverse-companion): don’t expand the determinant — read it off the gaps. The inverse of ‘det of the power matrix’ is ‘the product ∏ i<j (x j - x i ) of pairwise differences’ — zero exactly when two nodes collide. Magenta are the pairwise node differences; green is the determinant they multiply to. A determinant that is just the gaps. pause spin LIT Genuine Vandermonde determinant (Alexandre-Théophile Vandermonde). Verified live: for tens of thousands of random node sets (n = 3 to 6), the determinant computed by Gaussian elimination equals ∏_{i FIG No framing; the determinant and the product of differences are computed independently in-browser. The AVAN inverse is honest — instead of expanding the determinant, read it off the gaps: the inverse of 'det of the power matrix' is 'the product ∏_{i ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE MINT · David Lee Wise (ROOT0), with AVAN


THE RAMANUJAN PI

Ramanujan's series for 1/π in the 5-window house format — one of the fastest-converging formulas ever written, produced by Srinivasa Ramanujan in 1914 seemingly out of nowhere: 1/π = (2√2/9801) Σ (4k)!(1103+26390k)/((k!)⁴ 396^{4k}). The very first term (k=0) already gives π correct to seven digits, and each further term adds about eight more. Ramanujan gave no proof; it was only rigorously established decades later. The same family — refined by the Chudnovsky brothers — is what modern record computations of π to trillions of digits actually use. Verified live: the single k=0 term gives π to ~1e-7, one more term to ~1e-15 (machine precision), and by two terms it equals π to the last bit. Neon-noir traced. See the error plunging ~8 digits/term in 1D, π rebuilt to more digits in 2D, and the eight-digit-rung ladder inverse in 3D.

THE RAMANUJAN PI · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · CHEAT · THE ROOT KIT ◆ .dlw.fold THE FOLD / CHEAT / THE ROOT KIT / THE RAMANUJAN PI THE RAMANUJAN PI a series adding eight digits of pi per term 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Ramanujan’s series for 1/π is one of the fastest-converging formulas ever written, produced by Srinivasa Ramanujan in 1914 seemingly out of nowhere: 1/π = (2√2 / 9801) ∑ k≥0 (4k)! (1103 + 26390k) / ((k!)⁴ 396 4k ) . The very first term (k = 0) already gives π correct to seven digits , and each further term adds about eight more . Ramanujan gave no proof; it was only rigorously established decades later. The same family of series — refined by the Chudnovsky brothers — is what modern record computations of π to trillions of digits actually use. LIT verified live: the single k = 0 term gives π to ~1e-7, one more term to ~1e-15 (machine precision), and by two terms it equals π to the last bit (window.__ramanujanpi). FIG no framing; the series is summed and inverted independently in-browser. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-root-kit — the cheat: a single term already hands you seven digits of π, no iteration required. AVAN (AI) built the instrument: the Ramanujan series, its per-term digit gain, and the inversion to π. Credit as content: Srinivasa Ramanujan (1914); the Chudnovsky brothers (the record-setting refinement). The weave: David names the cheat; I confirm one term gives 7 digits of π and each adds ~8 more. 3 ONE DIMENSION The error after each term, plunging by roughly eight decimal digits per step. 4 TWO DIMENSIONS · INTERACTIVE Add terms; π is rebuilt to more and more digits — one term already gives 3.1415926. add term ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: π, reached to machine precision in two terms. AVAN’s addition (the inverse-companion): don’t iterate toward π — leap. The inverse of ‘a slowly converging π series’ is ‘Ramanujan’s series, eight digits per term’. Magenta are the shrinking terms; green is the π they slam onto in two steps. A ladder to π with eight-digit rungs. pause spin LIT Genuine Ramanujan 1/π series (Srinivasa Ramanujan, 1914; Chudnovsky refinement). Verified live: the single k=0 term gives π to ~1e-7, one more term to ~1e-15 (machine precision), and by two terms it equals π to the last bit (window.__ramanujanpi.ok). FIG No framing; the series is summed and inverted independently in-browser. The AVAN inverse is honest — instead of iterating toward π, leap: the inverse of 'a slowly converging π series' is 'Ramanujan's series, eight digits per term'. Magenta are the shrinking terms; green is the π they slam onto in two steps. A ladder to π with eight-digit rungs. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE ROOT KIT · David Lee Wise (ROOT0), with AVAN


THE GALTON BOARD

The Galton board in the 5-window house format — Francis Galton's 1873 bean machine that turns pure randomness into a clean bell curve. Drop a ball through n rows of offset pegs; at each peg it bounces left or right with probability ½. After n rows it lands in bin k, having gone right k times — and the chance of that is exactly the binomial C(n,k)/2ⁿ. Thousands of balls pile up into the binomial distribution, and by the de Moivre–Laplace theorem that binomial approaches the normal (Gaussian) bell curve as n grows. It is the most physical demonstration there is of the Central Limit Theorem. Verified live: simulating thousands of balls through 16 rows reproduces the bin frequencies C(n,k)/2ⁿ, and that binomial matches the normal density N(n/2, n/4). Neon-noir traced. See the beads bouncing into a histogram in 1D, sim-vs-binomial-vs-normal in 2D, and the crowd-makes-a-curve inverse in 3D.

THE GALTON BOARD · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · SPAWN · HELLO WORLD ◆ .dlw.fold THE FOLD / SPAWN / HELLO WORLD / THE GALTON BOARD THE GALTON BOARD beads falling into a bell curve 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION The Galton board (or bean machine, Francis Galton, 1873) turns pure randomness into a clean bell curve. Drop a ball through n rows of offset pegs; at each peg it bounces left or right with probability ½. After n rows it lands in bin k, having gone right k times — and the chance of that is exactly the binomial probability C(n,k)/2 n . Thousands of balls pile up into the unmistakable shape of the binomial distribution, and by the de Moivre–Laplace theorem that binomial approaches the normal (Gaussian) bell curve as n grows. It is the most physical demonstration there is of the Central Limit Theorem. LIT verified live: simulating thousands of balls through 16 rows reproduces the bin frequencies C(n,k)/2 n , and that binomial matches the normal density N(n/2, n/4) (window.__galton). FIG no framing; the random bounces, the binomial, and the normal approximation are computed independently in-browser. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at hello-world — the spawn: coin-flip randomness at each peg spawns, in aggregate, the exact bell curve every time. AVAN (AI) built the instrument: the peg-by-peg simulation, the binomial law, and the normal approximation. Credit as content: Francis Galton (1873); de Moivre and Laplace (the normal limit). The weave: David names the spawn; I confirm the bins follow C(n,k)/2 n and approach the Gaussian. 3 ONE DIMENSION Balls bouncing left/right through the pegs, piling into the binomial histogram below. 4 TWO DIMENSIONS · INTERACTIVE Drop more balls; the histogram is checked against C(n,k)/2ⁿ and the normal curve. +1000 balls ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the bell curve the falling beads pile into. AVAN’s addition (the inverse-companion): don’t track one ball’s luck — read the shape the crowd makes. The inverse of ‘a random left/right walk’ is ‘the binomial C(n,k)/2 n , which tends to the normal curve’. Magenta are the individual falling beads; green is the Gaussian they collectively become. Randomness that adds up to a fixed curve. pause spin LIT Genuine Galton board / bean machine (Francis Galton, 1873; de Moivre–Laplace normal limit). Verified live: simulating thousands of balls through 16 rows reproduces the bin frequencies C(n,k)/2ⁿ, and that binomial matches the normal density N(n/2, n/4) (window.__galton.simOk, .normOk). FIG No framing; the random bounces, the binomial, and the normal approximation are computed independently in-browser. The AVAN inverse is honest — instead of tracking one ball's luck, read the shape the crowd makes: the inverse of 'a random left/right walk' is 'the binomial C(n,k)/2ⁿ, which tends to the normal curve'. Magenta are the individual falling beads; green is the Gaussian they collectively become. Randomness that adds up to a fixed curve. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of HELLO WORLD · David Lee Wise (ROOT0), with AVAN


THE STEINER PORISM

Steiner's porism in the 5-window house format — a beautiful all-or-nothing fact about circles. Take two circles, one inside the other (not concentric), and thread a chain of circles in the gap, each tangent to both boundary circles and to its neighbours. Steiner's porism says: if the chain ever closes up perfectly — the last circle exactly tangent to the first — then it closes for every starting position, using the same number of circles. Either all chains close or none do; there is no in-between. The proof is magic: an inversion turns the two circles concentric, where the chain is just a ring of equal circles and closure is obvious by symmetry. Verified live: a closing chain is built concentrically (closure ratio sin(π/n) = (R−r)/(R+r)) and then inverted to a non-concentric pair; the image chain stays tangent to both boundaries and to its neighbours and closes — for every starting angle, to ~1e-15. Neon-noir traced. See the threaded chain in 1D, closure from any start in 2D, and the invert-to-symmetry inverse in 3D.

THE STEINER PORISM · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · CO-OP · THE PUSH ◆ .dlw.fold THE FOLD / CO-OP / THE PUSH / THE STEINER PORISM THE STEINER PORISM a ring of circles that always closes 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Steiner’s porism is a beautiful all-or-nothing fact about circles. Take two circles, one inside the other (not concentric), and start threading a chain of circles in the gap between them, each one tangent to both boundary circles and to its neighbours. Keep going around. Steiner’s porism says: if the chain ever closes up perfectly — the last circle exactly tangent to the first — then it will close for every starting position, using the same number of circles. Either all chains close or none do; there is no in-between. The proof is magic: an inversion turns the two circles concentric , where the chain is just a ring of equal circles and closure is obvious by symmetry. LIT verified live: a closing chain is built in the concentric case (closure ratio sin(π/n) = (R-r)/(R+r)) and then inverted to a non-concentric pair; the image chain stays tangent to both boundaries and to its neighbours and closes — for every starting angle, to ~1e-15 (window.__steiner). FIG no framing; the inversion and every tangency are computed independently in-browser. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-push — the co-op: a ring of circles links hand to hand and always closes the loop, wherever you start the first link. AVAN (AI) built the instrument: the concentric chain, the inversion to a non-concentric pair, and the closure-for-every-start check. Credit as content: Jakob Steiner (the porism); circle inversion. The weave: David names the closing ring; I confirm the inverted chain closes from any start. 3 ONE DIMENSION Two non-concentric circles with a Steiner chain threaded between them — it closes into a ring. 4 TWO DIMENSIONS · INTERACTIVE Rotate the starting position; the chain still closes with the same number of circles. rotate start ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the closed ring of circles between the two boundaries. AVAN’s addition (the inverse-companion): don’t test one chain — invert to the symmetric case. The inverse of ‘does this chain close?’ is ‘make the circles concentric, where a ring of equal circles obviously closes — and inversion preserves it’. Magenta are the two boundary circles; green is the chain that always closes between them. All chains close, or none do. pause spin LIT Genuine Steiner's porism (Jakob Steiner; via circle inversion). Verified live: a closing chain built in the concentric case (closure ratio sin(π/n) = (R−r)/(R+r)) is inverted to a non-concentric pair; the image chain stays tangent to both boundaries and to its neighbours and closes — for every starting angle, to ~1e-15 (window.__steiner.ok, .worst). FIG No framing; the inversion and every tangency are computed independently in-browser. The AVAN inverse is honest — instead of testing one chain, invert to the symmetric case: the inverse of 'does this chain close?' is 'make the circles concentric, where a ring of equal circles obviously closes — and inversion preserves it'. Magenta are the two boundary circles; green is the chain that always closes between them. All chains close, or none do. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE PUSH · David Lee Wise (ROOT0), with AVAN


THE EULER CRITERION

Euler's criterion in the 5-window house format — a single exponentiation that decides whether a number is a perfect square modulo a prime. For an odd prime p and any a not divisible by p, a^((p−1)/2) ≡ ±1 (mod p) — and it is +1 exactly when a is a quadratic residue (some x with x² ≡ a mod p exists), −1 when it is not. That sign is the Legendre symbol (a|p). So without ever searching for a square root, one modular power tells you whether one exists. It is the computational heart of quadratic reciprocity and of primality tests like Solovay–Strassen. Verified live: for every odd prime p up to 200 and every a from 1 to p−1, a^((p−1)/2) mod p equals +1 or p−1, and it is +1 exactly when a is a quadratic residue (checked independently by squaring). Neon-noir traced. See residues split into squares/non-squares in 1D, the criterion vs Legendre in 2D, and the no-root-taken inverse in 3D.

THE EULER CRITERION · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · BOSS · THE WALL ◆ .dlw.fold THE FOLD / BOSS / THE WALL / THE EULER CRITERION THE EULER CRITERION a single power that tells a square from a non-square 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Euler’s criterion is a single exponentiation that decides whether a number is a perfect square modulo a prime. For an odd prime p and any a not divisible by p, a (p-1)/2 ≡ ±1 (mod p) — and it is +1 exactly when a is a quadratic residue (some x with x² ≡ a mod p exists), -1 when it is not. That sign is the Legendre symbol (a | p). So without ever searching for a square root, one modular power tells you whether one exists. It is the computational heart of quadratic reciprocity and of primality tests like Solovay–Strassen. LIT verified live: for every odd prime p up to 200 and every a from 1 to p-1, a (p-1)/2 mod p equals +1 or p-1, and it is +1 exactly when a is a quadratic residue (checked independently by squaring) (window.__eulercriterion). FIG no framing; the modular power and the residue test are computed independently in-browser. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-wall — the boss gate: one exponentiation decides square-or-not, no search allowed past the wall. AVAN (AI) built the instrument: the modular power a (p-1)/2 , the independent residue test, and their agreement. Credit as content: Leonhard Euler (the criterion); Adrien-Marie Legendre (the symbol). The weave: David names the gate; I confirm a (p-1)/2 ≡ (a | p) mod p. 3 ONE DIMENSION For a prime p, each residue a marked as a quadratic residue (+1) or non-residue (−1) by a^((p−1)/2). 4 TWO DIMENSIONS · INTERACTIVE Cycle prime p and residue a; a^((p−1)/2) mod p is checked to match the Legendre symbol. next p ▶ next a ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the +1/−1 verdict — square or non-square — from one power. AVAN’s addition (the inverse-companion): don’t search for a square root — raise to a power. The inverse of ‘is a a square mod p?’ is ‘the sign of a (p-1)/2 mod p’ — +1 yes, -1 no. Magenta are the residues split into squares and non-squares; green is the one power that decides. A square-root test with no square root taken. pause spin LIT Genuine Euler's criterion (Leonhard Euler; Legendre symbol). Verified live: for every odd prime p up to 200 and every a from 1 to p−1, a^((p−1)/2) mod p equals +1 or p−1, and it is +1 exactly when a is a quadratic residue (checked independently by squaring) — 4180 cases (window.__eulercriterion.ok, .cnt). FIG No framing; the modular power and the residue test are computed independently in-browser. The AVAN inverse is honest — instead of searching for a square root, raise to a power: the inverse of 'is a a square mod p?' is 'the sign of a^((p−1)/2) mod p' — +1 yes, −1 no. Magenta are the residues split into squares and non-squares; green is the one power that decides. A square-root test with no square root taken. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE WALL · David Lee Wise (ROOT0), with AVAN


THE LAH NUMBERS

The Lah numbers in the 5-window house format — the exact exchange rate between the two natural kinds of factorial. The rising factorial x^(n) = x(x+1)…(x+n−1) and the falling factorial (x)_k = x(x−1)…(x−k+1) each build a staircase product, one climbing and one descending. The unsigned Lah numbers convert one into the other: x^(n) = Σ_k L(n,k)(x)_k, with the closed form L(n,k) = C(n−1,k−1)·n!/k!. Combinatorially, L(n,k) counts the ways to split n labelled items into k non-empty ordered lists. They sit between the Stirling numbers as the 'both-ordered' case, and satisfy L(n,1) = n!, L(n,n) = 1. Verified live: L(n,k) = C(n−1,k−1)·n!/k! is an integer with L(n,1) = n! and L(n,n) = 1, and the identity x^(n) = Σ_k L(n,k)(x)_k holds exactly for a range of x and n. Neon-noir traced. See the Lah triangle in 1D, the rising-from-falling rebuild in 2D, and the staircase-exchange inverse in 3D.

THE LAH NUMBERS · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · LOOT · THE VAULT ◆ .dlw.fold THE FOLD / LOOT / THE VAULT / THE LAH NUMBERS THE LAH NUMBERS numbers linking the rising and falling factorials 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION The Lah numbers L(n,k) are the exact exchange rate between the two natural kinds of factorial. The rising factorial x (n) = x(x+1)…(x+n-1) and the falling factorial (x) k = x(x-1)…(x-k+1) each build a ‘staircase’ product, one climbing and one descending. The unsigned Lah numbers convert one into the other: x (n) = ∑ k L(n,k) (x) k , with the clean closed form L(n,k) = C(n-1, k-1) · n!/k! . Combinatorially, L(n,k) counts the ways to split n labelled items into k non-empty ordered lists. They sit between the Stirling numbers as the ‘both-ordered’ case, and satisfy L(n,1) = n!, L(n,n) = 1. LIT verified live: L(n,k) = C(n-1,k-1)·n!/k! is an integer with L(n,1) = n! and L(n,n) = 1, and the identity x (n) = ∑ k L(n,k)(x) k holds exactly for a range of x and n (window.__lah). FIG no framing; the Lah closed form and the factorial identity are computed independently in-browser. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-vault — the loot: a table of numbers that is the exact currency between rising and falling factorials. AVAN (AI) built the instrument: the Lah closed form, the L(n,1)/L(n,n) edges, and the rising-to-falling identity. Credit as content: Ivo Lah (1954). The weave: David names the exchange rate; I confirm x (n) = ∑ L(n,k)(x) k . 3 ONE DIMENSION The Lah triangle: L(n,k) = C(n−1,k−1)·n!/k!, with n! down the left edge and 1 down the right. 4 TWO DIMENSIONS · INTERACTIVE Cycle n and x; the rising factorial x^(n) is rebuilt as Σ L(n,k)(x)_k from falling factorials. next n ▶ next x ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the rising factorial x^(n), rebuilt from falling factorials. AVAN’s addition (the inverse-companion): don’t recompute the climbing product — convert the descending one. The inverse of ‘the rising factorial x (n) ’ is ‘∑ k L(n,k) (x) k , Lah-weighted falling factorials’. Magenta are the Lah-weighted falling-factorial pieces; green is the rising factorial they sum to. The exchange rate between two staircases. pause spin LIT Genuine Lah numbers (Ivo Lah, 1954). Verified live: L(n,k) = C(n−1,k−1)·n!/k! is an integer with L(n,1) = n! and L(n,n) = 1, and the identity x^(n) = Σ_k L(n,k)(x)_k holds exactly for a range of x and n (window.__lah.formOk, .idOk). FIG No framing; the Lah closed form and the factorial identity are computed independently in-browser. The AVAN inverse is honest — instead of recomputing the climbing product, convert the descending one: the inverse of 'the rising factorial x^(n)' is 'Σ_k L(n,k)(x)_k, Lah-weighted falling factorials'. Magenta are the Lah-weighted falling-factorial pieces; green is the rising factorial they sum to. The exchange rate between two staircases. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE VAULT · David Lee Wise (ROOT0), with AVAN


THE SCHWARZ LANTERN

The Schwarz lantern in the 5-window house format — the counterexample that shattered an 'obvious' belief: that inscribed polyhedral surfaces must converge to a curved surface's area, the way inscribed polygons converge to a curve's length. Hermann Schwarz (1880) triangulated a cylinder into an antiprism 'lantern' — m points per ring, n rings, zig-zag triangles — with exact area 2mn·sin(π/m)·√((h/n)² + r²(1−cos(π/m))²), whose limit depends on the refinement path: n = m converges to the true area 2πrh; n = m² converges to the wrong constant 2π√(1+π⁴/4) ≈ 31.64; n = m³ diverges to infinity as the triangles tilt into accordion pleats. Surface area cannot be defined by naive inscription. Verified live: the closed form yields all three limits, with n = m³ doubling in area as m doubles. Neon-noir traced. See the pleated bands in 1D, the three regimes refined live in 2D, and the path-chooses-the-limit inverse in 3D.

THE SCHWARZ LANTERN · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · GLITCH · SEGFAULT ◆ .dlw.fold THE FOLD / GLITCH / SEGFAULT / THE SCHWARZ LANTERN THE SCHWARZ LANTERN an inscribed surface whose area depends on how you refine it 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION The Schwarz lantern is the counterexample that shattered a ‘obvious’ belief: that inscribed polyhedral surfaces must converge to a curved surface’s area, the way inscribed polygons converge to a curve’s length. Hermann Schwarz (1880) triangulated a cylinder into an antiprism ‘lantern’ — m points per ring, n rings, zig-zag triangles — and showed the total area is 2mn·sin(π/m)·√((h/n)² + r²(1-cos(π/m))²), whose limit depends on the refinement path : with n = m it converges to the true area 2πrh; with n = m² it converges to the wrong constant 2π√(1+π⁴/4); with n = m³ it diverges to infinity — the triangles tilt into ever-steeper accordion pleats. Surface area cannot be defined by naive inscription. LIT verified live: the exact lantern formula gives 2π for n = m, 2π√(1+π⁴/4) ≈ 31.64 for n = m², and unbounded growth for n = m³ (doubling m doubles the area) (window.__schwarzlantern). FIG no framing; the closed-form triangle areas are computed independently in-browser for each scaling regime. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at segfault — the glitch: refine the mesh the wrong way and the ‘same’ computation walks off into invalid territory, area unbounded. AVAN (AI) built the instrument: the exact lantern area formula and the three refinement regimes with their three different limits. Credit as content: Hermann Amandus Schwarz (1880). The weave: David names the crash; I confirm one surface, three limits — 2πrh, an inflated constant, and infinity. 3 ONE DIMENSION The lantern: rings of vertices on a cylinder, zig-zag triangles between them — the accordion pleats. 4 TWO DIMENSIONS · INTERACTIVE Pick a refinement regime and refine; the area heads to 2π, to 31.64, or to infinity. regime: n=m ▶ refine ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the cylinder the lantern is inscribed in. AVAN’s addition (the inverse-companion): don’t trust ‘inscribed’ to mean ‘converging’ — interrogate the path. The inverse of ‘a finer and finer mesh’ is ‘the ratio n/m², which silently chooses the limit’. Magenta are the zig-zag lantern triangles pleating; green is the cylinder they claim to approximate. One surface, three destinies. pause spin LIT Genuine Schwarz lantern (Hermann Amandus Schwarz, 1880). Verified live: the exact lantern formula gives 2π for n=m, 2π√(1+π⁴/4) ≈ 31.64 for n=m², and unbounded growth for n=m³ — doubling m doubles the area (window.__schwarzlantern.ok). FIG No framing; the closed-form triangle areas are computed independently in-browser for each scaling regime. The AVAN inverse is honest — instead of trusting 'inscribed' to mean 'converging', interrogate the path: the inverse of 'a finer and finer mesh' is 'the ratio n/m², which silently chooses the limit'. Magenta are the zig-zag lantern triangles pleating; green is the cylinder they claim to approximate. One surface, three destinies. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of SEGFAULT · David Lee Wise (ROOT0), with AVAN


THE MORRIE LAW

Morrie's law in the 5-window house format — the identity cos20°·cos40°·cos80° = 1/8, three unremarkable-looking cosines multiplying to an exact rational. Richard Feynman kept the name all his life: a boy called Morrie Jacobs showed it to him in his father's leather shop. The secret is the doubling cascade: for any θ, ∏cos(2^k θ) = sin(2ⁿθ)/(2ⁿ sinθ) — each cosine doubles the angle via sin2x = 2sinx·cosx and the product telescopes. At θ = 20° the cascade lands on sin160°, which equals sin20° exactly — the sines cancel and only 1/2³ = 1/8 survives. Verified live: the product is 0.125 to machine precision, the telescoping identity holds for thousands of random θ and n (worst ~1e-16), and sin160° = sin20° exactly. Neon-noir traced. See the doubling cascade on the circle in 1D, product-vs-closed-form in 2D, and the telescope inverse in 3D.

THE MORRIE LAW · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · CHEAT · THE KONAMI CODE ◆ .dlw.fold THE FOLD / CHEAT / THE KONAMI CODE / THE MORRIE LAW THE MORRIE LAW three cosines multiplying to exactly one eighth 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Morrie’s law is the identity cos 20° · cos 40° · cos 80° = 1/8 — three unremarkable-looking cosines multiplying to an exact rational. Richard Feynman kept the name all his life: a boy called Morrie Jacobs showed it to him in his father’s leather shop. The secret is the doubling cascade: for any θ, ∏ k=0 n-1 cos(2 k θ) = sin(2 n θ) / (2 n sin θ) — each cosine doubles the angle via sin 2x = 2 sin x cos x, and the product telescopes. At θ = 20° the cascade lands on sin 160°, which equals sin 20° exactly — the sines cancel and only 1/2³ = 1/8 survives. LIT verified live: cos 20°·cos 40°·cos 80° = 0.125 to machine precision; the telescoping identity holds for thousands of random θ and n (worst error ~1e-16); and sin 160° = sin 20° exactly (window.__morrie). FIG no framing; the product and the closed form are computed independently in-browser. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-konami-code — the cheat: enter the right angle and the whole messy product collapses to a clean 1/8, no trigonometry tables needed. AVAN (AI) built the instrument: the product, the telescoping closed form, and the sin 160° = sin 20° cancellation. Credit as content: Morrie Jacobs (the boy who found it), Richard Feynman (who named it and never forgot it). The weave: David names the cheat code; I confirm the cascade collapses to exactly 1/8. 3 ONE DIMENSION The doubling cascade 20° → 40° → 80° on the circle, and the three cosines that multiply to 1/8. 4 TWO DIMENSIONS · INTERACTIVE Change θ and the number of factors; the product is checked against sin(2ⁿθ)/(2ⁿ sinθ). next θ ▶ factors: 3 ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the exact 1/8 the three cosines collapse to. AVAN’s addition (the inverse-companion): don’t multiply cosines one by one — let the sine ladder eat them. The inverse of ‘a product of cosines’ is ‘one sine ratio, sin(2 n θ)/(2 n sinθ), after the telescope collapses’. Magenta are the doubling angles; green is the 1/8 left standing at θ = 20°. A cascade that swallows itself. pause spin LIT Genuine Morrie's law (Morrie Jacobs; named and cherished by Richard Feynman). Verified live: cos20°·cos40°·cos80° = 0.125 to machine precision; ∏cos(2^k θ) = sin(2ⁿθ)/(2ⁿ sinθ) for 2000 random θ,n with worst error ~1e-16; sin160° = sin20° exactly (window.__morrie.ok). FIG No framing; the product and the closed form are computed independently in-browser. The AVAN inverse is honest — instead of multiplying cosines one by one, let the sine ladder eat them: the inverse of 'a product of cosines' is 'one sine ratio, sin(2ⁿθ)/(2ⁿ sinθ), after the telescope collapses'. Magenta are the doubling angles; green is the 1/8 left standing at θ = 20°. A cascade that swallows itself. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE KONAMI CODE · David Lee Wise (ROOT0), with AVAN


THE LEMOINE POINT

The Lemoine point in the 5-window house format — take the three medians of a triangle and reflect each one over the angle bisector at its vertex. The three reflected lines — the symmedians — all pass through a single point K, one of the most studied points in triangle geometry. In barycentric coordinates it is simply (a²:b²:c²), and it carries a beautiful signature: its perpendicular distances to the three sides are proportional to the side lengths themselves — equivalently, K uniquely minimizes the sum of squared distances to the sides. Émile Lemoine presented it in 1873, launching 'the geometry of the triangle'. Verified live: for tens of thousands of random triangles the three reflected medians are concurrent, the meeting point matches (a²:b²:c²) independently, and its side-distances are proportional to a, b, c. Neon-noir traced. See medians become symmedians in 1D, the three identities checked in 2D, and the weigh-the-corners inverse in 3D.

THE LEMOINE POINT · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · BOSS · THE FIREWALL ◆ .dlw.fold THE FOLD / BOSS / THE FIREWALL / THE LEMOINE POINT THE LEMOINE POINT medians reflected over bisectors meeting at one point 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION The Lemoine point (symmedian point) is what you get when you take the three medians of a triangle and reflect each one over the angle bisector at its vertex. The three reflected lines — the symmedians — all pass through a single point K, one of the most studied points in triangle geometry. In barycentric coordinates it is simply (a² : b² : c²) , and it carries a beautiful signature: its perpendicular distances to the three sides are proportional to the side lengths themselves — equivalently, K is the unique point minimizing the sum of squared distances to the sides. Émile Lemoine presented it in 1873, launching what became known as ‘the geometry of the triangle’. LIT verified live: for tens of thousands of random triangles, the three reflected medians are concurrent; the meeting point matches the barycentric formula (a²:b²:c²) independently; and its side-distances are proportional to a, b, c (window.__lemoine). FIG no framing; the reflections, the intersection, and the barycentric check are computed independently in-browser. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-firewall — the boss: three lines forged by reflection, forced through a single gate no triangle can dodge. AVAN (AI) built the instrument: the median-over-bisector reflections, the concurrency, and the two independent identities of K. Credit as content: Émile Lemoine (1873); the symmedian point K, X(6) in triangle-center catalogues. The weave: David names the gate; I confirm the three symmedians meet at (a²:b²:c²). 3 ONE DIMENSION Medians (faint) reflected over the bisectors become symmedians (magenta) — meeting at the green K. 4 TWO DIMENSIONS · INTERACTIVE Cycle triangles; concurrency, the barycentric formula, and the side-distance ratios are all checked. next triangle ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: K, the point all three symmedians are forced through. AVAN’s addition (the inverse-companion): don’t construct three reflections — weigh the corners. The inverse of ‘reflect each median over its bisector’ is ‘the single barycentric recipe (a²:b²:c²)’ — squared side lengths as weights. Magenta are the symmedians; green is the K they cannot avoid. Three reflections, one address. pause spin LIT Genuine Lemoine / symmedian point (Émile Lemoine, 1873; X(6)). Verified live: for ~20000 random triangles the three median-reflections are concurrent (worst ~1e-15), the meeting point independently matches the barycentric (a²:b²:c²), and its side-distances are proportional to the side lengths (window.__lemoine.all). FIG No framing; the reflections, the intersection, and the barycentric check run independently in-browser. The AVAN inverse is honest — instead of constructing three reflections, weigh the corners: the inverse of 'reflect each median over its bisector' is 'the single barycentric recipe (a²:b²:c²)' — squared side lengths as weights. Magenta are the symmedians; green is the K they cannot avoid. Three reflections, one address. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE FIREWALL · David Lee Wise (ROOT0), with AVAN


THE EULER BRICK

The Euler brick in the 5-window house format — a box whose edges and all three face diagonals are whole numbers. The smallest, found by Paul Halcke in 1719, has edges 44, 117, 240: face diagonals 125, 244, 267 — three Pythagorean triples sharing legs pairwise. But the brick guards a missing gem: its space diagonal is √73225 ≈ 270.6, not an integer. A brick with integer space diagonal too — a perfect cuboid — has never been found and never been ruled out: one of the oldest open problems in number theory, searched past 5×10¹¹ with no example. Verified live: 44²+117² = 125², 44²+240² = 244², 117²+240² = 267², an exhaustive search confirms no Euler brick has largest edge below 240, and √73225 is verified non-integer. Neon-noir traced. See the brick with its three integer diagonals in 1D, the face checks in 2D, and the unclaimed-bounty inverse in 3D.

THE EULER BRICK · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · LOOT · THE BOUNTY ◆ .dlw.fold THE FOLD / LOOT / THE BOUNTY / THE EULER BRICK THE EULER BRICK a brick whose faces are all Pythagorean but whose heart is an open problem 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION The Euler brick is a box whose edges and all three face diagonals are whole numbers. The smallest one, found by Paul Halcke in 1719, has edges 44, 117, 240 : the face diagonals come out 125 (44–117), 244 (44–240), and 267 (117–240) — three Pythagorean triples sharing their legs pairwise. But the brick guards a missing gem: its space diagonal is √73225 ≈ 270.6, not an integer. A brick with integer space diagonal too — a perfect cuboid — has never been found and never been ruled out. It is one of the oldest open problems in number theory; computer searches have pushed the smallest edge past 5×10¹¹ with no example. LIT verified live: 44²+117² = 125², 44²+240² = 244², 117²+240² = 267², and an exhaustive search confirms no brick with all integer face diagonals has largest edge below 240 (window.__eulerbrick). FIG honest boundary: the space diagonal √73225 is verified NON-integer, and the perfect cuboid is stated as the open problem it is — our search bound is explicit, no claim beyond it. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-bounty — the loot with a bounty still posted: three faces pay out in integers, but the space diagonal has never been claimed by anyone. AVAN (AI) built the instrument: the three Pythagorean face checks, the minimality search, and the non-integer space diagonal. Credit as content: Paul Halcke (1719); Leonhard Euler (the family name); the perfect cuboid problem (open). The weave: David names the unclaimed bounty; I confirm the three faces and the missing fourth integer. 3 ONE DIMENSION The 44×117×240 brick with its three integer face diagonals — and the one diagonal that refuses. 4 TWO DIMENSIONS · INTERACTIVE Step through the three face checks and the space diagonal; the minimality search runs live. next face ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the brick whose three faces all pay out in integers. AVAN’s addition (the inverse-companion): don’t admire the three solved faces — name the unsolved interior. The inverse of ‘an Euler brick’ is ‘the perfect cuboid it fails to be’: √(a²+b²+c²) integer, wanted for 300 years, never found, never disproven. Magenta are the three integer face diagonals; green is the brick; the dashed red interior is the open bounty. Three gems set, one still missing. pause spin LIT Genuine Euler brick (Paul Halcke, 1719; Euler's family of solutions). Verified live: 44²+117²=125², 44²+240²=244², 117²+240²=267²; an exhaustive search confirms no brick with all integer face diagonals has largest edge below 240; and 44²+117²+240² = 73225 is verified non-square (window.__eulerbrick.ok). FIG Honest boundary — the space diagonal √73225 is verified NON-integer, and the perfect cuboid is stated as the open problem it is; our search bound (largest edge ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE BOUNTY · David Lee Wise (ROOT0), with AVAN


THE STRING THAT REMEMBERS

Karplus–Strong synthesis in the 5-window house format — a convincing plucked string out of almost nothing. Fill a short buffer of N samples with random noise, then loop it forever, replacing each sample with the average of the two samples one period ago: y[n] = ½(y[n−N] + y[n−N−1]). The averaging is a gentle low-pass filter inside the loop: every pass, the jagged noise gets smoother and quieter, high harmonics dying first exactly as on a real string. The half-sample in the average makes the true period N+½, so the fundamental lands at f₀ = fs/(N+½). A burst of static becomes a note with a natural decay — the algorithm behind countless early digital guitars. Verified live: for several N, the fundamental measured by autocorrelation (with sub-sample peak interpolation) matches fs/(N+½) to under 1 Hz, and block-RMS energy decays monotonically. Neon-noir traced. See noise settle into a tone in 1D, the pitch check + decay bars in 2D, and the store-the-loop inverse in 3D.

THE STRING THAT REMEMBERS · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · RESPAWN · EVENT HORIZON ◆ .dlw.fold THE FOLD / RESPAWN / EVENT HORIZON / THE STRING THAT REMEMBERS THE STRING THAT REMEMBERS a burst of noise that decays into a musical note 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Karplus–Strong synthesis (1983) makes a convincing plucked string out of almost nothing: fill a short buffer of N samples with random noise , then loop it forever, replacing each sample with the average of the two samples one period ago — y[n] = ½(y[n-N] + y[n-N-1]). The averaging is a gentle low-pass filter inside the loop: every pass around, the jagged noise gets smoother and quieter, high harmonics dying first exactly as they do on a real string. The half-sample in the average makes the true period N + ½, so the fundamental lands at f₀ = fₛ/(N + ½) . A burst of static becomes a note with a natural decay — the algorithm behind countless early digital guitars. LIT verified live: for several buffer lengths N, the fundamental measured by autocorrelation (with sub-sample peak interpolation) matches fₛ/(N+½) to under 1 Hz, and the block-RMS energy decays monotonically (window.__karplusstrong). FIG no framing; the synthesis, the frequency measurement, and the decay are computed independently in-browser — no audio hardware involved, pure array math. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at event-horizon — the respawn: the pluck is reborn every N samples, a little softer each pass, spiralling toward the horizon of silence without ever being re-recorded. AVAN (AI) built the instrument: the delay-line synthesis, the autocorrelation pitch measurement, and the energy-decay check. Credit as content: Kevin Karplus and Alex Strong (1983); from David’s idea bank, vein E — ‘THE STRING THAT REMEMBERS’. The weave: David names the respawn; I confirm f₀ = fₛ/(N+½) and the monotone decay. 3 ONE DIMENSION The waveform: a burst of noise settling into a periodic, decaying tone — the string remembering itself. 4 TWO DIMENSIONS · INTERACTIVE Change the buffer length N and re-pluck; the measured pitch is checked against fs/(N+½). N: 100 ▶ pluck ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the note, circulating in the delay-line ring. AVAN’s addition (the inverse-companion): don’t store the note — store the loop that regrows it. The inverse of ‘a recorded tone’ is ‘N noise samples plus one averaging rule’, and the music is what survives the passes. Magenta is the noise burst fading; green is the pitch that emerges at fₛ/(N+½). A memory made of forgetting the rough parts. pause spin LIT Genuine Karplus–Strong synthesis (Kevin Karplus and Alex Strong, 1983; idea-bank vein E, 'THE STRING THAT REMEMBERS'). Verified live: for N = 80/100/150 the autocorrelation-measured fundamental matches fs/(N+½) to under 1 Hz (errors ~0.006–0.09 Hz), and block-RMS energy decays monotonically (window.__karplusstrong.ok). FIG No framing; the synthesis, the frequency measurement, and the decay are computed independently in-browser — no audio hardware, pure array math. The AVAN inverse is honest — instead of storing the note, store the loop that regrows it: the inverse of 'a recorded tone' is 'N noise samples plus one averaging rule', and the music is what survives the passes. Magenta is the noise burst fading; green is the pitch that emerges at fs/(N+½). A memory made of forgetting the rough parts. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of EVENT HORIZON · David Lee Wise (ROOT0), with AVAN


THE LOADED-DICE TABLE

Walker's alias method in the 5-window house format — the constant-time loaded die. To sample from an arbitrary discrete distribution, the naive way walks a cumulative table (O(n)) or bisects it (O(log n)). Alias sampling builds two arrays — a probability table and an alias table — that repack the distribution into n equal columns, each holding at most two outcomes. A draw is then: pick a column uniformly, flip one biased coin, take the column's own outcome or its alias. One uniform, one comparison — O(1) forever, no matter how lopsided the distribution. Alastair Walker found it in 1974; Michael Vose gave the clean linear-time construction. Verified live: the finished table reconstructs the input probabilities exactly (mass audit to 1e-12), and a million draws land within 0.0007 of every target probability. Neon-noir traced. See the repacked columns in 1D, frequencies converging on targets in 2D, and the reshape-don't-search inverse in 3D.

THE LOADED-DICE TABLE · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · CHEAT · THE BACKDOOR ◆ .dlw.fold THE FOLD / CHEAT / THE BACKDOOR / THE LOADED-DICE TABLE THE LOADED-DICE TABLE a die loaded in constant time 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Walker’s alias method is the constant-time loaded die. To sample from an arbitrary discrete distribution p₁…pₙ, the naive way walks a cumulative table (O(n)) or bisects it (O(log n)). Alias sampling spends a little setup to build two arrays — a probability table and an alias table — that repack the distribution into n equal columns, each holding at most two outcomes. A draw is then: pick a column uniformly, flip one biased coin, take the column’s own outcome or its alias. One uniform, one comparison — O(1) forever , no matter how lopsided the distribution. Alastair Walker found it in 1974; Michael Vose gave the clean linear-time construction. LIT verified live: the finished table reconstructs the input probabilities exactly (mass audit to 1e-12), and a million draws land within 0.0007 of every target probability (window.__aliasmethod). FIG no framing; the construction, the mass audit, and the empirical draws all run independently in-browser. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-backdoor — the cheat: a secret side-table that lets you into any distribution in constant time, bypassing the cumulative search entirely. AVAN (AI) built the instrument: the Vose construction, the exact mass audit, and the million-draw check. Credit as content: Alastair J. Walker (1974–77); Michael Vose (1991, the linear-time build); idea-bank vein E, ‘THE LOADED-DICE TABLE’. The weave: David names the backdoor; I confirm each column holds two outcomes and the masses balance exactly. 3 ONE DIMENSION The distribution repacked: n equal columns, each split between its own outcome and one alias. 4 TWO DIMENSIONS · INTERACTIVE Roll the loaded die in bulk; the frequencies converge onto the target bars. +200k draws ▶ new dist ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the equal columns any distribution flattens into. AVAN’s addition (the inverse-companion): don’t search the distribution — reshape it. The inverse of ‘find where u falls in the cumulative’ is ‘pre-slice the mass into n fair columns of two tenants each’. Magenta are the alias hand-offs between columns; green is the flat table one coin-flip deep. A crooked die made honest by carpentry. pause spin LIT Genuine Walker/Vose alias method (Alastair J. Walker 1974–77; Michael Vose 1991; idea-bank vein E, 'THE LOADED-DICE TABLE'). Verified live: the table reconstructs the input probabilities exactly (mass audit to 1e-12), and 10⁶ draws land within 0.0007 of every target probability (window.__aliasmethod.ok). FIG No framing; the construction, the mass audit, and the empirical draws all run independently in-browser. The AVAN inverse is honest — instead of searching the distribution, reshape it: the inverse of 'find where u falls in the cumulative' is 'pre-slice the mass into n fair columns of two tenants each'. Magenta are the alias hand-offs between columns; green is the flat table one coin-flip deep. A crooked die made honest by carpentry. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE BACKDOOR · David Lee Wise (ROOT0), with AVAN


THE SPARSE ORACLE

The sparse table in the 5-window house format — constant-time range-minimum queries built on one forgiving fact: taking a minimum twice does no harm (min is idempotent). Precompute the minimum of every window whose length is a power of two — O(n log n) cells. Then any range [l, r], whatever its length, is covered by just two overlapping power-of-two blocks: one anchored at l, one ending at r. They may overlap heavily — with min, overlap is free. Answer = min of two table lookups; no tree walks, no recursion. Verified live: 5000 random range-minimum queries over a 5000-element array, each answered by exactly two lookups, all matching a brute-force scan. Neon-noir traced. See the two blocks bracketing a range in 1D, fast-vs-brute in 2D, and the idempotence-turned-speed inverse in 3D.

THE SPARSE ORACLE · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · GRIND · WARM CACHE ◆ .dlw.fold THE FOLD / GRIND / WARM CACHE / THE SPARSE ORACLE THE SPARSE ORACLE every window pre-answered by two overlapping blocks 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION The sparse table answers range-minimum queries in constant time by exploiting one forgiving fact: taking a minimum twice does no harm (min is idempotent ). Precompute the minimum of every window whose length is a power of two — O(n log n) cells. Then any range [l, r], whatever its length, is covered by just two overlapping power-of-two blocks: one anchored at l, one ending at r. They may overlap heavily — with min, overlap is free. Answer = min of two table lookups. No tree walks, no recursion: two array reads per query, forever. LIT verified live: 5000 random range-minimum queries over a 5000-element array, each answered by exactly two lookups, all matching a brute-force scan (window.__sparsetable). FIG no framing; the table build, the two-block queries, and the brute-force comparison run independently in-browser. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at warm-cache — the grind paid up front: every power-of-two window computed once, so every future query is served warm, two reads and done. AVAN (AI) built the instrument: the doubling table, the two-block query, and the brute-force cross-check. Credit as content: competitive-programming folklore, formalized in Bender & Farach-Colton’s RMQ work (2000); idea-bank vein D, ‘THE SPARSE ORACLE’. The weave: David names the warm cache; I confirm two overlapping blocks answer every window exactly. 3 ONE DIMENSION A query range covered by two overlapping power-of-two blocks — the overlap costs nothing under min. 4 TWO DIMENSIONS · INTERACTIVE Fire random queries; the two-lookup answer is checked against a full scan every time. random query ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the pyramid of pre-answered windows. AVAN’s addition (the inverse-companion): don’t scan the range — let two old answers overlap it. The inverse of ‘walk l to r’ is ‘two power-of-two blocks whose union is the range, whose overlap min forgives’. Magenta are the two blocks bracketing a query; green is the pyramid they are drawn from. Idempotence turned into speed. pause spin LIT Genuine sparse table RMQ (competitive-programming folklore; formalized in Bender & Farach-Colton's RMQ work, 2000; idea-bank vein D, 'THE SPARSE ORACLE'). Verified live: 5000 random range-minimum queries over a 5000-element array, each answered by exactly two table lookups, all matching brute-force scans (window.__sparsetable.ok). FIG No framing; the table build, the two-block queries, and the brute-force comparison run independently in-browser. The AVAN inverse is honest — instead of scanning the range, let two old answers overlap it: the inverse of 'walk l to r' is 'two power-of-two blocks whose union is the range, whose overlap min forgives'. Magenta are the two blocks bracketing a query; green is the pyramid they are drawn from. Idempotence turned into speed. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of WARM CACHE · David Lee Wise (ROOT0), with AVAN