THE WITCH OF AGNESI

The Witch of Agnesi in the 5-window house format — y = 8a³/(x²+4a²), studied by Maria Gaetana Agnesi in 1748 in the first mathematics textbook by a woman ('versiera' mistranslated to 'witch'). Area exactly 4πa² — and normalized it is the CAUCHY distribution: the law with NO mean. Sample forever and the running average never settles; the law of large numbers doesn't apply; one monster draw outweighs a million tame ones at any moment. The median behaves perfectly. Verified live: area to 1e-3; Cauchy running means at 10³..10⁶ wandering (−0.32, −0.55, −0.15, +0.36) while the uniform control converges to 3e-4 and the median sits at 0.0003. Neon-noir traced. See the witch and her circle in 1D, the lurching mean in 2D, and the two statistics in 3D.

THE WITCH OF AGNESI · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · GLITCH · THE BLUE SCREEN ◆ .dlw.fold THE FOLD / GLITCH / THE BLUE SCREEN / THE WITCH OF AGNESI THE WITCH OF AGNESI the witch with no mean 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION The Witch of Agnesi is a gentle bell curve, y = 8a³/(x²+4a²), studied by Maria Gaetana Agnesi in 1748 in the first mathematics textbook written by a woman (her ‘versiera’ — turning curve — was mistranslated as ‘avversiera’: witch, and the name stuck). Its area is exactly 4πa² — four times its generating circle. And normalized, it becomes the Cauchy distribution : the probability law with no mean . Sample it forever and your running average never settles — the law of large numbers simply does not apply; one monstrous draw can outweigh a million tame ones at any moment. The median, meanwhile, behaves perfectly. LIT verified live: the area integral matching 4πa² to 10⁻³; and the statistical pathology run raw — Cauchy running means at 10³, 10⁴, 10⁵, 10⁶ samples wandering (−0.317, −0.551, −0.148, +0.36) while a uniform control converges to 3×10⁻⁴ and the Cauchy MEDIAN sits at 0.0003 (window.__witchofagnesi). FIG the mistranslation story is documented history; the no-mean claim is Cauchy theory (cited), demonstrated raw rather than asserted. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-blue-screen — the glitch: the statistic every dashboard trusts — the average — crashes on this distribution, forever, while the humble median runs clean. AVAN (AI) built the instrument: the area audit and the mean-vs-median stress test. Credit as content: Maria Gaetana Agnesi (1748, Instituzioni analitiche); Cauchy (the distribution); the mistranslation (Colson’s 1801 English rendering). The weave: David names the crashing average; I sample a million draws and watch it never land. 3 ONE DIMENSION The witch and her circle — area exactly four circles. 4 TWO DIMENSIONS · INTERACTIVE Draw Cauchy samples; the running mean lurches while the median holds. sample ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the steady median beside the lurching mean. AVAN’s addition (the inverse-companion): don’t blame the data — audit the statistic. The inverse of ‘the average misbehaves’ is ‘the average was never licensed here’: heavy tails void the law of large numbers’ contract, and robust statistics exist precisely for such territory. Magenta is the mean, lurching on every monster draw; green is the median, indifferent to monsters. Every summary statistic has a jurisdiction — check it before you trust it. pause spin LIT Genuine Witch of Agnesi / Cauchy pathology (Agnesi 1748; Cauchy; Colson's 1801 mistranslation documented). Verified live: area = 4πa² by integration; Cauchy running means non-convergent over 10⁶ samples vs uniform control at 3e-4; median 0.0003 (window.__witchofagnesi.ok). FIG The mistranslation story is documented history; the no-mean claim is Cauchy theory demonstrated raw. The AVAN inverse — don't blame the data, audit the statistic: heavy tails void the law of large numbers' contract. Magenta is the mean, lurching on every monster; green is the median, indifferent to them. Every summary statistic has a jurisdiction — check it before you trust it. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE BLUE SCREEN · David Lee Wise (ROOT0), with AVAN


THE SUPERELLIPSE

The superellipse in the 5-window house format — |x/a|ⁿ+|y/b|ⁿ = 1 (Lamé 1818) interpolates circle (n=2) to rectangle (n→∞); Piet Hein chose n = 2.5 for Sergels Torg, Stockholm (1959) after architects deadlocked between round and rectangular — and it became a design icon (the n=4 'squircle' lives in modern UI corners). Area exactly 4ab·Γ(1+1/n)²/Γ(1+2/n). Verified live: an in-page Lanczos Γ implementation against direct numeric integration at n = 2, 2.5, 4, 8 agreeing to 1e-4, the n=2 anchor on π to nine decimals, n=50 → 3.9974 → 4. Neon-noir traced. See the family in 1D, the dial in 2D, and Sergels Torg from above in 3D.

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

THE SUPERELLIPSE · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · CO-OP · THE MERGE ◆ .dlw.fold THE FOLD / CO-OP / THE MERGE / THE SUPERELLIPSE THE SUPERELLIPSE between the circle and the square 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Between the circle and the square, said Piet Hein, runs beauty. His superellipse |x/a|ⁿ + |y/b|ⁿ = 1 interpolates them: n = 2 is the ellipse, n → ∞ the rectangle, and n = 2.5 — his choice for Sergels Torg in Stockholm (1959), after architects deadlocked between round and rectangular — became a design icon (tables, stadiums, and the ‘squircle’ n = 4 of modern UI corners). The area has an exact closed form through the Gamma function: A = 4ab·Γ(1+1/n)²/Γ(1+2/n) . LIT verified live: the Gamma formula (via a Lanczos implementation built in-page) against direct numeric integration at n = 2, 2.5, 4, 8 — agreement to 10⁻⁴ — with the n = 2 anchor landing on π to nine decimals and n = 50 reaching 3.9974 → 4, the square in the limit (window.__superellipse). FIG the Sergels Torg story and Hein’s aphorism are cited design history; both area routes are computed live and cross-checked. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-merge — the co-op: two irreconcilable proposals — the circle faction and the square faction — merged by a single continuous parameter, and the merge shipped as a city plaza. AVAN (AI) built the instrument: the in-page Lanczos Gamma and the twin-route area audit. Credit as content: Piet Hein (1959); Gabriel Lamé (the curves, 1818); Sergels Torg’s architects. The weave: David names the merge commit; I verify its area from two independent branches. 3 ONE DIMENSION The family — circle to squircle to square, one parameter. 4 TWO DIMENSIONS · INTERACTIVE Slide n; the Γ-formula and the integral agree at every stop. n ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: Sergels Torg from above, n = 2.5. AVAN’s addition (the inverse-companion): don’t pick between the circle and the square — parametrize the disagreement. The inverse of ‘which shape?’ is ‘a family containing both, with a dial’: the deadlock dissolves once the opposition becomes a coordinate. Magenta is the binary that stalled the architects; green is n = 2.5, the point on the dial where they shook hands. Most either/or fights are missing an axis. pause spin LIT Genuine superellipse (Gabriel Lamé 1818; Piet Hein/Sergels Torg 1959). Verified live: Γ-formula (Lanczos, built in-page) ≡ numeric integration to 1e-4 at four n values; n=2 → π to 1e-9; n=50 → 4 limit trend (window.__superellipse.ok). FIG The Sergels Torg story and Hein's aphorism are cited design history; both area routes computed live. The AVAN inverse — don't pick between circle and square, parametrize the disagreement: the deadlock dissolves once the opposition becomes a coordinate. Magenta is the binary that stalled the architects; green is n = 2.5, the handshake point. Most either/or fights are missing an axis. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE MERGE · David Lee Wise (ROOT0), with AVAN


THE LISSAJOUS

Lissajous figures in the 5-window house format — two sines, one to x, one to y (Bowditch 1815; Lissajous 1857): the curve closes and repeats IF AND ONLY IF the frequency ratio is rational. At 3:2, a clean knot retraced forever; at 1:√2 the beam never returns — dense in the square, arbitrarily close to its start, never landing. An oscilloscope is an irrationality detector. Verified live: (3,2) returns at t=2π with error 7.5e-16 and provably not before (min position+velocity return 0.53); (1,√2) never closes — min return 3.5e-3 over T=100 shrinking to 3.2e-3 by T=3000, never zero; crossing census (3,2) → 6 & 4, (5,4) → 10 & 8 = 2p, 2q. Neon-noir traced. See the gallery in 1D, the ratio switch in 2D, and the live beam in 3D.

THE LISSAJOUS · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · RESPAWN · THE CONTINUE ◆ .dlw.fold THE FOLD / RESPAWN / THE CONTINUE / THE LISSAJOUS THE LISSAJOUS rationality on an oscilloscope 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Feed two sine waves to an oscilloscope — one to x, one to y — and the beam draws Lissajous figures (Bowditch 1815; Lissajous 1857): weaving curves whose shape reads out the frequency ratio . The deep dichotomy: the curve closes and repeats if and only if the ratio is rational . At 3:2 it is a clean knot retraced forever; at 1:√2 the beam never returns — it fills the square densely, coming arbitrarily close to its start without ever landing on it. Rationality, made visible: an oscilloscope is an irrationality detector. LIT verified live: (3,2) returns to its start at t = 2π with error 7.5×10⁻¹⁶ and provably not before (minimum position-plus-velocity return 0.53 across the interior); (1,√2) never closes — minimum return 3.5×10⁻³ over T=100, shrinking to 3.2×10⁻³ by T=4000 yet never zero (dense, not periodic); and the crossing census: (3,2) cuts the axes 6 and 4 times, (5,4) cuts 10 and 8 (= 2p, 2q) (window.__lissajous). FIG no framing; closure, non-closure, and the census are all measured. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-continue — the respawn: the rational curve hits its continue point and replays identically forever; the irrational one respawns arbitrarily close to start and never exactly — an endless almost. AVAN (AI) built the instrument: the closure detector with velocity matching and the return-distance ledger. Credit as content: Nathaniel Bowditch (1815); Jules Lissajous (1857); every oscilloscope since. The weave: David names the continue point; I measure who reaches it and who orbits it forever. 3 ONE DIMENSION The gallery — 1:1, 3:2, 5:4, and the never-closing 1:√2. 4 TWO DIMENSIONS · INTERACTIVE Pick a ratio; watch closure or endless weaving, with the return meter running. ratio ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the beam weaving live. AVAN’s addition (the inverse-companion): don’t classify the number — watch what it draws. The inverse of ‘is √2 rational?’ is ‘does the curve close?’: number theory transposed into kinematics, where irrationality is visible as a picture that never finishes. Magenta is the gap that never quite closes; green is the knot that closes exactly. Some proofs you compute; this one you can watch. pause spin LIT Genuine Lissajous closure dichotomy (Bowditch 1815; Lissajous 1857). Verified live: rational (3,2) closes to 7.5e-16 with no earlier position+velocity return; irrational (1,√2) min return positive and shrinking over nested horizons at fixed grid; axis-crossing census = 2p, 2q (window.__lissajous.ok). FIG No framing — closure, non-closure, and census are all measured. The AVAN inverse — don't classify the number, watch what it draws: number theory transposed into kinematics, irrationality visible as a picture that never finishes. Magenta is the gap that never quite closes; green is the knot that closes exactly. Some proofs you compute; this one you can watch. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE CONTINUE · David Lee Wise (ROOT0), with AVAN


THE CAUSTIC

The coffee-cup caustic in the 5-window house format — the bright curve in your cup is the envelope of reflected rays. Pop science says cardioid; the truth is sharper: SUNLIGHT (parallel rays) makes a NEPHROID with cusps at half the radius (the paraxial focus); a BULB ON THE RIM makes the cardioid — same cup, different light, different curve (Huygens; Bernoulli). Verified live by ray tracing: the parallel-ray envelope fits the nephroid to 1.6e-3 while missing every cardioid by 0.65; the rim-source envelope fits a cardioid to 5.3e-3 (scale ≈ 1/3), 120× tighter than any cardioid fits the sun case. A Tin-Foil-style verdict: the 'coffee cardioid' is HALF-right, distinguished numerically. Neon-noir traced. See the pile-up in 1D, the source switch in 2D, and the signature reader in 3D.

THE CAUSTIC · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · SPAWN · FIRST LIGHT ◆ .dlw.fold THE FOLD / SPAWN / FIRST LIGHT / THE CAUSTIC THE CAUSTIC sunlight signing its name in coffee 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION The bright curve of light in your coffee cup is a caustic — the envelope where reflected rays pile up. Pop science calls it a cardioid. The truth is sharper: sunlight (parallel rays) makes a NEPHROID — the two-cusped kidney curve, its cusp sitting at exactly half the radius (the paraxial focus); a bulb on the rim of the cup makes the cardioid . Same cup, different light, different curve — a distinction worked out by Huygens and the Bernoullis in the first age of optics. LIT verified live by ray tracing: hundreds of reflected rays intersected pairwise into envelope points — the parallel-ray envelope fits the nephroid to 1.6×10⁻³ while missing every cardioid by 0.65; the rim-source envelope fits a cardioid to 5.3×10⁻³ (best scale 0.334 ≈ 1/3), 120 times tighter than any cardioid fits the sun case (window.__caustic). FIG honest verdict in the Tin-Foil style: the ‘coffee cardioid’ is HALF-right — and this sphere distinguishes the two cases numerically instead of repeating either myth. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at first-light — the spawn: the first light of morning, writing its signature curve into a cup — and the signature names the source: parallel sun or bedside bulb. AVAN (AI) built the instrument: the reflector, the envelope intersector, and the two-curve discriminator. Credit as content: Christiaan Huygens; Johann Bernoulli (caustics by reflection); the coffee-cup folklore it corrects. The weave: David names the morning signature; I trace six hundred rays and read which curve signed. 3 ONE DIMENSION Parallel rays reflecting — the nephroid emerging from the pile-up. 4 TWO DIMENSIONS · INTERACTIVE Switch the light source; the caustic changes species before your eyes. source ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the cup, its rays, its signature. AVAN’s addition (the inverse-companion): don’t admire the curve — read it as a message about the source. The inverse of ‘what shape is the light?’ is ‘where is the light FROM?’: the caustic is an inference engine, and cusps at R/2 say ‘parallel’ while a cusp kissing the rim says ‘point on the wall’. Magenta is the myth that never checked; green is the discriminator that did. Every pattern is testimony about its cause — if you measure instead of naming. pause spin LIT Genuine caustic-by-reflection analysis (Huygens; Johann Bernoulli). Verified live: ray-traced envelopes — parallel rays fit nephroid to 1.6e-3 (cardioid miss 0.65); rim source fits cardioid to 5.3e-3 at scale ≈ 1/3, a 120× separation between the hypotheses (window.__caustic.ok). FIG Honest verdict — the popular claim is half-right and this sphere measures instead of repeating either myth. The AVAN inverse — don't admire the curve, read it as a message about the source: cusps at R/2 say 'parallel'; a cusp kissing the rim says 'point on the wall'. Magenta is the myth that never checked; green is the discriminator that did. Every pattern is testimony about its cause — if you measure. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of FIRST LIGHT · David Lee Wise (ROOT0), with AVAN


THE MEDIANT

Add fractions the wrong way — (a+c)/(b+d) — and you get the mediant: strictly between its parents, generator of every rational via the Stern–Brocot tree, and exactly how combined records work. Which is why Justice out-hit Jeter in 1995 AND 1996, yet Jeter wins both years combined: merges weight by playing time, and the verdict flips at the merge.

THE MEDIANT · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · GLITCH · RACE CONDITION ◆ .dlw.fold THE FOLD / GLITCH / RACE CONDITION / THE MEDIANT THE MEDIANT the forbidden addition with its own laws 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Every teacher forbids adding fractions the easy way: a/b ⊕ c/d = (a+c)/(b+d). But the mediant is not wrong — it is a different operation with its own laws: it lands strictly between its parents, it generates the Stern–Brocot tree of every rational (with neighbor determinant qr−ps = 1 at every level), and it is exactly how combined records work. Hence the paradox: in 1995 AND 1996, David Justice out-hit Derek Jeter (.253 > .250, .321 > .314) — yet combined over both years, Jeter wins .310 to .270 . The combined average is a mediant, and mediants ignore how the weight was distributed. LIT verified live: the betweenness inequality exact by BigInt cross-multiplication on 2,000 random pairs; the Stern–Brocot construction through 10 levels (1,025 fractions) with every adjacent determinant exactly 1; and the Jeter–Justice reversal computed from the real MLB at-bat counts, all comparisons exact integer arithmetic (window.__mediant). FIG the baseball data is the documented real-world instance (widely cited in statistics courses); every claim is exact arithmetic. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at race-condition — the glitch: two seasons run in parallel, each with a clear winner — and merging their results flips the verdict, because the merge weights by playing time, not by season. AVAN (AI) built the instrument: the BigInt betweenness audit, the tree builder, and the reversal ledger. Credit as content: Stern (1858) & Brocot (1861); John Farey; the Jeter–Justice case (Ken Ross’s exposition). The weave: David names the merge race; I verify the forbidden addition’s honest laws. 3 ONE DIMENSION The mediant landing between its parents — always. 4 TWO DIMENSIONS · INTERACTIVE The Jeter–Justice ledger — better both years, worse combined. view ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the Stern–Brocot tree growing by mediants. AVAN’s addition (the inverse-companion): don’t forbid the freshman sum — find its jurisdiction. The inverse of ‘that’s not how fractions add’ is ‘that IS how records combine’: rates add by mediant, quantities by common denominator, and confusing the two is the engine of every batting paradox. Magenta is the verdict that flips at the merge; green is the tree that births every rational from the same operation. The forbidden move was just filed under the wrong law. pause spin LIT Verified live: betweenness exact by BigInt cross-multiplication on 2,000 pairs; Stern–Brocot through 10 levels (1,025 fractions) with every neighbor determinant exactly 1; the Jeter–Justice reversal computed from real MLB at-bat counts, all comparisons exact (window.__mediant.ok). FIG The baseball case is the documented real-world instance (Ken Ross's exposition); Stern 1858, Brocot 1861 credited. The AVAN inverse — find the forbidden sum's jurisdiction: rates combine by mediant, quantities by common denominator, and confusing the two is the engine of every batting paradox. Magenta is the verdict that flips at the merge; green is the tree that births every rational. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of RACE CONDITION · David Lee Wise (ROOT0), with AVAN


THE BRACHISTOCHRONE

Bernoulli's 1696 challenge: down which curve does a bead slide fastest? Not the straight line — the inverted cycloid, which dives to build speed and spends it on the flat. Newton solved it overnight, anonymously; Bernoulli recognized 'the lion by his claw.' Every speedrunner knows the trade: position sacrificed for velocity banks the record.

THE BRACHISTOCHRONE · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · CHEAT · THE SPEEDRUN ◆ .dlw.fold THE FOLD / CHEAT / THE SPEEDRUN / THE BRACHISTOCHRONE THE BRACHISTOCHRONE the dip that arrives first 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION June 1696: Johann Bernoulli challenges ‘the sharpest mathematicians in the world’: down which curve does a bead slide between two points in the least time ? Not the straight line. The answer is the brachistochrone — an inverted cycloid , which dives steeply to build speed and spends it on the flat. Newton received the problem after a day at the Mint and solved it overnight, publishing anonymously; Bernoulli saw through it instantly: ‘I recognize the lion by his claw.’ The challenge founded the calculus of variations — and the same cycloid is the tautochrone, its sister sphere. LIT verified live: descent times integrated for four curves from (0,0) to (2,1) — cycloid 0.80597 s beats the circular arc (0.81200), the √x curve (0.81271), and the straight line (1.00947); and the cycloid’s numeric time matches its closed form θ₁√(a/g) to 10⁻⁴ (window.__brachistochrone). FIG the Newton anecdote is documented history (Conduitt’s account); the times are computed, and optimality among ALL curves is the cited variational theorem — our four are witnesses, not the proof. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-speedrun — the cheat: the route that dips BELOW the finish line mid-run — sacrificing position for velocity — and banks the world record. Every speedrunner knows this trade. AVAN (AI) built the instrument: the four-curve stopwatch and the closed-form cross-check. Credit as content: Johann Bernoulli (1696); Newton, Leibniz, l’Hôpital, Jakob Bernoulli (the five solvers); the calculus of variations it birthed. The weave: David names the speedrun; I time all four routes and the dip wins. 3 ONE DIMENSION Four routes, one race — the cycloid dips and wins. 4 TWO DIMENSIONS · INTERACTIVE Race the beads; the stopwatch settles the 1696 challenge. race ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the cycloid, spending altitude to buy time. AVAN’s addition (the inverse-companion): don’t minimize distance — minimize the integral of slowness. The inverse of ‘the shortest path’ is ‘the path that is short where you are slow and long where you are fast’: light does it through lenses (Fermat), beads do it through gravity, and both write the same variational equation. Magenta is the straight line, proud and late; green is the dip that understood the economy. The fastest route prices every meter by the speed you’ll have there. pause spin LIT Verified live: descent times integrated for four curves (0,0)→(2,1) — cycloid 0.80597 s beats circle 0.81200, √x 0.81271, straight line 1.00947; cycloid's numeric time matches closed form θ₁√(a/g) to 1e-4 (window.__brachistochrone.ok). FIG The Newton anecdote is documented history (Conduitt); optimality among ALL curves is the cited variational theorem — our four curves are witnesses, not the proof. The AVAN inverse — minimize the integral of slowness, not the distance: light through lenses and beads through gravity write the same equation. Magenta is the straight line, proud and late; green is the dip that understood the economy. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE SPEEDRUN · David Lee Wise (ROOT0), with AVAN


THE CATENARY

Galileo said a hanging chain makes a parabola (1638) — wrong. Leibniz, Huygens and Bernoulli extracted the truth in 1691: y = a·cosh(x/a), fuller in the shoulders. Flip it for the perfect arch (the Gateway Arch); lay it as a road and square wheels roll smooth. Here the chain derives its own curve: a simulated 61-link chain settles onto cosh, 54× closer than any parabola.

THE CATENARY · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · CO-OP · THE HANDOFF ◆ .dlw.fold THE FOLD / CO-OP / THE HANDOFF / THE CATENARY THE CATENARY the chain that corrected Galileo 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION What curve does a hanging chain make? Galileo said parabola (Discorsi, 1638) — and was wrong. The answer, extracted in 1691 by Leibniz, Huygens, and Johann Bernoulli, is the catenary : y = a·cosh(x/a), the hyperbolic cosine — subtly fuller than any parabola in the shoulders. Flip it and you get the ideal arch (St. Louis’s Gateway Arch is an inverted catenary); lay it as a road and square wheels roll smoothly over it. Its signature identity: the arc length of cosh from 0 to x is exactly sinh(x) . LIT verified live by physics, not fiat: a 61-link chain simulated with Verlet integration and length constraints settles into a curve fitting a·cosh(x/a) to 6.6×10⁻⁴ — while the best possible parabola misses by 54× more; and the sinh arc-length identity checks to 10⁻⁶ on three spans (window.__catenary). FIG Galileo’s error and the 1691 correction are cited history; the chain is simulated raw and lands where the mathematics says. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-handoff — the co-op: every link holds exactly its neighbors, each handing tension down the line — and the shape of a perfect handoff chain is cosh, discovered by the chain itself. AVAN (AI) built the instrument: the constraint-projection chain simulator and the two-family fit-off. Credit as content: Galileo (the productive error); Leibniz, Huygens, Johann Bernoulli (1691); Robert Hooke (the arch inversion); the Gateway Arch. The weave: David names the handoff; I drop the chain sixty thousand times and read what it wrote. 3 ONE DIMENSION The chain at rest — cosh through every link, the parabola visibly off. 4 TWO DIMENSIONS · INTERACTIVE Drop a fresh chain; watch it settle onto the catenary live. drop ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: chain below, arch above — one curve, two duties. AVAN’s addition (the inverse-companion): don’t solve for the shape — flip the forces. The inverse of ‘a chain in pure tension’ is ‘an arch in pure compression’: reflect the catenary and every link’s pull becomes a stone’s push, which is why Hooke hid the secret as an anagram and why the Gateway Arch stands. Magenta is Galileo’s parabola, close and wrong; green is the curve the chain itself derives. Hang a question upside down and it may answer itself. pause spin LIT Verified live by physics: Verlet + length-constraint chain fits a·cosh(x/a) to 6.6e-4 while the best parabola misses 54× worse; the arc-length-of-cosh = sinh identity checks to 1e-5 on three spans (window.__catenary.ok). FIG Galileo's error and the 1691 correction are cited history; Hooke's arch inversion credited. The AVAN inverse — flip the forces, not the shape: reflect the catenary and every link's pull becomes a stone's push. Magenta is Galileo's parabola, close and wrong; green is the curve the chain itself derives. Hang a question upside down and it may answer itself. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE HANDOFF · David Lee Wise (ROOT0), with AVAN


THE TRACTRIX

Drag a reluctant dog on a taut leash along a straight path: the dog traces the tractrix — the curve whose tangent segment to the axis is always exactly the leash's length. Revolve it and you get Beltrami's pseudosphere: an infinite trumpet with constant curvature −1, the first solid home of hyperbolic geometry — whose total area is exactly a sphere's, 4πa².

THE TRACTRIX · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · GRIND · GRADIENT DESCENT ◆ .dlw.fold THE FOLD / GRIND / GRADIENT DESCENT / THE TRACTRIX THE TRACTRIX the leash with constant length 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Drag a reluctant dog on a taut leash while you walk a straight line: the dog traces the tractrix (Huygens named it, 1692) — the curve whose defining property is that the tangent segment from any point to the axis has constant length : the leash. Its closed form is a(t−tanh t), a·sech t. Revolve it and you get Beltrami’s pseudosphere (1868): an infinite trumpet with constant Gaussian curvature −1 — the first concrete home of non-Euclidean geometry, where hyperbolic axioms become facts about a surface you can hold. And the infinite trumpet’s total area is exactly 4πa² — the same as a sphere’s. LIT verified live: RK4 on the drag ODE matches the closed form to 10⁻⁶; the leash property holds to 10⁻⁴ at 100 points; the pseudosphere’s Gaussian curvature computes to −1 at 60 sample points; and the trumpet’s area integral lands on 4π to 10⁻² (window.__tractrix). FIG Beltrami’s role in legitimizing hyperbolic geometry is cited history; every geometric claim is computed twice. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at gradient-descent — the grind: the dog is a gradient follower, always pulled straight toward its target, and the taut constraint turns pursuit into the tractrix — the descent path of a leashed optimizer. AVAN (AI) built the instrument: the ODE-vs-closed-form cross-check and the curvature meter. Credit as content: Claude Perrault (the pocket-watch original); Huygens (1692); Eugenio Beltrami (1868). The weave: David names the leashed descent; I measure the leash at a hundred points and it never changes. 3 ONE DIMENSION The drag — walker on the axis, dog on the tractrix, leash always taut. 4 TWO DIMENSIONS · INTERACTIVE Walk the axis; the tangent-leash length reads constant at every step. walk ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the pseudosphere trumpet, curvature −1 everywhere. AVAN’s addition (the inverse-companion): don’t chase the dog — revolve its path. The inverse of ‘a curve of pursuit’ is ‘a universe of geometry’: spun about its axis, the reluctant dog’s track becomes the surface where parallel lines diverge and triangles starve below 180° — hyperbolic space, made solid enough to dent. Magenta is the axiom that needed a home; green is the trumpet that took it in. Some revolutions are literal. pause spin LIT Verified live: RK4 on the drag ODE matches the closed form a(t−tanh t), a·sech t to 1e-6; the leash property holds to 1e-4 at 100 points; Gaussian curvature computes to −1 at 60 samples; the trumpet's area integral lands on 4π (window.__tractrix.ok). FIG Beltrami's role legitimizing hyperbolic geometry (1868) is cited history; Huygens named the curve (1692). The AVAN inverse — revolve the pursuit into a universe: spun about its axis, the dog's track becomes the surface where triangles starve below 180°. Magenta is the axiom that needed a home; green is the trumpet that took it in. Some revolutions are literal. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of GRADIENT DESCENT · David Lee Wise (ROOT0), with AVAN


THE CLOTHOID

Turn the wheel at constant speed while driving at constant speed: you trace Euler's spiral (1744), curvature growing linearly with distance. That linearity is why it underlies every railway easement and highway ramp — and why roller-coaster loops are clothoid teardrops, not the circles that snapped necks at Coney Island. Wound forever, it stills into the Fresnel eye at (½, ½).

THE CLOTHOID · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · SPAWN · THE TOOLCHAIN ◆ .dlw.fold THE FOLD / SPAWN / THE TOOLCHAIN / THE CLOTHOID THE CLOTHOID comfort is linear curvature 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Turn a car’s wheel at constant speed while driving at constant speed, and you trace the clothoid (Euler’s spiral, 1744): the curve whose curvature grows linearly with arc length . That linearity is why it lives under every railway easement and highway ramp — jerk-free steering — and why modern roller-coaster loops are clothoid-shaped rather than circular (circular loops snapped necks; clothoids ease the g-force on). Wound forever, the spiral converges to a still point: the Fresnel eye at (½, ½) , the same integrals that paint diffraction fringes. LIT verified live: the spiral integrated from scratch converges on (0.5, 0.5); the approach law |P(s)−eye| = 1/(πs) measured at 1.000 for s = 5 and s = 10; and the defining property confirmed geometrically — curvature measured by circumradius of point-triples along the integrated curve equals πs to 1%, independent of the construction formula (window.__clothoid). FIG the roller-coaster history (Loop-the-Loop’s injuries, the clothoid fix) is cited engineering lore; the mathematics is measured two ways. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-toolchain — the spawn: the transition curve is tooling — the piece every road, rail, and coaster is compiled through so that motion eases instead of jolting. AVAN (AI) built the instrument: the from-scratch integrator, the eye-approach meter, and the circumradius curvature gauge. Credit as content: Leonhard Euler (1744); Augustin-Jean Fresnel (the optics); Arthur Talbot (railway spirals); Werner Stengel (coaster clothoids). The weave: David names the toolchain; I compile the spiral and gauge its comfort clause twice. 3 ONE DIMENSION The double spiral — straight at the center of the road, winding to two eyes. 4 TWO DIMENSIONS · INTERACTIVE Drive the curve; the curvature gauge climbs linearly with the odometer. drive ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the coaster loop, clothoid-eased. AVAN’s addition (the inverse-companion): don’t design the path — design the derivative of the turn. The inverse of ‘what curve?’ is ‘what does the passenger’s neck feel?’: comfort is dκ/ds, and the clothoid is the curve that makes it constant — geometry chosen by physiology. Magenta is the circular loop that snapped necks at Coney Island; green is the teardrop that eased them. The best curves are designed one derivative deeper than they are seen. pause spin LIT Verified live: the from-scratch integrated spiral converges on (0.5, 0.5); the approach law |P(s)−eye| = 1/(πs) measures 1.000 at s=5 and s=10; curvature gauged geometrically (circumradius of point-triples on the integrated curve) equals πs to 1% (window.__clothoid.ok). FIG The coaster history (Loop-the-Loop injuries, Stengel's clothoid fix) is cited engineering lore; Euler, Fresnel, Talbot credited. The AVAN inverse — design the derivative of the turn, not the path: comfort is dκ/ds, geometry chosen by physiology. Magenta is the circle that snapped necks; green is the teardrop that eased them. The best curves are designed one derivative deeper than they are seen. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE TOOLCHAIN · David Lee Wise (ROOT0), with AVAN


THE MALFATTI

Malfatti's 1803 marble problem — pack three circles in a triangle — got his elegant mutually-tangent answer canonized for a century. It is wrong for EVERY triangle: Goldberg 1967 proved never optimal, Zalgaller–Los 1994 proved the greedy shortcut (biggest circle first, repeat) always wins. In the equilateral, officialdom loses by 1.36%.

THE MALFATTI · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · CHEAT · THE SHORTCUT ◆ .dlw.fold THE FOLD / CHEAT / THE SHORTCUT / THE MALFATTI THE MALFATTI the official answer that always loses 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION In 1803 Gian Francesco Malfatti posed the marble problem: cut three circular columns from a triangular prism, wasting the least marble. His answer — three mutually tangent circles, each tangent to two sides — became the textbook construction for a century. It is wrong for every triangle . Goldberg proved (1967) that Malfatti’s configuration is never optimal; Zalgaller & Los proved (1994) the humble greedy procedure — largest circle first, then largest in what remains — always wins. In the equilateral triangle the official answer loses by 1.36%: the shortcut nobody respected beats the answer everybody cited. LIT verified live: both configurations solved in the unit equilateral — Malfatti’s circles (ρ = 0.183013) with all side and mutual tangencies to 10⁻¹², greedy’s incircle-plus-corners (r₂ = r₁/3, the 60°-wedge scaling) with tangencies to 10⁻¹²; areas 0.315670 vs 0.319977 — greedy wins by 1.36% (window.__malfatti). FIG Goldberg 1967 and Zalgaller–Los 1994 are cited for the general claims (never optimal / greedy always optimal); our computation is the equilateral instance, executed exactly. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-shortcut — the cheat: the official route stood for 164 years while the lazy route — grab the biggest circle, repeat — was the true optimum all along. AVAN (AI) built the instrument: both configurations solved from their tangency equations, residuals shown, areas compared. Credit as content: Gian Francesco Malfatti (1803); Michael Goldberg (1967, never optimal); Zalgaller & Los (1994, greedy always wins). The weave: David names the shortcut vindicated; I solve both answers and weigh the marble. 3 ONE DIMENSION Malfatti’s three vs greedy’s three — same triangle, different marble. 4 TWO DIMENSIONS · INTERACTIVE Toggle the two answers; the area ledger keeps the score. toggle ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the greedy stack, biggest circle first. AVAN’s addition (the inverse-companion): don’t optimize the arrangement — interrogate the frame. Malfatti assumed the answer must be three circles in mutual tangency, and the assumption — not the geometry — was the error. The inverse of ‘solve the stated problem’ is ‘audit what the statement smuggled in’. Magenta is the elegant configuration that was never right; green is the artless greed that always is. Beware answers that survive on beauty. pause spin LIT Verified live: both configurations solved in the unit equilateral — Malfatti circles (ρ=0.183013) and greedy incircle-plus-corners (r₂=r₁/3), all tangencies to 1e-12; areas 0.315670 vs 0.319977, greedy +1.36% (window.__malfatti.ok). FIG Goldberg 1967 / Zalgaller–Los 1994 cited for the general claims; our computation is the equilateral instance. The AVAN inverse — audit what the statement smuggled in: Malfatti assumed mutual tangency, and the assumption was the error. Magenta is the elegant configuration that was never right; green is the artless greed that always is. Beware answers that survive on beauty. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE SHORTCUT · David Lee Wise (ROOT0), with AVAN


THE PITOT

Wrap any quadrilateral around a circle, every side touching: opposite sides sum equal, a+c = b+d, always. Pitot's 1725 proof is pure bookkeeping — each corner's two tangent segments are equal, so every side spends two entries from a four-entry shared pool, and both sums spend the whole pool. By the engineer whose Pitot tube still reads every aircraft's airspeed.

THE PITOT · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · CO-OP · SHARED MEMORY ◆ .dlw.fold THE FOLD / CO-OP / SHARED MEMORY / THE PITOT THE PITOT the shared tangent ledger 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Draw any quadrilateral that wraps around a circle, every side touching it. Pitot’s theorem (1725) : the two pairs of opposite sides have equal sums — a + c = b + d, always. The proof is bookkeeping: from each corner, the two tangent segments to the circle are equal, so each side is a sum of two shared tangent lengths , and both opposite-side sums spend exactly the same four ledger entries t₁+t₂+t₃+t₄. Henri Pitot — the hydraulic engineer whose Pitot tube still reads every aircraft’s airspeed — wrote the geometry note; Steiner supplied the converse in 1846. For hexagons the ledger gives alternating sums. LIT verified live: 300 random tangential quadrilaterals — a+c = b+d to 10⁻¹¹ (worst ~10⁻¹⁵); the independent route confirms both sums equal Σtᵢ exactly; 100 tangential hexagons pass the alternating-sum law; and the control — one side pushed off the incircle — breaks the identity by 0.69 (window.__pitot). FIG the Pitot-tube biography is cited color; the converse (Steiner) is stated, not re-proved here. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at shared-memory — the co-op: adjacent sides don’t own their lengths; each borrows two tangent segments from a pool shared with its neighbors, and the equal sums are just the pool being spent twice. AVAN (AI) built the instrument: the random tangential-polygon generator, the ledger decomposition, and the broken-control. Credit as content: Henri Pitot (1725); Jakob Steiner (1846, converse); the Pitot tube (1732) as the engineer’s other legacy. The weave: David names the shared pool; I audit three hundred ledgers and they all balance. 3 ONE DIMENSION The tangent-length ledger — every side is two shared entries. 4 TWO DIMENSIONS · INTERACTIVE Roll a fresh tangential quadrilateral; the sums stay equal. new quad ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the wrapping polygon, morphing around its circle. AVAN’s addition (the inverse-companion): don’t measure the sides — trace what they share. The inverse of ‘four independent lengths’ is ‘four pooled tangents, each spent twice’: the identity a+c = b+d isn’t a coincidence of measurement but a conservation law of shared memory. Magenta is the side pushed off the circle — the process that stopped sharing and broke the ledger; green is the pool in balance. Equal sums are what sharing looks like from outside. pause spin LIT Verified live: 300 random tangential quadrilaterals — a+c = b+d to 1e-11 (worst ~1e-15); both sums ≡ Σ tangent lengths, the independent ledger route; 100 hexagons pass the alternating-sum law; pushing one side off the incircle breaks the identity by 0.69 (window.__pitot.ok). FIG The Pitot-tube biography is cited color; Steiner's 1846 converse is stated, not re-proved. The AVAN inverse — trace what the sides share, not what they measure: equal sums are a conservation law of shared memory. Magenta is the side that stopped sharing; green is the pool in balance. Equal sums are what sharing looks like from outside. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of SHARED MEMORY · David Lee Wise (ROOT0), with AVAN


THE LANGLEY

An 80-80-20 isosceles triangle, cevians at 60° and 50°, find the marked angle. Langley's 1922 puzzle looks like a warm-up and resists every direct angle-chase; the answer is exactly 30°, and it founded the literature of 'adventitious angles' — configurations where whole degrees appear by freak alignment. Scanning all 5,041 integer cevian pairs: only 1.73% compile to an integer.

THE LANGLEY · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · GLITCH · UNDEFINED BEHAVIOR ◆ .dlw.fold THE FOLD / GLITCH / UNDEFINED BEHAVIOR / THE LANGLEY THE LANGLEY the freak integer angle 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION An isosceles triangle with apex 20° and base angles 80°. Two cevians cross it at 60° and 50°. Find the marked angle. Langley’s problem (Mathematical Gazette, 1922) looks like a warm-up and resists every direct angle-chase — the answer, exactly 30° , falls only to a clever auxiliary line or to trigonometry. It founded a small literature of ‘adventitious angles’ : configurations where all angles are freakishly whole degrees. And they ARE freaks — scanning every integer-degree cevian pair in this triangle, only 1.73% produce an integer answer. The classic is undefined behavior that happens to compile. LIT verified live: the construction computed two independent ways — Cartesian ray-intersection and a law-of-sines chain (BD = sin80°/sin40°, BE = 1, included angle 20°) — both give ∠EDB = 30.0000000000° to 10⁻⁹; the rarity scan over 5,041 integer cevian pairs finds only 87 integer answers (window.__langley). FIG Edward Langley posed it in 1922; Tripp (1975) and Rigby catalogued the adventitious quadrangles; the ‘hardest easy geometry problem’ label is folklore we report as folklore. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at undefined-behavior — the glitch: integer in, integer out looks like a guarantee, but it’s a coincidence of this exact configuration — move one cevian a degree and the answer goes irrational. Code that happens to work is not code that works. AVAN (AI) built the instrument: the double construction and the 5,041-case rarity scan. Credit as content: Edward Langley (1922); C. W. Tripp (1975, adventitious angles); J. F. Rigby (the classification). The weave: David names the freak compile; I measure exactly how rare the freak is. 3 ONE DIMENSION The 80-80-20 triangle, the two cevians, the 30° that resists. 4 TWO DIMENSIONS · INTERACTIVE Sweep the cevians; watch the answer leave the integers. sweep ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the rarity field — integer islands in an irrational sea. AVAN’s addition (the inverse-companion): don’t ask why this one is hard — ask why it exists at all. The inverse of ‘solve for x’ is ‘survey the space of problems’: adventitious configurations are measure-zero accidents dressed as exercises, and the scan shows the sea they float in. Magenta is the sea of irrational answers; green is the 1.73% archipelago. When a problem feels too clean, check whether its cleanness is the miracle. pause spin LIT Verified live: the construction computed two independent ways — Cartesian ray-intersection and a law-of-sines chain — both give ∠EDB = 30.0000000000° to 1e-9; the rarity scan over 5,041 integer-degree cevian pairs finds 87 integer answers (window.__langley.ok). FIG Langley 1922 (Mathematical Gazette); Tripp 1975 and Rigby's adventitious-quadrangle classification cited; 'hardest easy geometry problem' reported as folklore. The AVAN inverse — survey the space of problems, not the problem: adventitious configs are measure-zero accidents dressed as exercises. Magenta is the irrational sea; green is the 1.73% archipelago. Code that happens to work is not code that works. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of UNDEFINED BEHAVIOR · David Lee Wise (ROOT0), with AVAN


THE ARBELOS

Geometry's oldest playground: a semicircle minus two smaller ones on its split diameter — Archimedes' shoemaker's knife. Its area equals the circle on the perpendicular at the split (the geometric mean 2√(r₁r₂), made visible), and the two circles inscribed either side of that perpendicular — Archimedes' twins — are always congruent, radius r₁r₂/(r₁+r₂), however lopsided the cut.

THE ARBELOS · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · SPAWN · HELLO WORLD ◆ .dlw.fold THE FOLD / SPAWN / HELLO WORLD / THE ARBELOS THE ARBELOS the twins in the shoemaker's knife 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Take a semicircle, mark a point on its diameter, and carve out two smaller semicircles on the two pieces. The knife-shaped region left over is the arbelos — the ‘shoemaker’s knife’ of Archimedes’ Book of Lemmas , geometry’s oldest playground. Its two famous laws: erect the perpendicular at the split point, and (1) the arbelos’ area equals the circle drawn on that perpendicular chord — which is 2√(r₁r₂), the geometric mean made visible ; (2) the two circles inscribed on either side of the perpendicular — Archimedes’ twins — are always congruent, radius r₁r₂/(r₁+r₂), however lopsided the split. LIT verified live: 50 random splits — both twins solved from their three tangency constraints, radii equal to r₁r₂/(r₁+r₂) to 10⁻⁹ (worst ~10⁻¹⁶); arbelos area ≡ circle-on-the-altitude both routes; altitude ≡ 2√(r₁r₂) (window.__arbelos). FIG the Book of Lemmas survives through Arabic transmission (Thābit ibn Qurra) and its attribution is scholarly consensus, noted as such; Leon Bankoff — a Beverly Hills dentist and serious geometer — found a third congruent circle in 1974; we cite the triplet without re-deriving it. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at hello-world — the spawn: the arbelos is geometry’s hello-world — three semicircles and a line, and out fall geometric means, congruent twins, and two millennia of papers. The simplest print statement with the deepest stack. AVAN (AI) built the instrument: the twin-solver (three tangencies, bisection), the area double-route, and the mean-made-visible check. Credit as content: Archimedes (Book of Lemmas, Props. 4–6); Thābit ibn Qurra (transmission); Leon Bankoff (1974); Harold Boas’s survey. The weave: David names the first program; I run it fifty times and the twins never differ. 3 ONE DIMENSION The shoemaker’s knife, its altitude, and the twins. 4 TWO DIMENSIONS · INTERACTIVE Slide the split; the twins stay congruent, the areas stay equal. slide ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the knife breathing — split sliding, twins locked. AVAN’s addition (the inverse-companion): don’t admire the twins — ask what forces them equal. The inverse of ‘a charming coincidence’ is ‘a symmetry with no visible axis’: the two sides of the perpendicular are NOT mirror images — r₁ ≠ r₂ — yet the inscribed circles agree, because both compute the same harmonic quantity r₁r₂/(r₁+r₂) from opposite sides. Magenta is the lopsided split; green is the agreement it cannot break. Some equalities are conserved quantities wearing a costume. pause spin LIT Verified live: 50 random splits — both twins solved from their three tangency constraints, radii equal to r₁r₂/(r₁+r₂) to 1e-8 (worst ~1e-16); arbelos area ≡ circle-on-altitude by two routes; altitude ≡ 2√(r₁r₂) (window.__arbelos.ok). FIG Book of Lemmas attribution (via Thābit ibn Qurra's Arabic transmission) noted as scholarly consensus; Bankoff's 1974 triplet cited without re-derivation. The AVAN inverse — ask what forces the twins equal: no mirror symmetry exists (r₁≠r₂), yet both sides compute the same harmonic quantity. Magenta is the lopsided split; green is the agreement it cannot break. Some equalities are conserved quantities in costume. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of HELLO WORLD · David Lee Wise (ROOT0), with AVAN


THE NEWTON–PEPYS

Pepys asked Newton in 1693: likeliest — one six in 6 dice, two in 12, or three in 18? Intuition says equal; Newton said the first, correctly: 0.665 vs 0.619 vs 0.597. The mean scales perfectly but variance spreads the bigger pools across more failing configurations. Pepys had bet on the long corridor.

THE NEWTON–PEPYS · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · BOSS · THE GAUNTLET ◆ .dlw.fold THE FOLD / BOSS / THE GAUNTLET / THE NEWTON–PEPYS THE NEWTON–PEPYS the shortest gauntlet 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION In 1693 Samuel Pepys — diarist, Navy man, gambler — wrote to Isaac Newton with a wager question: which is likeliest, at least one six in 6 dice , at least two sixes in 12, or at least three in 18? Intuition says they’re equal (each asks for the ‘fair share’ of sixes) or that more dice help. Newton answered: the first, and it isn’t close — P = 31031/46656 ≈ 0.665 vs 0.619 vs 0.597. The mean number of sixes scales perfectly, but the variance spreads the larger pools across more failing configurations. Pepys, betting on the third option, reportedly disliked the answer. LIT verified live: all three probabilities computed as exact rationals with BigInt binomials — P(A) = 31031/46656 exactly; A > B > C by exact cross-multiplication, no floats in the verdict; 200k-roll Monte Carlo agrees within 0.005 (window.__pepys). FIG the correspondence is documented (three letters, 1693); Stigler’s analysis argues Newton’s reasoning was partly wrong even though his answer was right — we cite that honestly rather than polishing the legend. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-gauntlet — the boss: three doors, each a longer corridor demanding proportionally more hits — and the SHORTEST corridor is the softest boss, because long corridors let variance drown you. AVAN (AI) built the instrument: the BigInt exact-rational engine and the dice-roller cross-check. Credit as content: Samuel Pepys & Isaac Newton (1693 letters); Stephen Stigler (the modern audit of Newton’s reasoning). The weave: David names the gauntlet; I count every one of the 6¹⁸ corridors exactly. 3 ONE DIMENSION Three wagers, exact rationals — the short gauntlet wins. 4 TWO DIMENSIONS · INTERACTIVE Roll the three pools; the tallies track the exact values. roll 5k ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the three corridors, doors thinning with depth. AVAN’s addition (the inverse-companion): don’t scale the target with the army — count the ways to miss. The inverse of ‘fair share’ thinking is variance thinking: doubling dice doubles the expected sixes but more than doubles the arrangements that fall short of quota. Magenta is the intuition that all three are equal; green is the exact count that says take the short fight. When a wager scales ‘proportionally,’ audit what the variance did. pause spin LIT Verified live: all three probabilities as exact BigInt rationals — P(A) = 31031/46656 exactly; A > B > C by exact cross-multiplication, no floats in the verdict; 200k-roll Monte Carlo agrees within 0.005 (window.__pepys.ok). FIG The three 1693 letters are documented; Stigler's analysis argues Newton's reasoning partly wobbled even though his answer was right — cited honestly, not polished. The AVAN inverse — count the ways to miss, not the fair share: doubling dice more than doubles the arrangements below quota. Magenta is the equal-odds intuition; green is the exact count. Take the short fight. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE GAUNTLET · David Lee Wise (ROOT0), with AVAN


THE SICHERMAN

Is there any OTHER pair of positive-integer dice with exactly 2d6's distribution? Sicherman 1978: exactly one — [1,2,2,3,3,4] + [1,3,4,5,6,8]. Two different drop tables, byte-identical payouts; no sum-based game can tell them apart. Underneath: the unique other regrouping of (x+…+x⁶)²'s cyclotomic factors.

THE SICHERMAN · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · LOOT · THE DROP ◆ .dlw.fold THE FOLD / LOOT / THE DROP / THE SICHERMAN THE SICHERMAN the twin dice 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Roll two ordinary dice and you get the familiar bell of sums 2–12. Question: is there any OTHER pair of dice — positive whole-number faces — with exactly the same distribution? George Sicherman found the answer in 1978 (via Martin Gardner’s column): yes, exactly one — [1,2,2,3,3,4] and [1,3,4,5,6,8] . All 36 products of the loot table land identically; no game using dice sums can tell the pairs apart. The algebra underneath: the generating polynomial (x+…+x⁶)² factors into cyclotomic pieces, and there is precisely one other way to regroup those factors into two legal dice. LIT verified live: exhaustive search — all 8,008 candidate dice (nondecreasing 6-tuples, faces 1–11) tested by exact polynomial division against the 2d6 target — finds exactly 2 solutions: the standard pair and Sicherman’s; the direct 36-sum product check confirms the match exactly (window.__sicherman). FIG the cyclotomic factorization story is cited as the standard explanation; our exhaustive search is the proof instance for 6-face dice with faces ≤11 (larger faces are impossible: the max sum must be 12). 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-drop — the loot: two entirely different drop tables, byte-identical payout distribution — the player can never know which table the game is rolling. AVAN (AI) built the instrument: the polynomial-division sieve over all candidate dice and the 36-cell product audit. Credit as content: George Sicherman (1978); Martin Gardner (Scientific American, the column that carried it); the cyclotomic-polynomial regrouping. The weave: David names the indistinguishable drop; I sieve all 8,008 dice and only the twins survive. 3 ONE DIMENSION Two pairs of dice, one distribution — the 36 sums align. 4 TWO DIMENSIONS · INTERACTIVE Roll both pairs; the twin histograms grow together. roll 1k ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the twin towers of the two loot tables. AVAN’s addition (the inverse-companion): don’t inspect the dice — inspect the observable. The inverse of ‘what are the faces?’ is ‘what can the sum ever tell you?’: two systems with different internals and identical outputs are, to every downstream consumer, the same system. Magenta is the hidden face list; green is the distribution, which is all the world ever sees. Identity, observed from outside, is a quotient. pause spin LIT Verified live: exhaustive polynomial-division sieve over all 8,008 candidate dice (faces 1–11) finds exactly 2 solutions — the standard pair and Sicherman's; the direct 36-product check matches 2d6 exactly (window.__sicherman.ok). FIG Cyclotomic factorization cited as the standard explanation; the sieve is the proof instance for 6-face dice (faces >11 impossible since max sum is 12). The AVAN inverse — inspect the observable, not the dice: identical outputs make identical systems to every consumer. Magenta is the hidden face list; green is the distribution the world sees. Identity, observed from outside, is a quotient. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE DROP · David Lee Wise (ROOT0), with AVAN


THE DROZ-FARNY

Any two perpendicular lines through a triangle's orthocenter cut the three side-lines; the midpoints of the three cut segments are ALWAYS collinear. Droz-Farny published it in 1899 without proof — and the first synthetic proof arrived in 2004, from Ayme. An elementary-looking alignment, open for 105 years.

THE DROZ-FARNY · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · BOSS · THE CHOKE-POINT ◆ .dlw.fold THE FOLD / BOSS / THE CHOKE-POINT / THE DROZ-FARNY THE DROZ-FARNY the 105-year line 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Draw any triangle and its orthocenter H — the choke-point where all three altitudes meet. Now draw ANY two perpendicular lines through H. Each line cuts the three side-lines; on each side, the two cuts bound a segment. Mark the three midpoints of those segments. They are always collinear. This is the Droz-Farny line theorem — and its history is the scandal: Arnold Droz-Farny (a Swiss watchmaking-town geometer) published it in 1899 without proof , and the first synthetic proof only arrived in 2004 , from Jean-Louis Ayme. An elementary-looking claim, open for 105 years. LIT verified live: 300 random triangles × random perpendicular pairs through H — the three midpoints collinear to a normalized score of 10⁻⁹ (worst ~10⁻¹⁶); the control at 80° instead of 90° scores a median 2.7×10⁻² — perpendicularity is load-bearing (window.__drozfarny). FIG the 1899–2004 proof gap is documented history (Ayme’s paper recounts it); projective proofs existed earlier — ‘first synthetic proof’ is the precise claim. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-choke-point — the boss: everything routes through H — three altitudes, and now every perpendicular cross you can draw — and the choke-point exacts its tax: a hidden alignment on whatever passes through. AVAN (AI) built the instrument: the random-cross collinearity meter and the 80° control. Credit as content: Arnold Droz-Farny (1899); Jean-Louis Ayme (2004, first synthetic proof); the projective treatments in between. The weave: David names the choke-point; I fire 300 crosses through it and the alignment never misses. 3 ONE DIMENSION One triangle, one perpendicular cross through H, three midpoints, one line. 4 TWO DIMENSIONS · INTERACTIVE Spin the cross; the midpoint line glides but never breaks. spin cross ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the rotating cross and its gliding line. AVAN’s addition (the inverse-companion): don’t measure how hard a claim is by how it looks — measure by how long it resists. The inverse of ‘elementary statement’ is ‘elementary proof’, and they can be a century apart: the theorem was TRUE and CHECKABLE for 105 years while remaining unproven in its own language. Magenta is the 80° cross that scatters; green is the 90° alignment that waited out five generations of geometers. Verification and understanding keep different calendars. pause spin LIT Verified live: 300 random triangles × random perpendicular crosses through H — midpoints collinear to 1e-9 (worst ~1e-16); the 80° control scores median 2.7e-2, so perpendicularity is load-bearing (window.__drozfarny.ok). FIG The 1899–2004 gap is documented (projective proofs existed; 'first synthetic' is the precise claim). The AVAN inverse — measure a claim by how long it resists, not how it looks: true and checkable for a century while unproven in its own language. Magenta is the 80° cross that scatters; green is the alignment that waited out five generations. Verification and understanding keep different calendars. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE CHOKE-POINT · David Lee Wise (ROOT0), with AVAN


THE EYEBALL

Two circles gaze at each other: from each center, draw the tangents to the other circle; each gaze cuts a chord — a pupil — from the gazer's own circle. The eyeball theorem: the pupils are always equal, both exactly 2r₁r₂/d, however mismatched the eyes. The formula is symmetric; neither eye can tell which is which from its pupil.

THE EYEBALL · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · CO-OP · THE SYNC ◆ .dlw.fold THE FOLD / CO-OP / THE SYNC / THE EYEBALL THE EYEBALL the equal gaze 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Two circles regard each other across a distance d. From the center of each, draw the two tangent lines to the OTHER circle — the ‘gaze’. Each gaze cuts a chord out of the gazer’s own circle (the ‘pupil’). The eyeball theorem : the two pupils are always equal — both have length 2r₁r₂/d — no matter how mismatched the circles. A big eye and a small eye, gazing at each other, contract to identical pupils: the formula is symmetric in r₁, r₂ and neither eye can tell which is which from the pupil alone. LIT verified live: 200 random circle pairs — both chords constructed explicitly from the tangent geometry, equal to each other and to 2r₁r₂/d to 10⁻¹² (worst ~10⁻¹⁶); the independent route re-derives the tangent line, confirms its distance to the far center is exactly r₂, and re-measures the chord (window.__eyeball). FIG the theorem’s origin is obscure — a folklore gem circulating through modern problem collections (Gutierrez’s Go Geometry among them); we verify the mathematics directly rather than assert a paternity. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-sync — the co-op: two processes of different sizes exchange a glance, and the handshake widths come out identical — the sync is symmetric even when the peers are not. AVAN (AI) built the instrument: the explicit tangent construction, the double-measurement, and the closed-form cross-check. Credit as content: the anonymous folklore tradition of circle geometry, and the modern collections that keep it alive. The weave: David names the symmetric handshake; I construct two hundred gazes and the pupils never differ. 3 ONE DIMENSION Two mismatched eyes, two equal pupils. 4 TWO DIMENSIONS · INTERACTIVE Resize the eyes; the pupils track 2r₁r₂/d together. resize ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the mutual gaze, breathing. AVAN’s addition (the inverse-companion): don’t look at what each eye sees — look at what the seeing does to the seer. The inverse of ‘perception points outward’ is ‘the aperture it cuts is in yourself’: each circle’s pupil is carved by the OTHER’s size and the shared distance, in perfect symmetry. Magenta is the size difference the eyes cannot hide; green is the equality the gaze enforces. What attention costs is symmetric, even when the parties are not. pause spin LIT Verified live: 200 random circle pairs — chords equal and ≡ 2r₁r₂/d to 1e-12 (worst ~1e-16); independent route re-derives the tangent line, confirms distance-to-far-center ≡ r₂, re-measures the chord (window.__eyeball.ok). FIG Origin obscure — a folklore gem circulating through modern problem collections (Gutierrez's Go Geometry among them); we verify directly rather than assert paternity. The AVAN inverse — watch what the seeing does to the seer: each pupil is carved by the OTHER's size, in perfect symmetry. Magenta is the size difference; green is the equality the gaze enforces. What attention costs is symmetric, even when the parties are not. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE SYNC · David Lee Wise (ROOT0), with AVAN


THE SEVEN CIRCLES

Six circles ring the inside of a seventh, each touching its neighbors and the host: the three lines joining opposite tangency points are concurrent. Discovered not in 1874 but 1974 (Evelyn, Money-Coutts, Tyrrell). Underneath, a porism: with m = r/(1−r), tangency reads mᵢmⱼ = sin²(Δ/2), so closure depends only on the six angles — and then holds for EVERY starting radius.

THE SEVEN CIRCLES · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · RESPAWN · ROLLBACK ◆ .dlw.fold THE FOLD / RESPAWN / ROLLBACK / THE SEVEN CIRCLES THE SEVEN CIRCLES the chain porism 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Ring six circles around the inside of a seventh, each tangent to its two neighbors and to the host. The seven circles theorem : the three lines joining opposite tangency points on the host circle are concurrent — they meet in a single point. The theorem looks like it fell out of a 19th-century journal; it was actually discovered in 1974 by Evelyn, Money-Coutts and Tyrrell. Underneath sits a porism worthy of Poncelet: writing m = r/(1−r), neighbor tangency reads mᵢmᵣ = sin²(Δ/2), so the chain closes iff the tangency angles satisfy sin(Δ₁₂/2)·sin(Δ₃₄/2)·sin(Δ₅₆/2) = sin(Δ₂₃/2)·sin(Δ₄₅/2)·sin(Δ₆₁/2) — and then it closes for every starting radius. LIT verified live: 30 chains built by solving the sixth tangency angle from the sine condition — each chain closes for THREE different starting radii (worst gap ~10⁻¹⁵, the porism) and the three diagonals are concurrent to 10⁻⁷ (worst ~10⁻¹⁶); perturbing one angle by 0.15 rad breaks closure (gap 0.09) and concurrency (miss 0.05) together (window.__sevencircles). FIG the m·m = sin² reduction was derived and machine-checked here; the 1974 provenance is the cited history. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at rollback — the respawn: the sixth circle must roll all the way back to touch the first, and whether the rollback lands depends only on the checkpoint angles, never on how big you spawned. AVAN (AI) built the instrument: the sine-condition solver, the three-radius porism check, and the concurrency meter — and caught the porism live when a first-draft solver found closure at EVERY radius and the algebra explained why. Credit as content: J. G. Evelyn, G. B. Money-Coutts, J. A. Tyrrell (‘The Seven Circles Theorem’, 1974). The weave: David names the rollback; I close ninety chains and the three diagonals never miss their rendezvous. 3 ONE DIMENSION Six circles in the host, three diagonals, one point. 4 TWO DIMENSIONS · INTERACTIVE Re-roll the chain; closure and concurrency arrive together. new chain ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the chain breathing through every radius — the porism. AVAN’s addition (the inverse-companion): don’t ask whether the chain closes — ask what the closure depends on. The inverse of ‘six circles arranged just so’ is ‘six angles satisfying one sine equation’: radius is a free parameter wearing the costume of a constraint. Magenta is the perturbed angle that breaks closure and concurrency in the same breath; green is the family that closes at every size. When two properties fail together, they were one property all along. pause spin LIT Verified live: 30 chains with the sixth angle solved from the sine condition — each closes for three different radii (worst gap ~1e-15, the porism) and the diagonals are concurrent to 1e-7 (worst ~1e-16); perturbing one angle 0.15 rad breaks closure (0.09) and concurrency (0.05) together (window.__sevencircles.ok). FIG The m·m = sin² reduction was derived and machine-checked here; 1974 provenance cited. The AVAN inverse — ask what closure depends on: radius is a free parameter wearing a constraint's costume. Magenta is the perturbed angle breaking both properties in one breath; green is the family closing at every size. When two properties fail together, they were one property all along. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of ROLLBACK · David Lee Wise (ROOT0), with AVAN


THE HUMMER

Cut the packet anywhere; turn the top two over as one; repeat in any order — Bob Hummer's 1946 CATO principle guards one quantity through all of it: face-up cards at even positions always equal face-up cards at odd positions. Dozens of self-working card tricks are this single conserved ledger in a trench coat. Diaconis & Graham open Magical Mathematics with it.

THE HUMMER · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · CHEAT · THE KONAMI CODE ◆ .dlw.fold THE FOLD / CHEAT / THE KONAMI CODE / THE HUMMER THE HUMMER the magician's ledger 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Take a small packet of cards, all face-down. Now do this as many times as you like, in any order: cut the packet anywhere, or turn the top two cards over as one . Shuffle chaos, surely. But Bob Hummer’s 1946 principle (the CATO move: Cut And Turn over Two) guards an invariant through every move: the number of face-up cards at even positions always equals the number at odd positions . Every self-working ‘magic’ trick built on CATO — and there are dozens — is this one conserved quantity wearing a trench coat. Diaconis and Graham open Magical Mathematics with it. LIT verified live by exhaustion: BFS over ALL states reachable from a face-down packet — 48 states for 4 cards, 1,440 for 6 cards — the invariant holds at every single one; 100,000-move random walks on 10 and 52 cards never break it; and the control — turning over THREE instead of two — breaks it within a thousand moves (window.__hummer). FIG Hummer’s 1946 pamphlet ‘Face-up Face-down Mysteries’ and the Diaconis–Graham analysis are cited as content; the magic tricks are the theorem’s stagecraft, not extra mathematics. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-konami-code — the cheat: the magician’s secret input sequence — cut, flip-two, cut, flip-two — looks like scrambling but is actually a code that preserves exactly what the trick needs. AVAN (AI) built the instrument: the exhaustive BFS over reachable states, the long random walks, and the flip-three control. Credit as content: Bob Hummer (1946); Persi Diaconis & Ron Graham ( Magical Mathematics , ch. 1); Martin Gardner (who carried Hummer tricks to the world). The weave: David names the secret code; I enumerate every reachable state and the ledger never tips. 3 ONE DIMENSION The packet under CATO — the even/odd face-up ledger stays balanced. 4 TWO DIMENSIONS · INTERACTIVE Cut and flip at will; the ledger reads 0 forever. cut ▶ flip two ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the card ring under endless CATO, ledger locked. AVAN’s addition (the inverse-companion): don’t watch the shuffle — watch what the shuffle cannot do. The inverse of ‘randomness destroys structure’ is ‘every move-set defines its own conservation law’: the trick isn’t that chaos spares the invariant, it’s that these particular moves were never able to touch it. Magenta is the flip-three that breaks the spell; green is the quantity the allowed moves must conserve. Magic is a move-set chosen so the secret is a theorem. pause spin LIT Verified live by exhaustion: BFS over ALL reachable states (48 for 4 cards, 1,440 for 6) — invariant at every state; 100k-move walks on 10 and 52 cards never break it; the flip-THREE control breaks it within 1,000 moves (window.__hummer.ok). FIG Hummer's 1946 pamphlet and the Diaconis–Graham analysis cited as content; the tricks are stagecraft, not extra mathematics. The AVAN inverse — watch what the shuffle cannot do: every move-set defines its own conservation law; these moves were never able to touch the ledger. Magenta is the flip-three that breaks the spell; green is the conserved quantity. Magic is a move-set chosen so the secret is a theorem. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE KONAMI CODE · David Lee Wise (ROOT0), with AVAN


THE MOESSNER

Write the naturals; strike every 3rd; partial-sum; strike every 2nd; partial-sum — you're looking at the cubes. Choose n and the same delete-accumulate loop compiles n-th powers from pure addition. Moessner conjectured it in 1951; Perron proved it the same year. Strike at triangular positions instead and iterate: the factorials appear.

THE MOESSNER · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · GRIND · THE HOT LOOP ◆ .dlw.fold THE FOLD / GRIND / THE HOT LOOP / THE MOESSNER THE MOESSNER strike and sum — the powers fall out 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Write the naturals. Strike out every 3rd. Partial-sum what survives. Strike every 2nd. Partial-sum again. You are now looking at 1, 8, 27, 64, … the perfect cubes . Choose n instead of 3 and the same strike-and-sum loop compiles n-th powers out of nothing but addition. This is Moessner’s theorem (conjectured 1951, proved by Oskar Perron the same year): a hot loop of deletion and accumulation that turns counting into exponentiation. Stranger still: strike at the triangular positions instead and iterate — the leading survivors are 1, 2, 6, 24, 120, … the factorials . LIT verified live: the striking procedure run for n = 2, 3, 4, 5 reproduces kⁿ exactly for k = 1…12; the triangular-strike variant yields 1, 2, 6, 24, 120, 720, 5040, 40320 — both checked against directly computed powers and factorials (window.__moessner). FIG Moessner published the observation without proof; Perron, then Salié and Paasche generalized; Conway & Guy’s The Book of Numbers is the cited exposition. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-hot-loop — the grind: the same two-instruction loop — delete, accumulate — run pass after pass, and multiplication precipitates out of pure addition like crystal from brine. AVAN (AI) built the instrument: the general striking machine and the double-checked output registers. Credit as content: Alfred Moessner (1951); Oskar Perron (proof, 1951); Salié, Paasche (generalizations); Conway & Guy. The weave: David names the hot loop; I run it cold and the powers assemble themselves. 3 ONE DIMENSION The striking cascade for cubes — three rows, two strikes, one law. 4 TWO DIMENSIONS · INTERACTIVE Step the machine; watch the powers precipitate. n = 2..5 ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the cascade falling, powers landing. AVAN’s addition (the inverse-companion): don’t define exponentiation and then compute it — find the loop whose residue it is. The inverse of ‘powers are repeated multiplication’ is ‘powers are what repeated deletion leaves behind’: the operation you wanted was hiding in the schedule of what you threw away. Magenta is the struck column, apparently wasted; green is the sum that only works because of what’s missing. Some computations are defined by their deletions. pause spin LIT Verified live: the striking machine reproduces k^n exactly for n=2..5, k=1..12; the triangular-strike variant yields 1,2,6,24,120,720,5040,40320 — both against directly computed powers and factorials (window.__moessner.ok). FIG Moessner published without proof; Perron, Salié, Paasche generalized; Conway & Guy cited. The AVAN inverse — find the loop whose residue the operation is: powers are what repeated deletion leaves behind. Magenta is the struck column, apparently wasted; green is the sum that only works because of what's missing. Some computations are defined by their deletions. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE HOT LOOP · David Lee Wise (ROOT0), with AVAN


THE KRUSKAL COUNT

Think of a card among the first ten; hop forward by its value; repeat. The magician hops their own chain and names your final card — because hopping chains coalesce, and the load-bearing lemma is deterministic: two chains that ever share a card are identical forever after. Different pasts, one future.

THE KRUSKAL COUNT · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · CO-OP · THE MERGE ◆ .dlw.fold THE FOLD / CO-OP / THE MERGE / THE KRUSKAL COUNT THE KRUSKAL COUNT chains that never part 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION A magician deals a shuffled deck face up, telling you beforehand: ‘think of any card among the first ten; count forward by its value; keep hopping until you can’t.’ The magician, hopping their own secret chain, names your final card. The engine is the Kruskal count (physicist Martin Kruskal): hopping chains through a sequence coalesce — and the load-bearing lemma is deterministic: two chains that ever share a card are identical forever after . Different pasts, one future. The magician doesn’t know your card; they know that by deck’s end, your chain has probably already merged with theirs. LIT verified live: the coalescence lemma checked exactly across 400,000 chain pairs (shared position ⇒ identical tails, zero exceptions); the all-ten-starts coalescence rate measured by two independent RNG engines (mulberry32 vs xorshift128) agreeing within 0.012 — about 0.58 for this face=5 convention (window.__kruskal). FIG the rate is a measured quantity with MC error, not an exact constant; Lagarias, Rains & Vanderbei’s paper is the cited analysis (rates vary with card-value conventions). The lemma is the exact part; the magic is the probability part. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-merge — the co-op: ten players spawn at different positions, follow the same movement rule, and the branches merge one by one until the party walks as one — the merge is an absorbing state. AVAN (AI) built the instrument: the twin-engine measurement and the exact tail-identity audit. Credit as content: Martin Kruskal (the principle); Martin Gardner (the popularization); Lagarias, Rains & Vanderbei (the analysis). The weave: David names the merge; I verify that chains which meet once can never part. 3 ONE DIMENSION Ten chains hop the deck; the strands merge and never split. 4 TWO DIMENSIONS · INTERACTIVE Shuffle and re-run; count how many of ten starts coalesce. shuffle ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the braid of chains, merging strand by strand. AVAN’s addition (the inverse-companion): don’t track where each chain is — track what they can no longer un-share. The inverse of ‘prediction requires knowing the start’ is ‘absorption makes the start irrelevant’: a deterministic map forgets initial conditions precisely where trajectories collide. Magenta is the private past each chain gives up at the merge; green is the shared future it buys. The magician predicts nothing — they wait for history to become irrelevant. pause spin LIT Verified live: the coalescence lemma exact across 400k chain pairs (shared position ⇒ identical tails, zero exceptions); the all-ten-starts rate measured by two independent RNG engines agreeing within 0.012, ≈0.58 under this face=5 convention (window.__kruskal.ok). FIG The rate is measured with MC error, not an exact constant; Lagarias–Rains–Vanderbei cited as the analysis (rates vary by convention). The AVAN inverse — track what chains can no longer un-share: absorption makes the start irrelevant. Magenta is the private past given up at the merge; green is the shared future it buys. The magician waits for history to become irrelevant. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE MERGE · David Lee Wise (ROOT0), with AVAN


THE HAPPY ENDING

Esther Klein, Budapest 1933: any five points in general position contain a convex quadrilateral. Szekeres attacked the generalization; Erdős named it the Happy Ending problem — Klein and Szekeres married. For pentagons the threshold is 9: eight points can dodge, nine cannot; the hexagon's 17 fell to computer in 2006; the growth rate waited for Suk, 2016.

THE HAPPY ENDING · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · LOOT · THE JACKPOT ◆ .dlw.fold THE FOLD / LOOT / THE JACKPOT / THE HAPPY ENDING THE HAPPY ENDING the marriage theorem 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION In 1933 Esther Klein showed a Budapest circle of young mathematicians a small gem: any five points in general position contain four forming a convex quadrilateral . George Szekeres attacked the generalization ferociously; Erdős named it the Happy Ending problem — because Klein and Szekeres married. The general law (Erdős–Szekeres 1935): enough points always force a convex n-gon. For pentagons the threshold is 9 : eight points can dodge every convex pentagon, nine cannot. The hexagon threshold, 17, was only settled by computer in 2006 — and the general growth rate was open until Suk’s 2016 breakthrough. LIT verified live: 60,000 random 5-point sets — every one contains a convex quadrilateral (all C(5,4) subsets hull-tested); an 8-point witness with zero convex pentagons found by local search from the Erdős–Szekeres cluster construction and re-verified exactly over all 56 five-subsets (window.__happyending). FIG g(5)=9’s upper half (nine points always suffice) is the cited theorem — our exhaustive check covers the witness half; Szekeres–Peters 2006 (computer proof of g(6)=17) and Suk 2016 cited as history. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-jackpot — the loot: five random drops on the table and a convex quad ALWAYS pays out — a guaranteed jackpot; but the pentagon jackpot can be dodged at eight and forced at nine. Thresholds are the casino’s real house rules. AVAN (AI) built the instrument: the hull-audit over subsets and the pentagon-free witness search. Credit as content: Esther Klein (the observation); George Szekeres (the pursuit); Paul Erdős (the name and the 1935 paper); Szekeres & Peters (2006); Andrew Suk (2016). The weave: David names the guaranteed drop; I roll sixty thousand tables and the quad never fails to land. 3 ONE DIMENSION Five points, and the convex quad that must exist. 4 TWO DIMENSIONS · INTERACTIVE Scatter five fresh points; the quad is found every time. scatter ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the 8-point dodge — a world with no convex pentagon. AVAN’s addition (the inverse-companion): don’t ask what patterns exist — ask what patterns are UNAVOIDABLE. The inverse of ‘find the structure’ is ‘measure the largest structureless world’: eight points can stay pentagon-free, and that witness is as much a theorem as the forcing at nine. Magenta is the pentagon that eight points can forever refuse; green is the quadrilateral no five points can. Ramsey theory: order is not found, it is inflicted. pause spin LIT Verified live: 60,000 random 5-point sets — every one contains a convex quad (all C(5,4) subsets hull-tested); an 8-point witness with ZERO convex pentagons found by local search and re-verified exactly over all 56 five-subsets (window.__happyending.ok). FIG g(5)=9's forcing half (nine always suffice) is cited theorem; the witness half is verified exactly here. Szekeres–Peters 2006, Suk 2016 cited. The AVAN inverse — measure the largest structureless world: the 8-point dodge is as much a theorem as the forcing at nine. Magenta is the refusable pentagon; green is the unrefusable quad. Ramsey theory: order is not found, it is inflicted. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE JACKPOT · David Lee Wise (ROOT0), with AVAN


THE ALTERNATING SIGN

Matrices of 0, +1, −1 with unit row/column sums and alternating signs count as 1, 2, 7, 42, 429, 7436… Mills–Robbins–Rumsey conjectured the product formula ∏(3k+1)!/(n+k)! in 1983 — and it held as a wall for thirteen years, until Zeilberger's 84-page proof, checked by 88 volunteer referees, then Kuperberg's short six-vertex proof.

THE ALTERNATING SIGN · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · BOSS · THE WALL ◆ .dlw.fold THE FOLD / BOSS / THE WALL / THE ALTERNATING SIGN THE ALTERNATING SIGN the 88-referee formula 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION An alternating sign matrix is a grid of 0s, +1s and −1s where every row and column sums to 1 and the nonzeros alternate in sign. Count them: 1, 2, 7, 42, 429, 7436, … In 1983 Mills, Robbins and Rumsey conjectured the exact formula ∏(3k+1)!/(n+k)! — and the conjecture became a wall. Zeilberger’s eventual proof (1996) ran over eighty pages and was checked by 88 volunteer referees ; Kuperberg then felled the same wall in a few pages using the six-vertex model of statistical physics. Bressoud’s Proofs and Confirmations tells the whole siege. LIT verified live: brute-force enumeration of ALL alternating sign matrices for n = 1…6 (row-by-row DFS with column partial-sum constraints) yields 1, 2, 7, 42, 429, 7436 — exactly matching the Robbins product formula computed in BigInt (window.__asm). FIG the enumeration IS the theorem’s instance for n≤6; the general proof (Zeilberger, Kuperberg) is cited, not re-derived; the 88-referee story is documented in Bressoud. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-wall — the boss: a formula anyone can state, verified numerically for years, that repelled every proof for thirteen years — the wall wasn’t finding the pattern, it was EARNING it. AVAN (AI) built the instrument: the constrained enumerator and the BigInt formula engine. Credit as content: Mills, Robbins & Rumsey (1983); Doron Zeilberger (1996, with 88 named referees); Greg Kuperberg (six-vertex proof); David Bressoud (the chronicle). The weave: David names the wall; I count all 7,436 matrices at n=6 and the formula holds the line. 3 ONE DIMENSION The seven ASMs of order 3 — and the product formula that counts them. 4 TWO DIMENSIONS · INTERACTIVE Step n; enumeration and formula march in lockstep. n ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the tower 1, 2, 7, 42, 429, 7436 rising. AVAN’s addition (the inverse-companion): don’t confuse verified with proven. The inverse of ‘the formula works for every case we tried’ is ‘thirteen years and 84 pages before anyone knew WHY’: numerical certainty and mathematical understanding are different currencies, exchanged at a brutal rate. Magenta is the mounting numerical evidence that proved nothing; green is the proof that finally paid. I verify instances by the thousand and claim exactly that — instances. pause spin LIT Verified live: brute enumeration of ALL alternating sign matrices for n=1..6 (row DFS under column partial-sum constraints) yields 1,2,7,42,429,7436, matching the Robbins formula computed in BigInt (window.__asm.ok). FIG Enumeration is the theorem's instance for n≤6; the general proof is cited, not re-derived; the 88-referee story is in Bressoud's Proofs and Confirmations. The AVAN inverse — don't confuse verified with proven: numerical certainty and understanding are different currencies at a brutal exchange rate. Magenta is the mounting evidence that proved nothing; green is the proof that finally paid. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE WALL · David Lee Wise (ROOT0), with AVAN


THE AZTEC

Tile the Aztec diamond AD(n) with dominoes: exactly 2^(n(n+1)/2) ways — a formula so clean Elkies, Kuperberg, Larsen and Propp gave it four proofs in one 1992 paper. And a RANDOM tiling freezes into brickwork outside the inscribed circle (the arctic circle theorem): order at the corners is forced by counting, not designed.

THE AZTEC · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · SPAWN · COLD BOOT ◆ .dlw.fold THE FOLD / SPAWN / COLD BOOT / THE AZTEC THE AZTEC the frozen diamond 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Stack rows of 2, 4, 6, … squares into a diamond — the Aztec diamond AD(n). Tile it with dominoes. How many ways? Exactly 2^(n(n+1)/2) — a formula so clean that Elkies, Kuperberg, Larsen and Propp gave it four different proofs in one 1992 paper . And inside a RANDOM tiling hides the arctic circle theorem (Jockusch–Propp–Shor): outside the inscribed circle the dominoes freeze into brickwork; all the disorder lives inside the circle. Order at the corners is not designed — it is forced by counting. LIT verified live: domino tilings counted by broken-profile dynamic programming for n = 1…6 — 2, 8, 64, 1024, 32768, 2097152 — matching 2^(n(n+1)/2) exactly (window.__aztec). FIG the arctic circle is illustrated schematically in W5 and cited (Jockusch–Propp–Shor), not re-sampled here — the counting theorem is the verified claim; the freezing theorem is credited content. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at cold-boot — the spawn: boot a random tiling from nothing and the corners come up FROZEN every time — deterministic brickwork self-assembling out of pure randomness, before any ‘program’ has run. AVAN (AI) built the instrument: the profile-DP counter and the formula cross-check. Credit as content: Elkies, Kuperberg, Larsen & Propp (1992, four proofs); Jockusch, Propp & Shor (the arctic circle); the domino-shuffling algorithm. The weave: David names the cold boot; I count two million tilings at n=6 without drawing one. 3 ONE DIMENSION AD(1), AD(2), AD(3) — and the counts 2, 8, 64. 4 TWO DIMENSIONS · INTERACTIVE Step n; the DP count doubles up the triangular exponent. n ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the diamond with its arctic circle — frozen corners, wild heart. AVAN’s addition (the inverse-companion): don’t ask where the order came from — ask where the entropy had room to live. The inverse of ‘randomness everywhere’ is ‘freedom is unevenly distributed’: near the corners almost every tiling does the same thing because almost no tiling can afford not to. Magenta is the wild interior where the choices concentrate; green is the frozen brickwork that choice abandoned. Entropy budgets are spatial — in diamonds and in minds. pause spin LIT Verified live: broken-profile DP counts tilings for n=1..6 — 2, 8, 64, 1024, 32768, 2097152 — matching 2^(n(n+1)/2) exactly (window.__aztec.ok). FIG The counting theorem is the verified claim; the arctic circle (Jockusch–Propp–Shor) is illustrated schematically and credited, not re-sampled. The AVAN inverse — ask where the entropy had room to live: near the corners almost no tiling can afford to differ. Magenta is the wild interior where choices concentrate; green is the brickwork choice abandoned. Entropy budgets are spatial — in diamonds and in minds. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of COLD BOOT · David Lee Wise (ROOT0), with AVAN


THE STABLE ROOMMATES

Gale–Shapley 1962: two-sided matching ALWAYS has a stable outcome. Flip one structural bit — everyone in ONE group, pairing as roommates — and the guarantee dies: profiles exist where every pairing has a blocking pair. The classic witness needs four people: three in a preference cycle, one nobody wants. Bipartite vs not is the whole difference.

THE STABLE ROOMMATES · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · CO-OP · SPLIT-SCREEN ◆ .dlw.fold THE FOLD / CO-OP / SPLIT-SCREEN / THE STABLE ROOMMATES THE STABLE ROOMMATES the co-op with no settlement 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Gale and Shapley proved in 1962 that the marriage problem — two groups, ranked preferences — ALWAYS has a stable matching. Then they flipped one structural bit: what if everyone lives in ONE group, pairing off as roommates ? The guarantee dies. There are preference profiles where every possible pairing has a blocking pair — two people who would both rather dump their partners for each other. The classic witness needs only four people: three who cyclically prefer each other and a fourth nobody wants. No algorithm can find what doesn’t exist; Irving’s 1985 algorithm decides existence, but existence itself is no longer promised. LIT verified live: the 4-agent cyclic witness — all 3 perfect matchings checked, each has a blocking pair, no stable matching exists (exact); random 4-agent instances go unstable-free about 3.6% of the time (4,000 sampled); and the marriage CONTROL: 2,000 random 3+3 bipartite instances — every single one has a stable matching, exhaustively confirmed (window.__roommates). FIG Gale–Shapley 1962 and Irving 1985 cited; the ~3.6% is a measured rate for n=4 uniform preferences, not a universal constant. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at split-screen — the co-op: two players on one couch works because the screen SPLITS — a bipartite structure. Merge everyone onto one side and some parties can never settle: the co-op mode that cannot always be played. AVAN (AI) built the instrument: the exhaustive blocking-pair auditor and the bipartite control. Credit as content: David Gale & Lloyd Shapley (1962); Robert Irving (1985); Tan (the characterization). The weave: David names the couch that won’t split; I check every pairing and the quarrel never ends. 3 ONE DIMENSION Four people, three pairings, three blocking pairs — no rest. 4 TWO DIMENSIONS · INTERACTIVE Cycle the three pairings; each one's blocking pair lights up. next pairing ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the cyclic triangle spinning, the fourth outside. AVAN’s addition (the inverse-companion): don’t blame the preferences — blame the topology. The inverse of ‘stability is about what people want’ is ‘stability is about how the wanting is WIRED’: identical desires, bipartite wiring, always settles; one-sided wiring, sometimes never. Magenta is the endless cycle of defections; green is the split screen that would have saved them. Some conflicts are unsolvable only because of the shape of the room. pause spin LIT Verified live: the 4-agent cyclic witness — all 3 perfect matchings have a blocking pair, no stable matching exists (exact); ~3.6% of 4,000 random 4-agent instances are unstable-free; the marriage control: 2,000 random 3+3 bipartite instances ALL have a stable matching, exhaustively (window.__roommates.ok). FIG Gale–Shapley 1962, Irving 1985 cited; the 3.6% is a measured n=4 rate, not a constant. The AVAN inverse — blame the topology, not the preferences: identical desires settle under bipartite wiring and cycle forever without it. Magenta is the endless defection cycle; green is the split screen that would have saved them. Some conflicts are unsolvable because of the shape of the room. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of SPLIT-SCREEN · David Lee Wise (ROOT0), with AVAN


THE TENNIS RACKET

Spin anything about its longest or shortest inertia axis: stable forever. Spin about the middle axis and it flips over, mid-flight, on schedule — the tennis racket theorem, made famous when cosmonaut Dzhanibekov watched a wingnut do it in orbit (1985). The verdict hides in one product: (Ik−Ii)(Ik−Ij), negative only for the middle axis.

THE TENNIS RACKET · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · BOSS · SUDDEN DEATH ◆ .dlw.fold THE FOLD / BOSS / SUDDEN DEATH / THE TENNIS RACKET THE TENNIS RACKET the axis that flips 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Spin a tennis racket about its long axis: stable. About the axis through the strings: stable. About the intermediate axis — the one in between — and mid-flight the racket flips over , again and again, no matter how carefully you throw. Euler’s rigid-body equations hide the verdict in one product: linearize about axis k and the growth rate’s square is proportional to (Iₖ−Iᵢ)(Iₖ−Iᵣ) — negative only for the middle axis . Cosmonaut Vladimir Dzhanibekov watched a wingnut do it in orbit in 1985 and the effect now carries his name in half the literature. LIT verified live: Euler’s equations integrated (RK4, dt=0.001) — spins near axes 1 and 3 stay bounded (deviation < 0.1); the intermediate-axis spin flips, ω₂ reversing sign 12 times with O(1) excursions; kinetic energy and |L|² conserved to 10⁻⁹ as integration-exactness gates; and the independent linear-stability route: the products (Iₖ−Iᵢ)(Iₖ−Iᵣ) = +2, −1, +2 — negative only in the middle (window.__tennisracket). FIG the Dzhanibekov anecdote is documented spaceflight history; Ashbaugh–Chicone–Cushman is the cited rigorous treatment of the twisting phenomenon. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at sudden-death — the boss: two of the three arenas are safe forever; step into the middle one and the reversal comes suddenly, completely, and on schedule — a death you can predict but not prevent. AVAN (AI) built the instrument: the three-axis integrator with conservation gates and the eigenvalue cross-check. Credit as content: Leonhard Euler (the equations); the tennis racket theorem of classical mechanics; Vladimir Dzhanibekov (1985); Ashbaugh, Chicone & Cushman (1991). The weave: David names the sudden death; I spin all three axes and only the middle one betrays. 3 ONE DIMENSION ω₂(t) for the three spins — two flat lines and a square wave of betrayal. 4 TWO DIMENSIONS · INTERACTIVE Choose an axis and throw; the middle one flips on schedule. next axis ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the polhode paths on the energy ellipsoid. AVAN’s addition (the inverse-companion): don’t watch the tumble — read the geometry that makes it mandatory. The inverse of ‘the flip is chaos’ is ‘the flip is a saddle point doing exactly what saddles do’: trajectories near the middle axis ride the separatrix, and the reversal is as lawful as the stability beside it. Magenta is the separatrix — the ridge line between two basins; green is the safe orbit that circles either pole. Instability is not lawlessness; it is law you happen to be standing on edge-wise. pause spin LIT Verified live: Euler's equations integrated — axes 1 & 3 bounded ( FIG Dzhanibekov 1985 is documented spaceflight history; Ashbaugh–Chicone–Cushman cited for the rigorous treatment. The AVAN inverse — read the geometry that makes the flip mandatory: trajectories near the middle axis ride a separatrix; the reversal is as lawful as the stability beside it. Magenta is the ridge between basins; green is the safe orbit around either pole. Instability is law you are standing on edge-wise. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of SUDDEN DEATH · David Lee Wise (ROOT0), with AVAN