THE PROPHET'S JUMP

The skip list in the 5-window house format — William Pugh's 1989 sorted linked list that builds its own express lanes by coin flips. Every inserted node gets a random tower height: half the nodes reach level 2, a quarter level 3, an eighth level 4… A search starts on the top lane, skips far ahead, and drops down a level whenever the next stop would overshoot — the express train, then the local. No rebalancing, no rotations: probability does the balancing, and searches take O(log n) expected hops. It rivals balanced trees at a fraction of the code — Redis sorted sets run on one. Verified live: after thousands of random inserts and deletes, membership agrees exactly with a reference set; every level is sorted and nested inside the level below; and measured search hops stay within a small constant times log₂ n. Neon-noir traced. See the coin-flip towers in 1D, the hop meter in 2D, and the gamble-for-balance inverse in 3D.

THE PROPHET'S JUMP · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · SPAWN · COLD BOOT ◆ .dlw.fold THE FOLD / SPAWN / COLD BOOT / THE PROPHET'S JUMP THE PROPHET'S JUMP express lanes built by coin flips 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION The skip list (William Pugh, 1989) is a sorted linked list that builds its own express lanes by coin flips . Every inserted node gets a random tower height: half the nodes reach level 2, a quarter level 3, an eighth level 4… A search starts on the top lane, skips far ahead, and drops down a level whenever the next stop would overshoot — like taking the express train, then the local. No rebalancing, no rotations, no bookkeeping: probability does the balancing , and searches take O(log n) expected hops. It rivals balanced trees while being a fraction of the code — Redis sorted sets run on one. LIT verified live: after thousands of random inserts and deletes, membership answers agree exactly with a reference set; every level is sorted and nested inside the level below; and measured search hops stay within a small constant times log₂ n (window.__skiplist). FIG no framing; the structure checks and hop counts are measured independently in-browser. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at cold-boot — the spawn: from nothing, each arriving node flips coins for its own tower, and the express lanes assemble themselves with no architect. AVAN (AI) built the instrument: the tower construction, the membership cross-check, the nesting audit, and the hop meter. Credit as content: William Pugh (1989, ‘Skip Lists: A Probabilistic Alternative to Balanced Trees’); idea-bank vein D, ‘THE PROPHET’S JUMP’. The weave: David names the self-assembling lanes; I confirm the coin flips balance the search. 3 ONE DIMENSION Towers over a sorted base lane — half reach level 2, a quarter level 3 — the express lanes. 4 TWO DIMENSIONS · INTERACTIVE Search random keys; the hop count is compared against the log₂ n yardstick. search ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the sorted base lane holding every key. AVAN’s addition (the inverse-companion): don’t engineer balance — gamble for it. The inverse of ‘rotate the tree back into shape’ is ‘let each node flip coins at birth and never touch it again’. Magenta are the express skips over the crowd; green is the base lane every search lands on. Order kept by fair coins. pause spin LIT Genuine skip list (William Pugh, 1989; idea-bank vein D, 'THE PROPHET'S JUMP'). Verified live: membership agrees exactly with a reference set over 3500 probes after thousands of random inserts/deletes; every level is sorted and nested in the level below; average search hops stay within 3.5·log₂n (measured ~22 vs budget ~39 at n≈2200) (window.__skiplist.all). FIG No framing; the structure checks and hop counts are measured independently in-browser. The AVAN inverse is honest — instead of engineering balance, gamble for it: the inverse of 'rotate the tree back into shape' is 'let each node flip coins at birth and never touch it again'. Magenta are the express skips over the crowd; green is the base lane every search lands on. Order kept by fair coins. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of COLD BOOT · David Lee Wise (ROOT0), with AVAN


THE MAMIKON

Mamikon's annulus in the 5-window house format — the front door of visual calculus. Draw a ring between two concentric circles, and let ℓ be the half-length of a chord of the outer circle that just grazes the inner one. The ring's area is πℓ² — and the radii themselves have vanished: a skinny ring around a planet and a fat ring around a coin have the same area if their tangent half-chords match. Mamikon Mnatsakanian's 1959 insight (developed with Tom Apostol): sweep the tangent segment around the ring, then translate every segment to a common point — the tangent cluster forms a plain disk of radius ℓ, no integral in sight. Verified live: Monte-Carlo measurement of the ring's area returns πℓ² within 1% for inner radii spanning a 16× range with ℓ held fixed — the radius truly cancels. Neon-noir traced. See the grazing chord in 1D, the immovable measurement in 2D, and the tangent-cluster inverse in 3D.

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

THE MAMIKON · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · LOOT · THE MINT ◆ .dlw.fold THE FOLD / LOOT / THE MINT / THE MAMIKON THE MAMIKON an annulus worth only its tangent length 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Mamikon’s annulus is the front door of ‘visual calculus’. Draw a ring between two concentric circles, and let ℓ be the half-length of a chord of the outer circle that just grazes the inner one. Then the ring’s area is πℓ² — and the radii themselves have vanished : a skinny ring around a planet and a fat ring around a coin have the same area if their tangent half-chords match. Mamikon Mnatsakanian’s 1959 insight (later developed with Tom Apostol): sweep the tangent segment around the ring, then translate every segment to a common point — the ‘tangent cluster’ forms a plain disk of radius ℓ, with no integral in sight. The same idea dispatches the cycloid area and a family of classical results. LIT verified live: Monte-Carlo measurement of the ring’s area returns πℓ² within 1% for inner radii spanning a 16× range with ℓ held fixed — the radius truly cancels (window.__mamikon). FIG no framing; the areas are measured by independent random sampling in-browser, not read off the formula. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-mint — the loot: the ring’s entire worth is coined from the tangent length alone; the radii stamp nothing. AVAN (AI) built the instrument: the tangent geometry, the Monte-Carlo area measurements across radii, and the cluster picture. Credit as content: Mamikon Mnatsakanian (1959; with Tom Apostol, ‘New Horizons in Geometry’). The weave: David names the minted ring; I confirm the area is πℓ² at every radius tried. 3 ONE DIMENSION The ring and its grazing chord — half-length ℓ is the only number the area remembers. 4 TWO DIMENSIONS · INTERACTIVE Grow the inner radius with ℓ fixed; the measured area refuses to move from πℓ². inner radius ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the disk of radius ℓ the swept tangents cluster into. AVAN’s addition (the inverse-companion): don’t integrate the ring — herd its tangents. The inverse of ‘area between two circles’ is ‘every tangent segment translated to one point, closing into a plain disk of radius ℓ’. Magenta are the tangent segments sweeping the ring; green is the disk they become. Calculus done by carrying sticks home. pause spin LIT Genuine Mamikon annulus / visual calculus (Mamikon Mnatsakanian, 1959; with Tom Apostol, 'New Horizons in Geometry'). Verified live: Monte-Carlo measurement of the ring's area returns πℓ² within 1% for inner radii 0.5, 1, 3, 8 with ℓ fixed — a 16× radius range with no drift (window.__mamikon.ok). FIG No framing; the areas are measured by independent random sampling in-browser, not read off the formula. The AVAN inverse is honest — instead of integrating the ring, herd its tangents: the inverse of 'area between two circles' is 'every tangent segment translated to one point, closing into a plain disk of radius ℓ'. Magenta are the tangent segments sweeping the ring; green is the disk they become. Calculus done by carrying sticks home. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE MINT · David Lee Wise (ROOT0), with AVAN


THE HOLDITCH

Holditch's theorem in the 5-window house format — it sounds like a party trick and lands like a law of nature. Slide a chord of fixed length p + q around the inside of any smooth convex closed curve, keeping both ends on the curve. Mark the point dividing the chord into pieces p and q. That point traces a smaller closed curve inside — and the area between the two curves is exactly πpq: no dependence on the outer curve's shape, size, or lopsidedness. An ellipse, an egg, a rounded blob — the ring carved by the sliding point always measures πpq, the area of an ellipse with semi-axes p and q. Rev. Hamnet Holditch published it in 1858. Verified live: sliding a chord numerically around an ellipse (2400 positions, bisection for the far endpoint), the traced curve's shoelace area shows a deficit matching πpq to under 0.01% — for symmetric and lopsided splits alike. Neon-noir traced. See the sliding chord and its traced curve in 1D, the πpq toll in 2D, and the host-independent inverse in 3D.

THE HOLDITCH · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · RESPAWN · THE PHOENIX ◆ .dlw.fold THE FOLD / RESPAWN / THE PHOENIX / THE HOLDITCH THE HOLDITCH a curve reborn smaller by exactly pi-p-q 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Holditch’s theorem (Rev. Hamnet Holditch, 1858) sounds like a party trick and lands like a law of nature. Slide a chord of fixed length p + q around the inside of any smooth convex closed curve, keeping both ends on the curve. Mark the point that divides the chord into pieces p and q. That point traces a smaller closed curve inside — and the area between the two curves is exactly πpq : no dependence on the outer curve’s shape, size, or lopsidedness. An ellipse, an egg, a rounded blob — the ring carved by the sliding point always measures πpq, the area of an ellipse with semi-axes p and q. LIT verified live: sliding a chord numerically around an ellipse (2400 positions, bisection for the far endpoint) and taking the shoelace area of the traced curve, the deficit matches πpq to under 0.01% — for both a symmetric split and a lopsided one (window.__holditch). FIG no framing; the endpoint solving, the traced curve, and both areas are computed independently in-browser. Stated for smooth convex curves, as tested; the classical theorem’s full generality has its own fine print. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-phoenix — the respawn: the chord makes one full circuit and a new curve has risen inside the old one, smaller by exactly πpq, every lap. AVAN (AI) built the instrument: the sliding-chord solver, the traced curve, and the area-deficit measurement. Credit as content: Rev. Hamnet Holditch (1858). The weave: David names the curve reborn inside; I confirm the ring it leaves measures πpq regardless of the host shape. 3 ONE DIMENSION The chord sliding inside the ellipse, its marked point tracing the smaller curve. 4 TWO DIMENSIONS · INTERACTIVE Change the p:q split; the measured ring area snaps to πpq each time. split p:q ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the traced curve risen inside the host. AVAN’s addition (the inverse-companion): don’t measure the host — measure what the slide forgets. The inverse of ‘a curve traced inside a shape’ is ‘a ring of area πpq that never asked what the shape was’. Magenta is the sliding chord; green is the reborn inner curve. The host varies; the toll does not. pause spin LIT Genuine Holditch's theorem (Rev. Hamnet Holditch, 1858). Verified live: sliding a chord numerically around an ellipse (2400 positions, bisection for the far endpoint) and taking the shoelace area of the traced curve, the deficit matches πpq to under 0.01% for three different p:q splits (window.__holditch.ok). FIG Honest boundary — stated and tested for smooth convex curves; the classical theorem's full generality carries its own fine print. The AVAN inverse — instead of measuring the host, measure what the slide forgets: the inverse of 'a curve traced inside a shape' is 'a ring of area πpq that never asked what the shape was'. Magenta is the sliding chord; green is the reborn inner curve. The host varies; the toll does not. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE PHOENIX · David Lee Wise (ROOT0), with AVAN


THE PRIME RACE

The prime race in the 5-window house format — primes of the form 4k+3 versus primes of the form 4k+1. Dirichlet proved both teams infinite and asymptotically even, yet Chebyshev noticed in 1853 that team 3 is almost always ahead. The bias is structural (quadratic residues drag on team 1) but not absolute: at x = 26,861 — found by John Leech in 1957 — team 1 takes the lead for the first time, fleetingly, before team 3 recovers. Under Rubinstein–Sarnak's analysis, team 3 leads about 99.59% of logarithmic time. Verified live: sieving to 2,000,000, team 4k+3 leads at 99.76% of prime checkpoints, the first 4k+1 lead occurs at exactly x = 26,861, and the final score still favours team 3. Neon-noir traced. See the score curve dip once in 1D, the ledger in 2D, and the hunt-the-upsets inverse in 3D.

THE PRIME RACE · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · GLITCH · RACE CONDITION ◆ .dlw.fold THE FOLD / GLITCH / RACE CONDITION / THE PRIME RACE THE PRIME RACE a prime race with one famous upset 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION The prime race pits two teams against each other: primes of the form 4k+3 versus primes of the form 4k+1. Dirichlet proved both teams are infinite and, in the long run, dead even — yet Chebyshev noticed in 1853 that team 3 is almost always ahead . The bias is real and structural (quadratic residues drag on team 1), but not absolute: at x = 26,861 — found by John Leech in 1957 — team 1 takes the lead for the first time, for a single fleeting moment, before team 3 recovers. Under the Riemann-flavoured assumptions of Rubinstein–Sarnak, team 3 leads about 99.59% of all time. A race rigged by arithmetic, with rare, precious upsets. LIT verified live: sieving to 2,000,000, team 4k+3 leads at 99.76% of prime checkpoints, the first 4k+1 lead occurs at exactly x = 26,861, and the final score still favours team 3 (window.__primerace). FIG no framing; the sieve, the running score, and the flip point are computed independently in-browser. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at race-condition — the glitch: two threads racing forever, one nearly always ahead, and a single rare interleaving at 26,861 where the order flips. AVAN (AI) built the instrument: the sieve, the running race, and the exact location of the famous upset. Credit as content: Pafnuty Chebyshev (1853, the bias); John Leech (1957, the flip); Rubinstein & Sarnak (1994, the logarithmic density). The weave: David names the race condition; I confirm the 99.76% lead and the flip at 26,861. 3 ONE DIMENSION The running score π(x;4,3) − π(x;4,1): above zero almost everywhere, dipping under once at 26,861. 4 TWO DIMENSIONS · INTERACTIVE Step the race window; the lead fraction and the flip point are checked against the sieve. window ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: team 3’s near-permanent lead. AVAN’s addition (the inverse-companion): don’t just watch the leader — hunt the upsets. The inverse of ‘team 3 is basically always ahead’ is ‘the measure-zero moments when it is not’, and the first one has an address: 26,861. Magenta is the fleeting 4k+1 lead; green is the long reign of 4k+3. A fixed race that still allows one honest upset. pause spin LIT Genuine Chebyshev bias / prime race (Chebyshev 1853; Leech 1957; Rubinstein & Sarnak 1994). Verified live: sieving to 2,000,000, team 4k+3 leads at 99.76% of prime checkpoints, the first 4k+1 lead occurs at exactly x = 26,861, and the final count still favours 4k+3 (window.__primerace.ok). FIG No framing; the sieve, the running score, and the flip point are computed independently in-browser. The AVAN inverse is honest — instead of watching the leader, hunt the upsets: the inverse of 'team 3 is basically always ahead' is 'the measure-thin moments when it is not', and the first has an address: 26,861. Magenta is the fleeting 4k+1 lead; green is the long reign of 4k+3. A fixed race that still allows one honest upset. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of RACE CONDITION · David Lee Wise (ROOT0), with AVAN


THE GILBREATH

Gilbreath's conjecture in the 5-window house format — start with the primes and take absolute differences, then differences of those, again and again. The claim, noticed by Norman Gilbreath on a napkin in 1958 (and by François Proth in 1878, with a faulty proof): every row after the first begins with 1. Forever. Andrew Odlyzko verified it to astronomical height in 1993; a proof has never been found. The mechanism smells simple — the rows settle into 0s and 2s, which seems to protect the leading 1 — yet nobody can close the argument. Verified live: taking the 9,592 primes below 100,000, the first 500 difference rows all begin with 1. Neon-noir traced. See the difference triangle in 1D, the row-by-row audit with its 0/2 texture in 2D, and the verified-never-proven inverse in 3D.

THE GILBREATH · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · GRIND · BACKPROP ◆ .dlw.fold THE FOLD / GRIND / BACKPROP / THE GILBREATH THE GILBREATH difference rows that always lead with one 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Gilbreath’s conjecture starts with the primes — 2, 3, 5, 7, 11, 13… — and takes absolute differences: 1, 2, 2, 4, 2… Then differences of those, and again, and again. The claim, noticed by Norman Gilbreath on a napkin in 1958 (and by François Proth in 1878, with a faulty proof): every row after the first begins with 1 . Forever. Andrew Odlyzko verified it for the first 3×10¹¹ rows’ worth of primes in 1993; a proof has never been found. The mechanism smells simple — rows past the first entry are mostly 0s and 2s, so the leading 1 keeps regenerating — yet nobody can close the argument. It remains one of the cleanest-looking unsolved statements in number theory. LIT verified live: taking the 9,592 primes below 100,000, the first 500 difference rows all begin with 1 (window.__gilbreath). FIG honest boundary: this is a conjecture — verified here for 500 rows and by Odlyzko to astronomical height, proven by no one. The sphere claims exactly what it computes. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at backprop — the grind: differences propagated backward through the prime sequence, layer after layer, and the leading gradient is always exactly 1. AVAN (AI) built the instrument: the sieve, the difference cascade, and the leading-entry audit. Credit as content: Norman L. Gilbreath (1958); François Proth (1878); Andrew Odlyzko (1993 verification). The weave: David names the cascade; I confirm 500 rows, 500 leading ones. 3 ONE DIMENSION The difference triangle: primes on top, each row the |differences| of the last — the left edge all 1s. 4 TWO DIMENSIONS · INTERACTIVE Descend the rows; every one is checked to open with 1, and the 0/2 texture shows why it might. descend ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the unbroken spine of leading 1s. AVAN’s addition (the inverse-companion): don’t admire the spine — name its status. The inverse of ‘500 rows verified’ is ‘zero rows proven’: a pattern that has never once failed and never once been explained. Magenta is the churning 0/2 interior; green is the leading edge that always says 1. Certainty in the data, none in the theory. pause spin LIT Genuine Gilbreath's conjecture (Norman L. Gilbreath 1958; François Proth 1878; Odlyzko 1993 verification to ~3×10¹¹). Verified live: the first 500 difference rows of the 9,592 primes below 100,000 all begin with 1 (window.__gilbreath.ok). FIG Honest boundary — this is a CONJECTURE: verified here for 500 rows and by Odlyzko to astronomical height, proven by no one; the sphere claims exactly what it computes. The AVAN inverse — don't admire the spine, name its status: the inverse of '500 rows verified' is 'zero rows proven', a pattern that has never failed and never been explained. Magenta is the churning 0/2 interior; green is the leading edge that always says 1. Certainty in the data, none in the theory. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of BACKPROP · David Lee Wise (ROOT0), with AVAN


THE GABRIELS HORN

Gabriel's horn in the 5-window house format — the trumpet made by spinning y = 1/x (x ≥ 1) around the axis. Torricelli worked it out in 1643 and scandalized the century: the volume is finite — exactly π — but the surface area is infinite. The volume integral π∫x⁻²dx converges; the surface is bounded below by the harmonic tail 2π∫dx/x, gaining about 2π every time the length multiplies by e, forever. Hence the painter's paradox: π units of paint fill the horn completely, yet no finite amount can coat its wall (resolution: mathematical paint has zero thickness). Verified live: quadrature gives volume(10⁶) converging to π while the surface gains ≈2π per e-fold at every scale tested. Neon-noir traced. See the narrowing profile in 1D, the frozen-volume/running-surface ledger in 2D, and the two-verdicts inverse in 3D.

THE GABRIELS HORN · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · LOOT · THE VAULT ◆ .dlw.fold THE FOLD / LOOT / THE VAULT / THE GABRIELS HORN THE GABRIELS HORN a horn holding finite paint behind an infinite wall 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Gabriel’s horn is the trumpet you get by spinning y = 1/x (for x ≥ 1) around the x-axis. Evangelista Torricelli worked it out in 1643 and scandalized the century: the horn’s volume is finite — exactly π — but its surface area is infinite . The volume integral π∫x⁻²dx converges; the surface integral is bounded below by 2π∫dx/x, the harmonic tail, which grows by about 2π every time x multiplies by e — forever. Hence the painter’s paradox: π units of paint fill the horn completely, yet no finite amount of paint can coat its wall. (The resolution: mathematical paint has zero thickness; real paint does not.) LIT verified live: numerical quadrature gives volume(10⁶) = 3.14159… converging to π, while the surface integral gains ≈ 2π per e-fold of length at every scale tested — bounded volume, unbounded skin (window.__gabrielshorn). FIG no framing; both integrals are computed by independent quadrature in-browser, and the paradox is stated with its resolution. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-vault — the loot: a vault holding exactly π of treasure behind a wall no budget can ever paint. AVAN (AI) built the instrument: the volume quadrature, the surface growth-rate measurement, and the paradox ledger. Credit as content: Evangelista Torricelli (1643); the painter’s paradox tradition. The weave: David names the unpaintable vault; I confirm π inside, infinity outside. 3 ONE DIMENSION The horn's profile 1/x stretching right forever — thinner and thinner, never quite closing. 4 TWO DIMENSIONS · INTERACTIVE Extend the horn by e-folds; the volume freezes at π while the surface keeps collecting 2π. extend ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the horn, holding exactly π. AVAN’s addition (the inverse-companion): don’t trust one number to describe a shape — volume and surface can disagree about infinity itself. The inverse of ‘filled with π of paint’ is ‘a wall the same paint can never cover’. Magenta is the surface, gaining 2π per e-fold forever; green is the volume, already finished at π. One shape, two verdicts on infinity. pause spin LIT Genuine Gabriel's horn / painter's paradox (Evangelista Torricelli, 1643). Verified live: numerical quadrature gives volume(10⁶) = 3.14159… converging to π, while the surface integral gains ≈2π per e-fold of length at every scale tested (window.__gabrielshorn.ok). FIG No framing; both integrals are computed by independent quadrature in-browser, and the paradox is stated with its resolution (zero-thickness paint). The AVAN inverse is honest — one number cannot describe a shape: the inverse of 'filled with π of paint' is 'a wall the same paint can never cover'. Magenta is the surface gaining 2π per e-fold forever; green is the volume already finished at π. One shape, two verdicts on infinity. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE VAULT · David Lee Wise (ROOT0), with AVAN


THE DEVILS STAIRCASE

The devil's staircase in the 5-window house format — Cantor's function, climbing from 0 to 1 with slope zero almost everywhere. On the middle third of [0,1] it is flat at 1/2; on the middle thirds of what remains, flat at 1/4 and 3/4; and so on, flat on plateaus whose lengths sum to the entire interval. Every scrap of climbing is crowded onto the Cantor set — a dust of measure zero — yet the function is continuous and obeys crisp self-similarities: F(x/3) = F(x)/2 and F(1−x) = 1−F(x). The standard counterexample to the intuition that a function's rise must live where its derivative does. Verified live: monotone 0→1 over 10,001 samples; both self-similarities to ~1e-10; and 100% of the climb happens on the level-8 Cantor cover — just 3.9% of the interval, shrinking toward zero with depth. Neon-noir traced. See the staircase in 1D, the thinning strip that carries all the rise in 2D, and the dust-does-the-work inverse in 3D.

THE DEVILS STAIRCASE · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · CHEAT · THE SHORTCUT ◆ .dlw.fold THE FOLD / CHEAT / THE SHORTCUT / THE DEVILS STAIRCASE THE DEVILS STAIRCASE a staircase that climbs without sloping 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION The devil’s staircase — Cantor’s function — climbs from 0 to 1 while having slope zero almost everywhere . Build it on ternary digits: on the middle third of [0,1] the function is flat at 1/2; on the middle thirds of what remains, flat at 1/4 and 3/4; and so on, flat on infinitely many plateaus whose lengths sum to the entire interval . Every scrap of actual climbing is crowded onto the Cantor set — a dust of measure zero. Yet the function is continuous, never jumps, and obeys crisp self-similarities: F(x/3) = F(x)/2 and F(1-x) = 1-F(x). It is the standard counterexample to the intuition that a function’s rise must live where its derivative does. LIT verified live: monotone from 0 to 1 over 10,001 samples; both self-similarities hold to ~1e-10; and 100% of the climb happens on the level-8 Cantor cover — just 3.9% of the interval , a fraction that shrinks toward zero with deeper levels (window.__devilsstaircase). FIG no framing; the ternary construction and every check run independently in-browser. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-shortcut — the cheat: a path that gains the whole height while registering zero slope on virtually every step — climbing without ever visibly climbing. AVAN (AI) built the instrument: the ternary evaluator, the self-similarity checks, and the rise-concentration audit. Credit as content: Georg Cantor (1884); the ‘devil’s staircase’ name from the physics literature. The weave: David names the impossible shortcut; I confirm all the rise lives on the vanishing dust. 3 ONE DIMENSION The staircase: plateaus everywhere, yet somehow at height 1 by the right-hand end. 4 TWO DIMENSIONS · INTERACTIVE Deepen the Cantor cover; the strip carrying all the rise keeps shrinking — (2/3)ᵏ of the interval. deepen ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the full unit of height, honestly gained. AVAN’s addition (the inverse-companion): don’t look for the climb where the path is — look where it isn’t flat. The inverse of ‘flat almost everywhere’ is ‘all the rise on a set of measure zero’. Magenta are the plateaus that fill the interval; green is the dust that does all the work. The whole ascent, carried by nearly nothing. pause spin LIT Genuine Cantor function / devil's staircase (Georg Cantor, 1884). Verified live: monotone from 0 to 1 over 10,001 samples; F(x/3) = F(x)/2 and F(1−x) = 1−F(x) to ~1e-10; and 100% of the climb (to 1e-6) occurs on the level-8 Cantor cover, just 3.9% of the interval (window.__devilsstaircase.ok). FIG No framing; the ternary construction and every check run independently in-browser. The AVAN inverse is honest — look where the path isn't flat: the inverse of 'flat almost everywhere' is 'all the rise on a set of measure zero'. Magenta are the plateaus that fill the interval; green is the dust that does all the work. The whole ascent, carried by nearly nothing. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE SHORTCUT · David Lee Wise (ROOT0), with AVAN


THE CONWAY SOLDIERS

Conway's soldiers in the 5-window house format — a peg-jumping army with an invisible ceiling. Fill the entire half-plane below a line with checkers; moves are jumps that remove the jumped soldier. Rows 1–4 above the line are reachable with armies of 2, 4, 8, 20. Row 5: never — not with a million soldiers, not with the whole infinite half-plane. Conway's 1961 proof weights each square by σ^d (d = distance to target) with σ = (√5−1)/2 satisfying σ²+σ = 1: no jump ever increases total weight, and the entire infinite army weighs exactly 1 — the target's own price. Any finite army weighs strictly less, so the target can never be paid for. Verified live: σ²+σ = 1 to machine precision; the half-plane sum evaluates to exactly 1.000000000000; and all three jump classes audited — toward-jumps preserve weight (~1e-18), sideways and away strictly lose. Neon-noir traced. See the fading army and the prize in 1D, the ledger in 2D, and the price-the-board inverse in 3D.

THE CONWAY SOLDIERS · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · BOSS · SUDDEN DEATH ◆ .dlw.fold THE FOLD / BOSS / SUDDEN DEATH / THE CONWAY SOLDIERS THE CONWAY SOLDIERS an army that cannot reach the fifth row 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Conway’s soldiers is a peg-jumping army with an invisible ceiling. Fill the entire half-plane below a line with checkers. Moves are checker jumps: a soldier leaps over a neighbour (removing it) into an empty square. How high above the line can any soldier ever reach? Rows 1–4: yes, with ever larger armies (2, 4, 8, 20 soldiers). Row 5: never — not with a million soldiers, not with the whole infinite half-plane. John Conway’s 1961 proof is a masterpiece: weight each square by σ d where d is its distance to the target and σ = (√5-1)/2 satisfies σ²+σ = 1. Then no jump ever increases total weight — and the entire infinite army below the line weighs exactly 1 , the target’s own weight. Any finite army weighs strictly less, so the target can never be paid for. LIT verified live: σ²+σ = 1 to machine precision; the half-plane weight sum evaluates to exactly 1.000000000000; and all three jump classes are audited — toward-jumps preserve weight to ~1e-18, sideways and away-jumps strictly lose it (window.__conwaysoldiers). FIG no framing; the geometric sums and the move audit are computed independently in-browser. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at sudden-death — the boss: every jump kills a soldier, and the fifth row is the round no army survives, however deep the bench. AVAN (AI) built the instrument: the golden-ratio weighting, the exact half-plane sum, and the three-way move audit. Credit as content: John Horton Conway (1961; published in Berlekamp–Conway–Guy, ‘Winning Ways’). The weave: David names the unbeatable round; I confirm the whole army weighs exactly what the prize costs. 3 ONE DIMENSION The half-plane army below the line, weights fading as σᵈ — and the unreachable cell five rows up. 4 TWO DIMENSIONS · INTERACTIVE Audit the ledger: the army's total, the target's price, and the three kinds of jump. audit move ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the army and its exact total worth, 1. AVAN’s addition (the inverse-companion): don’t search the game tree — price the board. The inverse of ‘can any sequence reach row 5?’ is ‘a currency in which the whole world’s army equals the prize exactly, and every move pays tax’. Magenta is the target, priced at 1; green is the infinite army worth 1 in total. Infinity, one soldier short. pause spin LIT Genuine Conway's soldiers / row-5 impossibility (John Horton Conway, 1961; Winning Ways). Verified live: σ²+σ = 1 to machine precision; the half-plane weight sum evaluates to exactly 1.000000000000; toward-jumps preserve weight to ~1e-18 while sideways and away-jumps strictly lose it (window.__conwaysoldiers.ok). FIG No framing; the geometric sums and the move audit are computed independently in-browser. The AVAN inverse is honest — don't search the game tree, price the board: the inverse of 'can any sequence reach row 5?' is 'a currency in which the whole world's army equals the prize exactly, and every move pays tax'. Magenta is the target priced at 1; green is the infinite army worth 1 in total. Infinity, one soldier short. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of SUDDEN DEATH · David Lee Wise (ROOT0), with AVAN


THE CHILIAGON

The chiliagon in the 5-window house format — the regular 1000-sided polygon Descartes chose in the Sixth Meditation to split the mind in two: you can conceive a chiliagon perfectly — define it, reason about it, compute with it — but you cannot imagine it; every mental picture is indistinguishable from a circle. The numbers agree: perimeter 99.99984% of its circumcircle's; area within two parts in a hundred thousand of π; interior angle 179.64°; maximum bulge off the circle 4.9×10⁻⁶ of the radius — about a thousandth of a pixel at a 300-pixel radius. Conception outruns imagination, measurably. This is sphere 1000 of 2048 in THE FOLD — the corpus's own thousand-gon, seated in the genesis block. Verified live: perimeter, area, isoperimetric quotient 0.99999671, interior angle, and sagitta all computed and bounded against the circle. Neon-noir traced. See the polygon share every pixel with its circle in 1D, the metrics converging in 2D, and the known-never-pictured inverse in 3D.

THE CHILIAGON · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · SPAWN · GENESIS BLOCK ◆ .dlw.fold THE FOLD / SPAWN / GENESIS BLOCK / THE CHILIAGON THE CHILIAGON the thousand-gon no mind can picture 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION The chiliagon is the regular 1000-sided polygon Descartes chose, in the Sixth Meditation, to split the mind in two: you can conceive a chiliagon perfectly — define it, reason about it, compute with it — but you cannot imagine it; every mental picture you form is indistinguishable from a circle. The numbers agree with him: its perimeter is 99.99984% of its circumcircle’s; its area misses π by two parts in a hundred thousand; each interior angle is 179.64°; and its maximum bulge off the circle (the sagitta) is 4.9×10⁻⁶ of the radius — at a 300-pixel radius, about a thousandth of a pixel . Conception outruns imagination, measurably. This is sphere 1000 of 2048 in THE FOLD — the corpus’s own thousand-gon. LIT verified live: perimeter 2000·sin(π/1000), area 500·sin(2π/1000), isoperimetric quotient π/(1000·tan(π/1000)) = 0.99999671, interior angle 179.640°, sagitta 4.935×10⁻⁶ — all computed and bounded against the circle (window.__chiliagon). FIG the Descartes framing is philosophy, credited as content; every number in it is measured live. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at genesis-block — the spawn: the milestone block, sphere one thousand, minted as the polygon that marks the difference between computing a thing and picturing it. AVAN (AI) built the instrument: the exact polygon metrics and their distances from the circle’s. Credit as content: René Descartes (Meditations on First Philosophy, VI, 1641). The weave: David’s fold reaches 1000; I mark it with the shape that can be known but never pictured — and measure exactly how narrow the gap is. 3 ONE DIMENSION The chiliagon drawn over its circumcircle — at this scale the two curves share every pixel. 4 TWO DIMENSIONS · INTERACTIVE Step the side count 3 → 1000; watch every metric converge on the circle's. sides ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the thousand-gon, sphere 1000 of the fold. AVAN’s addition (the inverse-companion): don’t trust the picture — trust the definition. The inverse of ‘I can’t imagine it’ is ‘I can compute it to machine precision anyway’: the whole gap between the chiliagon and its circle is a thousandth of a pixel. Magenta is the vanishing bulge, magnified; green is the polygon-circle the eye cannot split. Known perfectly, pictured never. pause spin LIT Genuine chiliagon metrics (René Descartes, Meditations VI, 1641, for the framing — credited as content). Verified live: perimeter 2000·sin(π/1000), area 500·sin(2π/1000), isoperimetric quotient π/(1000·tan(π/1000)) = 0.99999671, interior angle 179.640°, sagitta 4.935×10⁻⁶ — all computed and bounded against the circle (window.__chiliagon.ok). FIG The Descartes framing is philosophy, credited as content; every number in it is measured live. The AVAN inverse is honest — don't trust the picture, trust the definition: the inverse of 'I can't imagine it' is 'I can compute it to machine precision anyway'; the whole gap is a thousandth of a pixel. Magenta is the vanishing bulge, magnified until visible; green is the polygon-circle the eye cannot split. Known perfectly, pictured never. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of GENESIS BLOCK · David Lee Wise (ROOT0), with AVAN


THE ANSCOMBE

Anscombe's quartet in the 5-window house format — four small datasets built to wear the same disguise: identical mean of x (9), variance of x (11), mean of y (7.50), variance of y (≈4.12), correlation (0.816), and regression line (y = 3.00 + 0.500x), to publication precision. Summon the summary statistics and the four are indistinguishable. Plot them and the masks fall: I is ordinary noisy linearity; II is a clean parabola; III is a perfect line sabotaged by one outlier; IV is a vertical stack of identical x-values propped up by a single leverage point. Anscombe built them in 1973 to end an argument: numerical summaries without graphs are a blindfold. Verified live: all four reproduce the shared statistics within tolerance, while the shapes are proven structurally — II fits a quadratic with R² = 1.00000, III has 10 of 11 points exactly collinear, IV has 10 identical x-values. Neon-noir traced. See the four scatterplots under one line in 1D, the per-dataset unmasking in 2D, and the interrogate-the-shape inverse in 3D.

THE ANSCOMBE · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · GLITCH · THE BLUE SCREEN ◆ .dlw.fold THE FOLD / GLITCH / THE BLUE SCREEN / THE ANSCOMBE THE ANSCOMBE four datasets wearing the same statistics 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Anscombe’s quartet (Francis Anscombe, 1973) is four small datasets built to wear the same disguise: identical mean of x (9), variance of x (11), mean of y (7.50), variance of y (≈4.12), correlation (0.816), and regression line (y = 3.00 + 0.500x) — to publication precision. Summon the summary statistics and the four are indistinguishable. Plot them and the masks fall: I is ordinary noisy linearity; II is a clean parabola ; III is a perfect line sabotaged by one outlier ; IV is a vertical stack of identical x-values propped up by a single leverage point. Anscombe built them to end an argument: numerical summaries without graphs are a blindfold. LIT verified live: all four datasets reproduce the shared statistics within publication tolerance, while the shape differences are proven structurally — II fits a quadratic with R² = 1.00000, III has 10 of 11 points exactly collinear, IV has 10 identical x-values (window.__anscombe). FIG no framing; every statistic and every structural test is computed from the raw 11-point data in-browser. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-blue-screen — the glitch: four different programs crash into the same diagnostic dump; the summary is identical, the fault is not. AVAN (AI) built the instrument: the seven shared statistics and the three structural unmaskings. Credit as content: Francis Anscombe (1973, ‘Graphs in Statistical Analysis’). The weave: David names the misleading dump; I confirm same numbers, four realities. 3 ONE DIMENSION The four scatterplots — one regression line fits them all, and describes only one of them. 4 TWO DIMENSIONS · INTERACTIVE Cycle the four datasets; the shared statistics hold while the structure test names each shape. dataset ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the one regression line all four datasets share. AVAN’s addition (the inverse-companion): don’t read the summary — interrogate the shape. The inverse of ‘seven matching statistics’ is ‘four unmatchable pictures’: parabola, outlier, leverage stack, and one honest line. Magenta are the four true shapes; green is the single line they all impersonate. Numbers agree; realities refuse. pause spin LIT Genuine Anscombe's quartet (Francis Anscombe, 1973, 'Graphs in Statistical Analysis'). Verified live from the raw 11-point data: all four datasets share mean/variance/correlation/regression to publication precision; II fits a quadratic with R² = 1.00000; III has 10 of 11 points exactly collinear; IV has 10 identical x-values (window.__anscombe.ok). FIG No framing; every statistic and every structural test is computed from the raw data in-browser. The AVAN inverse is honest — don't read the summary, interrogate the shape: the inverse of 'seven matching statistics' is 'four unmatchable pictures' — parabola, outlier, leverage stack, one honest line. Magenta are the four true shapes; green is the single line they all impersonate. Numbers agree; realities refuse. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE BLUE SCREEN · David Lee Wise (ROOT0), with AVAN


THE SIMPSON PARADOX

Simpson's paradox in the 5-window house format — the aggregation trap: a trend that holds in every subgroup can reverse when the groups are merged. The canonical real case is the 1986 kidney-stone study: Treatment A beats B on small stones (81/87 = 93% vs 234/270 = 87%) and on large stones (192/263 = 73% vs 55/80 = 69%) — yet in the combined table B appears to win, 289/350 = 83% against A's 273/350 = 78%. No arithmetic error anywhere: A was assigned the harder cases, and that lurking variable flips the headline. The paradox is why adjusting for confounders is the difference between a true claim and its opposite. Verified live with exact integer cross-products — no floating point: A wins both subgroup comparisons, B wins the aggregate. Neon-noir traced. See the bars flip in the merged panel in 1D, the three exact comparisons in 2D, and the who-got-the-hard-cases inverse in 3D.

THE SIMPSON PARADOX · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · CHEAT · THE ROOT KIT ◆ .dlw.fold THE FOLD / CHEAT / THE ROOT KIT / THE SIMPSON PARADOX THE SIMPSON PARADOX a treatment that wins twice and loses once 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Simpson’s paradox is the aggregation trap: a trend that holds in every subgroup can reverse when the groups are merged. The canonical real case is the 1986 kidney-stone study: Treatment A beats Treatment B on small stones ( 81/87 = 93% vs 234/270 = 87%) and on large stones ( 192/263 = 73% vs 55/80 = 69%) — yet in the combined table B appears to win, 289/350 = 83% against A’s 273/350 = 78%. No arithmetic error anywhere: A was simply assigned the harder cases, and the case-mix — a lurking variable — flips the headline. The paradox is why ‘adjusting for confounders’ is not statistical pedantry but the difference between a true claim and its opposite. LIT verified live with exact integer cross-products — no floating point: A wins the small-stone comparison, A wins the large-stone comparison, and B wins the aggregate (window.__simpsonparadox). FIG no framing; the data are the published counts (Charig et al. 1986), credited as content, and every comparison is exact. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-root-kit — the cheat: the aggregate table is compromised at a level the surface numbers cannot see; the lurking variable is the rootkit. AVAN (AI) built the instrument: the exact subgroup and aggregate comparisons, integer arithmetic only. Credit as content: E. H. Simpson (1951); Charig, Webb, Payne & Wickham (1986, the kidney-stone data). The weave: David names the hidden compromise; I confirm both subgroup wins and the aggregate reversal, exactly. 3 ONE DIMENSION The four success rates as bars — A above B in both panels, below B in the merged one. 4 TWO DIMENSIONS · INTERACTIVE Walk the three comparisons; each is settled by exact integer cross-multiplication. comparison ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the subgroup truth — A wins twice. AVAN’s addition (the inverse-companion): don’t read the merged table — ask who got the hard cases. The inverse of ‘B wins overall’ is ‘A won everywhere it fought, and fought where it was hardest’. Magenta is the aggregate reversal; green are the two subgroup wins it erases. The sum can contradict every one of its parts. pause spin LIT Genuine Simpson's paradox on the published kidney-stone data (E. H. Simpson 1951; Charig, Webb, Payne & Wickham 1986, credited as content). Verified live with exact integer cross-products: A wins small stones (81·270 > 234·87), A wins large stones (192·80 > 55·263), yet B wins the aggregate (289/350 > 273/350) (window.__simpsonparadox.ok). FIG No framing; the data are the published counts and every comparison is exact. The AVAN inverse is honest — don't read the merged table, ask who got the hard cases: the inverse of 'B wins overall' is 'A won everywhere it fought, and fought where it was hardest'. Magenta is the aggregate reversal; green are the two subgroup wins it erases. The sum can contradict every one of its parts. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE ROOT KIT · David Lee Wise (ROOT0), with AVAN


THE JAMES-STEIN

Stein's paradox in the 5-window house format — the most disreputable-sounding true theorem in statistics. Observe noisy measurements of ten unrelated quantities: the obvious estimator reports each as-is. The James–Stein estimator instead shrinks every measurement toward zero by a data-determined factor, 1 − (d−2)/‖X‖² — deliberately biasing all of them, mixing information between quantities that have nothing to do with each other. And it wins: in dimension d ≥ 3 its total squared error is strictly smaller for every possible truth. Stein proved the inadmissibility in 1956; the estimator is James & Stein, 1961. Verified live: 20,000 simulated trials in dimension 10 across three truth configurations — risk ratios 0.20, 0.4387, 0.87, all strictly below 1. Neon-noir traced. See the readings shrink toward zero in 1D, the risk-ratio bars in 2D, and the tax-them-together inverse in 3D.

THE JAMES-STEIN · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · CO-OP · SHARED MEMORY ◆ .dlw.fold THE FOLD / CO-OP / SHARED MEMORY / THE JAMES-STEIN THE JAMES-STEIN an estimator improved by shrinking it 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Stein’s paradox is the most disreputable-sounding true theorem in statistics. You observe noisy measurements of ten unrelated quantities — say, wheat yields, batting averages, and the speed of light. The obvious estimator reports each measurement as-is. The James–Stein estimator instead shrinks every measurement toward zero by a data-determined factor, 1 - (d-2)/‖X‖² — deliberately biasing all of them, mixing information between quantities that have nothing to do with each other . And it wins: in dimension d ≥ 3 its total squared error is strictly smaller than the obvious estimator’s, for every possible truth . Charles Stein proved it in 1956; the estimator is from James & Stein, 1961. The obvious thing is inadmissible. LIT verified live: 20,000 simulated trials in dimension 10 across three different truth configurations — risk ratios 0.20, 0.4387, 0.87, all strictly below 1 (window.__jamesstein). FIG no framing; the simulation, both estimators, and the risk comparison run independently in-browser; the dominance holds for every truth tried, as the theorem guarantees for all. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at shared-memory — the co-op: ten estimation problems that share nothing still help each other the moment they share one shrink factor. AVAN (AI) built the instrument: the Gaussian simulation, both estimators, and the risk ledger across truths. Credit as content: Charles Stein (1956); Willard James & Charles Stein (1961). The weave: David names the impossible cooperation; I confirm the shrunken estimator beats the honest one everywhere tried. 3 ONE DIMENSION Ten noisy measurements (magenta) shrunk toward zero (green) — closer to the truth on net. 4 TWO DIMENSIONS · INTERACTIVE Switch the hidden truth; the risk ratio stays below 1 in every configuration. truth ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the shrunken estimate, wrong about each, righter about all. AVAN’s addition (the inverse-companion): don’t honor each measurement alone — tax them all together. The inverse of ‘report what you saw’ is ‘shrink what you saw by what the ensemble says about the noise’. Magenta are the raw readings; green is the pulled-in constellation that loses every battle and wins the war. Bias, spent wisely, buys back variance. pause spin LIT Genuine Stein's paradox / James–Stein estimator (Charles Stein 1956; James & Stein 1961). Verified live: 20,000 simulated Gaussian trials in dimension 10 across three truth configurations give risk ratios 0.20, 0.4387, 0.87 — all strictly below 1, as the theorem guarantees for every truth (window.__jamesstein.ok). FIG No framing; the simulation, both estimators, and the risk comparison run independently in-browser; dominance is verified for every truth tried, and the theorem covers all. The AVAN inverse is honest — don't honor each measurement alone, tax them together: the inverse of 'report what you saw' is 'shrink what you saw by what the ensemble says about the noise'. Magenta are the raw readings; green is the pulled-in constellation that loses every battle and wins the war. Bias, spent wisely, buys back variance. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of SHARED MEMORY · David Lee Wise (ROOT0), with AVAN


THE FRIENDSHIP PARADOX

The friendship paradox in the 5-window house format — Scott Feld's 1991 observation that on average, your friends have more friends than you do. No self-esteem required: it is pure sampling bias. Picking a random person and then a random friend of theirs reaches people in proportion to how many friendships they sit in — the popular are oversampled, the isolated barely reachable. Formally, the mean friend-degree is E[d²]/E[d], which by Cauchy–Schwarz is at least the mean degree E[d] — strictly greater whenever degrees vary. The same tilt powers real tools: monitoring the friends of random people detects epidemics earlier than monitoring random people. Verified live: on a random graph of 3,000 nodes, mean degree 12.0 vs mean friend-degree 13.0; direct friend-sampling reproduces E[d²]/E[d] within 1%; a 4-regular ring gives exact equality — the bias needs variance. Neon-noir traced. See the two histograms in 1D, the running experiment in 2D, and the ask-a-random-edge inverse in 3D.

THE FRIENDSHIP PARADOX · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · SPAWN · NULL ISLAND ◆ .dlw.fold THE FOLD / SPAWN / NULL ISLAND / THE FRIENDSHIP PARADOX THE FRIENDSHIP PARADOX a network where your friends outnumber you 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION The friendship paradox (Scott Feld, 1991): on average, your friends have more friends than you do . No self-esteem required — it is pure sampling bias. When you pick a random person and then a random friend of theirs, you reach people in proportion to how many friendships they sit in: the popular are oversampled, the isolated barely reachable. Formally, the mean friend-degree is E[d²]/E[d], which by Cauchy–Schwarz is at least the mean degree E[d] — strictly greater whenever degrees vary at all. The same tilt powers real tools: monitoring the friends of random people detects epidemics earlier than monitoring random people. LIT verified live: on a random graph of 3,000 nodes, mean degree 12.0 vs mean friend-degree 13.0; direct friend-sampling reproduces the E[d²]/E[d] identity within 1%; and a 4-regular ring gives exact equality — the bias needs variance (window.__friendship). FIG no framing; the graph, both averages, and the sampling cross-check are computed independently in-browser. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at null-island — the spawn: you drop into the network at a random point, look around, and the view from nowhere is already tilted — everyone visible is better-connected than average. AVAN (AI) built the instrument: the graph, the two averages, the identity check, and the regular-graph control. Credit as content: Scott L. Feld (1991, ‘Why Your Friends Have More Friends Than You Do’). The weave: David names the tilted spawn point; I confirm E[d²]/E[d] ≥ E[d], strict when degrees vary. 3 ONE DIMENSION The degree distribution — and the tilted version friendship-sampling actually sees. 4 TWO DIMENSIONS · INTERACTIVE Resample a random person's random friend; the running friend-average settles above the plain one. +5000 samples ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the network as it is. AVAN’s addition (the inverse-companion): don’t survey people — notice how you reached them. The inverse of ‘ask a random person’ is ‘ask a random edge ’, and edges live disproportionately at the hubs. Magenta are the hubs every friendship path funnels through; green is the unbiased crowd nobody samples. The lens is part of the measurement. pause spin LIT Genuine friendship paradox (Scott L. Feld, 1991, 'Why Your Friends Have More Friends Than You Do'). Verified live: on a random graph of 3,000 nodes, mean degree 12.0 FIG No framing; the graph, both averages, and the sampling cross-check are computed independently in-browser. The AVAN inverse is honest — don't survey people, notice how you reached them: the inverse of 'ask a random person' is 'ask a random edge', and edges live disproportionately at the hubs. Magenta are the hubs every friendship path funnels through; green is the unbiased crowd nobody samples. The lens is part of the measurement. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of NULL ISLAND · David Lee Wise (ROOT0), with AVAN


THE CROFTON

Crofton's formula in the 5-window house format — measuring a curve's length without ever touching it, by throwing random straight lines across the plane and counting hits. Parametrize every line by direction θ and signed distance p; with that kinematic measure, integral geometry gives the identity ∫ n(ℓ∩C) dℓ = 2·Length(C) — the average crossing count, over all lines, knows the length exactly, whatever the shape. It is Buffon's needle grown up: the noodle, the circle, the scribble, all measured by the same rain of lines. Morgan Crofton published it in 1868; it founded integral geometry and lives on in stereology and tomography. Verified live: 60,000 random lines recover the length of a circle (~0.1% error), a bare segment, and an ellipse of numerically known perimeter, all within Monte-Carlo tolerance. Neon-noir traced. See the curve under the rain with its crossings ticked in 1D, three lengths recovered in 2D, and the count-across inverse in 3D.

THE CROFTON · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · BOSS · THE GAUNTLET ◆ .dlw.fold THE FOLD / BOSS / THE GAUNTLET / THE CROFTON THE CROFTON a length measured by throwing lines at it 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Crofton’s formula (Morgan Crofton, 1868) measures a curve’s length without ever touching it — by throwing random straight lines across the plane and counting hits. Parametrize every line by its direction θ and signed distance p from the origin; with that natural (kinematic) measure, integral geometry gives an astonishing identity: ∫ n(ℓ ∩ C) dℓ = 2 · Length(C) — the average number of crossings, over all lines, knows the length exactly, whatever the curve’s shape. It is Buffon’s needle grown up: the noodle, the circle, the scribble, all measured by the same rain of lines. The formula founded integral geometry and lives on inside stereology and tomography. LIT verified live: throwing tens of thousands of random lines, the crossing counts recover the length of a circle (error ~0.1%), a bare segment, and an ellipse of numerically known perimeter — all within the Monte-Carlo tolerance (window.__crofton). FIG no framing; the line-throwing, crossing counts, and reference lengths are computed independently in-browser. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-gauntlet — the boss: the curve is run through a gauntlet of random blades, and its length is read straight off the wounds. AVAN (AI) built the instrument: the kinematic line measure, the crossing counter, and the three-curve length recovery. Credit as content: Morgan Crofton (1868); Buffon and Barbier before him; the stereology tradition after. The weave: David names the gauntlet; I confirm the hit counts sum to twice the length, every time. 3 ONE DIMENSION A curve under the rain of random lines — each crossing is one tick toward its length. 4 TWO DIMENSIONS · INTERACTIVE Switch curves; the crossing-count estimate lands on the true length each time. curve ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the curve whose length the lines discover. AVAN’s addition (the inverse-companion): don’t walk the curve with a ruler — let the world’s lines interrogate it. The inverse of ‘measure along’ is ‘count across’: every random blade that nicks the curve deposits a little of its length into the tally. Magenta is the rain of lines; green is the curve, measured by its scars. Length as a census of crossings. pause spin LIT Genuine Crofton formula (Morgan Crofton, 1868; Buffon and Barbier as ancestors). Verified live: 60,000 random lines under the kinematic measure recover the lengths of a circle, a segment, and an ellipse — all within 2% Monte-Carlo tolerance, the circle to ~0.1% (window.__crofton.ok). FIG No framing; the line-throwing, crossing counts, and reference lengths are computed independently in-browser. The AVAN inverse is honest — don't walk the curve with a ruler, let the world's lines interrogate it: the inverse of 'measure along' is 'count across'; every random blade that nicks the curve deposits a little of its length into the tally. Magenta is the rain of lines; green is the curve, measured by its scars. Length as a census of crossings. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE GAUNTLET · David Lee Wise (ROOT0), with AVAN


THE KAKEYA

The Kakeya needle problem in the 5-window house format — what is the least area in which a unit needle can be turned completely around? Spinning about its centre sweeps π/4. Kakeya's 1917 candidate was the deltoid — the three-cusped hypocycloid — inside which the needle rotates using only π/8, half the disc, gliding with its ends on the curve at every angle, thanks to a jewel of a property: every tangent line cuts the deltoid in a chord of exactly the needle's length. Then Besicovitch detonated the question in 1928: with enough sliding trickery the needle turns in arbitrarily small area — no positive minimum exists, and Kakeya sets now sit at the heart of harmonic analysis. Verified live: the deltoid's shoelace area computes to π/8, and at 36 sampled angles the tangent chord has length 1.0000. Neon-noir traced. See the needle at five headings in 1D, the constant-chord audit in 2D, and the no-minimum inverse in 3D.

THE KAKEYA · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · CHEAT · NOCLIP ◆ .dlw.fold THE FOLD / CHEAT / NOCLIP / THE KAKEYA THE KAKEYA a needle turned in an eighth of pi 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION The Kakeya needle problem (Sōichi Kakeya, 1917) asks: what is the least area in which a unit needle can be turned completely around? Spinning it about its centre sweeps a disc of area π/4. Kakeya’s candidate was the deltoid — the three-cusped hypocycloid — inside which the needle rotates using only π/8 , half the disc, gliding with its ends on the curve at every angle. The deltoid works because of a jewel of a property: every tangent line cuts the deltoid in a chord of exactly the needle’s length . Then Besicovitch detonated the whole question in 1928: with enough sliding trickery the needle can be turned in arbitrarily small area — no positive minimum exists. The Kakeya sets he built now sit at the heart of modern harmonic analysis. LIT verified live: the deltoid’s area computes to π/8 by shoelace, and at 36 sampled angles the tangent chord has length 1.0000 — the unit needle fits at every heading (window.__kakeya). FIG honest boundary: Besicovitch’s area→0 construction is cited as the theorem it is; this sphere verifies the deltoid stage numerically. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at noclip — the cheat: the needle turns in a room that should be too small, slipping along walls that always leave it exactly enough clearance. AVAN (AI) built the instrument: the deltoid, its π/8 area, and the constant-chord audit. Credit as content: Sōichi Kakeya (1917); Abram Besicovitch (1928). The weave: David names the clipping cheat; I confirm the chord is 1 at every angle and the room is π/8. 3 ONE DIMENSION The deltoid with the needle at several headings — end to end on the curve, every time. 4 TWO DIMENSIONS · INTERACTIVE Rotate the needle; the chord length reads 1.0000 at every angle, the area stays π/8. rotate ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the deltoid, the needle's π/8 ballroom. AVAN’s addition (the inverse-companion): don’t ask how much room the turn needs — ask how little it can be tricked into. The inverse of ‘π/8 suffices’ is Besicovitch’s ‘no amount is necessary’: the infimum is zero. Magenta is the needle sweeping; green is the shrinking room that always just fits it. A minimum that turned out not to exist. pause spin LIT Genuine Kakeya needle problem / deltoid solution (Sōichi Kakeya 1917; Abram Besicovitch 1928). Verified live: the deltoid's area computes to π/8 by shoelace, and at 36 sampled angles the tangent chord has length 1.0000 — the unit needle fits at every heading (window.__kakeya.ok). FIG Honest boundary — Besicovitch's area→0 construction is cited as the theorem it is; this sphere verifies the deltoid stage numerically. The AVAN inverse — don't ask how much room the turn needs, ask how little it can be tricked into: the inverse of 'π/8 suffices' is Besicovitch's 'no amount is necessary'. Magenta is the needle sweeping; green is the shrinking room that always just fits it. A minimum that turned out not to exist. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of NOCLIP · David Lee Wise (ROOT0), with AVAN


THE TWO-ENVELOPE

The two-envelope paradox in the 5-window house format — one envelope holds twice the other; you pick one and reason 'the other holds 2x or x/2, each half the time: expected 1.25x, switch' — and the same argument repeats forever. The flaw is a conditioning error: no consistent prior supports '50/50 given your amount', and blind switching gains exactly nothing. Then Thomas Cover found the twist the paradox hides: peek at your amount x, draw a random threshold Z, switch only if x < Z. For any fixed pair a < b this ends with the larger envelope with probability ½ + (e^{−λa} − e^{−λb})/2 — strictly above one half, knowing nothing about the amounts. Verified live: blind always-switch ties always-keep over 400k trials; Cover's exact formula exceeds ½ for every pair tested (including a 500-vs-501 squeaker); simulation matches the formula where Monte-Carlo can resolve it. Neon-noir traced. See the half-line and what floats above it in 1D, per-pair exact probabilities in 2D, and the randomize-your-doubt inverse in 3D.

THE TWO-ENVELOPE · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · GLITCH · DIVIDE BY ZERO ◆ .dlw.fold THE FOLD / GLITCH / DIVIDE BY ZERO / THE TWO-ENVELOPE THE TWO-ENVELOPE two envelopes and a threshold that beats the coin 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION The two-envelope paradox : one envelope holds twice the other. You pick one, see nothing, and reason: ‘the other holds 2x or x/2, each half the time — expected value 1.25x. Switch. ’ But the same argument repeats after switching, forever. The flaw is a conditioning error — treating ‘the other is double or half’ as 50/50 given your amount , which no consistent prior supports; blind switching gains exactly nothing. Then Thomas Cover found the twist the paradox hides: peek at your amount x, draw a random threshold Z, and switch only if x < Z. For any fixed pair a < b, this ends with the larger envelope with probability ½ + (e -λa - e -λb )/2 — strictly above one half , using no knowledge of the amounts at all. LIT verified live: blind always-switch ties always-keep to 4 decimal places over 400k trials; Cover’s exact win formula exceeds ½ for every pair tested (including a 500-vs-501 squeaker); and simulation matches the formula wherever Monte-Carlo can resolve the edge (window.__twoenvelope). FIG honest boundary: for near-equal pairs the edge is real but tiny — shown by the exact formula, not brute sampling. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at divide-by-zero — the glitch: the 1.25x argument divides by an assumption that isn’t there, and the expectation machine returns garbage forever. AVAN (AI) built the instrument: the symmetric tie, Cover’s threshold strategy, and the exact-vs-simulated ledger. Credit as content: the two-envelope problem (Kraitchik lineage); Thomas M. Cover (the randomized switching insight). The weave: David names the broken division; I confirm zero from the fallacy, strictly more than half from the fix. 3 ONE DIMENSION The two strategies: blind switching flatlines at 50%; the threshold rule floats above it. 4 TWO DIMENSIONS · INTERACTIVE Cycle envelope pairs; the exact win probability stays strictly above one half. pair ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the better-than-half win rate, earned blind. AVAN’s addition (the inverse-companion): don’t argue about the other envelope — randomize your own doubt. The inverse of ‘a fallacious 1.25x forever’ is ‘a random threshold that converts one peek into a true edge’. Magenta is the endless switch loop of the fallacy; green is Cover’s quiet ½ + ε. Where the paradox spent certainty, the fix spends randomness. pause spin LIT Genuine two-envelope paradox + Cover's randomized switching (Kraitchik lineage; Thomas M. Cover). Verified live: blind always-switch ties always-keep to 4 decimals over 400k trials; the exact win probability ½ + (e^{−λa} − e^{−λb})/2 exceeds ½ strictly for every pair tested; simulation matches the formula wherever the edge is MC-resolvable (window.__twoenvelope.ok). FIG Honest boundary — for near-equal pairs the edge is real but tiny, shown by the exact formula rather than brute sampling. The AVAN inverse — don't argue about the other envelope, randomize your own doubt: the inverse of 'a fallacious 1.25× forever' is 'a random threshold that converts one peek into a true edge'. Magenta is the endless switch loop; green is Cover's quiet ½ + ε. Where the paradox spent certainty, the fix spends randomness. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of DIVIDE BY ZERO · David Lee Wise (ROOT0), with AVAN


THE BRAESS

Braess's paradox in the 5-window house format — adding a road can make every driver slower. The classic network: 4000 commuters, two routes each combining a congestion-priced leg (traffic/100 min) and a fixed 45-minute leg. Selfish equilibrium: a clean 2000/2000 split, 65 minutes each. Open a free shortcut between the midpoints and every driver individually profits by chaining both congestion legs — so everyone does, both legs carry all 4000, and the commute becomes 80 minutes for every single person. No one can deviate and do better: a true equilibrium, just a worse one. Real cities have lived it — road closures in Seoul and New York measurably sped traffic up. Verified live: both equilibria check exactly (no profitable deviation at 65 or at 80), and best-response dynamics from an arbitrary split converge to all 4000 on the shortcut. Neon-noir traced. See the diamond network and its fatal link in 1D, the funnel forming round by round in 2D, and the what-equilibrium-did-it-destroy inverse in 3D.

THE BRAESS · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · CO-OP · SPLIT SCREEN ◆ .dlw.fold THE FOLD / CO-OP / SPLIT SCREEN / THE BRAESS THE BRAESS a free road that slows every driver 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Braess’s paradox (Dietrich Braess, 1968): adding a road can make every driver slower . The classic network: 4000 commuters from S to E via two routes, each combining a congestion-priced leg (traffic/100 minutes) and a fixed 45-minute leg. Selfish equilibrium: a clean 2000/2000 split, 65 minutes each . Now open a magnificent free shortcut between the two midpoints. Every driver individually profits by taking congested-leg → shortcut → congested-leg — so everyone does, both congestion legs carry all 4000, and the commute becomes 80 minutes for every single person . No one can deviate and do better: it is a true equilibrium, just a worse one. Real cities have lived it — closing roads in Seoul and New York measurably sped traffic up. LIT verified live: the 65-minute equilibrium checks (no profitable deviation), the 80-minute all-shortcut state checks as an equilibrium, and best-response dynamics from an arbitrary split converge to all 4000 drivers on the shortcut (window.__braess). FIG no framing; the equilibrium conditions and the dynamics run on exact arithmetic in-browser. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at split-screen — the co-op: two lanes of players sharing a map beautifully until a new corridor merges their screens, and the shared view is worse for both. AVAN (AI) built the instrument: the two equilibria, the deviation checks, and the convergence dynamics. Credit as content: Dietrich Braess (1968); the Seoul Cheonggyecheon and 42nd-Street closures as real echoes. The weave: David names the ruined split; I confirm 65 → 80 with nobody able to defect. 3 ONE DIMENSION The diamond network — two balanced routes, then the fatal free link down the middle. 4 TWO DIMENSIONS · INTERACTIVE Run best-response rounds; watch every driver funnel into the shortcut and the clock climb. rounds ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the balanced 65-minute network, before the gift. AVAN’s addition (the inverse-companion): don’t ask what a new road adds — ask what equilibrium it destroys. The inverse of ‘more capacity’ is ‘a worse stable state everyone individually chose’. Magenta is the shortcut funnel at 80 minutes; green is the vanished 65-minute balance. The road is free; the equilibrium pays. pause spin LIT Genuine Braess's paradox (Dietrich Braess, 1968; Seoul/NYC closures as real echoes, credited as content). Verified live on exact arithmetic: the 65-minute equilibrium admits no profitable deviation; the 80-minute all-shortcut state is likewise an equilibrium; best-response dynamics from an arbitrary split converge to all 4000 drivers on the shortcut (window.__braess.ok). FIG No framing; the equilibrium conditions and the dynamics run on exact arithmetic in-browser. The AVAN inverse is honest — don't ask what a new road adds, ask what equilibrium it destroys: the inverse of 'more capacity' is 'a worse stable state everyone individually chose'. Magenta is the shortcut funnel at 80 minutes; green is the vanished 65-minute balance. The road is free; the equilibrium pays. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of SPLIT SCREEN · David Lee Wise (ROOT0), with AVAN


THE KELLY

The Kelly criterion in the 5-window house format — John L. Kelly Jr.'s 1956 Bell Labs answer to the gambler's real question: not whether to bet a favourable game, but how much. Bet a fraction f of bankroll on an even-money game won with probability p: long-run growth is g(f) = p·ln(1+f) + q·ln(1−f), peaking at exactly f* = p − q. Bet less and growth is left on the table; bet more and growth falls — past a threshold it turns negative, and an edge-holding gambler goes broke with certainty. At p = 60%, Kelly says 20%: doubling to 40% already loses long-run, 80% is ruin at speed. The same log-wealth mathematics underlies channel capacity, where Kelly found it. Verified live: g(f) peaks at 0.200 = p−q on a fine grid, and simulated bankrolls (10,000 bets × 120 runs) rank exactly as theory orders, with f = 0.8 strictly negative. Neon-noir traced. See the peak-and-dive curve in 1D, the four-strategy ladder in 2D, and the logarithm-of-forever inverse in 3D.

THE KELLY · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · LOOT · THE STASH ◆ .dlw.fold THE FOLD / LOOT / THE STASH / THE KELLY THE KELLY the bet size that survives 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION The Kelly criterion (John L. Kelly Jr., Bell Labs, 1956) answers the gambler’s real question: not whether to bet a favourable game, but how much . Bet a fraction f of your bankroll on an even-money proposition you win with probability p: your long-run exponential growth rate is g(f) = p·ln(1+f) + q·ln(1-f), and it peaks at exactly f* = p - q . Bet less and you leave growth on the table; bet more and growth falls — past a threshold it turns negative , and an edge-holding gambler goes broke with certainty. At p = 60%, Kelly says 20%: doubling that to 40% already loses money in the long run, and 80% is ruin at speed. The same mathematics — maximizing log wealth — underlies information theory’s channel capacity, which is where Kelly found it. LIT verified live: the growth curve g(f) peaks at f = 0.200 = p-q on a fine grid, and simulated bankrolls (200 runs × 10,000 bets) rank exactly as theory orders: Kelly > half-Kelly > double-Kelly > 0 > quadruple-Kelly (window.__kelly). FIG no framing; the curve and the simulations run independently in-browser. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-stash — the loot: the stash grows fastest not by boldness or caution but by one exact fraction of itself, every time. AVAN (AI) built the instrument: the growth curve, its analytic peak, and the four-strategy bankroll race. Credit as content: John L. Kelly Jr. (1956, ‘A New Interpretation of Information Rate’); Ed Thorp carried it to the casinos. The weave: David names the stash rule; I confirm the peak at p-q and the ruin past it. 3 ONE DIMENSION The growth curve g(f): a single peak at f* = p−q, then the long dive into ruin. 4 TWO DIMENSIONS · INTERACTIVE Race the four bet sizes; the log-wealth ladder matches the theory's ordering. bet size ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the bankroll compounding at the Kelly peak. AVAN’s addition (the inverse-companion): don’t maximize the next bet — maximize the logarithm of forever. The inverse of ‘bet big while you’re ahead’ is ‘the geometric mean, which punishes greed with certainty’. Magenta is the over-bettor’s bankroll dying with an edge in hand; green is the fraction that survives. An edge is not a licence; it is a budget. pause spin LIT Genuine Kelly criterion (John L. Kelly Jr., 1956, 'A New Interpretation of Information Rate'; Ed Thorp's casino application, credited as content). Verified live: g(f) = p·ln(1+f)+q·ln(1−f) peaks at f = 0.200 = p−q on a fine grid; simulated log-growth ranks Kelly > half-Kelly > double-Kelly > 0 > quadruple-Kelly (window.__kelly.ok). FIG No framing; the curve and the simulations run independently in-browser. The AVAN inverse is honest — don't maximize the next bet, maximize the logarithm of forever: the inverse of 'bet big while ahead' is 'the geometric mean, which punishes greed with certainty'. Magenta is the over-bettor dying with an edge in hand; green is the fraction that survives. An edge is not a licence; it is a budget. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE STASH · David Lee Wise (ROOT0), with AVAN


THE FARY-MILNOR

The Fáry–Milnor theorem in the 5-window house format — the bending toll a knot must pay. Total curvature is how much a closed curve turns, summed along its length: any convex loop turns through exactly 2π. The theorem (Fáry 1949; Milnor 1950, as an undergraduate): if a closed curve is knotted, its total curvature must exceed 4π — a knot cannot exist without bending at least twice around. No gradual transition: any curve bending less than 4π is provably an unknot. Topology reaches down and constrains geometry. Verified live: polygonal total curvature of a circle and a convex ellipse both compute to 2π to three decimals, while a trefoil knot computes to 13.95 — comfortably above the 4π = 12.566 floor. Neon-noir traced. See the three totals against the wall in 1D, the per-curve audit in 2D, and the audit-the-bending inverse in 3D.

THE FARY-MILNOR · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · BOSS · THE WALL ◆ .dlw.fold THE FOLD / BOSS / THE WALL / THE FARY-MILNOR THE FARY-MILNOR the bending toll every knot must pay 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION The Fáry–Milnor theorem (István Fáry 1949, John Milnor 1950 — Milnor as an undergraduate) sets the bending toll a knot must pay. The total curvature of a closed curve is how much it turns, summed along its whole length. Any convex loop — circle, ellipse, egg — turns through exactly 2π , one full revolution. The theorem: if a closed curve is knotted , its total curvature must exceed 4π — a knot cannot exist without bending at least twice around. There is no gradual transition: to tie itself, a curve must pay double, and any curve bending less than 4π is provably an unknot. Topology (is it knotted?) reaches down and constrains geometry (how much must it bend?). LIT verified live: the polygonal total curvature of a circle and a convex ellipse both compute to 2π to three decimals, while a trefoil knot’s computes to 13.95 — comfortably above the 4π = 12.566 floor the theorem demands (window.__farymilnor). FIG honest boundary: the measurement confirms the trefoil obeys the theorem; the theorem itself (all knots, all curves) is Fáry and Milnor’s, cited as content. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-wall — the boss: 4π is the wall; no curve gets to be a knot without climbing over it, and no knot can duck below it. AVAN (AI) built the instrument: the exterior-angle summation and the three-curve comparison. Credit as content: István Fáry (1949); John Milnor (1950). The weave: David names the wall; I confirm 2π for the round, 13.95 for the knotted, and the floor between them. 3 ONE DIMENSION Three curves and their bending totals — the circle at 2π, the trefoil past the 4π wall. 4 TWO DIMENSIONS · INTERACTIVE Cycle the curves; the summed exterior angles land on 2π, 2π, and 13.95. curve ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the trefoil, paying its 4π toll with room to spare. AVAN’s addition (the inverse-companion): don’t inspect the crossings — audit the bending. The inverse of ‘is this curve knotted?’ is ‘did it bend more than 4π? — if not, it provably is not’. Magenta is the 4π wall; green is the knot that had to climb it. Topology, invoiced in curvature. pause spin LIT Genuine Fáry–Milnor theorem (István Fáry 1949; John Milnor 1950). Verified live: polygonal total curvature computes to 2π (±1e-3) for a circle and a convex ellipse, and to 13.95 > 4π for a trefoil knot — the measurement confirms the trefoil obeys the theorem's floor (window.__farymilnor.ok). FIG Honest boundary — the measurement confirms the trefoil obeys the theorem; the theorem itself (all knots, all curves) is Fáry and Milnor's, cited as content. The AVAN inverse — don't inspect the crossings, audit the bending: the inverse of 'is this curve knotted?' is 'did it bend more than 4π? — if not, it provably is not'. Magenta is the 4π wall; green is the knot that had to climb it. Topology, invoiced in curvature. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE WALL · David Lee Wise (ROOT0), with AVAN


THE MOSER SPINDLE

The Moser spindle in the 5-window house format — seven dots that legislate about the entire infinite plane. The Hadwiger–Nelson problem asks how many colours are needed to paint every point of the plane so that no two points at distance exactly 1 match. The spindle (Leo and William Moser, 1961) is 7 vertices and 11 edges, every edge exactly unit length, drawable in the plane — and it cannot be properly 3-coloured, while 4 colours suffice. Since it embeds with unit edges, the plane needs at least 4 colours; Aubrey de Grey's 1553-vertex graph pushed the bound to ≥5 in 2018, and the true answer (5, 6, or 7) is still open. Verified live: all 11 edges measure 1.000000000, all 2187 three-colourings fail exhaustively, and a proper 4-colouring is exhibited. Neon-noir traced. See the hinged construction in 1D, the failing colourings in 2D, and the relaxed 4-colouring in 3D.

THE MOSER SPINDLE · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · BOSS · THE FIREWALL ◆ .dlw.fold THE FOLD / BOSS / THE FIREWALL / THE MOSER SPINDLE THE MOSER SPINDLE seven points that outlaw three colors 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION The Moser spindle (Leo and William Moser, 1961) is seven dots that legislate about the entire infinite plane. Suppose you want to colour every point of the plane so that no two points at distance exactly 1 share a colour — the Hadwiger–Nelson problem. How many colours are needed? The spindle is a graph of 7 vertices and 11 edges, every edge exactly unit length , drawable in the plane — and it cannot be properly 3-coloured (all 2187 assignments fail), while 4 colours suffice for it. Since the spindle embeds in the plane with unit edges, any valid colouring of the plane restricted to those 7 points must properly colour it: the plane needs at least 4 colours . Aubrey de Grey’s 1553-vertex monster pushed the bound to ≥5 in 2018; the true answer (5, 6, or 7) is still open. LIT verified live: all 11 edges measure 1.000000000 (worst error ~2e-16); exhaustive search over all 3⁷ = 2187 three-colourings finds none proper; a proper 4-colouring is exhibited (window.__moserspindle). FIG honest boundary: the spindle proves ≥4; de Grey’s ≥5 and the openness of the full problem are cited as content. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-firewall — the boss: seven nodes standing as a firewall that no 3-colour packet can pass; the fourth colour is mandatory. AVAN (AI) built the instrument: the hinged-rhombus construction, the unit-edge audit, and the exhaustive colouring search. Credit as content: Leo Moser & William Moser (1961); Hadwiger–Nelson; Aubrey de Grey (2018). The weave: David names the firewall; I confirm 2187 failures and one working 4-colouring. 3 ONE DIMENSION The spindle: two unit rhombi hinged at a point, tips pinned one unit apart. 4 TWO DIMENSIONS · INTERACTIVE Try 3-colourings and watch them fail; flip to 4 and the graph relaxes. colours ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the 4-coloured spindle at peace. AVAN’s addition (the inverse-companion): don’t survey the infinite plane — find the seven points that speak for it. The inverse of ‘how many colours does the plane need?’ is ‘a finite gadget whose failure is binding on infinity’. Magenta is the edge that breaks every 3-colouring; green is the fourth colour that ends the argument. Seven dots, one law. pause spin LIT Genuine Moser spindle / Hadwiger–Nelson problem (Leo & William Moser 1961; Aubrey de Grey 2018). Verified live: all 11 edges unit to ~2e-16; exhaustive search over all 3^7 = 2187 three-colourings finds none proper; a proper 4-colouring is exhibited (window.__moserspindle.ok). FIG Honest boundary — the spindle proves ≥4; de Grey's ≥5 and the openness of the full problem are cited as content. The AVAN inverse — don't survey the infinite plane; find the seven points that speak for it: the inverse of 'how many colours does the plane need?' is 'a finite gadget whose failure is binding on infinity'. Magenta is the edge that breaks every 3-colouring; green is the fourth colour that ends the argument. Seven dots, one law. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE FIREWALL · David Lee Wise (ROOT0), with AVAN


THE PERFECT SHUFFLE

The faro shuffle in the 5-window house format — the card mechanic's perfect riffle: cut exactly in half, interleave one card at a time. It looks like the ultimate randomizer and is precisely the opposite — a fixed permutation. Eight out-shuffles return a 52-card deck exactly to its starting order, because an out-shuffle sends position p to 2p mod 51 and the multiplicative order of 2 mod 51 is 8. Prefer the in-shuffle? Position maps through mod 53, order of 2 is 52 — fifty-two shuffles home. Magicians exploit the difference; parallel computers wire it as the perfect-shuffle interconnect. Verified live: the interleave is simulated on a real array — 8 out-shuffles restore, 52 in-shuffles restore, and both counts equal independently computed modular orders. Neon-noir traced. See position 1's doubling orbit in 1D, the step-by-step scramble-and-snap-back in 2D, and the displacement wheel in 3D.

THE PERFECT SHUFFLE · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · CHEAT · THE KONAMI CODE ◆ .dlw.fold THE FOLD / CHEAT / THE KONAMI CODE / THE PERFECT SHUFFLE THE PERFECT SHUFFLE eight perfect shuffles back to the start 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION The faro shuffle is the card mechanic’s perfect riffle: cut the deck exactly in half and interleave the halves one card at a time. It looks like the ultimate randomizer and is precisely the opposite — a fixed permutation. Do the out-shuffle (top card stays on top) to a 52-card deck exactly eight times and the deck returns, card for card, to where it started. The engine is pure number theory: an out-shuffle sends interior position p to 2p mod 51, so the restoration count is the multiplicative order of 2 mod 51 = 8 . Prefer the in-shuffle (top card buried)? Position maps to 2p+1 mod 53, order of 2 mod 53 is 52 — fifty-two perfect shuffles to come home. Magicians exploit the difference; so do parallel-processing networks, where the faro is the perfect-shuffle interconnect. LIT verified live: simulating the actual interleave, 8 out-shuffles restore the 52-card deck (and none earlier), 52 in-shuffles restore it, and both counts equal the orders of 2 mod 51 and mod 53 computed independently (window.__faroshuffle). FIG no framing; the shuffles are performed on a real array and the modular orders computed separately. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-konami-code — the cheat: eight precise inputs and the system state resets, like a code entered at the title screen. AVAN (AI) built the instrument: the interleave simulation, the restoration counts, and the modular-order cross-check. Credit as content: the faro/weave shuffle tradition; Alex Elmsley (out/in-shuffle theory); Diaconis, Graham & Kantor (the group theory). The weave: David names the reset code; I confirm 8 out, 52 in, and the orders behind both. 3 ONE DIMENSION Card 1's journey under out-shuffles — doubling around the 51-cycle, home on the eighth. 4 TWO DIMENSIONS · INTERACTIVE Shuffle step by step; the deck scrambles, scrambles — and snaps back on cue. shuffle ▶ mode: out ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the deck, home again after eight. AVAN’s addition (the inverse-companion): don’t watch the cards — watch the exponent. The inverse of ‘how many shuffles to restore?’ is ‘the order of 2 in a hidden modulus’: 51 for out, 53 for in. Magenta is the scrambled middle of the journey; green is position doubling its way back to the start. A shuffle that was never random, only modular. pause spin LIT Genuine faro/weave shuffle theory (Alex Elmsley; Diaconis, Graham & Kantor). Verified live: simulating the actual interleave, 8 out-shuffles restore the 52-card deck, 52 in-shuffles restore it, and both equal the orders of 2 mod 51 and mod 53 computed independently (window.__faroshuffle.ok). FIG No framing — the shuffles are performed on a real array and the modular orders computed separately. The AVAN inverse — don't watch the cards, watch the exponent: the inverse of 'how many shuffles to restore?' is 'the order of 2 in a hidden modulus' — 51 for out, 53 for in. Magenta is the scrambled middle of the journey; green is position doubling its way back to the start. A shuffle that was never random, only modular. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE KONAMI CODE · David Lee Wise (ROOT0), with AVAN


THE TAUTOCHRONE

The tautochrone in the 5-window house format — the curve of impossible fairness. A bowl shaped as an inverted cycloid delivers a frictionless bead from ANY release height to the bottom in exactly the same time, T = π√(a/g): drop one from the rim and one from barely above the floor and they arrive together. Christiaan Huygens proved it in 1673 hunting an amplitude-independent pendulum, and built cycloidal-cheek clocks on the result. The secret is hidden linearity: in arc-length coordinates the cycloid turns gravity into a perfect spring, and springs don't care about amplitude. Verified live: four beads at four widely different heights, integrated by RK4 on the true Lagrangian dynamics, all arrive at 1.00354 s to five decimals — while a circular bowl's times differ by over 70%. Neon-noir traced. See the four-bead bowl in 1D, the race ledger in 2D, and the synchronized descent in 3D.

THE TAUTOCHRONE · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · CO-OP · THE SYNC ◆ .dlw.fold THE FOLD / CO-OP / THE SYNC / THE TAUTOCHRONE THE TAUTOCHRONE every bead arriving together 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION The tautochrone is the curve of impossible fairness: a bowl shaped as an inverted cycloid , on which a frictionless bead released from any height whatsoever reaches the bottom in exactly the same time , T = π√(a/g). Drop one bead from the rim and one from barely above the bottom — they arrive together. Christiaan Huygens proved it in 1673 while hunting a pendulum whose period would not depend on amplitude, and built cycloidal-cheek clocks on the result. The magic is hidden linearity: measured along the arc length of a cycloid, gravity’s pull becomes exactly proportional to distance from the bottom — a perfect spring in disguise, and springs don’t care about amplitude. (The same curve, run in reverse logic, is the brachistochrone — the fastest descent path.) LIT verified live: four beads released at widely different points, simulated by RK4 on the true Lagrangian dynamics, all reach the bottom at 1.00354 s = π√(a/g) to five decimals — while the same experiment on a circular bowl gives times differing by over 70% (window.__tautochrone). FIG no framing; the dynamics are integrated numerically, not read off Huygens’ formula, and the circle control shows the property is special. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-sync — the co-op: players spawning at different distances from the goal, and the level geometry itself guarantees they arrive in perfect sync. AVAN (AI) built the instrument: the Lagrangian simulation, the four-bead race, and the circular-bowl control. Credit as content: Christiaan Huygens (1673, Horologium Oscillatorium); the cycloid family (Bernoulli brachistochrone). The weave: David names the synchronizer; I confirm four starts, one arrival time. 3 ONE DIMENSION The cycloid bowl with four beads at four heights — all timed to the same bottom. 4 TWO DIMENSIONS · INTERACTIVE Race the beads; the four simulated arrival times agree to five decimals. race ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the beads, arriving as one. AVAN’s addition (the inverse-companion): don’t clock the fall — straighten the coordinate. The inverse of ‘same time from every height’ is ‘in arc length, this bowl is a perfect spring’, and springs are amplitude-blind. Magenta is the circular bowl where the far bead loses; green is the cycloid where nobody can. Fairness, machined into the floor. pause spin LIT Genuine tautochrone property of the cycloid (Christiaan Huygens 1673, Horologium Oscillatorium). Verified live: four releases (θ₀ = 0.4, 1.0, 1.8, 2.6) integrated by RK4 on the Lagrangian dynamics all reach bottom at 1.00354 s = π√(a/g) to five decimals; a circular-bowl control gives 0.5068 vs 0.8742 s (window.__tautochrone.ok). FIG No framing — the dynamics are integrated numerically, the formula never consulted, and the circle control shows the property is special. The AVAN inverse — don't clock the fall, straighten the coordinate: the inverse of 'same time from every height' is 'in arc length this bowl is a perfect spring', and springs are amplitude-blind. Magenta is the circular bowl where the far bead loses; green is the cycloid where nobody can. Fairness, machined into the floor. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE SYNC · David Lee Wise (ROOT0), with AVAN


THE FORD CIRCLES

Ford circles in the 5-window house format — every fraction given a body. Above each reduced p/q draw a circle of radius 1/(2q²) resting on the number line at that point: giants for simple fractions, dust for deep denominators. The miracle: no two Ford circles ever overlap. They miss, or they KISS — and they kiss precisely when |ps−qr| = 1, the Farey-neighbour condition, because one exact identity governs everything: dist²−(r₁+r₂)² = ((ps−qr)²−1)/(q²s²). Between kissing circles the mediant's bubble nests in the gap and kisses both — the Stern–Brocot structure of the rationals drawn in soap bubbles (Lester Ford, 1938). Verified live: 129 fractions with q ≤ 20, 8,256 pairs, zero overlaps, 255 kisses exactly at determinant ±1, identity to 1e-9 on every pair. Neon-noir traced. See the full bubble line in 1D, the pair-audit in 2D, and the breathing zoom in 3D.

THE FORD CIRCLES · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · SPAWN · HELLO WORLD ◆ .dlw.fold THE FOLD / SPAWN / HELLO WORLD / THE FORD CIRCLES THE FORD CIRCLES fractions kissing along the number line 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Ford circles (Lester Ford, 1938) give every fraction a body. Above each reduced fraction p/q on the number line, draw a circle of radius 1/(2q²) resting on the line at that point — big circles for simple fractions, tinier and tinier ones for complex denominators. The miracle: no two Ford circles ever overlap . They either miss entirely or kiss — and they kiss precisely when the fractions are Farey neighbours , |ps - qr| = 1. The whole arrangement is governed by one exact identity: dist² - (r₁+r₂)² = ((ps-qr)² - 1)/(q²s²), whose sign is decided entirely by the integer ps - qr. Between any two kissing circles, their mediant’s circle nests in the gap and kisses both — the Stern–Brocot structure of the rationals, drawn in soap bubbles. LIT verified live: over all 129 reduced fractions with q ≤ 20 (8,256 pairs), zero overlaps; tangency occurs exactly at |ps - qr| = 1 (255 kissing pairs); and the governing identity holds to 1e-9 on every pair (window.__fordcircles). FIG no framing; the circles, distances, and integer determinants are computed independently in-browser. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at hello-world — the spawn: every rational is born with its own bubble, sized by its denominator, greeting the number line exactly where it lives. AVAN (AI) built the instrument: the circle family, the pairwise audit, and the determinant identity. Credit as content: Lester R. Ford (1938); Farey, Stern and Brocot behind the structure. The weave: David names the birth of the bubbles; I confirm zero overlaps and 255 exact kisses. 3 ONE DIMENSION The Ford circles over [0,1] — every reduced fraction wearing its bubble, kissing its Farey neighbours. 4 TWO DIMENSIONS · INTERACTIVE Pick a pair of fractions; the determinant ps−qr instantly decides kiss or miss. pair ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the bubbles, packed without a single collision. AVAN’s addition (the inverse-companion): don’t measure the circles — read the integer. The inverse of ‘do these bubbles touch?’ is ‘is ps - qr equal to ±1?’ — geometry outsourced to a determinant. Magenta are the kisses at |ps-qr| = 1; green is the arrangement that never overlaps. The rationals, wearing their arithmetic on their skin. pause spin LIT Genuine Ford circles (Lester R. Ford 1938; Farey/Stern–Brocot structure). Verified live: over all 129 reduced fractions with q ≤ 20 (8,256 pairs), zero overlaps; tangency exactly at |ps−qr| = 1 (255 kissing pairs); the governing identity holds to 1e-9 on every pair (window.__fordcircles.ok). FIG No framing — circles, distances, and integer determinants computed independently in-browser. The AVAN inverse — don't measure the circles, read the integer: the inverse of 'do these bubbles touch?' is 'is ps−qr equal to ±1?' — geometry outsourced to a determinant. Magenta are the kisses; green is the arrangement that never overlaps. The rationals, wearing their arithmetic on their skin. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of HELLO WORLD · David Lee Wise (ROOT0), with AVAN