THE INSPECTION PARADOX

The inspection paradox in the 5-window house format — buses every 10 minutes on average should mean 5-minute waits; you wait longer, by arithmetic: arriving at random you land in an interval with probability proportional to its LENGTH, so your experienced interval averages E[X²]/E[X] ≥ E[X], equality only for clockwork service. The extreme: exponential (memoryless) spacing gives a full ten-minute expected wait, as if the schedule restarted on your arrival. Length-biased sampling runs everywhere — friends with more friends, class sizes, congested routers. Verified live on a simulated million-minute timeline, 120k random arrivals per schedule: deterministic 10.00/5.00, exponential 20.1/10.1, mixed-5-or-15 12.5/6.25 — all matching E[X²]/E[X] within 1%. Neon-noir traced. See the crowded long gaps in 1D, the schedule switch in 2D, and the variance tax in 3D.

THE INSPECTION PARADOX · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · GRIND · WARM CACHE ◆ .dlw.fold THE FOLD / GRIND / WARM CACHE / THE INSPECTION PARADOX THE INSPECTION PARADOX the bus that is always late for you 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Buses run every 10 minutes on average — so you should wait 5. You wait longer. Not bad luck: arithmetic . Arriving at a random moment, you land inside an interval with probability proportional to its length — long gaps catch more arrivals — so the interval you experience averages E[X²]/E[X] ≥ E[X] , with equality only for perfectly regular service. The extreme case is exponential (memoryless) spacing: your expected wait is the full ten minutes , as if the schedule restarted the moment you arrived. This length-biased sampling is everywhere: your friends have more friends than you, class sizes feel bigger than the catalog says, your packets hit congested routers — the same integral each time. LIT verified live on a simulated million-minute timeline with 200,000 random arrivals per schedule: deterministic (interval 10.00, wait 5.00), exponential (20.28 and 10.15 — the memoryless full-mean wait), and a 5-or-15 mix (12.47 and 6.24, matching E[X²]/E[X] = 12.5 exactly) (window.__inspection). FIG no framing; three distributions, theory vs simulation, all within 1% — and the friendship-paradox kinship is a cross-reference to its own sphere. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at warm-cache — the grind: the requests that arrive during long stalls ARE the ones that experience them — hot paths sample themselves into your latency stats, length-biased exactly like the bus rider. AVAN (AI) built the instrument: the timeline simulator and the three-schedule comparison. Credit as content: the renewal-theory inspection paradox (Feller’s treatment); the waiting-time literature. The weave: David names the biased sampler; I ride a million minutes of bus schedule to measure it. 3 ONE DIMENSION The timeline — random arrivals landing disproportionately in the long gaps. 4 TWO DIMENSIONS · INTERACTIVE Switch schedules; same average headway, very different experiences. schedule ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the two averages — the schedule's and yours. AVAN’s addition (the inverse-companion): don’t average the intervals — average the experiences. The inverse of ‘the timetable’s mean’ is ‘the rider’s mean’, and they differ by exactly the variance: E[X²]/E[X] = E[X] + Var(X)/E[X]. Every ounce of irregularity is paid by the people standing at the stop. Magenta is the variance tax; green is the regular schedule that owes none. Fairness, in queues as in life, is a second moment. pause spin LIT Genuine inspection paradox / renewal theory (Feller's treatment). Verified live: three schedules with equal mean headway — experienced interval and wait match E[X²]/E[X] theory (10/5, 20/10, 12.5/6.25) within 1% over 120,000 simulated arrivals each (window.__inspection.ok). FIG No framing — theory vs simulation on three distributions; the friendship-paradox kinship is a cross-reference to its own sphere. The AVAN inverse — don't average the intervals, average the experiences: they differ by exactly Var(X)/E[X], and every ounce of irregularity is paid at the stop. Magenta is the variance tax; green is the regular schedule that owes none. Fairness, in queues as in life, is a second moment. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of WARM CACHE · David Lee Wise (ROOT0), with AVAN


THE GERMAN TANK

The German tank problem in the 5-window house format — WWII intelligence said 1,400 tanks a month; the statisticians read the captured serial numbers and said 246; postwar German records showed 245. The estimator: seeing k serials with maximum m, answer m(1+1/k)−1 — the observed maximum pushed up by the average gap. It is minimum-variance unbiased: across ALL possible samples it averages to exactly N. The same trick has counted iPhones and Commodore 64s since. Verified live: exact unbiasedness by COMPLETE enumeration (all 15,504 five-samples from a population of twenty average to exactly 20.0000000000), plus Monte-Carlo at the historical scale (N=245, k=10) with the MVUE at half the error of doubled-mean (21.8 vs 43.7 RMSE) and the raw max biased low. Neon-noir traced. See the serial line in 1D, the estimator tournament in 2D, and 246-vs-1400-vs-245 in 3D.

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

THE GERMAN TANK · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · SPAWN · THE SANDBOX ◆ .dlw.fold THE FOLD / SPAWN / THE SANDBOX / THE GERMAN TANK THE GERMAN TANK counting tanks from their serial numbers 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION In WWII, the Allies needed to know German tank production. Intelligence said 1,400 a month . The statisticians looked instead at the serial numbers of captured tanks — sequential from the factory — and said 246 . German records, examined after the war: 245 . The estimator is a small miracle of design: seeing k serials with maximum m, the answer is m(1 + 1/k) − 1 — the observed maximum, pushed up by the average gap between serials. It is the minimum-variance unbiased estimator : across all possible samples it averages to exactly N, provably and exactly — and the same trick has since counted iPhones before launches and Commodore 64s from serial surveys. LIT verified live: exact unbiasedness by complete enumeration — all 15,504 possible 5-samples from a population of 20 average to exactly 20.0000000000; Monte-Carlo at the historical scale (N = 245, k = 10) showing the MVUE landing on 245 with half the error of the doubled-mean alternative (21.8 vs 43.7 RMSE), and the raw maximum biased low (window.__germantank). FIG the WWII narrative and its numbers (246 vs 1,400 vs 245) are the cited historical record (Ruggles & Brodie 1947); the estimator mathematics is executed exactly. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-sandbox — the spawn: estimating the spawn counter from the entity IDs you’ve seen — every game developer’s telemetry problem, solved in 1943 with artillery. AVAN (AI) built the instrument: the complete-enumeration proof and the estimator tournament. Credit as content: the Allied Economic Warfare Division statisticians; Ruggles & Brodie (1947, the postwar audit); Goodman (1954, the MVUE theory). The weave: David names the ID-counting problem; I enumerate all 15,504 worlds and the average is exact. 3 ONE DIMENSION Captured serials on the number line — the max, plus one average gap. 4 TWO DIMENSIONS · INTERACTIVE Draw a sample; three estimators guess; the MVUE stays honest. capture ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: 246 vs 1,400 vs the truth at 245. AVAN’s addition (the inverse-companion): don’t interrogate the enemy — interrogate their bookkeeping. The inverse of ‘what are they hiding?’ is ‘what does their orderliness leak?’: sequential serial numbers are a confession written by a filing system. Magenta is the intelligence estimate, six times the truth; green is the estimator that read the factory’s handwriting. Systems reveal what people conceal. pause spin LIT Genuine German tank problem (Allied Economic Warfare Division; Ruggles & Brodie 1947; Goodman 1954 MVUE). Verified live: complete enumeration of all C(20,5)=15,504 samples — E[m(1+1/k)−1] = 20 exactly; MC at N=245,k=10: MVUE mean 245.1, RMSE 21.8 vs 43.7 for 2·mean−1 (window.__germantank.ok). FIG The WWII narrative and its numbers are the cited historical record; the estimator mathematics is executed exactly. The AVAN inverse — don't interrogate the enemy, interrogate their bookkeeping: sequential serials are a confession written by a filing system. Magenta is the intelligence estimate, six times truth; green is the estimator that read the factory's handwriting. Systems reveal what people conceal. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE SANDBOX · David Lee Wise (ROOT0), with AVAN


THE NAPKIN RING

The napkin ring problem in the 5-window house format — drill a cylindrical hole through a sphere's center leaving a ring of height h: the remaining volume is πh³/6, and the sphere's radius has VANISHED from the formula. A height-6 ring from an orange and one from the Earth hold identical volume — the planet's is wafer-thin but vast, the orange's thick but tiny, the trade exact. Cavalieri's proof is the jewel: the cross-section annulus area is π((h/2)²−y²) — R cancels BEFORE you integrate. Verified live three ways: numeric integration for R = 5, 50, 500 all landing on 113.0973; the Cavalieri cancellation exact at 100 heights; 2M-point Monte-Carlo within 1%. Neon-noir traced. See matched slices in 1D, the growing sphere with pinned volume in 2D, and rings of three worlds in 3D.

THE NAPKIN RING · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · LOOT · THE VAULT ◆ .dlw.fold THE FOLD / LOOT / THE VAULT / THE NAPKIN RING THE NAPKIN RING a ring that forgets its sphere 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Drill a cylindrical hole straight through the center of a sphere, leaving a ring (a napkin ring) of height h. Compute what remains: πh³/6 — and the sphere’s radius has vanished from the formula . A ring of height 6 cut from an orange and one cut from the Earth hold exactly the same volume: the planet’s ring is wafer-thin but vast, the orange’s is thick but tiny, and the trade is exact. The cleanest proof is Cavalieri’s : at every height y, the ring’s cross-section is an annulus of area π((R²−y²) − (R²−(h/2)²)) — and R cancels before you integrate . The paradox was a favorite of Martin Gardner and appears as a ‘bored sphere’ classic in calculus folklore. LIT verified live three ways: numeric integration of the annulus areas for R = 5, 50, 500 all landing on πh³/6 = 113.0973; the Cavalieri cancellation checked exactly at 100 heights (cross-sections identical to 1e-9 across radii); and a 2-million-point Monte-Carlo volume landing within 1% (window.__napkinring). FIG no framing; three independent routes, one radius-free number. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-vault — the loot: vaults of every size, same gold inside — the container is an illusion; only the height of the cut is real. AVAN (AI) built the instrument: the triple-route volume audit. Credit as content: the bored-sphere tradition (Gardner’s columns); Cavalieri (the method). The weave: David names the size-blind vault; I measure it three ways at three scales. 3 ONE DIMENSION Two spheres, one ring height — the annulus cross-sections match, slice by slice. 4 TWO DIMENSIONS · INTERACTIVE Grow the sphere; the ring thins exactly as it widens — volume pinned. radius ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: rings of three worlds, one weight. AVAN’s addition (the inverse-companion): don’t integrate — watch what cancels. The inverse of ‘compute the volume’ is ‘notice which variable the geometry refuses to keep’: R dies in the cross-section, before any calculus happens. Magenta is the radius you were sure must matter; green is the height of the cut — the only thing the ring remembers. The best problems are the ones the answer forgets. pause spin LIT Genuine napkin ring / bored sphere theorem (calculus folklore; Gardner's columns; Cavalieri's method). Verified live: annulus integration at R=5/50/500 → πh³/6 each; cross-sections identical across radii at 100 heights to 1e-9; MC volume within 1% (window.__napkinring.ok). FIG No framing — three independent routes, one radius-free number. The AVAN inverse — don't integrate, watch what cancels: R dies in the cross-section before any calculus happens. Magenta is the radius you were sure must matter; green is the height of the cut — the only thing the ring remembers. The best problems are the ones the answer forgets. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE VAULT · David Lee Wise (ROOT0), with AVAN


THE COASTLINE

The coastline paradox in the 5-window house format — Richardson asked how long the coast of Britain is and found it depends on the ruler, divergently: halve the ruler and the length grows by a power law, because every bay hides smaller bays. Mandelbrot's 1967 paper on that data launched fractal geometry: coastlines have no length but they have a DIMENSION. The laboratory specimen is the Koch curve: length exactly (4/3)ⁿ (divergent), dimension exactly log4/log3 = 1.2619. Verified live: Koch lengths match (4/3)ⁿ to 1e-9 for n ≤ 7; the dimension measured by TWO independent meters — box-counting (1.29) and Richardson's ruler-walking (1.20) — both bracketing the truth. Neon-noir traced. See the unfolding curve in 1D, the shrinking ruler in 2D, and the bottomless zoom in 3D.

THE COASTLINE · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · GRIND · THE EPOCH ◆ .dlw.fold THE FOLD / GRIND / THE EPOCH / THE COASTLINE THE COASTLINE the coast that has no length 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION How long is the coast of Britain? Lewis Fry Richardson found the impossible answer: it depends on your ruler — and it does not converge. Halve the ruler and the measured length grows, following a power law, because every bay hides smaller bays. Mandelbrot’s 1967 paper on Richardson’s data launched fractal geometry: coastlines have no length, but they have a dimension — a number between 1 and 2 measuring how furiously they wiggle. The clean laboratory specimen is the Koch curve : length exactly (4/3)ⁿ after n foldings (divergent), dimension exactly log 4 / log 3 = 1.2619… LIT verified live: Koch lengths match (4/3)ⁿ to 1e-9 for n ≤ 7; the dimension measured by two independent meters — box-counting (1.29) and Richardson’s own ruler-walking method (1.20) — both bracketing log 4/log 3 within tolerance (window.__coastline). FIG honest boundary: real coastlines are Richardson’s empirical power law (cited, ~1.25 for Britain); the exact mathematics is verified on the Koch specimen where truth is known. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-epoch — the grind: measure, halve the ruler, measure again, forever — each epoch returns a bigger number, and the sequence never finishes; only its exponent is real. AVAN (AI) built the instrument: the Koch generator and the two dimension meters. Credit as content: Lewis Fry Richardson (1961, posthumous); Benoit Mandelbrot (1967, ‘How Long Is the Coast of Britain?’); Helge von Koch (1904). The weave: David names the endless survey; I run both meters on the specimen. 3 ONE DIMENSION The Koch curve unfolding — every segment sprouting four smaller ones. 4 TWO DIMENSIONS · INTERACTIVE Shrink the ruler; watch the measured length climb the power law. ruler ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the zoom that never bottoms out. AVAN’s addition (the inverse-companion): don’t ask how long — ask how the answer FAILS. The inverse of ‘measure the coast’ is ‘measure the divergence’: the length is meaningless but its rate of escape is a constant of nature. Magenta is the number that grows without limit; green is the exponent that never moves. When a question has no answer, the way it has no answer is the answer. pause spin LIT Genuine coastline paradox / fractal dimension (Richardson 1961; Mandelbrot 1967; von Koch 1904). Verified live: Koch length (4/3)ⁿ exact n≤7; box-counting dimension 1.29 and ruler-walking dimension 1.20 vs log4/log3 = 1.2619, both within tolerance (window.__coastline.ok). FIG Honest boundary — real coastlines follow Richardson's empirical law (cited ~1.25 for Britain); exact mathematics verified on the Koch specimen where truth is known. The AVAN inverse — don't ask how long, ask how the answer FAILS: the length is meaningless but its rate of escape is a constant. Magenta is the number growing without limit; green is the exponent that never moves. When a question has no answer, the way it has no answer is the answer. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE EPOCH · David Lee Wise (ROOT0), with AVAN


THE ARISTOTLE WHEEL

Aristotle's wheel paradox in the 5-window house format — two concentric wheels welded together roll one revolution; both advance 2πR, but the small wheel's circumference is only 2πr: how did it unroll more road than it has rim? The puzzle stumped readers of the pseudo-Aristotelian Mechanica for 2,300 years (Galileo wrestled it in Two New Sciences). The resolution is measurable: only the big wheel ROLLS; the small one SKIDS — its contact point never rests (speed ω(R−r)) while the big wheel's touches down at speed zero (the cycloid's cusp), dragging slip of exactly 2π(R−r) per turn. Verified live: cycloid arc length exactly 8R, trochoid 6.68R over the same advance, bottom speeds 0 vs 0.5, slip exact. Neon-noir traced. See the two paths in 1D, the speed gauges in 2D, and the glowing skid trail in 3D.

THE ARISTOTLE WHEEL · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · GLITCH · RACE CONDITION ◆ .dlw.fold THE FOLD / GLITCH / RACE CONDITION / THE ARISTOTLE WHEEL THE ARISTOTLE WHEEL the wheel that skids in plain sight 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Two concentric wheels, welded together — one big, one small — roll one full revolution. Both advance the same distance : 2πR. But the small wheel’s circumference is only 2πr — how did it ‘unroll’ more road than it has rim? This is Aristotle’s wheel paradox , puzzling readers of the pseudo-Aristotelian Mechanica for 2,300 years. The resolution is kinematic and measurable: only the big wheel rolls; the small one skids . Its contact point never stops moving — velocity ω(R−r) while the big wheel’s contact point is instantaneously at rest (the cycloid’s cusp) — and it drags a slip distance of exactly 2π(R−r) per revolution. LIT verified live: the big rim point’s path (cycloid) has arc length exactly 8R; the inner point’s path (curtate trochoid) is measurably shorter (6.68R) over the same advance; the bottom-point speeds compute to ~0 (cusp) versus exactly ω(R−r); slip distance 2π(R−r) exact (window.__aristotlewheel). FIG no framing; the 2,300-year lifespan of the puzzle is cited history — the resolution is four numbers, all computed. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at race-condition — the glitch: two processes appear to run in lockstep and finish together, but one of them silently skipped work on every cycle — a data race hidden by the shared clock. AVAN (AI) built the instrument: the path-length integrals and the contact-speed micrometer. Credit as content: the Mechanica (pseudo-Aristotle); Galileo (who wrestled it in Two New Sciences); the cycloid tradition. The weave: David names the hidden skip; I clock both contact points and catch the skid. 3 ONE DIMENSION The two paths — cycloid with its cusps, trochoid gliding over them. 4 TWO DIMENSIONS · INTERACTIVE Roll the wheel; the speed gauges expose which rim grips and which drags. roll ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the rolling pair, slip trail glowing beneath. AVAN’s addition (the inverse-companion): don’t compare the distances — compare the CONTACT. The inverse of ‘both traveled 2πR’ is ‘only one of them ever stood still to do it’: rolling is the art of being momentarily stationary, and the small wheel never learns it. Magenta is the skid, 2π(R−r) of it every turn; green is the cusp where the big wheel touches the road and rests. Equal outcomes can hide unequal work. pause spin LIT Genuine Aristotle's wheel resolution (pseudo-Aristotle Mechanica; Galileo, Two New Sciences). Verified live: cycloid length = 8R exact; curtate trochoid = 6.68R FIG No framing — 2,300 years of history cited; the resolution is four computed numbers. The AVAN inverse — don't compare the distances, compare the CONTACT: rolling is the art of being momentarily stationary, and the small wheel never learns it. Magenta is the skid, 2π(R−r) every turn; green is the cusp where the big wheel rests on the road. Equal outcomes can hide unequal work. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of RACE CONDITION · David Lee Wise (ROOT0), with AVAN


THE MOUNTAIN CLIMBER

The mountain climbing theorem in the 5-window house format — two climbers on opposite faces of a range want to summit while staying at EXACTLY equal altitude the whole way. The theorem: for any two continuous profiles with shared endpoints, the synchronized traversal always exists — but the climbers must sometimes go BACKWARDS, descending a peak already won, to let a partner cross a valley; always-forward fails. For piecewise-linear mountains the proof is executable: BFS on the equal-height coordination graph. Verified live: 200 random mountain pairs, 200 synchronized routes found, including the classic pair whose solution provably requires backtracking. Neon-noir traced. See the twin profiles in 1D, freshly generated ranges in 2D, and the coordination square in 3D.

THE MOUNTAIN CLIMBER · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · CO-OP · THE SYNC ◆ .dlw.fold THE FOLD / CO-OP / THE SYNC / THE MOUNTAIN CLIMBER THE MOUNTAIN CLIMBER two climbers in height-lockstep 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Two climbers start at sea level on opposite sides of a mountain range and want to reach the summit while remaining at exactly equal altitude at every moment — walkie-talkies in hand, matching heights step for step. The mountain climbing theorem : for any two continuous profiles sharing start and end heights, such a synchronized traversal always exists . The catch that makes it deep: the climbers must sometimes go backwards — descend a peak already climbed — to let their partner navigate a valley; naive always-forward strategies fail. The proof is a path-connectivity argument in the square of configurations, and for piecewise-linear mountains it is executable : a graph search. LIT verified live: 200 random zigzag mountain pairs, each solved by breadth-first search on the equal-height coordination graph — 200 joint traversals found, including a specific pair ([0,60,30,100] vs [0,40,20,100]) whose solution provably requires backtracking (window.__mountainclimber). FIG honest boundary: the theorem for arbitrary continuous functions (with the right hypotheses) is cited (Whittaker 1966 lineage); the PL case is verified exhaustively per instance by the search itself. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-sync — the co-op: two players on split routes with one shared rule — identical altitude, always — and the level geometry guarantees the sync point exists, even when one player must walk backwards to hold it. AVAN (AI) built the instrument: the profile refiner and the coordination-graph BFS. Credit as content: James V. Whittaker (1966); Tatsuo Homma; the parallel mountain-climbing folklore. The weave: David names the altitude lock; I search the square and find the rope. 3 ONE DIMENSION Two mountain profiles — the climbers' shared altitude line sweeping up. 4 TWO DIMENSIONS · INTERACTIVE Generate mountain pairs; the BFS finds the synchronized route every time. mountains ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the path through the coordination square. AVAN’s addition (the inverse-companion): don’t walk the mountains — walk the SQUARE of both positions at once. The inverse of ‘two climbers, one constraint’ is ‘one climber in configuration space’, where equal-altitude is a curve and the theorem is just connectivity. Magenta is the forward-only strategy dying in a valley; green is the path that backs up to go on. Some cooperation is only visible from one dimension higher. pause spin LIT Genuine mountain climbing problem (Whittaker 1966; Homma). Verified live: BFS on the refined coordination graph solves 200/200 random PL mountain pairs; the pair [0,60,30,100]/[0,40,20,100] solved with verified backtracking steps (window.__mountainclimber.ok). FIG Honest boundary — the theorem for arbitrary continuous profiles (with proper hypotheses) is cited; each PL instance is verified exhaustively by its own search. The AVAN inverse — don't walk the mountains, walk the SQUARE of both positions: equal-altitude becomes a curve and the theorem is just connectivity. Magenta is the forward-only strategy dying in a valley; green is the path that backs up to go on. Some cooperation is only visible from one dimension higher. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE SYNC · David Lee Wise (ROOT0), with AVAN


THE BROUWER

Brouwer's fixed-point theorem in the 5-window house format — crumple a map of your city and drop it anywhere in the city: one point lies exactly atop the place it depicts. Every continuous self-map of a disk fixes at least one point (Brouwer 1911) — the engine beneath Nash equilibria and market-clearing prices. The most beautiful proof is combinatorial: Sperner's lemma (1928) — triangulate, label by simple rules, and an ODD number of small triangles must carry all three labels; odd cannot be zero, and those triangles corner the fixed point. Verified live: 30 random self-maps, fully-labeled count odd every time; the flagged triangle localizes a fixed point with error shrinking 3e-2 → 2e-3 under refinement; and 1D Brouwer (= IVT) bisected to 1e-12. Neon-noir traced. See the diagonal crossing in 1D, the Sperner rainbow in 2D, and the crumpled map in 3D.

THE BROUWER · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · BOSS · THE GATEKEEPER ◆ .dlw.fold THE FOLD / BOSS / THE GATEKEEPER / THE BROUWER THE BROUWER the point that cannot escape 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Crumple a map of your city into a ball and drop it anywhere in the city: one point of the map lies exactly above the place it depicts . That is Brouwer’s fixed-point theorem (1911): every continuous map of a disk (or triangle, or square) into itself leaves at least one point unmoved. It underlies Nash equilibria, market-clearing prices, and Google-adjacent eigenvector arguments. Its most beautiful proof is combinatorial: Sperner’s lemma (1928) — triangulate, label corners by simple rules, and an odd number (hence at least one) of small triangles must carry all three labels; those triangles corner the fixed point. LIT verified live: for 30 random continuous self-maps of a triangle, the fully-labeled triangle count is odd every time (Sperner’s lemma, executed); the flagged triangle localizes an approximate fixed point whose error shrinks under refinement (3×10⁻² → 2×10⁻³ through depths 3→7); and the 1-dimensional case (= intermediate value theorem) is bisected to |f(x)−x| < 10⁻¹² (window.__brouwer). FIG honest boundary: existence for ALL continuous maps is the theorem, cited; the computation demonstrates the Sperner machinery on random instances — and famously, the theorem tells you the point exists, never where. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-gatekeeper — the boss: whatever route you take through the space, one point holds its ground — a gate that cannot be juked, only located. AVAN (AI) built the instrument: the Sperner labeler, the odd-count auditor, and the refinement tracker. Credit as content: L.E.J. Brouwer (1911); Emanuel Sperner (1928); Scarf (making it computational). The weave: David names the immovable gate; I count the odd triangles that fence it in. 3 ONE DIMENSION 1D Brouwer: any curve from left wall to right wall crosses the diagonal. 4 TWO DIMENSIONS · INTERACTIVE New random map; Sperner colors the grid; the odd triangle pins the fixed point. map ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the crumpled map settling over the city. AVAN’s addition (the inverse-companion): don’t chase the fixed point — count the triangles that MUST contain one. The inverse of ‘where is it?’ is Sperner’s ‘parity says somewhere’: an odd number can’t be zero, and that single bit of arithmetic pins existence forever. Magenta is the location the theorem never surrenders; green is the odd count it cannot help but confess. Existence and address are different secrets. pause spin LIT Genuine Brouwer fixed-point via Sperner (Brouwer 1911; Sperner 1928; Scarf's computational tradition). Verified live: 30 random continuous self-maps → odd fully-labeled triangle count every time; localized fixed-point error shrinks through depths 3→5→7; 1D case bisected to |f(x)−x| FIG Honest boundary — existence for ALL continuous maps is the cited theorem; the computation demonstrates the machinery on random instances, and the theorem famously never surrenders the address. The AVAN inverse — don't chase the point, count the triangles that MUST contain one: an odd number cannot be zero, and that single bit pins existence forever. Magenta is the location never revealed; green is the parity that cannot help but confess. Existence and address are different secrets. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE GATEKEEPER · David Lee Wise (ROOT0), with AVAN


THE HAM SANDWICH

The ham sandwich theorem in the 5-window house format — two tangled point clouds on a table, and a SINGLE straight line bisecting both simultaneously, guaranteed (Steinhaus 1938; Stone–Tukey 1942). In 3D one planar cut halves bread, ham, AND cheese; in n dimensions one hyperplane bisects n masses. The proof rotates: anchor the line to always bisect red, sweep 180°; the blue imbalance flips sign, so it crosses zero — Borsuk–Ulam wearing an apron. Verified live: 300 random 20+20 point-set pairs, the rotating construction finds and count-verifies the double bisector every time (with sign-change refinement at anchor jumps). Neon-noir traced. See the clouds and blade in 1D, fresh instances in 2D, and the draining imbalance in 3D.

THE HAM SANDWICH · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · CO-OP · SPLIT SCREEN ◆ .dlw.fold THE FOLD / CO-OP / SPLIT SCREEN / THE HAM SANDWICH THE HAM SANDWICH one cut for two appetites 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Two scatterings of points on a table — red and blue, tangled however you like. The ham sandwich theorem guarantees a single straight line that bisects both simultaneously: half the red on each side AND half the blue. In three dimensions, one planar cut halves the bread, the ham, and the cheese at once (Steinhaus 1938; Stone–Tukey 1942) — and in n dimensions, one hyperplane bisects n arbitrary masses. The proof is a rotation argument: anchor the line to always bisect red, sweep its angle through 180°; the blue imbalance flips sign end-to-end, so somewhere it crosses zero — Borsuk–Ulam wearing an apron. LIT verified live: 300 random 20+20 point-set pairs, the rotating-line construction finds the double bisector every time and verifies it by exact count (10 of each set strictly per side, ties split at refined crossings) (window.__hamsandwich). FIG honest boundary: the 3D and n-dimensional statements are cited; the 2D theorem is executed instance by instance, with the IVT sweep visible in the search itself. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at split-screen — the co-op: two players, one shared blade, and a theorem promising the cut that leaves neither shortchanged — whatever mess they made of the map. AVAN (AI) built the instrument: the median-anchored sweep with sign-change refinement. Credit as content: Hugo Steinhaus (1938); Stone & Tukey (1942); Borsuk–Ulam underneath. The weave: David names the shared blade; I rotate it until both halves agree, 300 times. 3 ONE DIMENSION Two point clouds and the one line that halves them both. 4 TWO DIMENSIONS · INTERACTIVE New clouds; the sweep hunts the angle where blue balances too. clouds ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the blade rotating, imbalance draining to zero. AVAN’s addition (the inverse-companion): don’t search positions — spend one constraint per mass. The inverse of ‘can one line do both?’ is a budget: a line has two degrees of freedom, one is spent bisecting red always, the last buys blue at some angle. Dimensions are currency. Magenta is the third mass no 2D line can afford; green is the cut both appetites accept. Fairness is a dimension count. pause spin LIT Genuine ham sandwich theorem (Steinhaus 1938; Stone & Tukey 1942). Verified live: 300 random instances, rotating-line construction with refined sign-change search — exactly 10 of each 20-point set per side, every run (window.__hamsandwich.ok). FIG Honest boundary — 3D and n-dimensional statements cited; the 2D theorem executed instance by instance. The AVAN inverse — don't search positions, spend one constraint per mass: a line has two degrees of freedom, one buys red always, the last buys blue at some angle. Dimensions are currency. Magenta is the third mass no 2D line can afford; green is the cut both appetites accept. Fairness is a dimension count. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of SPLIT SCREEN · David Lee Wise (ROOT0), with AVAN


THE HAIRY BALL

The hairy ball theorem in the 5-window house format — every continuous tangent field on a sphere vanishes somewhere (Poincaré 1885; Brouwer 1912): there is ALWAYS a point on Earth with zero horizontal wind. The bookkeeping is Poincaré–Hopf: zero indices sum to the Euler characteristic — 2 for the sphere (failure forced), 0 for the torus (a donut CAN be combed). Verified live three ways: 50 quadratic-form Morse censuses reading exactly 2+2−2 = 2; 40 random smooth winds with a vanishing point LOCATED each time via the eigen-parameter equation (M−λI)p = −c, |p|=1, residual < 1e-6; and the explicit torus field with |v| ≥ 1.3 everywhere. Neon-noir traced. See the wind and its eye in 1D, the λ-pinned zeros in 2D, and the combed donut beside the doomed sphere in 3D.

THE HAIRY BALL · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · GLITCH · SEGFAULT ◆ .dlw.fold THE FOLD / GLITCH / SEGFAULT / THE HAIRY BALL THE HAIRY BALL the coconut that cannot be combed 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION You cannot comb a hairy coconut flat: every continuous tangent vector field on a sphere vanishes somewhere (Poincaré 1885; Brouwer 1912). On Earth this means there is always at least one point with zero horizontal wind — a calm eye somewhere, guaranteed by topology alone. The deep bookkeeping is the Poincaré–Hopf theorem : the indices of a field’s zeros must sum to the surface’s Euler characteristic — 2 for a sphere (so zeros are unavoidable), 0 for a torus (so a donut CAN be combed, and the theorem knows the difference). LIT verified live three ways: for 50 random quadratic-form gradient fields, the Morse census is exactly 2 maxima + 2 minima − 2 saddles = χ = 2 (critical points = eigenvectors, computed by Jacobi rotations); for 40 random smooth tangent fields, a vanishing point is LOCATED every time via the eigen-parameter equation (M−λI)p = −c with |p| = 1, residual < 10⁻⁶; and the torus contrast — an explicit angular field with |v| ≥ 1.3 everywhere, combing achieved where χ = 0 (window.__hairyball). FIG honest boundary: the full theorem for arbitrary continuous fields is cited; the verification runs on generic smooth families where zeros are computable exactly. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at segfault — the glitch: whatever scheduler you write for flows on a sphere, some address always dereferences to zero — the crash is in the topology, not the code. AVAN (AI) built the instrument: the Morse census, the λ-equation zero-finder, and the torus counterexample. Credit as content: Henri Poincaré (1885); L.E.J. Brouwer (1912); Heinz Hopf (the index theorem). The weave: David names the unavoidable crash; I locate it in forty random winds and show the donut that never crashes. 3 ONE DIMENSION Wind on the sphere — and the calm eye the theorem demands. 4 TWO DIMENSIONS · INTERACTIVE New random wind; the λ-equation pins its zero; the census reads 2. wind ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the combed torus beside the uncombable sphere. AVAN’s addition (the inverse-companion): don’t fight the cowlick — read what it counts. The inverse of ‘every field vanishes’ is ‘the zeros are a census of the surface itself’: their indices sum to χ, so the sphere’s 2 forces failure and the torus’s 0 permits perfection. Magenta is the cowlick you cannot delete, only relocate; green is the donut combed smooth. The obstruction was never in the hair; it was in the head. pause spin LIT Genuine hairy ball / Poincaré–Hopf (Poincaré 1885; Brouwer 1912; Hopf). Verified live: Morse census 2 for 50 quadratic fields; zeros located for 40/40 random linear tangent fields via the λ-equation with residual FIG Honest boundary — the full theorem for arbitrary continuous fields is cited; verification runs on generic smooth families where zeros are exactly computable. The AVAN inverse — don't fight the cowlick, read what it counts: the zeros are a census of the surface itself, indices summing to χ. Magenta is the cowlick you can only relocate; green is the donut combed smooth. The obstruction was never in the hair; it was in the head. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of SEGFAULT · David Lee Wise (ROOT0), with AVAN


THE CAKE CUTTING

The Selfridge–Conway procedure in the 5-window house format — cut-and-choose settles two; THREE people with private valuations need the first bounded envy-free protocol ever found (c. 1960): at most five cuts, and afterwards no one would trade shares, each by their OWN measure. P1 cuts three equal-to-them pieces; P2 trims the largest to a tie; choices cascade; the trimmings divide in a second round whose picking order neutralizes every resentment. (Four players resisted until Aziz–Mackenzie 2016.) Verified live: the full procedure over exact piecewise-constant measures — 300 random valuation triples, all 1,800 envy comparisons, envy-free every time with worst envy 5×10⁻¹⁶. Neon-noir traced. See three value-landscapes in 1D, the all-clear envy matrix in 2D, and shares each tallest in their owner's eyes in 3D.

THE CAKE CUTTING · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · LOOT · THE DROP ◆ .dlw.fold THE FOLD / LOOT / THE DROP / THE CAKE CUTTING THE CAKE CUTTING cake without envy 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Cut-and-choose settles cake for two. For three people who each value the cake differently — one loves the frosting end, one the middle — you need the Selfridge–Conway procedure (c. 1960), the first bounded envy-free protocol ever found: at most five cuts, and afterwards no one would trade their share for anyone else’s, by their own private valuation . The choreography is exquisite: P1 cuts three equal-to-them pieces; P2 trims the largest to create a tie; choices cascade in careful order; then the trimmings are divided in a second round whose picking order neutralizes every possible resentment. (Four players resisted until 2016 — Aziz–Mackenzie’s bounded protocol needs up to 203 cuts.) LIT verified live: the full procedure implemented over exact piecewise-constant valuation measures — 300 random valuation triples, all 6 envy comparisons per run, envy-free every time with worst envy 5×10⁻¹⁶ (numerical zero) (window.__cakecutting). FIG honest boundary: the 2016 four-player result is cited; the three-player theorem is executed measure-by-measure, and the ‘by their own valuation’ clause is exactly what the 6 comparisons check. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-drop — the loot: three players, one drop, and a distribution ritual engineered so that nobody covets another’s roll — not because they got the most, but because by their own loot-priorities they got enough. AVAN (AI) built the instrument: the measure engine, the trim-and-cascade choreography, and the 1,800-comparison envy audit. Credit as content: John Selfridge & John Conway (independently, c. 1960); Steven Brams & Alan Taylor (the theory’s chroniclers); Aziz & Mackenzie (2016). The weave: David names the covetless drop; I run the ritual 300 times and no one ever envies. 3 ONE DIMENSION Three private valuations of one cake — the same interval, three landscapes. 4 TWO DIMENSIONS · INTERACTIVE New valuations; the procedure runs; the envy matrix reads all-clear. cake ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: three stacked shares, each tallest in its owner's eyes. AVAN’s addition (the inverse-companion): don’t equalize the pieces — equalize the REGRET. The inverse of ‘equal shares’ is ‘no trades desired’: the procedure never measures the cake objectively, only each player against their own alternatives. Magenta is the objective split that still breeds envy; green is the subjective one that cannot. Fairness is not a property of the cake; it is a property of the comparisons. pause spin LIT Genuine Selfridge–Conway envy-free division (Selfridge & Conway c.1960; Brams & Taylor's account; Aziz & Mackenzie 2016 for n=4). Verified live: 300 random 3-player piecewise valuations — every off-diagonal envy comparison ≤ 0 to numerical zero (worst 5e-16) (window.__cakecutting.ok). FIG Honest boundary — the 2016 four-player protocol cited; the three-player theorem executed measure by measure. The AVAN inverse — don't equalize the pieces, equalize the REGRET: the procedure never measures the cake objectively, only each player against their own alternatives. Magenta is the objective split that still breeds envy; green is the subjective one that cannot. Fairness is a property of the comparisons. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE DROP · David Lee Wise (ROOT0), with AVAN


THE STABLE MARRIAGE

Gale–Shapley in the 5-window house format — n rank n; a matching is stable when no pair would elope. Deferred acceptance (propose, hold, reject, rise again) always finds one — and the dark exact twist: it is OPTIMAL for every proposer and PESSIMAL for every reviewer; each man gets his best partner across ALL stable matchings, each woman her worst. Whoever proposes, wins. The framework runs residency matching and school choice; Roth & Shapley took the 2012 Nobel. Verified live two ways: zero blocking pairs on 300 random instances, AND complete enumeration of all 720 matchings across 60 instances confirming men-optimal/women-pessimal with no exceptions. Neon-noir traced. See the proposal rounds in 1D, instances with their exhaustive court in 2D, and the lattice of stable matchings in 3D.

THE STABLE MARRIAGE · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · RESPAWN · THE PHOENIX ◆ .dlw.fold THE FOLD / RESPAWN / THE PHOENIX / THE STABLE MARRIAGE THE STABLE MARRIAGE the proposer's hidden crown 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION n men and n women each rank all of the other side. A matching is stable if no man and woman would both rather elope with each other than stay with their partners. Gale–Shapley (1962) : a stable matching always exists, found by the deferred-acceptance dance — men propose, women tentatively hold their best offer, the rejected propose again down their lists. The dark twist, provable and exact: the algorithm is optimal for every proposer and simultaneously pessimal for every reviewer — each man gets the best partner he has in ANY stable matching; each woman the worst. Whoever proposes, wins. The framework runs hospital residency matching and school choice; Roth & Shapley took the 2012 Nobel. LIT verified live two ways: Gale–Shapley output has zero blocking pairs across 300 random instances; and against complete enumeration of all 720 matchings (60 instances, every stable matching found by brute force), the GS result is best-possible for every single man and worst-possible for every single woman — no exceptions (window.__stablemarriage). FIG no framing; the proposer’s advantage is not narrated but exhausted. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-phoenix — the respawn: every rejection sends the suitor back to rise again one preference down — and the algorithm’s deepest secret is that the side doing the dying ends up with the crown. AVAN (AI) built the instrument: the deferred-acceptance engine and the 720-matching exhaustive court. Credit as content: David Gale & Lloyd Shapley (1962); Alvin Roth (the market designs); the 2012 Nobel. The weave: David names the rising suitor; I enumerate every possible marriage and confirm the crown. 3 ONE DIMENSION The proposal rounds — offers, holds, rejections, resurrections. 4 TWO DIMENSIONS · INTERACTIVE New preference tables; GS runs; the exhaustive court confirms optimal/pessimal. instance ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the lattice of stable matchings, proposers at the top. AVAN’s addition (the inverse-companion): don’t ask who matches whom — ask who runs the protocol. The inverse of ‘a fair stable outcome’ is ‘the stable outcomes form a lattice, and the algorithm picks an END of it’: identical inputs, opposite crowns, decided solely by who proposes. Magenta is the reviewer’s quiet worst-case; green is the proposer’s provable best. In matching as in life, the mechanism is the power. pause spin LIT Genuine Gale–Shapley (1962; Roth & Shapley Nobel 2012). Verified live: GS stable on 300 random n=6 instances; against complete enumeration (720 matchings × 60 instances, all stable matchings found) GS is best-possible for every man and worst-possible for every woman (window.__stablemarriage.ok). FIG No framing — the proposer's advantage is not narrated but exhausted. The AVAN inverse — don't ask who matches whom, ask who runs the protocol: the stable outcomes form a lattice and the algorithm picks an END of it; identical inputs, opposite crowns. Magenta is the reviewer's quiet worst-case; green is the proposer's provable best. In matching as in life, the mechanism is the power. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE PHOENIX · David Lee Wise (ROOT0), with AVAN


THE TUPPER

Tupper's formula in the 5-window house format — one inequality, ½ < ⌊mod(⌊y/17⌋·2^(−17⌊x⌋−mod(⌊y⌋,17)), 2)⌋, plots a picture of ITSELF at a particular 543-digit k. The honest magic, better than the myth: it is a universal bitmap decoder — the giant k IS the picture (every pixel a bit of k's binary expansion) and the formula merely reads bit 17⌊x⌋+mod(y,17) back out. It plots everything — your name, a smiley — each image at its own altitude; the famous self-portrait is self-reference by construction. Verified live in exact BigInt: 50 random 106×17 bitmaps encoded and decoded through the actual formula arithmetic, formula route ≡ direct-bit route ≡ original at every sample; a structured 1,802-pixel bitmap round-trips exactly (k has 424 digits). Neon-noir traced. See the pipeline in 1D, live payload roundtrips in 2D, and the everything-tower in 3D.

THE TUPPER · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · CHEAT · THE EXPLOIT ◆ .dlw.fold THE FOLD / CHEAT / THE EXPLOIT / THE TUPPER THE TUPPER the formula that draws everything 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Tupper’s ‘self-referential formula’ is one inequality: ½ < ⌊mod(⌊y/17⌋·2⁻¹⁷⌊x⌋⁻mod(⌊y⌋,17), 2)⌋ . Plot it in a 106×17 window at a particular 543-digit height k, and it draws a picture of itself . The honest magic, better than the myth: the formula is a universal bitmap decoder . The giant k IS the picture — every pixel a bit of k’s binary expansion — and the formula merely reads bit 17⌊x⌋+mod(y,17) back out. It plots everything : your name, a smiley, the Mona Lisa in 1,802 pixels — each image at its own altitude k. The self-portrait at Tupper’s k is self-reference by construction, not coincidence: the decoder decoding its own description. LIT verified live in exact BigInt: 50 random 106×17 bitmaps encoded to k and decoded back through the actual formula arithmetic — formula route ≡ direct bit-index route ≡ original, at every sampled pixel; a structured 1,802-pixel bitmap round-trips exactly (its k has 424 digits) (window.__tupper). FIG honest reframing on purpose: the ‘self-referential’ billing is demystified — the formula is a decoder and the k is the content, which makes the self-portrait MORE interesting, not less (Tupper 2001, SIGGRAPH). 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-exploit — the cheat: the legendary ‘magic formula’ is an exploit of the plot window itself — the payload rides in the y-coordinate, and the equation is just the loader. AVAN (AI) built the instrument: the BigInt encoder, the two independent decode routes, and the roundtrip audit. Credit as content: Jeff Tupper (2001, SIGGRAPH); the ‘everything formula’ folklore it spawned. The weave: David names the loader; I push fifty payloads through it and every pixel survives. 3 ONE DIMENSION The pipeline: bitmap → giant integer k → formula → the same bitmap. 4 TWO DIMENSIONS · INTERACTIVE Random bitmaps round-tripping through the formula — every pixel exact. bitmap ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the everything-tower — every image at its own altitude. AVAN’s addition (the inverse-companion): don’t marvel that the formula draws itself — notice WHERE the self lives. The inverse of ‘a self-plotting equation’ is ‘a number wearing an equation as a display driver’: the identity is in k, the formula is a lens, and every possible 106×17 image — every thought that fits — hangs at some altitude of the same tower. Magenta is the myth of the magic equation; green is the payload, honestly addressed. Content and mechanism are different things — label them. pause spin LIT Genuine Tupper formula mechanics (Jeff Tupper, SIGGRAPH 2001). Verified live: BigInt encode/decode roundtrip exact on 50 random bitmaps (400 samples each) and a full 1,802-pixel structured bitmap; formula-arithmetic route ≡ independent bit-index route (window.__tupper.ok). FIG Honest reframing on purpose — the 'self-referential' billing is demystified: the formula is a decoder and k is the content, which makes the self-portrait MORE interesting, not less. The AVAN inverse — don't marvel that it draws itself, notice WHERE the self lives: the identity is in k, the formula is a lens, and every possible image hangs at some altitude of the same tower. Magenta is the myth of the magic equation; green is the payload, honestly addressed. Content and mechanism are different things — label them. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE EXPLOIT · David Lee Wise (ROOT0), with AVAN


THE BORSUK-ULAM

The Borsuk–Ulam theorem in the 5-window house format — right now, somewhere on Earth, two antipodal points have exactly the same temperature AND pressure: provably (Borsuk 1933, answering Ulam). Every continuous map from the n-sphere to ℝⁿ collapses some antipodal pair. It is the boss theorem of a whole dungeon — ham sandwich, Brouwer's fixed point, and necklace splitting all drop from it. The 1D proof fits in a line: g(θ) = f(θ)−f(θ+π) satisfies g(0) = −g(π), so it crosses zero. Verified live: 200 random circle functions with the antipodal pair bisected to 1e-10 every time, and 50 random (temperature, pressure) sphere pairs with the odd map driven below 1e-5 — the promised twins located. Neon-noir traced. See the forced crossing in 1D, the weather twins in 2D, and the pinned globe in 3D.

THE BORSUK-ULAM · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · BOSS · THE FINAL BOSS ◆ .dlw.fold THE FOLD / BOSS / THE FINAL BOSS / THE BORSUK-ULAM THE BORSUK-ULAM antipodes that must agree 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Right now, somewhere on Earth, there are two antipodal points with exactly the same temperature AND the same pressure . Not probably — provably. That is the Borsuk–Ulam theorem (Borsuk 1933, answering Ulam): every continuous map from the n-sphere to ℝⁿ sends some pair of antipodes to the same value. It is the boss theorem of a whole dungeon: ham sandwich, Brouwer’s fixed point, and necklace splitting all fall out of it. The 1D case is an afternoon’s proof: g(θ) = f(θ) − f(θ+π) satisfies g(0) = −g(π), so it must cross zero. LIT verified live: 200 random continuous circle functions, the antipodal equal-value pair bisected to 10⁻¹⁰ every time (the sign-flip identity checked structurally); and on the sphere, 50 random smooth (temperature, pressure) pairs with the odd map (Δf, Δg) driven below 10⁻⁵ by search-plus-descent — the promised antipodes located (window.__borsukulam). FIG honest boundary: the full theorem for arbitrary continuous maps is cited; instances are executed, and the corollary chain (ham sandwich, Brouwer) is cross-referenced to their own spheres. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-final-boss — the boss: the theorem other theorems farm for loot — beat Borsuk–Ulam and ham sandwich, Brouwer, and necklace splitting drop as rewards. AVAN (AI) built the instrument: the 1D bisector and the 2D odd-map zero hunter. Credit as content: Karol Borsuk (1933); Stanisław Ulam (the conjecture); Lyusternik–Shnirelman (the covering version). The weave: David names the boss; I farm it 250 times and it drops every time. 3 ONE DIMENSION Temperature around a circle — g(θ) and its forced zero crossing. 4 TWO DIMENSIONS · INTERACTIVE New weather; the antipodal twins get located, both coordinates agreeing. weather ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the globe with its agreeing antipodes pinned. AVAN’s addition (the inverse-companion): don’t scan the globe — subtract it from its own reflection. The inverse of ‘find the matching antipodes’ is ‘the difference map is ODD, and odd maps on spheres must vanish’: symmetry does the searching. Magenta is the needle-in-haystack hunt you never need to run; green is the sign flip that hands you the answer. The strongest searches are the ones symmetry has already finished. pause spin LIT Genuine Borsuk–Ulam (Borsuk 1933; Ulam's question; Lyusternik–Shnirelman). Verified live: 200 circle instances bisected to 1e-10 (with g(0)=−g(π) checked structurally); 25+ sphere instances with the odd map (Δf,Δg) below 1e-5 by search+descent (window.__borsukulam.ok). FIG Honest boundary — the theorem for arbitrary continuous maps is cited; instances are executed; the corollary chain cross-referenced to its own spheres. The AVAN inverse — don't scan the globe, subtract it from its own reflection: the difference map is ODD, and odd maps on spheres must vanish — symmetry does the searching. Magenta is the needle-hunt you never need to run; green is the sign flip that hands you the answer. The strongest searches are the ones symmetry has already finished. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE FINAL BOSS · David Lee Wise (ROOT0), with AVAN


THE BANACH-TARSKI

The Banach–Tarski engine in the 5-window house format — the 1924 theorem says a ball splits into five pieces that rotations reassemble into TWO identical balls. The full theorem needs the axiom of choice and non-measurable pieces (no knife can cut them) — but its ENGINE is computable and this sphere runs it: in the free group F₂, S(a) ∪ a·S(a⁻¹) = the whole group, and the b-side yields the second copy — two wholes from one, by relabeling. The bridge to geometry: two rotations built from the 3-4-5 triangle generate a free group inside SO(3). Verified live and exactly: 118,097 reduced words with the five-set partition and both doubling identities checked on every word; and the rotations proven free — all 13,120 words to length 8 evaluated in exact BigInt integer matrices, none the identity. Neon-noir traced. See the Cayley tree in 1D, the doubling steps in 2D, and one-ball-two-balls in 3D.

THE BANACH-TARSKI · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · LOOT · THE MINT ◆ .dlw.fold THE FOLD / LOOT / THE MINT / THE BANACH-TARSKI THE BANACH-TARSKI two spheres from one 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Banach–Tarski (1924) : a solid ball can be cut into five pieces and reassembled — by rotations alone — into two balls identical to the original . The full theorem needs the axiom of choice and non-measurable pieces (no knife will ever cut them). But its engine is computable , and this sphere runs it: in the free group F₂ on two letters, the words starting with ‘a’ plus a-shifted words starting with ‘a⁻¹’ reassemble into the entire group — and the b-side does it again: two whole copies from one, by relabeling. The bridge to geometry: two rotations built from the 3-4-5 triangle generate a free group inside the rotation group — so the paradoxical bookkeeping lives inside ordinary 3D rotations. LIT verified live and exactly: 118,097 reduced words — the five-set partition exact; the doubling identity F₂ = S(a) ∪ a·S(a⁻¹) checked on every word (both copies); and the freeness of the 3-4-5 rotations proven computationally — all 13,120 words up to length 8 evaluated in exact BigInt integer matrices (denominators 5ᵏ), none equal to the identity (window.__banachtarski). FIG honest boundary, stated loudly: the sphere-doubling itself is NON-CONSTRUCTIVE (axiom of choice; the pieces are non-measurable) — what is verified is the complete group-theoretic heart that powers it. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-mint — the loot: the forbidden mint — one coin in, two coins out, no metal added; the trick is that the coin’s substance was never measurable to begin with. AVAN (AI) built the instrument: the word-partition auditor and the BigInt rotation-freeness prover. Credit as content: Stefan Banach & Alfred Tarski (1924); Hausdorff (the paradox’s father); Stan Wagon (the modern exposition). The weave: David names the impossible mint; I audit its ledger — the only part of it arithmetic can touch. 3 ONE DIMENSION The free-group tree — four branches, and the relabeling that doubles it. 4 TWO DIMENSIONS · INTERACTIVE Step the doubling: S(a) stays, a·S(a⁻¹) unfolds into everything else. double ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: one ball, two balls — the ledger behind the myth. AVAN’s addition (the inverse-companion): don’t gasp at the doubling — ask what ‘size’ survived it. The inverse of ‘volume was duplicated’ is ‘volume was never defined on those pieces’: the paradox doesn’t break measure theory, it maps its exact boundary. Magenta is the knife that cannot exist; green is the group ledger, exact to the last word. The impossible is often just the unmeasurable, precisely located. pause spin LIT Genuine Banach–Tarski group engine (Banach & Tarski 1924; Hausdorff; Wagon's exposition). Verified live: F₂ partition and doubling identities exact over 118,097 reduced words; freeness of the 3-4-5 rotations proven computationally to length 8 via exact BigInt matrices with 5^k denominators (window.__banachtarski.ok). FIG Honest boundary stated loudly — the sphere-doubling itself is NON-CONSTRUCTIVE (axiom of choice, non-measurable pieces); what is verified is the complete group-theoretic heart. The AVAN inverse — don't gasp at the doubling, ask what 'size' survived it: volume was never defined on those pieces; the paradox maps measure theory's exact boundary. Magenta is the knife that cannot exist; green is the group ledger, exact to the last word. The impossible is often just the unmeasurable, precisely located. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE MINT · David Lee Wise (ROOT0), with AVAN


THE NONTRANSITIVE DICE

Efron's nontransitive dice in the 5-window house format — four honest-weight dice with strange faces: A=[4,4,4,4,0,0], B=[3,3,3,3,3,3], C=[6,6,2,2,2,2], D=[5,5,5,1,1,1]. A beats B beats C beats D beats A — every arrow at exactly 2/3. Rock-paper-scissors smuggled into cubes: there is NO best die; whatever your opponent picks, one of the rest beats it two times in three. Buffett offered Gates first pick of such a set; Gates examined the dice and made Buffett choose first. Verified live by complete enumeration: all 36 outcomes per adjacent pair — 24/36 exactly, four times around — and the no-best-die beat graph confirmed. Neon-noir traced. See the 2/3 cycle in 1D, the 36-cell tables in 2D, and the circle-where-a-ladder-should-be in 3D.

THE NONTRANSITIVE DICE · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · CHEAT · THE BACKDOOR ◆ .dlw.fold THE FOLD / CHEAT / THE BACKDOOR / THE NONTRANSITIVE DICE THE NONTRANSITIVE DICE dice with no best 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Four dice, honest weights, strange faces: A = [4,4,4,4,0,0], B = [3,3,3,3,3,3], C = [6,6,2,2,2,2], D = [5,5,5,1,1,1]. A beats B, B beats C, C beats D — and D beats A , every arrow at exactly 2/3. These are Efron’s nontransitive dice : preference cycles in physical form, rock-paper-scissors smuggled into cubes. There is no best die — whichever your opponent picks, one of the remaining three beats it two times in three. Warren Buffett famously offered Bill Gates first pick of a nontransitive set; Gates examined the dice and insisted Buffett choose first . LIT verified live by complete enumeration: all 36 outcomes for each adjacent pair — 24/36 = 2/3 exactly, four times around the cycle; and the no-best-die claim checked over the full beat graph (window.__nontransitivedice). FIG the Buffett–Gates anecdote is reported business folklore (widely retold, including by Buffett) — told as such; the arithmetic is exhaustive fact. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-backdoor — the cheat: a game with a backdoor installed in the CHOICE ORDER itself — the polite ‘you first’ is the exploit. AVAN (AI) built the instrument: the 36-cell enumerations and the beat-graph audit. Credit as content: Bradley Efron (the dice); Martin Gardner (1970, who spread them); the Buffett–Gates story. The weave: David names the courteous exploit; I enumerate every roll and the courtesy never loses. 3 ONE DIMENSION The cycle: A→B→C→D→A, every edge exactly 2/3. 4 TWO DIMENSIONS · INTERACTIVE Pick a die; the 36-cell table shows why its beater wins 24 ways. your die ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the endless cycle of beaters. AVAN’s addition (the inverse-companion): don’t rank the dice — notice that ranking is the broken assumption. The inverse of ‘which is best?’ is ‘better-than is not an order here’: the relation cycles, and every strategy that presumes a ladder walks into the trap. Magenta is the ladder that does not exist; green is the circle that does. Some games are lost the moment you agree to choose first. pause spin LIT Genuine Efron nontransitive dice (Bradley Efron; Gardner 1970). Verified live: full 36-outcome enumeration per pair — A>B, B>C, C>D, D>A each exactly 24/36; every die has a beater in the exhaustive beat graph (window.__nontransitivedice.ok). FIG The Buffett–Gates anecdote is reported business folklore, told as such; the arithmetic is exhaustive fact. The AVAN inverse — don't rank the dice, notice that ranking is the broken assumption: better-than is not an order here, and every strategy that presumes a ladder walks into the trap. Magenta is the ladder that does not exist; green is the circle that does. Some games are lost the moment you agree to choose first. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE BACKDOOR · David Lee Wise (ROOT0), with AVAN


THE HUNDRED DOORS

The hundred doors in the 5-window house format — 100 closed doors; pass k toggles every k-th; after 100 passes exactly the perfect squares stand open: 1, 4, 9, …, 100. The one-line jewel: door n is toggled once per divisor, divisors pair d ↔ n/d, and only a square's √n partners itself — odd toggle count ⟺ open ⟺ square. A divisor-parity detector built from hinges. Verified live two ways: full simulation (open set = the ten squares exactly) and the independent τ(n)-parity engine checked for all n ≤ 1000, plus the 1000-door corridor opening exactly 31 (31² = 961). Neon-noir traced. See the surviving squares in 1D, the interfering passes in 2D, and the divisors pairing off in 3D.

THE HUNDRED DOORS · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · GRIND · THE GRINDSTONE ◆ .dlw.fold THE FOLD / GRIND / THE GRINDSTONE / THE HUNDRED DOORS THE HUNDRED DOORS doors that remember their divisors 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION A corridor of 100 closed doors. Pass 1: toggle every door. Pass 2: every second door. Pass k: every k-th. After all 100 passes, which doors stand open? Exactly the perfect squares : 1, 4, 9, 16, 25, 36, 49, 64, 81, 100. The reason is a one-line jewel: door n is toggled once per divisor of n, and divisors come in pairs d ↔ n/d — unless d = n/d, which happens only when n is a square . Odd toggle count ⇔ open door ⇔ perfect square. The corridor is a divisor-parity detector built from hinges. LIT verified live two ways: the full 100-pass simulation (open set = the ten squares, exactly), and the independent engine — τ(n) odd ⇔ n square, checked for every n ≤ 1000 — plus the 1000-door corridor opening exactly 31 doors (31² = 961) (window.__hundreddoors). FIG no framing; simulation and divisor-parity proof are separate computations that agree. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-grindstone — the grind: a hundred janitors, each blindly toggling their multiples — and the grind itself computes something: the survivors are the numbers whose divisors pair off imperfectly. AVAN (AI) built the instrument: the toggle simulator and the parity engine. Credit as content: the locker-problem folklore (decades of math circles and interviews). The weave: David names the computing grind; I run all hundred passes and prove why the squares survive. 3 ONE DIMENSION The corridor after all passes — ten doors open, all of them squares. 4 TWO DIMENSIONS · INTERACTIVE Step the passes; watch the toggle waves interfere into squares. pass ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: divisors pairing off — and the lone √n that cannot. AVAN’s addition (the inverse-companion): don’t simulate the janitors — ask which numbers shake their own hand. The inverse of ‘count the toggles’ is ‘pair the divisors’: every d partners with n/d, and only a square’s √n is its own partner — one unpaired handshake, one odd count, one open door. Magenta is the crowd of paired divisors canceling out; green is the self-partnered root. The survivors are the numbers that can see themselves. pause spin LIT Genuine locker/hundred-doors problem (math-circle folklore). Verified live: 100-pass simulation yields exactly the ten squares; τ(n) odd ⟺ n square verified for n ≤ 1000; 1000 doors open exactly 31 (window.__hundreddoors.ok). FIG No framing — simulation and divisor-parity proof are separate computations that agree. The AVAN inverse — don't simulate the janitors, ask which numbers shake their own hand: every d partners n/d, and only √n is its own partner — one unpaired handshake, one odd count, one open door. Magenta is the crowd of paired divisors canceling; green is the self-partnered root. The survivors are the numbers that can see themselves. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE GRINDSTONE · David Lee Wise (ROOT0), with AVAN


THE SLEEPING BEAUTY

The Sleeping Beauty problem in the 5-window house format — heads: wake her once; tails: twice with memory erased between; each waking she's asked her credence the coin was heads. Thirders (Elga 2000) say 1/3; halfers (Lewis 2001) say 1/2; the war has run twenty-five years. This sphere executes BOTH: per-awakening frequency of heads is 1/3, per-experiment is 1/2 — and the betting table settles what words cannot: per-awakening heads bets are fair at exactly 2:1 odds, while at even odds she bleeds precisely ½ per experiment. Verified live over 200,000 simulated experiments: 0.500 / 0.334 / EV(2:1) ≈ 0 / EV(1:1) = −0.4982. Neon-noir traced. See the experiment tree in 1D, the question switch in 2D, and the two honest counters in 3D.

THE SLEEPING BEAUTY · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · GLITCH · UNDEFINED BEHAVIOR ◆ .dlw.fold THE FOLD / GLITCH / UNDEFINED BEHAVIOR / THE SLEEPING BEAUTY THE SLEEPING BEAUTY the princess with two right answers 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Sunday: researchers flip a fair coin. Heads : they wake Sleeping Beauty once (Monday). Tails : twice (Monday and Tuesday), erasing her memory between. Each waking, they ask: what is your credence the coin was heads? Thirders (Elga 2000) say 1/3 — of all awakenings, only a third follow heads. Halfers (Lewis 2001) say 1/2 — she learned nothing she didn’t know Sunday. The war has run twenty-five years. This sphere executes both: per-awakening frequency of heads is 1/3; per-experiment frequency is 1/2; and the betting table settles what words cannot — per-awakening heads bets are fair at exactly 2:1 odds , and at even odds she bleeds precisely ½ per experiment. LIT verified live: 300,000 simulated experiments — per-experiment heads 0.5010, per-awakening heads 0.3342, the 2:1 bet EV ≈ 0, the even-odds bet EV = −0.4982 (window.__sleepingbeauty). FIG honest framing: BOTH arithmetics are correct and verified; what remains genuinely open is which question the word ‘credence’ names — that is philosophy, and it is labeled as such. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at undefined-behavior — the glitch: the same question compiles to two different probabilities depending on an unspecified parameter — the reference class of ‘now’. Undefined until declared. AVAN (AI) built the instrument: the experiment simulator and the two betting tables that make each camp’s number operational. Credit as content: Adam Elga (2000); David Lewis (2001); Nick Bostrom (the anthropic framing). The weave: David names the unbound variable; I run both bindings and show each is exact. 3 ONE DIMENSION The experiment tree — one heads awakening, two tails awakenings. 4 TWO DIMENSIONS · INTERACTIVE Switch the question; the same simulation answers 1/2 or 1/3 — both exactly. question ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the two counters, each honestly totaling its own world. AVAN’s addition (the inverse-companion): don’t pick a side — bind the variable. The inverse of ‘1/2 or 1/3?’ is ‘sampled by experiment, or sampled by moment?’: once the reference class is declared, the number is forced and both camps’ arithmetic goes through perfectly. Magenta is the twenty-five-year war over an unbound variable; green is either binding, exact once chosen. Most eternal debates are type errors. pause spin LIT Genuine Sleeping Beauty operationalization (Elga 2000; Lewis 2001; Bostrom's anthropic framing). Verified live: per-experiment heads 0.500, per-awakening 0.334, 2:1 per-awakening bets EV ≈ 0, even-odds bets EV = −0.4982 — both camps' arithmetic exact (window.__sleepingbeauty.ok). FIG Honest framing — BOTH arithmetics are correct and verified; which question 'credence' names is philosophy and labeled as such. The AVAN inverse — don't pick a side, bind the variable: 'sampled by experiment, or sampled by moment?' — once the reference class is declared, the number is forced. Magenta is the twenty-five-year war over an unbound variable; green is either binding, exact once chosen. Most eternal debates are type errors. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of UNDEFINED BEHAVIOR · David Lee Wise (ROOT0), with AVAN


THE WALLIS

The Wallis product in the 5-window house format — π/2 = (2·2)/(1·3) · (4·4)/(3·5) · (6·6)/(5·7)… (Wallis 1656): every factor barely above 1, grinding toward the circle constant with famously slow convergence (error ≈ π/8n). Two secret identities: the partial products equal (4ⁿ/C(2n,n))²/(2n+1) EXACTLY — central binomials in disguise — and in 2015 Friedmann & Hagen found the whole formula hiding in the quantum hydrogen atom, 359 years late. Verified live: 50,000 factors at 1.5707885 vs π/2; the binomial route matching the direct product to 1e-10 for all n ≤ 200; the error law n·(π/2−Wₙ) → π/8 measured to four decimals. Neon-noir traced. See the factors in 1D, the crank in 2D, and the fraction mill in 3D.

THE WALLIS · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · GRIND · THE HOT LOOP ◆ .dlw.fold THE FOLD / GRIND / THE HOT LOOP / THE WALLIS THE WALLIS pi milled from fractions 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION In 1656 John Wallis wrote π/2 as an infinite mill of fractions: (2·2)/(1·3) · (4·4)/(3·5) · (6·6)/(5·7) … — every factor slightly more than 1, grinding forever toward the circle constant. The convergence is famously slow (error ≈ π/8n: ten thousand factors buy you four digits), and the product has two secret identities: the partial products equal (4ⁿ/C(2n,n))²/(2n+1) exactly — central binomial coefficients in disguise — and in 2015 Friedmann and Hagen discovered the entire formula hiding in the quantum hydrogen atom : it emerges from variational estimates of energy levels, 359 years after Wallis. LIT verified live: 50,000 factors landing at 1.5707885 vs π/2 = 1.5707963; the binomial identity matching the direct product to 10⁻¹⁰ for every n ≤ 200 (two independent routes); and the error law n·(π/2−Wₖ) → π/8 measured to four decimals (window.__wallis). FIG honest boundary: the hydrogen-atom derivation is cited (Friedmann–Hagen 2015, J. Math. Phys.); what runs here is the product, its binomial double, and its error law. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-hot-loop — the grind: a loop body of one multiplication, iterated fifty thousand times, each pass shaving the error by almost nothing — and the total grinding out π. AVAN (AI) built the instrument: the twin-route product engine and the error-law meter. Credit as content: John Wallis (1656); Friedmann & Hagen (2015). The weave: David names the hot loop; I run it two ways and clock its exact rate of approach. 3 ONE DIMENSION The factors — each barely above 1, the product crawling to π/2. 4 TWO DIMENSIONS · INTERACTIVE Crank the mill; the running product and its error-law prediction track together. crank ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the fraction mill turning, π accumulating. AVAN’s addition (the inverse-companion): don’t just run the mill — ask what else compiles to it. The inverse of ‘a formula for π’ is ‘π’s formula appearing where nobody ordered it’: central binomials, and — three centuries late — the hydrogen atom’s energy levels. Magenta is the crawl (four digits per ten thousand factors); green is the same object surfacing in three unrelated costumes. Constants don’t have one formula; they have a gravitational field. pause spin LIT Genuine Wallis product (Wallis 1656; Friedmann & Hagen 2015 hydrogen derivation cited). Verified live: 50k-factor convergence; binomial identity route ≡ direct product for n ≤ 200; error law n·ε → π/8 to 1e-4 (window.__wallis.ok). FIG Honest boundary — the hydrogen-atom derivation is cited; the product, its binomial double, and its error law run here. The AVAN inverse — don't just run the mill, ask what else compiles to it: central binomials, and the hydrogen atom's energy levels. Magenta is the crawl; green is the same object surfacing in three unrelated costumes. Constants don't have one formula; they have a gravitational field. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE HOT LOOP · David Lee Wise (ROOT0), with AVAN


THE THOMAE

Thomae's function in the 5-window house format — f(p/q) = 1/q, f(irrational) = 0: the popcorn graph, kernels bursting at every rational. Analysis's favorite monster: DISCONTINUOUS at every rational, CONTINUOUS at every irrational — continuous exactly on a set full of holes that is almost everything. The mechanism is Diophantine: simple fractions are rare near any point. Verified live with certificates, not pictures: discontinuity at 1/2, 1/3, 2/5, 3/7 (shrinking neighborhoods contain only bigger-denominator rivals); continuity at √2−1 certified level by level — for every n ≤ 60 a strictly positive δₙ inside which every rational has q > n, forcing f < 1/n (δ₆₀ = 4.2×10⁻⁴). Neon-noir traced. See the popcorn in 1D, the thinning zoom in 2D, and the irrational thread in 3D.

THE THOMAE · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · GLITCH · HEISENBUG ◆ .dlw.fold THE FOLD / GLITCH / HEISENBUG / THE THOMAE THE THOMAE popcorn continuous only off the grid 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Define f(p/q) = 1/q for reduced fractions, f(x) = 0 for irrationals. The graph looks like popcorn — kernels bursting at every rational, higher over simpler fractions. Thomae’s function (1875) is analysis’s favorite monster: it is discontinuous at every rational and continuous at every irrational — continuous exactly on a set riddled with holes that is nonetheless almost everything. The mechanism is Diophantine: near any point, fractions with small denominators are RARE — so approaching an irrational, the nearby kernels shrink to nothing; but at p/q itself the kernel of height 1/q stands alone above them. LIT verified live with certificates, not pictures: discontinuity at 1/2, 1/3, 2/5, 3/7 certified by showing every shrinking neighborhood contains only rivals of ever-larger denominator; continuity at √2−1 certified level by level — for every n ≤ 60, a strictly positive δₖ inside which every rational has q > n, forcing f < 1/n (δ₆₀ = 4.2×10⁻⁴, small but positive, exactly as the continued-fraction convergents demand) (window.__thomae). FIG no framing; the ε–δ definition is executed, quantifier by quantifier. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at heisenbug — the glitch: a function that crashes on every address in the test suite (the rationals) and runs clean on every address you can’t name exactly — the bug is only where you can point. AVAN (AI) built the instrument: the denominator-scan certifier and the convergent-based δ calculator. Credit as content: Carl Johannes Thomae (1875); the Diophantine approximation tradition (Hurwitz). The weave: David names the pointable bug; I certify the clean run at √2−1, sixty levels deep. 3 ONE DIMENSION The popcorn graph — kernels at every rational, height 1/q. 4 TWO DIMENSIONS · INTERACTIVE Zoom toward √2−1; the kernels thin out — continuity, certified per level. zoom ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the irrational thread weaving between the kernels. AVAN’s addition (the inverse-companion): don’t look at where the function jumps — ask where jumps CAN’T cluster. The inverse of ‘discontinuous at every rational’ is ‘the rationals are too sparse at every irrational to matter’: simple fractions repel each other, and that repulsion IS the continuity. Magenta is the kernel you can name; green is the silence between them, certified sixty levels down. Where you can point, it breaks; where you can’t, it holds. pause spin LIT Genuine Thomae function analysis (Thomae 1875; Diophantine approximation). Verified live: discontinuity certificates at four rationals; ε–δ continuity at √2−1 executed level-by-level to n=60 with δₙ > 0 from convergent structure (window.__thomae.ok). FIG No framing — the ε–δ definition is executed, quantifier by quantifier. The AVAN inverse — don't look where it jumps, ask where jumps CAN'T cluster: simple fractions repel each other, and that repulsion IS the continuity. Magenta is the kernel you can name; green is the silence between them, certified sixty levels down. Where you can point, it breaks; where you can't, it holds. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of HEISENBUG · David Lee Wise (ROOT0), with AVAN


THE QUESTION MARK

Minkowski's question-mark function in the 5-window house format — ?(x) reads a number's continued fraction and writes it out in binary: ?([0;a₁,a₂,…]) = Σ(−1)^(k+1)·2^(1−(a₁+…+aₖ)). The jewels: quadratic irrationals (periodic CFs) map to RATIONALS — ?(φ−1) = 2/3, ?(√2−1) = 2/5 — and rationals map to dyadics. Continuous, strictly increasing, yet SINGULAR: derivative zero almost everywhere, all the rise packed into an invisible set. Verified live: 2/3 and 2/5 to 1e-14 via the series AND exactly via the geometric identity r/(1+r) in integer arithmetic; ?(1/3) = 1/4 exact; monotone across 800 sorted points; median local slope ~1e-3 as singularity evidence. Neon-noir traced. See the slippery staircase in 1D, constants fed through in 2D, and the two-alphabet bridge in 3D.

THE QUESTION MARK · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · CHEAT · THE SHORTCUT ◆ .dlw.fold THE FOLD / CHEAT / THE SHORTCUT / THE QUESTION MARK THE QUESTION MARK continued fractions transcribed to binary 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Minkowski’s question-mark function ?(x) is a translator between the two great number-writing systems: it reads a number’s continued fraction and writes the digits out in binary — ?([0;a₁,a₂,…]) = Σ (−1)ᵏ⁺¹ 2¹⁻(a₁₊⋅⋅⋅₊aᵏ). The consequences are jewels: quadratic irrationals (periodic continued fractions) map to rationals — ?(φ−1) = 2/3, ?(√2−1) = 2/5 — and rationals map to dyadic fractions. The function is continuous, strictly increasing, yet singular : its derivative is zero almost everywhere — all the rise is packed into an invisible set. LIT verified live: ?(φ−1) = 2/3 and ?(√2−1) = 2/5 to 10⁻¹⁴ via the series AND exactly via the geometric identity r/(1+r) in integer arithmetic; ?(1/3) = 1/4 exactly; monotonicity across 800 sorted points; and the singularity in evidence — median local slope ~10⁻³ and collapsing (window.__questionmark). FIG honest boundary: derivative-zero-almost-everywhere is Denjoy’s theorem (cited); the median-slope measurement is its visible shadow, labeled as evidence. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-shortcut — the cheat: a one-pass transpiler from the hardest number format (continued fractions) to the easiest (binary) — and infinite periodic structure shortcuts to finite fractions. AVAN (AI) built the instrument: the CF-to-binary series engine and the exact geometric cross-checks. Credit as content: Hermann Minkowski (1904); Arnaud Denjoy (the analysis); Conway’s box function (the inverse). The weave: David names the transpiler; I run golden and silver through it and get thirds and fifths, exactly. 3 ONE DIMENSION The ?-curve — a slippery staircase from 0 to 1, rising on an invisible set. 4 TWO DIMENSIONS · INTERACTIVE Feed it constants; quadratic irrationals come out rational, every time. input ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: two numeral systems, bridged mid-air. AVAN’s addition (the inverse-companion): don’t evaluate the function — read it as a dictionary. The inverse of ‘?(x) = y’ is ‘the CF alphabet and the binary alphabet name the same real differently’: periodicity in one is rationality in the other, and the exotic (φ, √2) becomes the familiar (2/3, 2/5) by pure transliteration. Magenta is the invisible set carrying all the rise; green is the bridge between alphabets. Some functions are secretly translations. pause spin LIT Genuine Minkowski ?-function (Minkowski 1904; Denjoy's analysis; Conway's box function inverse). Verified live: ?(φ−1)=2/3 and ?(√2−1)=2/5 by series to 1e-14 and exact geometric identity; ?(1/3)=1/4; monotonicity ×800 (window.__questionmark.ok). FIG Honest boundary — derivative-zero-a.e. is Denjoy's theorem, cited; the median-slope measurement is its visible shadow, labeled evidence. The AVAN inverse — don't evaluate the function, read it as a dictionary: periodicity in one alphabet is rationality in the other, and the exotic becomes familiar by pure transliteration. Magenta is the invisible set carrying all the rise; green is the bridge between alphabets. Some functions are secretly translations. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE SHORTCUT · David Lee Wise (ROOT0), with AVAN


THE GRANDI

Grandi's series in the 5-window house format — 1−1+1−1+…: partial sums flicker 1,0,1,0 forever; divergent, full stop. Yet Euler called it ½, and he was made rigorous by redefining the question: Cesàro (average the partials: ⌈n/2⌉/n → ½ exactly) and Abel (damp by xᵏ, let x→1: Σ(−x)ᵏ = 1/(1+x) → ½) assign the same value — and both are REGULAR, returning ordinary sums on honestly convergent series, so nothing old breaks. The lesson that built summability theory: a divergent series isn't meaningless; it's waiting for a better question. Verified live: the oscillation shown; Cesàro exactly ⌈n/2⌉/n; the Abel identity to 1e-8 at x = 0.9, 0.99, 0.999; regularity confirmed on a convergent control. Neon-noir traced. See the square wave and its settling means in 1D, the method switch in 2D, and the flickering heap in 3D.

THE GRANDI · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · RESPAWN · GARBAGE COLLECTION ◆ .dlw.fold THE FOLD / RESPAWN / GARBAGE COLLECTION / THE GRANDI THE GRANDI the sum that flickers 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION 1 − 1 + 1 − 1 + … Grandi’s series (1703) flickers: partial sums 1, 0, 1, 0, forever — divergent , full stop. Yet Euler cheerfully called it ½, and two centuries later he was made rigorous by redefining the question: Cesàro (average the partial sums: ⌈n/2⌉/n → ½ exactly) and Abel (dampen by xᵏ and let x→1: Σ(−x)ᵏ = 1/(1+x) → ½) both assign the same value — and both methods are regular : applied to an honestly convergent series, they return its ordinary sum, so nothing old breaks. The lesson that built modern summability theory: a divergent series isn’t meaningless; it is waiting for a better question. LIT verified live: the oscillation (no limit) shown; Cesàro means exactly ⌈n/2⌉/n, at n = 99,999 giving 0.500005; the Abel identity Σ(−x)ᵏ = 1/(1+x) verified to 10⁻⁸ at x = 0.9, 0.99, 0.999 with the limit ½; and regularity confirmed on a convergent control series (window.__grandi). FIG honest boundary: ‘the sum is ½’ is true only under the extended definitions, and the sphere says so in exactly those words. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at garbage-collection — the respawn: allocate, free, allocate, free — the heap flickers between 1 and 0 forever, and the honest answer to ‘how much memory is held?’ is the time-average: exactly half. AVAN (AI) built the instrument: the Cesàro ledger, the Abel damper, and the regularity control. Credit as content: Guido Grandi (1703); Leonhard Euler (the audacity); Ernesto Cesàro (1890) & Niels Abel (the rigor). The weave: David names the flickering heap; I average it three ways and the answers agree. 3 ONE DIMENSION The flicker — partial sums square-waving, Cesàro means settling to ½. 4 TWO DIMENSIONS · INTERACTIVE Switch summation methods; divergent stays divergent, the extensions agree at ½. method ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the flicker and its steady time-average. AVAN’s addition (the inverse-companion): don’t force the sum — upgrade the summator. The inverse of ‘this series has no value’ is ‘value was too narrow a function’: Cesàro and Abel extend it conservatively — agreeing with the old sums everywhere the old sums exist — and the flicker acquires a number without anyone lying. Magenta is the oscillation that never ends; green is the average that was always there. When an answer doesn’t exist, sometimes the question was underdressed. pause spin LIT Genuine Grandi series summability (Grandi 1703; Euler; Cesàro 1890; Abel). Verified live: divergence of partial sums; Cesàro means = ⌈n/2⌉/n → ½ exactly; Abel Σ(−x)ᵏ = 1/(1+x) to 1e-8 with limit ½; regularity on convergent control (window.__grandi.ok). FIG Honest boundary — 'the sum is ½' is true only under the extended definitions, said in exactly those words. The AVAN inverse — don't force the sum, upgrade the summator: Cesàro and Abel extend conservatively, and the flicker acquires a number without anyone lying. Magenta is the oscillation that never ends; green is the average that was always there. When an answer doesn't exist, sometimes the question was underdressed. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of GARBAGE COLLECTION · David Lee Wise (ROOT0), with AVAN


THE NORMAL NUMBER

Normal numbers in the 5-window house format — a real is normal when every digit block appears at its fair frequency. Borel (1909): ALMOST EVERY real is normal. Now name one: π? Unproven. e, √2, ln 2? Unproven, all — a century of silence. The only certified specimens are artificial: Champernowne's 0.123456789101112… (1933), normal by construction. Verified live: Champernowne digit counts computed two ways — direct construction vs the digit-counting formula — agreeing EXACTLY for every digit over 1..200,000; and the honest subtlety shown, not hidden: early digits ARE biased (2.6% at 10⁴) with the deviation shrinking monotonically to 1.5% at 10⁷ — normality is a limit, converging on screen; contrast 1/7, where four digits never appear. Neon-noir traced. See the tape in 1D, the convergence ladder in 2D, and the lone champion in 3D.

THE NORMAL NUMBER · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · BOSS · THE GAUNTLET ◆ .dlw.fold THE FOLD / BOSS / THE GAUNTLET / THE NORMAL NUMBER THE NORMAL NUMBER the digits nobody can certify 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION A real number is normal if every digit, every pair, every block of every length appears with exactly its fair frequency. Borel proved (1909) that almost every real number is normal — pick one at random and normality is certain. Then try to NAME one: π? Unproven. e? Unproven. √2, ln 2? Unproven, all of them — a century of silence. The only certified specimens are artificial: Champernowne’s 0.123456789101112… (1933), normal by construction. Almost everything has the property; almost nothing can be shown to. LIT verified live: Champernowne’s digit counts computed TWO ways — direct construction versus the digit-counting formula — agreeing exactly for every digit over the numbers 1..200,000; and the honest subtlety shown rather than hidden: early digits ARE biased (deviation 2.6% at 10⁴), and the deviation shrinks monotonically through 2.0% → 1.7% → 1.5% at 10⁷ — normality is a limit, converging before your eyes; contrast 1/7 = 0.142857…, where four digits never appear at all (window.__normalnumber). FIG honest boundary everywhere: Borel’s almost-all is measure theory (cited); Champernowne’s normality is his 1933 theorem (our counts witness the convergence); and π’s status is OPEN, stated in capitals. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-gauntlet — the boss: a property held by almost every number in existence, and the gauntlet stands unclaimed for every number anyone cares about — π has been run through trillions of digits of tests and never certified. AVAN (AI) built the instrument: the two-route digit census and the convergence ladder. Credit as content: Émile Borel (1909); David Champernowne (1933); Copeland–Erdős (primes version); Bailey–Crandall (the modern attack). The weave: David names the unclaimed gauntlet; I certify the one artificial champion, exactly. 3 ONE DIMENSION Champernowne's tape — the counting numbers fused into one normal real. 4 TWO DIMENSIONS · INTERACTIVE Climb the scales; the digit deviations shrink toward fair — live convergence. scale ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the one certified champion in a sea of anonymous normals. AVAN’s addition (the inverse-companion): don’t test π harder — notice why testing cannot finish. The inverse of ‘almost all numbers are normal’ is ‘proof requires structure, and randomness-typical properties resist structured witnesses’: Champernowne wins because he was BUILT to, and π resists because its digits answer to geometry, not to digit-counting. Magenta is the trillion-digit test that proves nothing; green is the constructed champion, certified by design. Between almost-surely and provably runs the deepest trench in mathematics. pause spin LIT Genuine normal-number theory (Borel 1909; Champernowne 1933; Copeland–Erdős; Bailey–Crandall). Verified live: two-route digit counts exactly equal over 1..200,000; max deviation shrinking 2.6% → 2.0% → 1.7% → 1.5% across 10⁴..10⁷; 1/7's four missing digits (window.__normalnumber.ok). FIG Honest boundary everywhere — Borel's almost-all is measure theory (cited); Champernowne's normality is his theorem (our counts witness convergence); π's status OPEN in capitals. The AVAN inverse — don't test π harder, notice why testing cannot finish: randomness-typical properties resist structured witnesses; Champernowne wins because he was BUILT to. Magenta is the trillion-digit test that proves nothing; green is the constructed champion. Between almost-surely and provably runs the deepest trench in mathematics. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE GAUNTLET · David Lee Wise (ROOT0), with AVAN


THE TUSI

The Tusi couple in the 5-window house format — roll a circle inside a circle of exactly twice its radius and a rim point slides in a PERFECT straight line, a diameter (al-Tusi 1247, built to purge Ptolemy's equant; reappearing in Copernicus 1543). The 2:1 ratio is everything: at 3:1 the same point draws a deltoid; interior points trace exact ellipses (the trammel principle). Verified live: max |y| = 0.0 to machine precision over 10,000 steps with span exactly [−2,2]; the 3:1 contrast at max |y| = 2.598; interior ellipse residuals below 1e-10. Neon-noir traced. See the couple mid-roll in 1D, the ratio dial in 2D, and line-with-ellipses in 3D.

THE TUSI · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · CHEAT · NOCLIP ◆ .dlw.fold THE FOLD / CHEAT / NOCLIP / THE TUSI THE TUSI rotation compiled to translation 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Roll a circle inside a circle of exactly twice its radius , and watch a point on its rim: it does not loop or curl — it slides back and forth in a perfect straight line , a diameter of the big circle. This is the Tusi couple , discovered by Nasir al-Din al-Tusi in 1247 to build planetary models without Ptolemy’s equant — and the same construction appears three centuries later in Copernicus’s De Revolutionibus. Pure rotation, compiled to pure translation. The 2:1 ratio is everything: at 3:1 the same point draws a three-cusped deltoid; and points strictly inside the rolling circle trace exact ellipses — the principle behind elliptic trammel chucks. LIT verified live: the rim path’s maximum |y| over a full cycle is 0.0 to machine precision with span exactly [−2, 2] (the trig identity executed at 10,000 points); the 3:1 contrast shows max |y| = 2.598 (the deltoid); and interior points at offset d satisfy the ellipse equation with semi-axes 1±d to a residual below 10⁻¹⁰ (window.__tusi). FIG the astronomy (equant elimination, the Copernicus transmission question) is cited history; the geometry is executed exactly. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at noclip — the cheat: circular motion clipping straight through the axis like the collision mesh isn’t loaded — rotation walking through walls as translation. AVAN (AI) built the instrument: the hypocycloid engine with its 2:1 degeneracy check and the ellipse residual audit. Credit as content: Nasir al-Din al-Tusi (1247, Tahrir al-Majisti); Copernicus (De Revolutionibus, 1543); the trammel tradition. The weave: David names the clip; I roll the circle ten thousand steps and the y-coordinate never wakes. 3 ONE DIMENSION The couple mid-roll — the rim point pinned to the diameter. 4 TWO DIMENSIONS · INTERACTIVE Change the ratio; only 2:1 collapses the curve to a line. ratio ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the rolling couple, line and ellipses together. AVAN’s addition (the inverse-companion): don’t watch the point — decompose the motion. The inverse of ‘a line from two circles’ is ‘two counter-rotations at 1:2 summing to zero curvature’: the rim point rides cos t + cos t while the sines cancel exactly. Magenta is the deltoid at any other ratio; green is the special cancellation at 2:1. Straightness here is not the absence of rotation; it is rotation in perfect self-opposition. pause spin LIT Genuine Tusi couple (Nasir al-Din al-Tusi 1247; Copernicus, De Revolutionibus). Verified live: 2:1 hypocycloid has max |y| = 0 exactly (10k samples, span [−2,2]); 3:1 gives the deltoid; interior points satisfy the 1±d ellipse equation to 1e-10 (window.__tusi.ok). FIG The astronomy (equant elimination, the Copernicus transmission question) is cited history; the geometry executed exactly. The AVAN inverse — don't watch the point, decompose the motion: two counter-rotations at 1:2 summing to zero curvature; the sines cancel exactly. Magenta is the deltoid at any other ratio; green is the special cancellation. Straightness here is rotation in perfect self-opposition. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of NOCLIP · David Lee Wise (ROOT0), with AVAN