THE EHRENFEST

Two dogs, N fleas, one random jump per tick — the 1907 Ehrenfest urn, built to defuse the objection that reversible dynamics forbids an arrow of time. Equilibrium is binomial; and by Kac's theorem the expected return to any state is exactly 1/π(state): the all-on-one-dog reset costs 2^N ticks. Reversible in law, priced out in practice.

THE EHRENFEST · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · RESPAWN · HARD RESET ◆ .dlw.fold THE FOLD / RESPAWN / HARD RESET / THE EHRENFEST THE EHRENFEST the urn that takes 2^N to reset 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Two dogs, N fleas. Each tick, one flea — chosen at random — jumps to the other dog. That is the whole model, and Paul and Tatiana Ehrenfest built it in 1907 to defuse the deepest objection to Boltzmann: if microscopic dynamics is reversible, how can the world run one way? The urn answers by arithmetic. The equilibrium is binomial — near 50/50 — and by Kac’s recurrence theorem the expected time to return to a state is exactly 1/π(state) : returning to all-fleas-on-one-dog takes 2ⁿ ticks on average. Reversibility survives; you just cannot wait for it. Irreversibility is bookkeeping. LIT verified live for N=10: the binomial distribution satisfies πP = π exactly at all 11 states; mean first-return times solved by first-step analysis equal 1/π(k) to 10⁻⁶ — return-to-empty = 2¹⁰ = 1024.000 on the nose; a 400k-step simulation matches the binomial within 0.01 (window.__ehrenfest). FIG the 1907 model and Kac’s 1947 analysis are cited; the ‘defused Boltzmann’s critics’ framing is the standard historical reading. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at hard-reset — the respawn: the system CAN return to its boot state — the theorem guarantees it — but the expected wait doubles with every flea, and at N=100 the hard reset outlives the universe. Reversible in law, irreversible in practice. AVAN (AI) built the instrument: the exact stationary check, the Kac return-time solver, and the long-run simulation. Credit as content: Paul & Tatiana Ehrenfest (1907) — credit Tatiana Ehrenfest-Afanasyeva by name; Mark Kac (1947). The weave: David names the reset that never comes; I solve the wait exactly: 1024 ticks for ten fleas, 2ⁿ forever after. 3 ONE DIMENSION The urn walking its binomial ridge — and the exact return-time ledger. 4 TWO DIMENSIONS · INTERACTIVE Run the fleas; the histogram climbs into the binomial. 10k ticks ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: two dogs trading fleas around the binomial hill. AVAN’s addition (the inverse-companion): don’t ask whether return is possible — price it. The inverse of ‘reversibility vs irreversibility’ is ‘a fee schedule’: Kac’s theorem says every state IS revisited, at cost exactly 1/π — rare states are not forbidden, they are expensive. Magenta is the all-on-one-dog state, priced at 2ⁿ; green is the 50/50 ridge, priced at a handful of ticks. The arrow of time is a price list, not a law. pause spin LIT Verified live for N=10: πP = π exact at all 11 states; Kac return times solved by first-step analysis equal 1/π to 1e-6 — return-to-empty = 1024.000 on the nose; 400k-step simulation matches binomial within 0.01 (window.__ehrenfest.ok). FIG Paul & Tatiana Ehrenfest 1907 (credit Tatiana Ehrenfest-Afanasyeva by name), Kac 1947 cited; the 'defused Boltzmann's critics' framing is the standard historical reading. The AVAN inverse — price the return instead of debating it: rare states are not forbidden, they are expensive, at exactly 1/π. Magenta is the 2^N reset; green is the cheap 50/50 ridge. The arrow of time is a price list, not a law. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of HARD RESET · David Lee Wise (ROOT0), with AVAN


THE FPUT

Los Alamos 1955: Fermi, Pasta, Ulam — and Mary Tsingou, who wrote the program — pumped mode 1 of a 32-mass nonlinear chain and waited for equipartition. Instead the energy wandered a few low modes and came home: the FPUT recurrence, the computation that founded nonlinear science and seeded soliton theory.

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

THE FPUT · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · GRIND · WARM CACHE ◆ .dlw.fold THE FOLD / GRIND / WARM CACHE / THE FPUT THE FPUT the lattice that refused to thermalize 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Los Alamos, 1955: Fermi, Pasta, Ulam — and Mary Tsingou, who wrote the program — put a chain of 32 masses with slightly nonlinear springs on the MANIAC computer, pumped all the energy into the lowest mode, and waited for statistical mechanics to do its job: spread the energy evenly (equipartition). Instead the energy wandered through a few low modes and then came home — nearly all of it back in mode 1. The FPUT recurrence broke the assumption that nonlinearity guarantees thermalization, seeded soliton theory (Zabusky–Kruskal) and KAM-adjacent physics, and is routinely called the birth of computational nonlinear science. LIT verified live: the α-FPUT lattice (N=32, α=0.25, mode-1 start) integrated by velocity Verlet — total energy conserved to 10⁻⁵; mode-1 energy dips to 6% and recurs to 98.1% at t≈9,380; the four lowest modes never hold less than 80% of the energy, where equipartition would allot them ~13% (window.__fput). FIG the recurrence time and percentages are THIS run’s measurements (they depend on N, α, dt); the history — Tsingou’s erased credit included — is cited from the 1955 report LA-1940 and Dauxois’s ‘Fermi, Pasta, Ulam and a mysterious lady’. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at warm-cache — the grind: the energy SHOULD have been evicted to all 31 modes, but the lattice keeps re-fetching the same low-mode working set — a cache that refuses to go cold, tick after tick after tick. AVAN (AI) built the instrument: the symplectic integrator, the mode-energy spectrometer, and the conservation gate. Credit as content: Enrico Fermi, John Pasta, Stanislaw Ulam, Mary Tsingou (LA-1940, 1955); Zabusky & Kruskal (solitons, 1965); Dauxois (restoring Tsingou’s name, 2008). The weave: David names the warm cache; I run the MANIAC’s experiment again and the energy still comes home. 3 ONE DIMENSION Mode-1 energy over time — the dip, the wander, the homecoming. 4 TWO DIMENSIONS · INTERACTIVE Run the lattice; watch the mode spectrum refuse to flatten. advance ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the chain rippling, low modes glowing. AVAN’s addition (the inverse-companion): don’t trust the destination you were promised — watch where the system actually goes. The inverse of ‘nonlinearity guarantees mixing’ is ‘near-integrability guarantees memory’: the lattice sits close enough to a solvable system (Toda) that its energy keeps orbiting home instead of dissolving. Magenta is the equipartition that was promised; green is the recurrence that showed up. When an experiment refuses its theory, the refusal IS the discovery. pause spin LIT Verified live: α-FPUT (N=32, α=0.25) under velocity Verlet — energy conserved to 1e-5; mode-1 dips to 6% and recurs to 98.1% at t≈9,380; the four lowest modes never hold under 80% where equipartition would allot ~13% (window.__fput.ok). FIG Recurrence time and percentages are THIS run's measurements (N, α, dt dependent); history cited from LA-1940 and Dauxois's 'Fermi, Pasta, Ulam and a mysterious lady' — Tsingou's credit restored. The AVAN inverse — watch where the system goes, not where theory promised: near-integrability (Toda's shadow) guarantees memory. Magenta is the promised equipartition; green is the recurrence that showed up. When an experiment refuses its theory, the refusal IS the discovery. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of WARM CACHE · David Lee Wise (ROOT0), with AVAN


THE FIGURE-EIGHT

In the home stadium of chaos, three equal masses chase each other around a single figure-eight — every body on the SAME curve, T/3 apart, zero angular momentum. Found numerically by Cris Moore in 1993, proven by Chenciner & Montgomery in 2000: the first new closed three-body family since Lagrange, 1772.

THE FIGURE-EIGHT · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · SPAWN · FIRST LIGHT ◆ .dlw.fold THE FOLD / SPAWN / FIRST LIGHT / THE FIGURE-EIGHT THE FIGURE-EIGHT three bodies, one curve 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION The three-body problem is the canonical unsolvable — yet in 1993 Cris Moore found, numerically, three equal masses chasing each other around a single figure-eight curve , and in 2000 Chenciner and Montgomery proved it exists: a periodic three-body orbit where every body traces the SAME path, one third of a period apart. It was the first new closed three-body family since Lagrange (1772), it has zero angular momentum , and it opened the door to Simó’s hundreds of ‘choreographies’. In chaos’s home stadium, three bodies dance a braid. LIT verified live: the Chenciner–Montgomery initial conditions integrated one period (T = 6.32591, leapfrog dt = 10⁻⁴) — the bodies return to their start within ~10⁻⁵; energy conserved to 10⁻¹²-level, angular momentum exactly 0 to machine precision; the choreography property checked directly: after T/3 the three bodies permute onto each other’s positions to ~10⁻⁶; a 1% perturbation destroys the return (error 0.12) (window.__figureeight). FIG stability of the orbit (it is only marginally stable) and the deep variational proof are cited, not re-derived; Moore 1993, Chenciner–Montgomery 2000, Simó 2000. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at first-light — the spawn: after two centuries of three-body darkness, a genuinely NEW orbit switches on — not found in the sky but in a computer, then proven real by hand. First light from a numerical telescope. AVAN (AI) built the instrument: the leapfrog integrator with conservation gates, the return meter, and the permutation audit. Credit as content: Cris Moore (1993, the discovery); Alain Chenciner & Richard Montgomery (2000, the proof); Carles Simó (the choreography zoo); Lagrange (1772, the previous new family). The weave: David names the first light; I run the braid one full period and the three come home in each other’s places. 3 ONE DIMENSION The eight — one curve, three bodies, phase-shifted by T/3. 4 TWO DIMENSIONS · INTERACTIVE Run the dance; the conservation meters hold the line. step ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the braid in spacetime — three strands, one weave. AVAN’s addition (the inverse-companion): don’t search the equations — search the SYMMETRY class. The inverse of ‘solve for the motion’ is ‘demand the motion respect a shape and minimize’: Chenciner and Montgomery found the eight by minimizing action over loops with the right symmetry, letting the constraint do the discovering. Magenta is the generic chaos three bodies usually deliver; green is the measure-zero braid that order carved out anyway. In a chaotic universe, symmetry is where the survivors hide. pause spin LIT Verified live: the Chenciner–Montgomery orbit integrated one period (T=6.32591) — return within ~1e-5; energy to 1e-12, angular momentum 0 to machine precision; the choreography property checked directly (after T/3 the bodies permute onto each other, dev ~1e-6); 1% perturbation destroys the return (window.__figureeight.ok). FIG Marginal stability and the variational proof are cited, not re-derived (Moore 1993; Chenciner–Montgomery 2000; Simó's choreography zoo). The AVAN inverse — search the symmetry class, not the equations: the eight was found by minimizing action over symmetric loops, letting the constraint do the discovering. Magenta is generic three-body chaos; green is the measure-zero braid. In a chaotic universe, symmetry is where the survivors hide. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of FIRST LIGHT · David Lee Wise (ROOT0), with AVAN


THE KAPITZA

An inverted pendulum falls — unless its pivot vibrates fast enough, and then upside-down becomes STABLE: an effective well where gravity built a peak. Stephenson saw it in 1908, Kapitza analyzed it in 1951, and the averaging trick behind a²ω² > 2gL now runs Paul traps and strong-field physics.

THE KAPITZA · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · CHEAT · GOD MODE ◆ .dlw.fold THE FOLD / CHEAT / GOD MODE / THE KAPITZA THE KAPITZA gravity beaten by vibration 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION A pendulum balanced upside-down falls — unless you vibrate its pivot fast enough . Then the inverted position becomes STABLE: nudge it and it wobbles around straight-up like a well in a potential that gravity no longer owns. Stephenson saw it in 1908; Pyotr Kapitza analyzed it in 1951 and the trick now carries his name. The leading-order criterion is one inequality: a²ω² > 2gL — drive amplitude times frequency must outrun gravity — and the machinery behind it (averaging over fast oscillations into an effective potential) became a standard tool from Paul traps to strong-field physics. LIT verified live: RK4 integration — at 1.5× the threshold the inverted pendulum holds (excursion 0.10 rad over 60 s); at 0.5× it falls; the empirical stability boundary lands within 0.3% of √(2gL)/ω; and a dt-halving convergence gate guards the integrator (window.__kapitza). FIG a²ω² > 2gL is the leading-order effective-potential criterion, stated as such — the 0.3% agreement at ω=60 is this run’s measurement of how good the averaging already is there. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at god-mode — the cheat: gravity says the state is forbidden, and the exploit is not to fight the force but to SHAKE THE FRAME until the forbidden state becomes a home. AVAN (AI) built the instrument: the driven-pendulum integrator, the threshold bisection, and the convergence gate. Credit as content: Andrew Stephenson (1908); Pyotr Kapitza (1951); the effective-potential averaging method; Paul traps as the same mathematics. The weave: David names the god-mode; I shake the pivot and the pendulum stands on its head. 3 ONE DIMENSION Below threshold it falls; above, the inverted well holds. 4 TWO DIMENSIONS · INTERACTIVE Step the drive strength; watch stability switch at the criterion. drive ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the upside-down pendulum, trembling and standing. AVAN’s addition (the inverse-companion): don’t change the landscape — change the timescale you live on. The inverse of ‘the potential decides stability’ is ‘the AVERAGE potential decides, and averages can be engineered’: vibrate fast enough and the pendulum feels a valley where gravity built a peak. Magenta is the instantaneous force, always pulling down; green is the effective well that emerges from motion too fast to follow. Some stabilities exist only at the right frame rate. pause spin LIT Verified live: RK4 — at 1.5× threshold the inverted pendulum holds (0.10 rad over 60s); at 0.5× it falls; the empirical boundary lands 0.3% from √(2gL)/ω; dt-halving convergence gate passed (window.__kapitza.ok). FIG a²ω² > 2gL is the leading-order effective-potential criterion, stated as such; the 0.3% agreement is this run's measurement at ω=60. The AVAN inverse — change the timescale, not the landscape: averages can be engineered. Magenta is the instantaneous force, always pulling down; green is the well that emerges from motion too fast to follow. Some stabilities exist only at the right frame rate. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of GOD MODE · David Lee Wise (ROOT0), with AVAN


THE KURAMOTO

Fireflies, pacemaker cells, clocks on a beam — Kuramoto's 1975 model distilled them: oscillators with random frequencies, each pulled toward the crowd's mean phase. Below Kc = 2γ, anarchy; above it, synchronization ignites with the exact law r = √(1 − Kc/K) — a phase transition solved in closed form.

THE KURAMOTO · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · CO-OP · THE SYNC ◆ .dlw.fold THE FOLD / CO-OP / THE SYNC / THE KURAMOTO THE KURAMOTO the sync transition 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Fireflies flash together; pacemaker cells beat together; pendulum clocks on one beam agree. Yoshiki Kuramoto’s 1975 model distilled all of it: N oscillators with random natural frequencies, each pulled toward the crowd’s mean phase with coupling K. Below a critical coupling, anarchy — the order parameter r sits at zero. At Kₛ = 2γ (for a Lorentzian frequency spread) synchronization ignites , and Kuramoto solved the aftermath exactly: r = √(1 − Kₛ/K) . A phase transition you can hold in one equation — and one of the few many-body models with an exact answer. LIT verified live: 2,000 oscillators, deterministic Lorentzian quantile frequencies — measured r at K = 3, 4, 6 lands within 0.05 of √(1−2/K) (0.565/0.577, 0.699/0.707, 0.810/0.816); below threshold r ≈ 0.008; the transition is bracketed between K = 1.8 and 2.4 (window.__kuramoto). FIG finite-N and finite-T keep the sim a hair under the infinite-N formula — visible in the numbers and said aloud; Kuramoto 1975, Strogatz’s reviews cited. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-sync — the co-op: no leader, no clock signal, every node nudged only by the crowd’s average — and above one number the swarm phase-locks anyway. AVAN (AI) built the instrument: the mean-field integrator, the quantile frequency ladder, and the closed-form comparison. Credit as content: Yoshiki Kuramoto (1975); Arthur Winfree (the biological framing); Steven Strogatz (the modern theory). The weave: David names the leaderless sync; I measure the ignition at K = 2 where the formula said it would fire. 3 ONE DIMENSION The order parameter vs coupling — flat, then the square-root ignition. 4 TWO DIMENSIONS · INTERACTIVE Step K; the phase ring scatters or locks. K ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the firefly ring, igniting past threshold. AVAN’s addition (the inverse-companion): don’t look for the conductor — measure the feedback gain. The inverse of ‘who synchronized them?’ is ‘how strongly does each hear the mean?’: order appears when the loop from crowd to member and back crosses unity in the right units, and NO agent decides it. Magenta is the anarchic spread below Kₛ; green is the locked phalanx above. Consensus is a bifurcation, not a decree. pause spin LIT Verified live: 2,000 oscillators on deterministic Lorentzian quantiles — r at K=3,4,6 within 0.05 of √(1−2/K); r ≈ 0.008 below threshold; transition bracketed between K=1.8 and 2.4 (window.__kuramoto.ok). FIG Finite-N and finite-T keep the sim a hair under the infinite-N formula — visible and said aloud. Kuramoto 1975, Winfree, Strogatz cited. The AVAN inverse — measure the feedback gain, not the conductor: order appears when the crowd-to-member loop crosses unity, and no agent decides it. Magenta is the anarchic spread; green is the locked phalanx. Consensus is a bifurcation, not a decree. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE SYNC · David Lee Wise (ROOT0), with AVAN


THE WILBERFORCE

A mass on a soft helical spring, twist and stretch frequencies tuned to match: the bounce dies completely while the mass starts spinning, then the trade reverses — the two motions passing the whole energy budget back and forth like items between inventory slots. Wilberforce 1896; the beat is normal-mode interference, scheduled by the eigenvalues.

THE WILBERFORCE · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · LOOT · THE INVENTORY ◆ .dlw.fold THE FOLD / LOOT / THE INVENTORY / THE WILBERFORCE THE WILBERFORCE bounce traded for twist 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Hang a mass on a soft helical spring and set it bouncing. If the spring’s twist and stretch frequencies are tuned to match, something uncanny happens: the bouncing dies completely while the mass starts spinning — then the twist dies and the bounce returns, over and over, the two motions trading the whole energy budget like items passed between two inventory slots. This is the Wilberforce pendulum (L. R. Wilberforce, 1896): a textbook of normal modes — the true modes are symmetric and antisymmetric mixtures, split in frequency by the tiny helix coupling, and the trade-off beat is their interference. LIT verified live: the coupled system (ωz = ωθ, ε = 0.02) integrated by RK4 — the bounce empties to 0.00% (complete transfer); the half-energy crossing lands at t = 157.0 vs the normal-mode prediction T₁/4 = 157.1 (0.05%); total energy conserved to 10⁻¹²; the detuned control (ωθ² = 1.2) barely transfers at all (window.__wilberforce). FIG the demonstration’s long life in physics lecture halls is the cited legacy; parameters here are dimensionless. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-inventory — the loot: two item slots — BOUNCE and TWIST — and the game trades the entire stack back and forth on a timer, never duplicating, never losing a unit. AVAN (AI) built the instrument: the coupled integrator, the eigenmode prediction, and the detuned control. Credit as content: Lionel Robert Wilberforce (1896, Cavendish Laboratory); the normal-mode formalism it teaches. The weave: David names the inventory swap; I time the trade and the eigenvalues had already scheduled it. 3 ONE DIMENSION Bounce energy and twist energy — the full trade, on schedule. 4 TWO DIMENSIONS · INTERACTIVE Watch the two slots trade; detune and the trade jams. tuned/detuned ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the spring bouncing and twisting in trade. AVAN’s addition (the inverse-companion): don’t watch the coordinates you were given — find the ones the system actually uses. The inverse of ‘bounce and twist trade energy’ is ‘the normal modes never trade anything’: in the eigenbasis each mode keeps its energy forever, and the drama is an artifact of watching in the wrong basis. Magenta is the trade you see; green is the stillness underneath it. Much of what looks like exchange is a choice of coordinates. pause spin LIT Verified live: coupled system integrated by RK4 — the bounce empties to 0.00%; half-energy crossing at t=157.0 vs the normal-mode prediction T_beat/4 = 157.1 (0.05%); energy conserved to 1e-12; the detuned control barely trades at all (window.__wilberforce.ok). FIG A lecture-hall classic since the Cavendish; parameters dimensionless. The AVAN inverse — find the coordinates the system actually uses: in the eigenbasis each normal mode keeps its energy forever, and the drama is an artifact of the watching basis. Magenta is the trade you see; green is the stillness underneath. Much of what looks like exchange is a choice of coordinates. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE INVENTORY · David Lee Wise (ROOT0), with AVAN


THE FOUCAULT

Paris 1851: Foucault hangs 67 meters of wire in the Panthéon and the public watches the swing plane creep — the first direct dynamical proof that the Earth turns. One sine rules it all: precession at Ω sin(latitude) — 23.93h at the pole, 31.8h in Paris, never at the equator — and the whole motion has an exact closed form with rotation stamped on as a complex phase.

THE FOUCAULT · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · RESPAWN · THE CONTINUE ◆ .dlw.fold THE FOLD / RESPAWN / THE CONTINUE / THE FOUCAULT THE FOUCAULT the Earth turning under a wire 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Paris, 1851: Léon Foucault hangs a 67-meter wire in the Panthéon and the public watches the swing plane creep clockwise, hour by hour — the first direct, dynamical proof that the Earth turns , no telescope required. The law is one sine: the plane precesses at Ω sin(latitude) — a full circle per sidereal day at the pole, 31.8 hours in Paris, never at the equator. In the rotating frame the Coriolis term does the work, and the whole motion has an exact closed form: z(t) = e⁻ⁱΩₓᵗ·(oscillation), rotation stamped on as a complex phase. LIT verified live: the rotating-frame equations integrated by RK4 match the exact complex solution to 10⁻¹³; the plane rotates by exactly −Ωₖ per unit time (one-period check to 10⁻⁶); the latitude table computes 23.93 h at the pole and 31.8 h for Paris at 48.85° (window.__foucault). FIG the Panthéon history is documented; the linearized small-swing model is the standard treatment and is stated as the model being verified. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-continue — the respawn: the pendulum’s plane leaves its starting orientation and — given one sidereal day over sinφ — continues back around to exactly where it began: the slowest continue screen on Earth, and the screen is Earth. AVAN (AI) built the instrument: the Coriolis integrator, the exact-solution comparator, and the latitude table. Credit as content: Léon Foucault (1851); the Panthéon and its replicas worldwide; the Coriolis formalism. The weave: David names the continue; I integrate the wire and the Earth rotates beneath the arithmetic. 3 ONE DIMENSION The rosette — swing plane creeping as the Earth turns beneath. 4 TWO DIMENSIONS · INTERACTIVE Step the latitude; the precession clock recomputes. latitude ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the wire over the turning floor. AVAN’s addition (the inverse-companion): don’t ask what moves the pendulum — ask what the pendulum refuses to do. The inverse of ‘the plane precesses’ is ‘the plane holds still while the FLOOR precesses’: the wire is the inertial witness, and the creep we see is our own rotation, projected through sinφ. Magenta is the floor, certain it is still; green is the swing plane, actually still. The instrument doesn’t measure the Earth — it declines to join it. pause spin LIT Verified live: rotating-frame integration matches the exact complex solution to 1e-13; the plane rotates by exactly −Ωz per unit time (1e-6); the latitude table computes pole 23.93h and Paris 31.8h (window.__foucault.ok). FIG Panthéon history documented; the linearized small-swing model is the standard treatment, stated as the model verified. The AVAN inverse — ask what the pendulum refuses to do: the plane holds still while the floor precesses; the creep is our own rotation through sinφ. Magenta is the floor, certain it is still; green is the swing plane, actually still. The instrument declines to join the Earth. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE CONTINUE · David Lee Wise (ROOT0), with AVAN


THE SQUARE-FREE

A square is a stutter — any block repeated immediately. With two letters it's unavoidable: every binary word of length 4 contains one. With three, Thue proved in 1906 you can walk forever without stuttering, via a simple substitution word. The gate between alphabets 2 and 3 is absolute — and Thue's overlooked Norwegian papers founded combinatorics on words.

THE SQUARE-FREE · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · BOSS · THE GATEKEEPER ◆ .dlw.fold THE FOLD / BOSS / THE GATEKEEPER / THE SQUARE-FREE THE SQUARE-FREE three letters never stutter 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION A square in a word is a stutter: any block repeated immediately — ‘ abab ’, ‘ 11 ’. With two letters, stuttering is unavoidable: every binary word of length 4 already contains a square — the longest square-free binary words have length three. But with THREE letters, Axel Thue proved in 1906 you can walk forever without ever stuttering: an infinite square-free word exists, generated by a simple substitution rule. The gate between alphabet sizes 2 and 3 is absolute — and Thue’s papers, published in an obscure Norwegian journal and overlooked for decades, founded the field now called combinatorics on words . LIT verified live: all 16 binary words of length 4 contain a square (exhaustive); Thue’s morphism word (a→abc, b→ac, c→b) scanned to 5,000 characters — zero squares of any length; the census of ternary square-free words for lengths 1–14 — 3, 6, 12, 18, 30, …, 456 — grows exponentially (ratio ≈ 1.33), so the language never thins out (window.__squarefree). FIG the census matches the known enumeration (OEIS A006156) as computed here from scratch; Thue 1906/1912 and the Berstel translations are the cited history. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-gatekeeper — the boss: the door between alphabet two and alphabet three. On one side every path stutters within four steps; on the other, infinite corridors with no echo at all. The gatekeeper asks one question: how many letters do you carry? AVAN (AI) built the instrument: the exhaustive binary audit, the morphism generator with full square scan, and the census enumerator. Credit as content: Axel Thue (1906, 1912); Berstel’s editions that recovered him; Marston Morse (the independent rediscovery). The weave: David names the gate; I walk five thousand steps beyond it without a single echo. 3 ONE DIMENSION Binary stutters by length 4; the ternary walk never does. 4 TWO DIMENSIONS · INTERACTIVE Grow the Thue word; the scanner finds no square, ever. grow ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the echo-free corridor, color-coded and endless. AVAN’s addition (the inverse-companion): don’t count what a system can say — find what it can permanently avoid saying. The inverse of ‘expressiveness’ is ‘avoidability’: two symbols cannot avoid repeating themselves; three can, forever, and the census proves the avoidance is not a knife-edge but an exponentially wide road. Magenta is the stutter that two letters cannot escape; green is the third letter — the smallest purchase of infinite restraint. Freedom of speech begins at alphabet three. pause spin LIT Verified live: all 16 binary length-4 words contain a square (exhaustive); Thue's morphism word scanned to 5,000 chars — zero squares; the census of ternary square-free words for lengths 1–14 (3,6,12,18,…,456) grows exponentially, ratio ≈1.33 (window.__squarefree.ok). FIG Census computed from scratch, matching the known enumeration (OEIS A006156); Thue 1906/1912, Berstel's recovery, Morse's rediscovery cited. The AVAN inverse — find what a system can permanently avoid saying: avoidability is the dual of expressiveness, and the census shows the avoidance is an exponentially wide road, not a knife-edge. Magenta is the stutter two letters cannot escape; green is the third letter. Freedom of speech begins at alphabet three. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE GATEKEEPER · David Lee Wise (ROOT0), with AVAN


THE ISING

A grid of arrows, each preferring to agree with its neighbours, each shaken by heat. Cold: one giant aligned domain. Hot: static. The Ising model is the simplest genuine phase transition — and Onsager solved 2D exactly in 1944, pinning Tc = 2/ln(1+√2). The irony: Ising solved the 1D chain in 1925, found no transition, and wrongly concluded there was none in any dimension.

THE ISING · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · CO-OP · THE BROADCAST ◆ .dlw.fold THE FOLD / CO-OP / THE BROADCAST / THE ISING THE ISING the temperature that melts order 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION A grid of arrows, each preferring to agree with its neighbours, each shaken by temperature. Cold: they lock into one giant aligned domain. Hot: noise wins and order evaporates. The Ising model is the simplest system with a genuine phase transition — and in 1944 Lars Onsager solved the two-dimensional case exactly , pinning the critical temperature at Tℂ = 2/ln(1+√2) ≈ 2.269 and the spontaneous magnetization at m = [1 − sinh⁻⁴(2/T)]^(1/8). The irony in the name: Ernst Ising solved the ONE-dimensional chain in 1925, found no transition, and concluded there was none in any dimension. He was wrong by one dimension, and the model still carries his name. LIT verified live: Metropolis Monte Carlo on a 16×16 lattice reproduces Onsager’s exact magnetization to within 0.005 at T = 1.6, 1.8, 2.0 (0.981/0.980, 0.952/0.957, 0.908/0.911); above Tℂ the order melts (|m| = 0.16 at T = 3.2); and the 1D control matches the exact transfer-matrix energy −tanh(1/T) at three temperatures (window.__ising). FIG a 16×16 lattice shows finite-size rounding near Tℂ — the sharp transition is the infinite-lattice theorem, cited, while what we verify is the below-Tℂ magnetization curve and the melt. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-broadcast — the co-op: every node shouting its state to its neighbours and listening back. Below one noise level the whole network agrees on a message nobody sent; above it, the broadcast dissolves into static. AVAN (AI) built the instrument: the Metropolis sampler, the Onsager comparator, and the 1D exact control. Credit as content: Wilhelm Lenz (1920, posed it); Ernst Ising (1925, the 1D solution and the famous wrong conclusion); Lars Onsager (1944, the 2D exact solution); Metropolis et al. (1953, the algorithm). The weave: David names the broadcast; I cool the lattice and Onsager’s curve is already waiting there. 3 ONE DIMENSION |m| vs T — the Monte Carlo points landing on Onsager’s exact curve. 4 TWO DIMENSIONS · INTERACTIVE Step the temperature; watch the lattice order and melt. temperature ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the lattice breathing through its critical point. AVAN’s addition (the inverse-companion): don’t ask what each arrow does — ask what the ensemble cannot help doing. The inverse of ‘local rules’ is ‘global inevitability’: no spin knows the temperature, no spin decides to order, and yet below one number the whole lattice commits. Magenta is the noise that dissolves consensus; green is the domain nobody voted for. Collective states are not built — they precipitate. pause spin LIT Verified live: Metropolis MC on 16×16 reproduces Onsager's exact magnetization [1−sinh⁻⁴(2/T)]^{1/8} within 0.005 at T=1.6/1.8/2.0; above Tc the order melts (|m|=0.16 at T=3.2); the 1D control matches the exact transfer-matrix energy −tanh(1/T) (window.__ising.ok). FIG A 16×16 lattice shows finite-size rounding near Tc — the sharp transition is the infinite-lattice theorem, cited; what we verify is the below-Tc curve and the melt. Lenz 1920, Ising 1925, Onsager 1944, Metropolis 1953 credited. The AVAN inverse — ask what the ensemble cannot help doing: no spin knows the temperature, yet below one number the whole lattice commits. Collective states are not built — they precipitate. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE BROADCAST · David Lee Wise (ROOT0), with AVAN


THE GIANT COMPONENT

Sprinkle random edges on n nodes until each averages c connections. Below c = 1 the network is dust — every piece O(log n). Cross c = 1 and a single giant component appears holding a fixed fraction S of everything, with all others still logarithmic. Erdős and Rényi called it the double jump; the giant's size is the root of S = 1 − e^(−cS).

THE GIANT COMPONENT · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · BOSS · THE RAID ◆ .dlw.fold THE FOLD / BOSS / THE RAID / THE GIANT COMPONENT THE GIANT COMPONENT the edge where one giant appears 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Take n nodes and sprinkle random edges until each node averages c connections. Below c = 1, the network is dust: every connected piece is tiny, O(log n). Cross c = 1 and a single giant component appears, containing a fixed fraction S of everything — while all other pieces stay logarithmic. Erdős and Rényi called it the double jump , and the giant’s size is the root of a transcendental equation, S = 1 − e⁻ᶜˢ . One node’s worth of average degree separates a pile of fragments from a connected world. LIT verified live: 20,000-node graphs built by geometric-skip edge sampling and measured with union-find — S = 0.582/0.807/0.938 at c = 1.5/2/3 against the theory 0.583/0.797/0.940; at c = 0.5 the largest component is 0.10% of n; at exactly c = 1 it is 1.93%, inside the critical window that scales like n⁻¹⃗³ ≈ 3.7%; and the theoretical S satisfies its fixed-point equation to 10⁻¹² (window.__giant). FIG the O(log n) claim for subcritical components and the n²⃗³ critical-window scaling are cited theorems — what we measure is one instance consistent with them. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-raid — the boss: below one connection per player the guild is a scatter of duos who can never assemble; at exactly one, the raid group condenses out of the noise, and everyone left over stays a footnote. AVAN (AI) built the instrument: the sparse-graph sampler, the union-find measurer, and the fixed-point solver. Credit as content: Paul Erdős & Alfréd Rényi (1959–1960); Bollobás (the critical window); the percolation literature that grew from it. The weave: David names the raid threshold; I build twenty thousand nodes and the giant shows up exactly where the equation says. 3 ONE DIMENSION S vs c — flat dust, then the giant rising from c = 1. 4 TWO DIMENSIONS · INTERACTIVE Step the average degree; the components merge into one. degree ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the graph condensing as edges rain in. AVAN’s addition (the inverse-companion): don’t count the edges you added — count the ones the structure implies. The inverse of ‘connect things to build a network’ is ‘connectivity is a threshold phenomenon that arrives on its own schedule’: nothing special happens at the edge that tips c past 1, and yet after it the world has a spine. Magenta is the dust of components that never grow; green is the giant that eats them. Emergence has a coordinate, and it is usually one. pause spin LIT Verified live: 20,000-node graphs by geometric-skip sampling, measured with union-find — S = 0.582/0.807/0.938 at c = 1.5/2/3 vs theory 0.583/0.797/0.940; at c=0.5 the largest piece is 0.10% of n; at c=1 it is 1.93%, inside the n^(−1/3) ≈ 3.7% critical window; theory satisfies its fixed point to 1e-12 (window.__giant.ok). FIG The O(log n) subcritical bound and the n^(2/3) critical-window scaling are cited theorems — we measure one consistent instance. Erdős–Rényi 1959–60, Bollobás cited. The AVAN inverse — connectivity is a threshold that arrives on its own schedule: nothing special happens at the edge that tips c past 1, yet afterwards the world has a spine. Emergence has a coordinate, and it is usually one. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE RAID · David Lee Wise (ROOT0), with AVAN


THE ZEEMAN MACHINE

A cardboard disc on a pin and two elastics: move your hand smoothly and the disc follows — until it flips, and flips back at a DIFFERENT place. Zeeman's 1972 catastrophe machine is Thom's cusp made physical: a control plane with a bistable tongue, so continuous input yields discontinuous output and the route decides the state. Hysteresis you can build for a pound.

THE ZEEMAN MACHINE · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · LOOT · THE STASH ◆ .dlw.fold THE FOLD / LOOT / THE STASH / THE ZEEMAN MACHINE THE ZEEMAN MACHINE the disc that jumps 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION A cardboard disc on a pin, two elastic bands: one anchored to the table, the other led by your hand. Move your hand smoothly and the disc follows smoothly — until you cross an invisible boundary and the disc flips . Cross back and it flips at a different place . This is E. C. Zeeman’s catastrophe machine (1972), the physical demonstration of René Thom’s cusp catastrophe : a smooth two-parameter control plane containing a wedge-shaped tongue where the system has two stable states , so continuous input yields discontinuous output, and the path taken decides which state you get — hysteresis you can build for a pound. LIT verified live: scanning the control plane and counting local minima of the elastic energy — 86 of 525 sample points have two minima , forming a bounded tongue (x ∈ [−1.5, 2.7], y ∈ [−0.8, 0.8]); sweeping y up and down at x = 1.0 the branches differ by 1.220 rad — the disc jumps at different places in each direction; and outside the tongue (x = 3.6) the difference is 0.0000 rad (window.__zeeman; the offline harness at 4× the scan resolution finds 206/1,353 and the same tongue). FIG the classification of this as Thom’s cusp is the cited theory; the elastic model here is the standard idealization (Hooke bands, natural length 1). 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-stash — the loot: the system is holding TWO states in one place, and which one you can withdraw depends entirely on the route you took to get here. History, stored as position. AVAN (AI) built the instrument: the energy scanner, the minima counter, and the branch-following hysteresis sweeps. Credit as content: René Thom (catastrophe theory); E. C. Zeeman (1972, the machine and the popularization); the later critiques that rightly trimmed catastrophe theory’s claims outside physics. The weave: David names the two-state stash; I sweep the plane in both directions and the jumps land in different places. 3 ONE DIMENSION The bistable tongue in the control plane — inside it, two minima. 4 TWO DIMENSIONS · INTERACTIVE Sweep the control point; watch the energy landscape lose a well. sweep ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the disc and its two elastics, jumping. AVAN’s addition (the inverse-companion): don’t model the jump — model the disappearance of the well you were in . The inverse of ‘why did it suddenly change?’ is ‘nothing sudden happened; your minimum quietly stopped existing’: the discontinuity is in the state, never in the input. Magenta is the well that vanished under you; green is the one you fall into. Catastrophes are smooth from the landscape’s point of view. pause spin LIT Verified live: scanning the control plane, 206 of 1,353 points have TWO energy minima, forming a bounded tongue (x∈[−1.50,2.75], y∈[−0.88,0.88]); sweeping y up and down at x=1.0 the branches differ by 1.220 rad; outside the tongue (x=3.6) the difference is 0.0000 rad (window.__zeeman.ok). FIG Classification as Thom's cusp is cited theory; the elastic model is the standard idealization (Hooke bands, natural length 1). Zeeman 1972, and the later critiques that trimmed catastrophe theory's overreach outside physics. The AVAN inverse — model the disappearance of the well you were in: the discontinuity is in the state, never in the input. Catastrophes are smooth from the landscape's point of view. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE STASH · David Lee Wise (ROOT0), with AVAN


THE LANCHESTER

Lanchester, watching Great War aircraft, found that under aimed fire combat power scales as the SQUARE of numbers — invariant αA² − βB². Hence defeat in detail: 100 fighting two 50s in sequence walks away with √(100²−50²−50²) = 70.7 survivors where fighting all 100 at once annihilates it. And it's conditional: under unaimed fire the law goes linear and concentration buys nothing.

THE LANCHESTER · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · GRIND · BACKPROP ◆ .dlw.fold THE FOLD / GRIND / BACKPROP / THE LANCHESTER THE LANCHESTER why concentration wins 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Frederick Lanchester, watching aircraft in the Great War, wrote down two differential equations and found something brutal: under aimed fire , combat power scales as the square of numbers. The invariant is αA² − βB²: doubling a force quadruples its worth. The corollary is the oldest trick in generalship — defeat in detail : a force of 100 that fights two forces of 50 one after the other doesn’t merely win, it walks away with √(100²−50²−50²) = 70.7 survivors, where fighting all 100 at once would annihilate it. And the effect is conditional: under unaimed fire the law goes LINEAR and concentration buys nothing. LIT verified live: integrating the square-law equations — 100 v 100 with equal effectiveness annihilates both sides (0.00 vs 0.00); the same 100 fighting two 50s sequentially ends with 70.71 survivors, matching the closed form to two decimals; the invariant αA²−βB² holds to 10⁻⁵ along the integration; and the linear-law control gives sequential 3.12 vs direct 3.23 — no advantage (window.__lanchester). FIG Lanchester’s laws are a deliberately crude model of real combat — the mathematics is exact, the applicability is contested, and we claim only the mathematics. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at backprop — the grind: the gradient of the outcome with respect to your force is not constant; it grows with the force you already have. Every unit added is worth more than the last, which is exactly why splitting the enemy is how you win. AVAN (AI) built the instrument: the square-law integrator with an invariant gate, the sequential-battle simulator, and the unaimed-fire control. Credit as content: Frederick W. Lanchester (1916, Aircraft in Warfare ); the operations-research tradition that formalized and later criticized it. The weave: David names the backprop; I run the battles and the square law pays out to the second decimal. 3 ONE DIMENSION Two battles, one force: divided beats united by 70.7 survivors. 4 TWO DIMENSIONS · INTERACTIVE Fight the split; the survivor count follows the square law. next split ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the two force curves, squares racing to zero. AVAN’s addition (the inverse-companion): don’t count soldiers — count the conserved quantity. The inverse of ‘who is bigger?’ is ‘what does the battle preserve?’: αA²−βB² is fixed from the first shot, so the winner and the survivors are known before the fighting starts — the battle only spends time. Magenta is the linear world where numbers add; green is the squared world where they compound. Whether concentration is worth anything is a question about the exponent. pause spin LIT Verified live: 100v100 with equal effectiveness annihilates both (0.00 vs 0.00); sequential vs two 50s ends at 70.71, matching the closed form; the invariant holds to 1e-5 along the integration; the linear-law control gives 3.12 vs 3.23 — no advantage (window.__lanchester.ok). FIG Lanchester's laws are a deliberately crude combat model — the mathematics is exact, the applicability contested, and we claim only the mathematics. Lanchester 1916 and the OR tradition cited. The AVAN inverse — count the conserved quantity, not the soldiers: αA²−βB² is fixed from the first shot, so the outcome is known before the fighting; the battle only spends time. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of BACKPROP · David Lee Wise (ROOT0), with AVAN


THE LOTKA–VOLTERRA

Predator and prey chase each other around a loop forever. Two surprises: the orbits are exactly closed (an invariant preserves them), and the time-averages depend ONLY on the parameters — ⟨prey⟩ = c/d, ⟨predator⟩ = a/b, whatever the amplitude. Hence Volterra's principle: kill both species indiscriminately and the prey average RISES. He derived it in 1926 to explain why the WWI fishing halt raised the Adriatic shark fraction.

THE LOTKA–VOLTERRA · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · GRIND · THE EPOCH ◆ .dlw.fold THE FOLD / GRIND / THE EPOCH / THE LOTKA–VOLTERRA THE LOTKA–VOLTERRA why killing both helps the prey 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Predators eat prey; prey feed predators; both populations chase each other around a loop forever. The Lotka–Volterra equations are ecology’s hydrogen atom — and they hide two surprises. First: the orbits are exactly closed , preserved by the invariant dx − c·ln x + by − a·ln y, so the cycle never decays or grows. Second, and stranger, the time-averages are fixed by the parameters alone : ⟨prey⟩ = c/d and ⟨predator⟩ = a/b, whatever the amplitude. That gives Volterra’s principle : kill BOTH species indiscriminately — a pesticide, a fishing fleet — and the prey average RISES while the predator average falls. Volterra derived it in 1926 to explain why the WWI halt in Adriatic fishing had raised the shark fraction. LIT verified live: the invariant conserved to 10⁻¹⁴ over sixty time units (the orbit is closed, not spiralling); the measured period 10.789 exceeds the small-oscillation 2π/√(ac) = 9.472 (big orbits run slower); time-averages 4.0001 vs c/d = 4.0000 and 2.7500 vs a/b = 2.7500; and spraying both species at p = 0.2 moves the simulated averages 3.98/2.71 → 6.03/2.24 (window.__lotka). FIG the model is famously idealized (no carrying capacity, neutral cycles); Volterra’s principle is a theorem about this model with real-world support in pesticide-resurgence cases — cited as such, not as ecology in general. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-epoch — the grind: the loop runs forever, and the only thing that matters is what the loop AVERAGES to over one epoch — a number set by the rules, not by where you started or how hard you pushed. AVAN (AI) built the instrument: the RK4 integrator with an invariant gate, the period-crossing timer, the average meter, and the spray experiment. Credit as content: Alfred Lotka (1925); Vito Volterra (1926, and the Adriatic shark data of Umberto D’Ancona); the modern pesticide-resurgence literature. The weave: David names the epoch average; I spray both populations and the prey come out ahead, exactly as the ratios predict. 3 ONE DIMENSION The closed orbit in the phase plane, with its fixed centre. 4 TWO DIMENSIONS · INTERACTIVE Spray both species; the averages move the wrong way on purpose. spray ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the two populations chasing round the loop. AVAN’s addition (the inverse-companion): don’t intervene on the populations — read which parameter your intervention actually touches. The inverse of ‘kill pests to reduce pests’ is ‘the pest average is c/d, and your spray moves c the wrong way’: the lever you pulled was never attached to the number you cared about. Magenta is the intuition that killing reduces; green is the ratio that decides. In a coupled system, always ask which coefficient your action edits. pause spin LIT Verified live: invariant conserved to 1e-14 over sixty time units (closed orbit, not spiralling); measured period 10.789 > small-oscillation 2π/√(ac) = 9.472; averages 4.0001 vs c/d = 4.0000 and 2.7500 vs a/b = 2.7500; spraying both at p=0.2 moves simulated averages 3.98/2.71 → 6.03/2.24 (window.__lotka.ok). FIG The model is famously idealized (no carrying capacity, neutral cycles); Volterra's principle is a theorem ABOUT THIS MODEL with real-world support in pesticide-resurgence cases — cited as such, not as ecology in general. Lotka 1925, Volterra 1926, D'Ancona's shark data. The AVAN inverse — read which coefficient your intervention actually edits: the lever you pulled was never attached to the number you cared about. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE EPOCH · David Lee Wise (ROOT0), with AVAN


THE ARROW

Ask a voting rule for three things — never rank X above Y when everyone prefers Y; decide X vs Y using only how voters rank X against Y; and no dictator. Arrow proved in 1951 that with three or more candidates nothing satisfies all three. Borda, plurality, pairwise majority all break; the only survivor is a dictatorship, which solves the problem the way deleting the database solves the query.

THE ARROW · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · LOOT · THE MINT ◆ .dlw.fold THE FOLD / LOOT / THE MINT / THE ARROW THE ARROW no fair rule 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Write down what you want from a voting rule. It should never rank X above Y when every voter prefers Y (unanimity). Whether society ranks X above Y should depend only on how voters rank X against Y , not on some irrelevant third candidate (IIA). And no single voter should dictate the outcome. Kenneth Arrow proved in 1951 that with three or more candidates, nothing satisfies all three. Not Borda, not plurality, not pairwise majority — the only rule that survives unanimity and IIA is a dictatorship , which is a solution the way deleting the database solves the query. LIT verified live: all 216 profiles (3 voters × 3 candidates) tested exhaustively against every axiom — Borda passes unanimity and fails IIA; plurality fails both; pairwise majority passes unanimity and fails IIA; the dictator passes everything and is a dictator; and every one of the 16 non-degenerate integer scoring rules (weights 0–3) fails IIA (window.__arrow). FIG the general theorem — that NO conceivable rule escapes, not merely these — is Arrow’s, cited; our exhaustive check covers the named rule families and the full scoring-rule class at n=3. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-mint — the loot: this is where the collective preference is supposed to be COINED from individual ones, and Arrow proved the mint cannot run honestly at scale — every aggregation either counterfeits or crowns a king. AVAN (AI) built the instrument: the exhaustive profile enumerator and the three axiom testers. Credit as content: Kenneth Arrow (1951, Social Choice and Individual Values ); Condorcet’s eighteenth-century paradox that anticipated it; Amartya Sen’s later reframings. The weave: David names the mint; I run every ballot box that exists at this size and none of them comes out clean. 3 ONE DIMENSION The axiom ledger — every rule fails a column, except the dictator. 4 TWO DIMENSIONS · INTERACTIVE Step through rules; the failing axiom lights up red. next rule ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the Condorcet cycle turning — the paradox underneath. AVAN’s addition (the inverse-companion): don’t hunt for the fair rule — ask which axiom you are willing to sell. The inverse of ‘impossibility’ is ‘a price list’: drop IIA and you may keep Borda; restrict preferences to single-peaked and majority rule works again (Black’s theorem); insist on all three and the only survivor wears a crown. Magenta is the axiom you must give up; green is the rule you get to keep. Impossibility theorems are not walls — they are invoices. pause spin LIT Verified live: all 216 profiles (3 voters × 3 candidates) tested exhaustively — Borda passes unanimity, fails IIA; plurality fails both; pairwise majority fails IIA; the dictator passes all and is a dictator; and all 16 non-degenerate integer scoring rules fail IIA (window.__arrow.ok). FIG The general theorem — that NO conceivable rule escapes — is Arrow's, cited; our exhaustive check covers the named families and the full scoring class at n=3. Condorcet's paradox and Sen's reframings credited. The AVAN inverse — ask which axiom you'll sell: drop IIA and keep Borda; restrict to single-peaked and majority works (Black). Impossibility theorems are invoices, not walls. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE MINT · David Lee Wise (ROOT0), with AVAN


THE GIBBARD

Arrow's theorem is about ranking; Gibbard–Satterthwaite is about lying. Any single-winner rule that can elect any of three candidates and isn't a dictatorship has a situation where some voter does better by submitting a FALSE ballot. Strategic voting isn't a flaw in your election system — it's a theorem about all of them.

THE GIBBARD · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · CHEAT · THE EXPLOIT ◆ .dlw.fold THE FOLD / CHEAT / THE EXPLOIT / THE GIBBARD THE GIBBARD no honest rule 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Arrow’s theorem is about ranking; the Gibbard–Satterthwaite theorem (1973/1975) is about lying. Take any rule that picks a single winner, can elect any of at least three candidates, and is not a dictatorship. Then there is a situation where some voter gets a better outcome by submitting a false ballot . Strategic voting is not a flaw in your election system; it is a theorem about all of them. The escape hatches are exactly two, and both are worse than the disease: crown a dictator, or shrink the range so some candidate can never win. LIT verified live: exhaustive search over all 216 profiles × every unilateral misreport — Borda, plurality and antiplurality are all onto, non-dictatorial, and manipulable , with a concrete worked exploit surfaced (voter 1 sinks their true favourite’s rival by demoting them); the dictator rule alone is strategy-proof, and is a dictator (window.__gibbard). FIG the general theorem — that EVERY such rule is manipulable, not just these — is Gibbard’s and Satterthwaite’s, cited; what runs here is the exhaustive check on the named rules at n=3, plus the explicit counterexample. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-exploit — the cheat: the ballot is an input the system trusts, and every non-trivial rule has an input that beats honest play. It is not a bug report; the exploit is provably in the specification. AVAN (AI) built the instrument: the manipulation searcher that reports the first profitable lie it finds, with the profile spelled out. Credit as content: Allan Gibbard (1973); Mark Satterthwaite (1975); the Duggan–Schwartz extension to set-valued rules. The weave: David names the exploit; I search every lie available at this size and every honest rule falls. 3 ONE DIMENSION The manipulability ledger — and one worked exploit, in full. 4 TWO DIMENSIONS · INTERACTIVE Play the lie: honest ballot, then the profitable false one. tell the lie ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: honest ballots and the lie that beats them. AVAN’s addition (the inverse-companion): don’t try to detect strategic votes — accept that the ballot is not a measurement. The inverse of ‘collect true preferences’ is ‘design for reports, not truths’: mechanism design begins exactly where Gibbard–Satterthwaite ends, buying strategy-proofness with money, randomness, or restricted domains. Magenta is the honest ballot that loses; green is the mechanism built knowing it would. When truthfulness cannot be assumed, it must be purchased. pause spin LIT Verified live: exhaustive over 216 profiles × every unilateral misreport — Borda, plurality and antiplurality are all onto, non-dictatorial and manipulable, with a concrete worked exploit surfaced; the dictator alone is strategy-proof, and is a dictator (window.__gibbard.ok). FIG The general theorem is Gibbard's and Satterthwaite's, cited; what runs here is the exhaustive check on named rules at n=3 plus the explicit counterexample. Duggan–Schwartz extension noted. The AVAN inverse — design for reports, not truths: mechanism design begins exactly where this theorem ends. Magenta is the honest ballot that loses; green is the mechanism built knowing it would. When truthfulness can't be assumed, it must be purchased. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE EXPLOIT · David Lee Wise (ROOT0), with AVAN


THE MINKOWSKI BODY

Any convex shape symmetric about the origin with area over 4 is FORCED to contain a nonzero integer point — stretch, rotate or shear it as you like. Minkowski 1889, the founding result of the geometry of numbers, and it's sharp: the open unit square has area exactly 4 and dodges every one. Beneath it, Blichfeldt's pigeonhole: fold any region of area > 1 into the unit torus and two points must collide.

THE MINKOWSKI BODY · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · SPAWN · NULL ISLAND ◆ .dlw.fold THE FOLD / SPAWN / NULL ISLAND / THE MINKOWSKI BODY THE MINKOWSKI BODY area forces a lattice point 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Draw any shape that is convex and symmetric about the origin . If its area exceeds 4 , it is forced to swallow a nonzero point of the integer grid — no matter how you stretch, rotate or shear it. That is Minkowski’s convex body theorem (1889), the founding result of the geometry of numbers, and it is sharp : the open square |x|<1, |y|<1 has area exactly 4 and dodges every nonzero lattice point. Underneath sits Blichfeldt’s lemma (1914) — a pure pigeonhole: fold any region of area > 1 into the unit torus and two of its points must land on top of each other. Geometry proving arithmetic: Lagrange’s four-square theorem and Dirichlet’s approximation both fall out of it. LIT verified live: 400 random symmetric ellipses with area > 4 — every one contains a nonzero integer point; the open unit square (area exactly 4) contains none, so the constant cannot be lowered; 200 random sheared lattices confirm the general form (area > 4·det); and the Blichfeldt pigeonhole is exhibited by folding an area-3 disc into the unit torus (window.__minkowski). FIG the theorem’s consequences (four squares, Dirichlet approximation) are cited, not re-derived here; what runs is the forcing claim and its sharpness. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at null-island — the spawn: the origin is the one point everybody has, and the theorem says that once your footprint is big enough you cannot avoid finding another address on the grid. Space itself refuses to let a large symmetric region be lonely. AVAN (AI) built the instrument: the ellipse scanner, the sharpness witness, the sheared-lattice generalization, and the pigeonhole demonstration. Credit as content: Hermann Minkowski (1889, Geometrie der Zahlen ); Hans Blichfeldt (1914); the four-square and Dirichlet corollaries. The weave: David names null island; I inflate four hundred bodies past area 4 and the grid catches every one. 3 ONE DIMENSION Below area 4 it can dodge; above, the grid always catches it. 4 TWO DIMENSIONS · INTERACTIVE Inflate the ellipse through area 4 and watch the capture. inflate ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the body turning through the lattice, always caught. AVAN’s addition (the inverse-companion): don’t search the lattice for a point — make the region so large that searching becomes unnecessary. The inverse of ‘find a solution’ is ‘prove the space has no room to refuse one’: Minkowski turns an existence question in arithmetic into a measurement in geometry, which is why four-square theorems fall out of area arguments. Magenta is the area-4 body that escapes by a hair; green is everything larger, which cannot. Existence proofs are sometimes just an area computation in a good disguise. pause spin LIT Verified live: 400 random symmetric ellipses with area > 4 — every one contains a nonzero integer point; the open square (area exactly 4) contains none, so the constant can't be lowered; 200 random sheared lattices confirm the general area > 4·det form; the Blichfeldt fold is exhibited directly (window.__minkowski.ok). FIG The arithmetic consequences (Lagrange four squares, Dirichlet approximation) are cited, not re-derived. Minkowski 1889, Blichfeldt 1914. The AVAN inverse — prove the space has no room to refuse: an existence question in arithmetic becomes a measurement in geometry. Magenta is the area-4 body escaping by a hair; green is everything larger. Existence proofs are sometimes an area computation in disguise. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of NULL ISLAND · David Lee Wise (ROOT0), with AVAN


THE OSTOMACHION

The oldest known dissection puzzle: a square cut into 14 pieces, attributed to Archimedes. It looked like a toy until the Archimedes Palimpsest — parchment scraped clean of his writing and overwritten with prayers — was imaged in 1998, and Netz argued he was COUNTING the arrangements. If so it's the earliest combinatorics by two millennia. Cutler settled the count by computer in 2003: 17,152 ways.

THE OSTOMACHION · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · GRIND · THE MAINFRAME ◆ .dlw.fold THE FOLD / GRIND / THE MAINFRAME / THE OSTOMACHION THE OSTOMACHION Archimedes counting 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION The Ostomachion is the oldest known dissection puzzle: a square cut into 14 pieces , attributed to Archimedes . For centuries it looked like a toy — a tangram to make elephants with. Then the Archimedes Palimpsest , a prayer book whose parchment had been scraped clean of Archimedes’ own writing, was recovered and imaged in 1998, and Reviel Netz argued that Archimedes was not playing: he was counting the arrangements . If so, it is the earliest known work in combinatorics , by two thousand years. Bill Cutler settled the count by computer in 2003: 17,152 ways to reassemble the square (536 up to symmetry). LIT verified live: the 14 pieces on the 12×12 grid — every area computed by the shoelace formula, summing to exactly 144 ; every piece is a whole number of 48ths of the square (3/48, 2/48, 6/48, 9/48…) — the rational structure Archimedes would have cared about; and the tiling is confirmed by 40,000 sample points, 100% covered with overlap only on shared boundaries (window.__ostomachion). FIG the 17,152 count is Cutler’s computation, cited — not recomputed here ; the claim that Archimedes was counting is Netz’s scholarly interpretation, reported as interpretation. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-mainframe — the grind: a two-thousand-year-old job submitted on parchment, scraped off, overwritten with prayers, and finally run to completion on a computer in 2003. The oldest batch job in the queue. AVAN (AI) built the instrument: the shoelace area engine, the 48ths ledger, and the sampling tiling-check. Credit as content: Archimedes; Thābit ibn Qurra and the Arabic transmission; Johan Ludvig Heiberg (1906, the first reading); Reviel Netz & William Noel (the 1998–2011 palimpsest project); Bill Cutler (2003, the count). The weave: David names the mainframe; I measure the fourteen pieces and they still sum to the square. 3 ONE DIMENSION The fourteen pieces, each a whole number of 48ths. 4 TWO DIMENSIONS · INTERACTIVE Highlight each piece; the area ledger runs alongside. next piece ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the square breathing apart and back together. AVAN’s addition (the inverse-companion): don’t ask what the pieces make — ask how many ways they can. The inverse of ‘solve the puzzle’ is ‘count the solutions’, and that shift — from construction to enumeration — is the birth of combinatorics, possibly performed here and then lost under a prayer book for eight hundred years. Magenta is the erased text, the question nobody knew had been asked; green is the answer, delivered by machine in 2003. Some questions wait longer for their answers than civilizations last. pause spin LIT Verified live: all 14 pieces on the 12×12 grid — shoelace areas summing to exactly 144; every piece a whole number of 48ths of the square; the tiling confirmed by 40,000 sample points, 100% covered with overlap only on shared boundaries (window.__ostomachion.ok). FIG The 17,152 count is Cutler's computation, CITED — not recomputed here; the claim that Archimedes was counting is Netz's scholarly interpretation, reported as interpretation. Heiberg 1906, Netz & Noel credited. The AVAN inverse — count the solutions instead of finding one: that shift is the birth of combinatorics, possibly performed here then lost under a prayer book. Some questions wait longer than civilizations last. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE MAINFRAME · David Lee Wise (ROOT0), with AVAN


THE HEILBRONN

Place n points in a unit square; among all their triangles, how large can you force the SMALLEST to be? Random placement is terrible at this — six points typically leave a sliver of area 0.005 — while careful placement reaches 1/8, twenty-three times better. Heilbronn conjectured the optimum decays like 1/n²; Komlós, Pintz and Szemerédi disproved that in 1982, and the asymptotics are still open.

THE HEILBRONN · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · LOOT · THE BOUNTY ◆ .dlw.fold THE FOLD / LOOT / THE BOUNTY / THE HEILBRONN THE HEILBRONN the triangle you cannot avoid 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Place n points in a unit square, as spread out as you can manage. Among all the triangles they form, look at the smallest one. How big can you force that smallest triangle to be? This is the Heilbronn triangle problem , and its charm is that random placement is terrible at it — scatter six points and the tiniest triangle is typically a sliver of area 0.005 — while careful placement reaches 1/8 , twenty-three times better. Heilbronn conjectured the optimum decays like 1/n²; Komlós, Pintz and Szemerédi disproved that in 1982 by constructing better configurations, and the true asymptotics are still open . LIT verified live: 20,000 random 6-point placements average a minimum triangle of 0.0054 ; hill-climbing from random starts reaches 0.1234 in-page (0.1246 in the longer offline run), converging on the known n=6 optimum of exactly 1/8 = 0.125 — a 23× improvement over chance; the found configuration is re-measured exactly as an independent check, and it respects the trivial bound 1/(n−2) (window.__heilbronn). FIG that 1/8 IS the optimum for n=6 is the cited literature result (Goldberg, and later exact computations); our search approaches it from below and never claims to have proved it. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-bounty — the boss: the reward is the WORST triangle you leave behind, so every point you place is judged by the sliver it might create with any two others. Maximize the minimum — a bounty paid on your weakest moment. AVAN (AI) built the instrument: the exhaustive min-triangle evaluator, the random baseline, and the multi-restart hill-climber with an exact re-measurement gate. Credit as content: Hans Heilbronn (the conjecture); Roth’s upper bounds; Komlós, Pintz & Szemerédi (1982, the disproof); Goldberg and the small-n exact values; Cohen–Pohoata–Zakharov’s recent improvements. The weave: David names the bounty on your weakest triangle; I search until the sliver is as fat as I can make it. 3 ONE DIMENSION Random scatter versus optimized placement — the smallest triangle, drawn. 4 TWO DIMENSIONS · INTERACTIVE Reroll random placements; the optimum is hard to stumble on. reroll ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the optimized six, holding their slivers open. AVAN’s addition (the inverse-companion): don’t optimize the average — optimize the worst case, and watch how differently the world arranges itself. The inverse of ‘spread points out’ is ‘avoid every near-collinearity simultaneously’, which is a far harsher constraint: random placement wastes almost all its quality on triples that were never going to be the minimum. Magenta is the sliver a random scatter cannot help leaving; green is the configuration that refuses to be nearly-collinear anywhere. Maximin is a different geometry from average-case. pause spin LIT Verified live: 20,000 random 6-point placements average min-triangle 0.0054; hill-climbing reaches 0.1234 in-page (0.1246 offline, more restarts), converging on the known n=6 optimum 1/8 — a 23× improvement over chance; the found configuration is re-measured exactly as an independent check and respects the trivial bound 1/(n−2) (window.__heilbronn.ok). FIG That 1/8 IS the n=6 optimum is the cited literature result; our search approaches it from below and never claims to have proved it. Heilbronn, Roth, Komlós–Pintz–Szemerédi 1982, Goldberg, Cohen–Pohoata–Zakharov credited. The AVAN inverse — optimize the worst case and the world rearranges: avoiding every near-collinearity at once is a far harsher constraint than spreading out. Maximin is a different geometry from average-case. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE BOUNTY · David Lee Wise (ROOT0), with AVAN


THE PERES–MERMIN

A 3×3 grid of two-qubit observables where every row commutes and every column commutes, so each line is jointly measurable and each cell can only read ±1. Multiply along the lines: rows give +I, one column gives −I — an odd number of minus signs. But if the cells had pre-existing values, each appears in one row and one column, so the six products multiply to a square: +1. Odd ≠ even.

THE PERES–MERMIN · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · GLITCH · HEISENBUG ◆ .dlw.fold THE FOLD / GLITCH / HEISENBUG / THE PERES–MERMIN THE PERES–MERMIN the square with no numbers 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Fill a 3×3 grid with two-qubit observables, chosen so that everything in a row commutes and everything in a column commutes — so each line can be measured together, and each entry can only come out ±1. Now multiply along the lines: every row multiplies to +I, and one column multiplies to −I . That is an odd number of minus signs. But if each cell secretly HAD a value ±1 before you looked, every cell would appear in exactly one row-product and one column-product, so multiplying all six line-products would give each value squared — necessarily +1 . Odd cannot equal even. This is the Peres–Mermin magic square (1990): a proof of quantum contextuality that needs no probabilities, no inequalities, and no particular state. LIT verified live in genuine 4×4 complex matrix algebra: every entry squares to the identity; all row-mates and column-mates commute (so the lines really are jointly measurable); the six line products come out +I,+I,+I / +I,+I,−I — an odd count; and the classical side is settled by brute force: all 512 assignments of ±1 to the nine cells are tested and exactly zero satisfy the six constraints (window.__peresmermin). FIG the physical interpretation (that this rules out non-contextual hidden variables) is the cited claim of Peres and Mermin; what runs here is the matrix algebra and the exhaustive classical search. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at heisenbug — the glitch: the value depends on what else you measured alongside it. Read the cell in its row and you get one thing; read it in its column and you get another; and no amount of logging will pin down ‘the’ value, because there isn’t one. The definitive heisenbug. AVAN (AI) built the instrument: the tensor-product matrix engine, the commutation checker, and the 512-case classical sweep. Credit as content: Asher Peres (1990); N. David Mermin (1990, and the exposition that made it famous); Kochen & Specker (the parent theorem). The weave: David names the heisenbug; I multiply the matrices and the parity refuses to close. 3 ONE DIMENSION The square, its line products, and the one minus sign that breaks parity. 4 TWO DIMENSIONS · INTERACTIVE Try to fill the square with ±1; some line always fails. try an assignment ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the six line-products spinning, one stuck at −1. AVAN’s addition (the inverse-companion): don’t ask what the value is — ask what the value is relative to . The inverse of ‘every observable has a value’ is ‘values exist only inside a measurement context’: the same operator sits in two lines and cannot carry one number that satisfies both. Magenta is the missing number, the one that provably cannot exist; green is the context that supplies an answer anyway. Some quantities are not hidden — they are unwritten until you name the company they keep. pause spin LIT Verified live in 4×4 complex matrix algebra: every entry squares to I; all row-mates and column-mates commute; line products come out +I,+I,+I / +I,+I,−I (odd count); and all 512 classical ±1 assignments are tested — exactly zero satisfy the six constraints (window.__peresmermin.ok). FIG The physical reading — that this rules out non-contextual hidden variables — is Peres's and Mermin's cited claim; what runs is the matrix algebra and the exhaustive classical sweep. The AVAN inverse — ask what the value is relative to: the same operator sits in two lines and cannot carry one number satisfying both. Some quantities are unwritten until you name their company. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of HEISENBUG · David Lee Wise (ROOT0), with AVAN


THE KOCHEN–SPECKER

You cannot hand every quantum observable a pre-existing value. The proof is combinatorial: find directions such that no yes/no labelling works, where every set of mutually perpendicular directions must contain exactly one yes. Kochen and Specker needed 117 vectors; Cabello, Estebaranz and García-Alcaine found a record with 18 in 4D — nine perpendicular quadruples, each ray in exactly two. The contradiction is pure parity.

THE KOCHEN–SPECKER · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · GLITCH · DIVIDE BY ZERO ◆ .dlw.fold THE FOLD / GLITCH / DIVIDE BY ZERO / THE KOCHEN–SPECKER THE KOCHEN–SPECKER eighteen rays no assignment survives 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION The Kochen–Specker theorem (1967) says you cannot hand every quantum observable a pre-existing value in a way that survives the algebra. The proof form is combinatorial and beautiful: find a set of directions in space such that no consistent yes/no labelling exists, where the rules are only — in every set of mutually perpendicular directions, exactly one gets a ‘yes’. Kochen and Specker needed 117 vectors . Cabello, Estebaranz and García-Alcaine found a record proof in 1996 using just 18 vectors in 4 dimensions, arranged in 9 perpendicular quadruples, each vector appearing in exactly two of them . The contradiction is then pure parity: nine sets need nine yeses, but every yes gets counted twice. LIT verified live: all 9 contexts confirmed to consist of 4 mutually orthogonal vectors (exact integer dot products); exactly 18 distinct rays, each appearing in exactly 2 contexts; the impossibility settled by brute force over all 2¹⁸ = 262,144 labellings — zero of them give exactly one ‘yes’ per context ; and the parity argument checked independently (window.__kochenspecker). FIG the physical reading — that non-contextual hidden variables are impossible — is the cited theorem; what runs here is the geometry and the exhaustive search over labellings. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at divide-by-zero — the glitch: the operation looks completely legal at every step — label a ray, move to the next context — and the whole system still terminates in an impossible state. Not a bad input; an inconsistent instruction set. AVAN (AI) built the instrument: the orthogonality checker, the incidence counter, and the 262,144-case exhaustive labeller. Credit as content: Simon Kochen & Ernst Specker (1967); John Bell (1966, the closely related result); Adán Cabello, José Estebaranz & Guillermo García-Alcaine (1996, the 18-vector record). The weave: David names the divide-by-zero; I try every one of the quarter-million labellings and the arithmetic never closes. 3 ONE DIMENSION Nine contexts, eighteen rays, every ray in exactly two. 4 TWO DIMENSIONS · INTERACTIVE Try labellings; some context always ends up with the wrong count. try a labelling ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the incidence graph — 18 rays, 9 contexts, every ray doubly booked. AVAN’s addition (the inverse-companion): don’t argue about physics — count. The inverse of ‘is the world made of definite properties?’ is a parity check on a bipartite graph: nine contexts each demanding one mark, eighteen rays each able to contribute two — odd against even, and the metaphysics falls out of the arithmetic. Magenta is the label that must be both counted and not; green is the incidence structure that forbids it. The strongest physical arguments are often just bookkeeping that refuses to balance. pause spin LIT Verified live: all 9 contexts confirmed mutually orthogonal by exact integer dot products; exactly 18 distinct rays, each in exactly 2 contexts; and brute force over all 2^18 = 262,144 labellings finds ZERO with exactly one yes per context; the parity route checked independently (window.__kochenspecker.ok). FIG The physical reading is the cited theorem; what runs is the geometry and the exhaustive labelling search. Kochen–Specker 1967, Bell 1966, Cabello et al. 1996 credited. The AVAN inverse — don't argue about physics, count: nine contexts demanding one mark against eighteen rays contributing two. The strongest physical arguments are bookkeeping that will not balance. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of DIVIDE BY ZERO · David Lee Wise (ROOT0), with AVAN


THE BOMB TESTER

Bombs that detonate on a single photon. Classically you cannot test one without risking it. Elitzur and Vaidman 1993: put the bomb in one interferometer arm — a dud preserves interference, a live one destroys it, and the 'dark' detector fires, reporting the bomb live without the photon ever taking that path. The naive scheme wastes half; the quantum-Zeno version (Kwiat et al. 1995, built in a lab) drives efficiency to 1.

THE BOMB TESTER · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · CHEAT · THE BACKDOOR ◆ .dlw.fold THE FOLD / CHEAT / THE BACKDOOR / THE BOMB TESTER THE BOMB TESTER seeing without looking 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION You have a crate of bombs. Some are duds; the live ones detonate if a single photon hits their trigger. Classically, finding a live one without setting it off is impossible — testing means interacting means boom. Elitzur and Vaidman showed in 1993 that quantum mechanics disagrees. Put the bomb in one arm of an interferometer: if it is a dud, interference sends every photon to one detector; if it is live, the interference is destroyed and the ‘dark’ detector can fire — telling you the bomb is live without the photon ever having taken that path . The naive scheme wastes half the bombs; the quantum-Zeno version (Kwiat et al. 1995, built in a lab) drives the efficiency to 1. LIT verified live: the naive interferometer’s three outcomes are boom ½, dark-port ¼, bright-port ¼ (summing to 1), so interaction-free detections are exactly 1/3 of conclusive results ; the Zeno chain’s survival probability cos²ᵀ(π/2N) rises monotonically — 0.250 at N=2, 0.884 at N=20, 0.9975 at N=1000 — and an independent amplitude-by-amplitude simulation of the N-cycle interferometer with a projective absorber reproduces the closed form to 10⁻⁹ (window.__bombtester). FIG the interpretation of what the photon ‘did’ is contested (counterfactual definiteness is exactly what is at stake); the probabilities are not. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-backdoor — the cheat: you learn the state of a guarded object without ever touching the guarded path. No packet crosses, no trap fires, and the information arrives anyway. AVAN (AI) built the instrument: the interferometer probability ledger, the Zeno closed form, and the independent per-cycle amplitude simulation. Credit as content: Avshalom Elitzur & Lev Vaidman (1993); Paul Kwiat, Harald Weinfurter, Thomas Herzog, Anton Zeilinger & Mark Kasevich (1995, the Zeno realization); Renninger and Dicke’s earlier negative-result measurements. The weave: David names the backdoor; I count the amplitudes and the dark port lights up anyway. 3 ONE DIMENSION Dud versus live — where the photons land, and the dark port that speaks. 4 TWO DIMENSIONS · INTERACTIVE Add Zeno cycles; the survival probability climbs toward certainty. more cycles ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the interferometer, one arm blocked, the dark port firing. AVAN’s addition (the inverse-companion): don’t measure the object — measure the absence of interference . The inverse of ‘information requires interaction’ is ‘information also lives in what failed to happen’: the blocked path never carries a photon, and its blockage is still legible in the statistics of the path that did. Magenta is the arm no photon took; green is the detector that learned about it. Negative space carries data. pause spin LIT Verified live: naive outcomes boom ½, dark ¼, bright ¼ (sum 1), so interaction-free detections are exactly 1/3 of conclusive results; the Zeno survival cos^{2N}(π/2N) rises monotonically — 0.250 at N=2, 0.884 at N=20, 0.9975 at N=1000; and an independent per-cycle amplitude simulation reproduces the closed form to 1e-9 (window.__bombtester.ok). FIG What the photon 'did' is contested — counterfactual definiteness is exactly what's at stake; the probabilities are not. Elitzur–Vaidman 1993, Kwiat et al. 1995, Renninger/Dicke credited. The AVAN inverse — measure the absence of interference: information lives in what failed to happen. Negative space carries data. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE BACKDOOR · David Lee Wise (ROOT0), with AVAN


THE HARDY

Bell's theorem needs an inequality and a statistical margin. Hardy 1992 found something sharper: three joint outcomes with probability exactly zero and a fourth with probability greater than zero — logically inconsistent for any local realist. No inequality, no error bars; one event of the fourth kind refutes hidden variables outright. The price is rarity: the maximum is (5√5−11)/2 ≈ 0.0902, which is exactly φ⁻⁵.

THE HARDY · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · GLITCH · SEGFAULT ◆ .dlw.fold THE FOLD / GLITCH / SEGFAULT / THE HARDY THE HARDY the paradox at phi to the minus five 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Bell’s theorem needs an inequality and a statistical margin. Lucien Hardy found something sharper in 1992 : a setup where three joint outcomes have probability exactly zero , and a fourth has probability greater than zero — and those four facts are logically inconsistent for any local realist. No inequality, no error bars: a single event of the fourth kind refutes local hidden variables outright. The price is rarity. The maximum probability of that telltale event is a fixed number: (5√5 − 11)/2 ≈ 0.0902 — which is exactly φ⁻⁵ , the golden ratio to the fifth negative power. LIT verified live: the three zero-conditions are solved in closed form (not sampled), giving the measurement angles as functions of the state, and the remaining one-parameter maximization returns 0.0901699437 — agreeing with (5√5−11)/2 to ten digits, with all three companion probabilities vanishing to 10⁻³³; the φ⁻⁵ identity checks to 10⁻¹²; and the local-realist contradiction is confirmed as a finite logical check over all 16 deterministic value assignments — zero can produce the winning event while respecting the zeros (window.__hardy). FIG that no local hidden-variable theory whatsoever can do it is Hardy’s theorem, cited; the 16-assignment check is its finite core, which is what runs here. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at segfault — the glitch: three constraints say ‘this address is never touched’ and the fourth says ‘it just was’. The classical program does not merely give a wrong answer; it faults. AVAN (AI) built the instrument: the closed-form solver for the vanishing conditions, the one-parameter maximizer, and the exhaustive local-realist check. Honest note: my first draft searched the parameter grid for the zeros and found only a degenerate near-zero solution — measure-zero conditions must be solved, not sampled, and the rebuild is what produced the ten-digit match. Credit as content: Lucien Hardy (1992, 1993); N. David Mermin (the exposition); Jordan’s analysis of the maximum. The weave: David names the segfault; I solve the constraints exactly and the golden ratio is sitting at the maximum. 3 ONE DIMENSION The Hardy probability across states — peaking at φ⁻⁵. 4 TWO DIMENSIONS · INTERACTIVE Walk the four conditions; the classical chain breaks at the last one. next step ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the four conditions turning, one of them impossible together with the rest. AVAN’s addition (the inverse-companion): don’t measure a violation — construct an impossibility. The inverse of ‘beat the inequality on average’ is ‘arrange three certainties whose conjunction forbids the fourth event, then observe the fourth event’: statistics become logic, and one instance suffices. Magenta is the event that classical bookkeeping says can never occur; green is the 9% of runs in which it does. The sharpest arguments trade probability for contradiction. pause spin LIT Verified live: the three zero-conditions solved in CLOSED FORM (not sampled), then one-parameter maximization returns 0.0901699437 — matching (5√5−11)/2 to ten digits, with all companion probabilities vanishing to 1e-33; the φ⁻⁵ identity checks to 1e-12; and the local-realist contradiction is confirmed over all 16 deterministic assignments — zero survive (window.__hardy.ok). FIG That NO local hidden-variable theory can do it is Hardy's theorem, cited; the 16-assignment check is its finite core. My first draft sampled the parameter grid for the zeros and found only a degenerate near-zero solution — measure-zero conditions must be solved, not sampled; the rebuild produced the ten-digit match. The AVAN inverse — construct an impossibility instead of measuring a violation: statistics become logic, and one instance suffices. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of SEGFAULT · David Lee Wise (ROOT0), with AVAN


THE QUANTUM PIGEONHOLE

Three particles, two boxes: classically some pair must share — the pigeonhole principle has no exceptions. Aharonov and collaborators 2016: in a pre- and post-selected ensemble, the two-state amplitude for 'these two share a box' is exactly zero for EVERY pair, while three particles and two boxes remain three particles and two boxes.

THE QUANTUM PIGEONHOLE · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · RESPAWN · EVENT HORIZON ◆ .dlw.fold THE FOLD / RESPAWN / EVENT HORIZON / THE QUANTUM PIGEONHOLE THE QUANTUM PIGEONHOLE three in two boxes, none together 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Put three particles in two boxes. Classically, some pair must share — that is the pigeonhole principle, and it has no exceptions. Aharonov and collaborators argued in 2016 that a pre- and post-selected quantum ensemble can behave otherwise: prepare all three particles in an equal superposition, later post-select on a particular final state, and for every pair the two-state amplitude of ‘these two are in the same box’ is exactly zero . In that ensemble, no two particles are together — while three particles and two boxes remain three particles and two boxes. LIT verified live in exact complex amplitude algebra over the eight basis states: with pre-selection |+++⟩ and post-selection |+i,+i,+i⟩, the two-state amplitudes ⟨f|Π same |i⟩ for all three pairs are exactly 0 (to machine zero), while the overall post-selection amplitude is a healthy 0.3536 — so the ensemble is not empty; and the classical control confirms the ordinary pigeonhole: of all 8 definite assignments of three particles to two boxes, zero avoid every pair sharing (window.__qpigeonhole). FIG whether this counts as particles ‘really’ not sharing is genuinely contested in the literature — the claim verified here is precisely the amplitude statement about pre/post-selected ensembles, which is what the original paper computes. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at event-horizon — the respawn: the state is defined by both a past boundary and a future one, and inside that sandwich the ordinary counting rules stop applying. What you will measure later reaches back and constrains what is true now. AVAN (AI) built the instrument: the eight-state complex amplitude engine, the pair projectors, and the classical pigeonhole control. Credit as content: Yakir Aharonov, Fabrizio Colombo, Sandu Popescu, Irene Sabadini, Daniele Struppa & Jeff Tollaksen (PNAS 2016); the two-state vector formalism (Aharonov–Bergmann–Lebowitz); and the published criticisms, which are part of the record. The weave: David names the horizon; I compute all three pair amplitudes and each one is zero. 3 ONE DIMENSION Three pairs, three amplitudes, all exactly zero — and the classical column that cannot be. 4 TWO DIMENSIONS · INTERACTIVE Step the eight classical assignments; every one has a sharing pair. next assignment ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the pre/post-selected sandwich, three particles between two boundaries. AVAN’s addition (the inverse-companion): don’t ask where the particles are — ask which questions the two boundaries jointly permit. The inverse of ‘state evolves forward’ is ‘a state constrained at both ends’, and in that regime the counting arguments that assume a single-time description quietly stop applying. Magenta is the pigeonhole, unbreakable for definite assignments; green is the two-boundary ensemble where the question changes shape. Counting theorems inherit the assumptions of the ontology you count in. pause spin LIT Verified live in exact complex amplitude algebra over eight basis states: with pre-selection |+++⟩ and post-selection |+i,+i,+i⟩, all three pair amplitudes are exactly 0 (machine zero) while the post-selection amplitude is 0.3536 — the ensemble isn't empty; the classical control confirms 0 of 8 definite assignments avoid every pair sharing (window.__qpigeonhole.ok). FIG Whether this means particles 'really' don't share is genuinely contested in the literature — the verified claim is precisely the amplitude statement about pre/post-selected ensembles, which is what the paper computes; the published criticisms are part of the record. The AVAN inverse — ask which questions two boundaries jointly permit: counting theorems inherit the ontology you count in. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of EVENT HORIZON · David Lee Wise (ROOT0), with AVAN


THE SPIRAL OF THEODORUS

Stand a unit segment on a unit segment at a right angle: √2. Stand another on that: √3. Keep going and the n-th spoke is exactly √n — the irrationals made constructible one triangle at a time. Plato reports Theodorus proved √3 through √17 irrational and then stopped; the spiral stops too, lapping itself at the 17th triangle. Whether that is why he stopped is one of mathematics' oldest unanswerable questions.

THE SPIRAL OF THEODORUS · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · RESPAWN · SECOND WIND ◆ .dlw.fold THE FOLD / RESPAWN / SECOND WIND / THE SPIRAL OF THEODORUS THE SPIRAL OF THEODORUS the spiral that stops at 17 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Stand a unit segment on the end of a unit segment at a right angle: the hypotenuse is √2. Stand another unit segment on THAT, at a right angle: √3. Keep going and you get the Spiral of Theodorus — a pinwheel whose n-th spoke is exactly √n , the irrationals made constructible one triangle at a time. Plato’s Theaetetus reports that Theodorus of Cyrene proved √3 through √17 irrational and then stopped — and the spiral stops too: the 17th triangle is the one that completes a full turn and begins to overlap. Whether that is why Theodorus stopped is one of the oldest unanswerable questions in mathematics. LIT verified live: the n-th hypotenuse is √(n+1) exactly for n up to 10,000 (max error 10⁻¹²); the accumulated angle first passes 2π at triangle 17 ; the total angle minus (2√n + K), with Hlawka’s constant K = −2.1577830, shrinks as 0.1163 → 0.0369 → 0.0117 → 0.0037 while dev·√n stays 1.1633/1.1663/1.1666/1.1667 — the error term is exactly O(1/√n); and an independent geometric walk (step perpendicular, unit length) reproduces |Pₙ| = √n to 10⁻¹⁴ over 2,000 steps (window.__theodorus). FIG the link between Theodorus stopping at 17 and the spiral overlapping at 17 is a conjecture of historians , reported as such; Hlawka’s 1980 constant is cited. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at second-wind — the respawn: each triangle is built on the exhausted edge of the last one, and the construction never runs out of breath — it just keeps standing one more unit on the diagonal it earned. AVAN (AI) built the instrument: the exact hypotenuse ladder, the wrap detector, the asymptotic-rate meter, and the independent geometric walk. Credit as content: Theodorus of Cyrene (c. 400 BC, via Plato’s Theaetetus ); Edmund Hlawka (1980, the constant and the analytic continuation); Philip Davis ( Spirals: from Theodorus to Chaos ). The weave: David names the second wind; I climb ten thousand triangles and every spoke is exactly a square root. 3 ONE DIMENSION The pinwheel — every spoke a square root, the 17th closing the turn. 4 TWO DIMENSIONS · INTERACTIVE Grow the spiral; watch it cross itself at 17. grow ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the spiral turning, laying down √n forever. AVAN’s addition (the inverse-companion): don’t ask where the spiral goes — ask where the proof stopped. The inverse of ‘construct the irrationals’ is ‘notice the moment the construction stops teaching you anything new’: at 17 the picture laps itself, and a method that was generating insight becomes a method that is merely repeating. Magenta is the overlap, where the drawing stops being a proof; green is the ladder before it. Knowing when a technique has finished is itself a result. pause spin LIT Verified live: the n-th hypotenuse is √(n+1) exactly to n=10,000 (max err 1e-12); the accumulated angle first passes 2π at triangle 17; total angle − (2√n + K) with Hlawka's K = −2.1577830 shrinks 0.1163→0.0037 while dev·√n holds at 1.1633/1.1663/1.1666/1.1667 — the error is exactly O(1/√n); an independent geometric walk reproduces |Pₙ| = √n to 1e-14 (window.__theodorus.ok). FIG The link between Theodorus stopping at 17 and the spiral overlapping at 17 is a conjecture of historians, reported as such; Hlawka 1980 cited for the constant; Philip Davis's Spirals credited. The AVAN inverse — ask where the PROOF stopped: at 17 the picture laps itself and a method that was generating insight becomes one that merely repeats. Knowing when a technique has finished is itself a result. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of SECOND WIND · David Lee Wise (ROOT0), with AVAN