Chirikov standard map

A kicked rotor — the simplest Hamiltonian system that can go chaotic — keeps its momentum forever bounded by invariant KAM tori at weak kicks. At exactly what kick strength does the LAST torus die, opening the whole phase space to chaotic transport, and can that threshold be measured from the map itself with no formula to plug in (none exists — no closed form for K_c is known to science)?

Units: dimensionless kick strength (Greene 1979: 0.971635 ± 3e-6; MacKay 1983 renormalization: 0.97163540631)

▶ Run this simulationRead how it works

The finding

Chirikov standard map — the last torus dies at Greene's K_c, and the death threshold emerges from periodic-orbit stability alone: with NOTHING coded but p' = p + K·sinθ, θ' = θ + p' and its tangent map, the chained residue criterion returns K̂_c = 0.971635132 ± 7.9e-8 vs MacKay's 0.971635406 (rel 2.8e-7), the convergence ratio lands on MacKay's renormalization eigenvalue as 1/δ² to 4% (never fed), the critical residue R∞ = 0.2500888 emerges to 2.6e-4 (never fed), probe levels 0.2/0.3 straddle and converge to the same point (0.25 is a probe, not an input), 24 chaotic-layer orbits confirm the torus is an ABSOLUTE barrier below (0/24 escapes in 1e7 steps) and gone above (every seed's crossing K > K̂_c, one-sided by theorem), and Chirikov's resonance-overlap rival π²/4 ≈ 2.47 is falsified 743 SE above the measured transport threshold — the barrier is already gone at K = 1.2, 51% below the rival's prediction

Method

Greene's residue criterion, chained and unprejudiced: periodic orbits approximating the golden-mean torus (rotation number F_n/F_{n+1} → (√5−1)/2, orders 5/8 up to 2584/4181) are found on the symmetry line θ = π by bisection in p0; each orbit's residue R = (2 − Tr M)/4 comes from the product of the map's per-step tangent matrices [[1+K·cosθ, 1],[K·cosθ, 1]]; K_n solves R_n(K) = 0.25 by bisection in K, the chain starting from a WIDE bracket K ∈ [0.5, 1.5] at the lowest order (no 0.9716 seeded anywhere) and warm-starting each order from the last. K̂_c = mean of the 5-order plateau q = 610…4181 (4181/6765 excluded a priori — below the double-precision orbit-closure noise floor). Independent confirmation: 24 mulberry32-seeded orbits jittered around the HYPERBOLIC fixed point (0,0) — not the elliptic (0,π) period-2 island, which traps seeds even above K_c — must never change p by 2π below K̂_c (an invariant circle is an absolute barrier) and must all cross above; per-seed bisected crossing K at a 1e6-step budget gives the live measurement 1.0273 ± 0.0019. 8 gates in scripts/stdmap-derisk.mjs (~22 s); known value, R∞ and δ in scripts/oracles/stdmap.reference.json, loaded ONLY to score; tamper ⇒ exit 1. ?world=stdmap.

Measurements, controls & cross-checks

Recovered se

7.9000e-8

Plateau

Orders q

• 610
• 987
• 1597
• 2584
• 4181

Spread

4.0000e-7

Worst order rel

5.0000e-7

Note

five independent Fibonacci orders agree to 4e-7; the residual 2.8e-7 mean offset is the un-extrapolated alternating tail, within ~3.5 SE of the plateau scatter

Renormalization scaling

Measured ratio gm

0.3929

Delta sq inv

0.3773

Rel

0.041

Note

the K_n sequence converges alternating-geometrically; its ratio matching 1/δ² for MacKay's δ = 1.6280 (two Fibonacci orders per oscillation period) is EMERGENT — δ appears only in the scoring

Critical residue

Measured

0.249834

Mackay

0.2500888

Abs err

0.000255

Note

residues of the q = 987…2584 orbits evaluated AT the recovered K̂_c hover at MacKay's universal critical residue — a second renormalization constant recovered, never fed

Probe independence

Probe 0 2

dist to K̂_c shrinks 1.44e-3 → 4.58e-5 (q = 144 → 4181), from below

Probe 0 3

dist shrinks 1.16e-3 → 3.97e-5, from above

Note

rerunning the ENTIRE chain with the criterion's threshold at 0.2 and 0.3 converges to the same K_c from opposite sides — Greene's 0.25 is a probe level, not an input; the recovered number is a property of the map

Dichotomy

Below

|R| collapses 9.5e-5 → 1.3e-7 (q = 144 → 987) at K̂_c − 0.05

Above

|R| explodes 6.6e+2 → 9.8e+19 at K̂_c + 0.05

Note

the recovered point is genuinely critical — residues die on one side and diverge on the other, like ising/percolation/logistic/kuramoto

Transport

Seeds

24

Below

0/24 orbits change p by 2π in 1e7 steps at K̂_c − 0.01 — the torus is an ABSOLUTE barrier

Above

24/24 cross at K = 1.2 within 1e6 steps

Crossing K

1.0273 ± 0.0019 (SE), min 1.0109

Note

independent dynamics-only confirmation: every seed's crossing K exceeds K̂_c as the theorem demands (one-sided bound — an invariant circle cannot be crossed), and the +5.6% mean bias at the 1e6-step budget is the MacKay–Meiss–Percival cantorus critical slowing, disclosed and gated inside (0, 0.08)

Rival chirikov overlap

Prediction

2.4674

Rel error

1.54

Z above transport

743

Note

Chirikov's simple resonance-overlap criterion (primary separatrix half-widths 2√K touch ⇒ K_ov = π²/4) overpredicts the barrier's death by 154%: the same machinery shows all 24 seeds crossing at K = 1.2 (51% below the rival) and the golden convergents' residues exploded there (|R(q=144)| ≈ 2e13). Overlap counts only primary-resonance widths; it misses the secondary layers and the self-similar noble-torus breakup that Greene's criterion resolves — historically why Chirikov's own refined estimates moved toward ~1

What it reduces to

Greene's residue criterion for the destruction of the golden-mean KAM torus of the Chirikov standard map (Greene 1979, J. Math. Phys. 20, 1183; MacKay 1983, Physica D 7, 283: K_c = 0.97163540631, R∞ ≈ 0.2500888, δ = 1.6280). It VALIDATES, not derives: no closed form for K_c exists anywhere in science, so nothing CAN be coded in the generator — the recovery path is the operational definition itself (the K at which golden-convergent periodic orbits are neither asymptotically stable nor unstable), built from the bare map and its tangent map only. Non-circularity is checked four ways in-script: the chain starts from an unprejudiced K ∈ [0.5, 1.5] bracket at the lowest Fibonacci order; the probe-level sweep shows the 0.25 threshold is not load-bearing (0.2 and 0.3 converge to the same answer from opposite sides); the critical residue R∞ and the renormalization ratio 1/δ² are scored but never fed, and both emerge; and the fully independent transport gate — pure orbit iteration, no residues, no periodic orbits — brackets the same threshold with the theorem-mandated one-sidedness (no seed crosses below, every seed's crossing K above). The decisive control is historical: Chirikov's 1979 resonance-overlap estimate π²/4 ≈ 2.47, the criterion Greene's method superseded, is falsified by 154% / 743 SE with the same map. This is the lab's fifth exactly-known critical point (after ising, percolation, logistic, kuramoto) and its FIRST Hamiltonian-chaos/KAM oracle — the first time the lab measures the death of an invariant torus rather than the onset of an attractor's chaos (logistic/lorenz are dissipative; this transition has no attractor at all).

Module systematics

HONEST-MODULE CERTIFICATE (refiner 2026-07-24, ZERO module edits — 6th zero-edit certificate; gates H..N in scripts/stdmap-derisk.mjs, pins in reference module_certificate): the derisk EXECUTES StandardMapModule.ts's own compute path — eight sha256-pinned slices (module TWO_PI, the 10 static constants, the _K = 1.0 default, the orbitLyapunov body with Benettin tangent map + both torus wraps, the chaoticFraction body, init's destructure, init's classification loop, the HUD textContent arithmetic) mechanically stripped (every replacement count asserted) and run via new Function with a 3-field Vector3 stub; static reads resolve through a shared C object so budget probes patch the module's OWN function. Pinned strict-===: the executed phase portrait at the shipped default K = 1.0 (29/99 chaotic orbits, the 99-λ grid sha256-pinned with λ ∈ [−1.85e-17, 0.245132…] to the ulp, both thin-instance point clouds — 42000 regular + 17400 chaotic points — sha256-pinned, the display identity fracHere === chaoticFraction(K)) and the FULL 204-char HUD string from the module's own arithmetic including the executed chaoticFraction at every K_SCAN value (0/13/29/77/93%, each pinned exactly; a one-character HUD tamper exits 1). Screen↔oracle reconciliation: the 'K_c ≈ 0.972 (last torus)' line remains a labeled literature constant, but the certificate proves |K_C − known| = 4.06e-7 < 5e-7 (it is the CORRECTLY-ROUNDED 6-dp literature value, not an arbitrary overlay) AND that the oracle's own non-circular recovery K̂_c = 0.971635132 rounds to the SAME 3-digit display string '0.972' — the screen agrees with the lab's measurement at full display precision. Instrument calibration on executed floats: seeded exactly on the hyperbolic fixed point (0,0), where the module's Benettin sum telescopes to ln‖Mᴺv₀‖ of the constant tangent matrix [[1+K,1],[K,1]], Richardson (600,1200) on the module's OWN loop returns the closed-form eigenvalue ln((2+K+√(K(K+4)))/2) to 5.6e-14 (K=1) and 6.2e-15 (K=4), λ600/λ1200 themselves pinned; at K = 0 the executed loop returns λ === 0 BIT-EXACTLY for all 99 grid orbits ((1,0) is the shear's neutral tangent vector) and chaoticFraction(0) === 0 — no kicks, no chaos, to the last ulp. Display systematics QUANTIFIED (the old 'several percent' is now exact): 16× the iteration budget flips exactly 7/99 classifications (29% → 32% chaotic — finite-time sticky orbits); halving/doubling the λ > 0.06 threshold moves the fraction to 38%/21% (the classifier band); the mean chaotic λ at K = 8 hits Chirikov's never-fed ln(K/2) asymptote to 1.27% (< 3% gate) while at K = 4 the same executed instrument shows the asymptote fails by +18.4% (pinned, disclosed — a large-K formula whose finite-K breakdown the module's own instrument resolves). Remaining honest limitation, unchanged: the module measures local chaos (Lyapunov), not transport or residues — the quantitative K_c lives in the oracle, and the certificate now proves screen and oracle agree at every digit the screen shows.

Notes

0 retries — 8/8 gates on the first full run (~22 s), scratchpad-prototype-first recipe (4 prototype scripts before authoring). Two traps caught AT DESIGN TIME by prototyping: (1) K-solving at high Fibonacci order needs CHAINING — a fixed wide secant bracket works to q = 377 then lands on wrong roots (residues scale exponentially in q off-criticality), so each order bisects K inside an adaptive bracket 4× the last inter-order step, warm-started from the previous order; (2) the transport gate's seeds must jitter around the HYPERBOLIC fixed point (0,0) — a first draft seeded near p = π with random θ and silently trapped seeds inside the ELLIPTIC period-2 island of the 1/2 resonance ((0,π): R = K²/4, stable for K < 2), which never escapes even far above K_c, faking a surviving barrier. Order 4181/6765 is excluded from the plateau a priori: at q ≈ 6765 the orbit-closure sensitivity amplifies the 1e-15 p0 resolution past the K_n signal (measured drift 3e-6). The known value's last digits were cross-checked against the plateau itself (mean 2.8e-7 below MacKay's 9-digit value, consistent within the alternating-tail systematic); R∞ and δ enter only two loose scoring gates (1e-3, 20%) so their 4th-digit provenance is not load-bearing. REFINER 2026-07-24 (validated → validated + honest-module): appended the module-honesty certificate (gates H..N, 16/16 first attempt, ~23 s total); zero module edits; both tamper self-tests exit 1 (known_value → gates A+J fail with recovery unchanged; one hex digit of a slice sha → gate H fails). Micro-lesson: a pure-computation module (everything in init(), no accumulator, no fixedUpdate) certifies with a much shorter slice set than a stepped one — 8 slices vs coupled's 9 with no schedule gate needed; the two structural anchors are the class-static indirection (destructures like `const { ITERS } = StandardMapModule` become reads off a shared C object, which lets the certificate Richardson-patch the module's own budget constants) and module-level consts (TWO_PI) which must be sliced like constants or execution throws ReferenceError.

Confidence & reproduction

Confidence

high

Validation

derisk-pass

Re-run the check

npm run derisk -- stdmap (scripts/stdmap-derisk.mjs)

Oracle

scripts/oracles/stdmap.reference.json

Sources

J. M. Greene, 'A method for determining a stochastic transition', J. Math. Phys. 20, 1183 (1979); R. S. MacKay, 'A renormalisation approach to invariant circles in area-preserving maps', Physica D 7, 283 (1983); B. V. Chirikov, 'A universal instability of many-dimensional oscillator systems', Phys. Rep. 52, 263 (1979); R. S. MacKay, J. D. Meiss, I. C. Percival, 'Transport in Hamiltonian systems', Physica D 13, 55 (1984).