Accelerator-mode windows weighed on the standard map

Where exactly does chaotic transport in the standard map turn anomalous — can the accelerator-mode windows the D(K) curve's superdiffusive spike lives in be measured from the raw dynamics, edge by edge, and is the ballistic velocity inside them really quantized?

Units: kick strength K (dimensionless) — upper edge of the m = 1 period-1 accelerator-mode stability window, √(4π²+16)

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The finding

Accelerator-mode windows weighed on the standard map: the parameter edges where diffusion turns ballistic are recovered from the raw map by numerical bifurcation analysis — birth edge K_low = 6.283185307179586 (2π to float64 exactness), doubling edge K_up = 7.448383556 ± 7.4e-9 vs √(4π²+16) (worst-variant rel 1.2e-9), the m = 2 window [12.566370614, 13.187633241] and the width ratio 1.8755325 to 4.4e-9 all tracked, and orbits trapped on the mode stream at the QUANTIZED velocity 2π per kick (global-IC median off by 7.7e-7, torus-circumference quantization) while at K = 6.1, 0.18 below the window, no orbit of 12000 sustains ballistic drift (vmax 0.35, diffusive); the pendulum-width rival (window = [2πm, 2πm+4], m-independent) misses the upper edge by 38% (~10⁶× the measurement error) and the width ratio by 88%; the module's on-screen anomalous spike lands inside the measured window at every tested seed (mirror 1–3 + executed twins 7, 8) with mean/robust split 3.4–7.0 (gate floor ≥ 2); the shipped ChaoticTransportModule.ts is CERTIFIED BY EXECUTION (gates J–R, every comparison Object.is/byte-exact, no numeric tolerance): the &seed=7 screen 'clean sea K=5: mean D≈5.92 · sharp 2π spike (K≈6.5): mean ~×13 vs robust ×1.8' is pinned executed output, the shipped sweep is bit-identical to the oracle's mirror at seeds 1–3 (zero instrument gap), and the empty fixedUpdate/render are proven dead by 480-call execution

Method

Generator: the raw Chirikov standard map p' = p + K sin θ, θ' = θ + p' only. (1) Birth edges: bisect K on whether the map's maximum one-step momentum gain (golden-section search on the raw step, 200 iterations) reaches the torus quantum 2πm. (2) Doubling edges: locate the accelerating point θ* by bisection root-finding on the raw one-step gain, then probe stability purely dynamically — iterate the TORUS map (p mod 2π too: the mode is a genuine fixed point) from (θ*+ε, 0) and classify elliptic vs hyperbolic by whether the deviation escapes; bisect K on that boundary over a 2×2 grid of (ε ∈ {1e-9, 3e-10}) × (threshold ∈ {1e-4, 3e-5}) instrument settings, N = 30000 steps, every variant gated. No trace formula, no Jacobian, no √((2πm)²+16) anywhere in the generator. (3) Drift quantization: seeded ensembles — 3 × 4000 global random ICs at K = 6.6 (selection threshold 0.95 × the ensemble's own fastest orbit, the known quantum enters no selection), 3 × 4000 at K = 6.1 (null), 3 × 400 boxed ICs around the measured m = 2 point at K = 12.9 (disclosed: the m = 2 island is too small a target for a global 4000-orbit ensemble, so that channel gates the quantum, not global abundance). (4) Verbatim mirror of ChaoticTransportModule's compute-once D(K) sweep, seeds 1–3. (5) Module certificate: the shipped ChaoticTransportModule.ts strip-paired to ESM and its init() EXECUTED in Node against recorder stubs, every observable Object.is/byte-exact vs an independent replica and pinned in the reference. Gates A–R, all seeded and deterministic, ~4.6 s.

The law it recovers

Period-1 accelerator modes (θ fixed, p gaining exactly 2πm per kick — a fixed point of the torus map) are born at K = 2πm, where the kick first reaches the resonance momentum, and stay stable until K = √((2πm)²+16), where |2 + K cos θ*| crosses 2 and the mode period-doubles. Measured edges: m = 1 [6.283185307179586, 7.448383556 ± 7.4e-9], m = 2 [12.566370614359172, 13.187633241 ± 6.2e-9]; width ratio w₁/w₂ = 1.8755325 vs exact 1.8755325125 (rel 4.4e-9) — windows shrink as ~8/(2πm) because the resonance eats most of the kick (effective kick K cos θ* = −√(K²−(2πm)²), not K). Inside the window the trapped orbits' sustained speed is the QUANTIZED 2πm per kick: global-IC median 6.283187–6.283190 vs 2π (worst rel 7.7e-7, wobble-limited), boxed m = 2 median 12.566372 vs 4π (worst rel 1.1e-7); the ballistic sliver is 0.7–0.95% of random ICs at K = 6.6 and absent (vmax 0.31–0.35, diffusive scale) at K = 6.1 below the birth edge.

Measurements, controls & cross-checks

Recovered half range

7.4000e-9

Rival

Name

pendulum-width rival: every resonance island inherits the primary island's stability range — the m = 0 fixed point is stable for 0 ≤ K ≤ 4, so each accelerator window is [2πm, 2πm+4] (upper edge 10.283 at m = 1), with m-INDEPENDENT width (ratio 1)

Falsified

measured K_up = 7.4484 misses the rival's 10.283 by rel 0.381 — ~3×10⁸ times the worst-variant measurement error of 1.2e-9 — and the measured width ratio 1.876 rejects the rival's 1.0 (gate ≥ 1.5). The below-window null is the second control: no orbit of 12000 at K = 6.1 sustains ballistic drift, so the quantized transport exists only inside the measured window.

Module systematic

Spike locations

• 6.5
• 6.75
• 6.75

Mean robust split

• 6.98
• 3.84
• 4

Pooled D5

5.944

Note

the module's on-screen anomalous spike (grid resolution 0.25) lands inside the measured window [6.283, 7.448] at every tested seed (mirror 1–3 and executed twins 7, 8), and its mean/robust-median split — the on-screen anomaly signature — spans 3.4–7.0 across those seeds vs ≈ 1 for a Gaussian sea (the original 'split ≥ 3.8' was a 3-mirror-seed statistic: the executed seed-8 twin gives 3.39, so the honest floor is gate I's ≥ 2). The on-screen 'sharp 2π spike (K≈6.5)' is therefore the oracle's window, seen at grid resolution. Module K = 5 sea value: pooled D(5) = 5.944 vs K²/4 = 6.25 (rel 4.9%, within the 3.5-SE band of its M = 600 statistics; the module's own docstring already discloses that its spike HEIGHT is outlier-driven and seed-dependent, which execution confirms: mean/QL ratio 5.8–12.5 across executed seeds 1–3, 7, 8 — the static banner tail '~order ×10' is the docstring's disclosed order-of-magnitude claim, not a measured constant). Module untouched — no on-screen bias to correct.

Module certificate

Gates

J–R in scripts/transport-derisk.mjs; pins in scripts/oracles/transport.reference.json module_certificate

Certified boot

?world=transport&seed=7 — banner 'clean sea K=5: mean D≈5.92 (K²/4=6.25) · sharp 2π spike (K≈6.5): mean ~×13 · vs robust ×1.8' pinned as executed output (spike ratio 12.508337480405938, robust 1.7909984569972308, D(5) 5.918157194332758); seed-8 twin 5.73/×6/×1.7 — every displayed digit moves, both spikes inside the measured window

Shape

the FIRST fully dead-dynamics cert: fixedUpdate() and render() are EMPTY (compute-once world) — proven by executing 480 calls (0 field mutations, 0 scene/DOM ops), so there is no lockstep and no clock systematic by construction; the whole measurement is one executed init(): 37200-draw mulberry32 sweep (4×31 arrays bit-exact vs independent replica), deterministic 400-orbit portrait (74100+1900 Float32 buffer entries bit-exact, seed-INVARIANT by execution), banner/chart/style byte-equal, camera/env pinned, dispose restores; default boot's Math.random fallback executed (seeded 0 draws, unseeded exactly 1, seed === (draw·0xffffffff)>>>0); closure: shipped sweep Object.is-identical to the oracle's gate-I mirror at seeds 1–3 — zero instrument gap

What it reduces to

Chirikov's period-1 accelerator modes (Phys. Rep. 52, 263 (1979) §5; Meiss, Rev. Mod. Phys. 64, 795 (1992); Lichtenberg & Lieberman §4.1b): the standard map's superdiffusive windows are the stability intervals 2πm ≤ K ≤ √((2πm)²+16) of torus-map fixed points that stream at the quantized velocity 2πm per kick. Non-circular because the generator contains only the raw map plus generic numerical search (golden-section maximization, bisection, an ε-perturbation escape probe): the birth edge emerges from where the map's own maximum one-step gain reaches the torus circumference, the doubling edge from where the iterated dynamics switches from bounded to escaping, and the quantum from the median sustained speed of ensemble orbits selected relative to the ensemble's own fastest — the closed forms are computed only in the scorer. DISTINCT from the chaosdiff oracle by construction: chaosdiff gates the diffusion coefficient D(K) with the accelerator windows deliberately excluded from its sweep (its single inside-window point is gated only as 'anomalous, ratio ≥ 1.5'); no window edge, no width ratio, and no drift quantum is measured there. The two oracles share the map but score disjoint quantities.

Confidence & reproduction

Confidence

high

Validation

derisk-pass

Re-run the check

npm run derisk -- transport (scripts/transport-derisk.mjs — 18/18 gates, deterministic, ~4.6 s: A–I physics (birth edge m=1, doubling edge headline across 4 probe variants, m=2 edges, width ratio, quantized drift global/null/boxed, rival falsified, module mirror) + J–R module certificate (sha pin; strip+load, 38 pairs hit-count asserted; executed seed-7 state, portrait and display bit/byte-exact vs independent replica AND reference pins; 480-call dead-dynamics + dispose proof; Math.random fallback-seed site executed; shipped ≡ gate-I mirror closure at seeds 1–3; seed-8 twin + window tie-in). Tamper self-tests, all exit 1: known_value → 7.3 flips B ONLY with the recovered value unchanged; sha-pin flip → J only; sweep-kick ×0.9 in the shipped file → J+L+N+Q+R while ALL physics gates AND the hand mirror stay green (the cert reads the code, the oracle reads the physics); banner-text tamper → J+N only; module and reference restored byte-identical (cmp))

Oracle

scripts/oracles/transport.reference.json

Sources

B. V. Chirikov, Phys. Rep. 52, 263 (1979); J. D. Meiss, Rev. Mod. Phys. 64, 795 (1992); A. J. Lichtenberg & M. A. Lieberman, Regular and Chaotic Dynamics, 2nd ed. (Springer 1992)