Measured by the lab
6.325914
Does the Chenciner–Montgomery figure-8 — three equal masses chasing each other around a single figure-eight — really close with the period the discoverers computed, and is its stability the exception (an island of λ ≈ 0) in a three-body problem that is generically chaotic?
Units: dimensionless time (G = m = 1, standard figure-8 normalization)
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The figure-8 three-body choreography weighed blind: raw RK4 of pure pairwise Newtonian gravity — no period value, no choreography theory, no orbit formula in the recovery path — returns T̂ = 6.3259139869 ± 4.7e-8 vs Chenciner–Montgomery/Simó's T = 6.32591398 (rel 1.1e-9, 12 seeds; noiseless 4.95e-9 = the 8-decimal IC-truncation floor, dt-independent to 7e-14) from blind phase-space recurrence whose integer-multiple structure t_k = k·T̂ holds to 1.3e-10; mechanical similarity T ∝ s^{3/2} emerges with fitted exponent 1.500000000 over 16× in T; the stability dichotomy is MEASURED — fig-8 Benettin λ(100) = 0.054 and decaying (regular) vs λ ≥ 0.46 for every resolved free-fall scalene triple — and under a = −r̂/r^{2.5} the same ICs NEVER return (closest approach 1.31 vs recurrence depth 1.8e-5)
Generator: classical RK4 on the 12-D state of three unit masses under pure pairwise Newtonian 1/r² (no softening; min pairwise separation on the eight is ~0.6), dt = 5e-4, from the standard 8-decimal figure-8 ICs (identical to ThreeBodyModule._initState). Recovery is BLIND phase-space recurrence: d(t) = normalized 12-D distance to the initial state, every local minimum below 0.05 after a t > 1 guard recorded with parabolic refinement on d² (d has a linear cusp at a transversal recurrence), and T̂ = the origin-line fit t_k = k·T̂ — the period value appears ONLY in the scorer (scripts/oracles/threebody.reference.json). Gates: (A) noiseless rel ≤ 2e-8 with integer-multiple residual ≤ 1e-8 over 11 recurrences; (B) 12-seed σ = 1e-8 IC ensemble, mean rel ≤ 3e-8, SE_rel ≤ 3e-8, worst ≤ 3e-7; (C) σ = 1e-4 ensemble (10⁴× the IC truncation) degrades gracefully, rel ≤ 4e-4; (D) mechanical-similarity sweep r → s·r, v → v/√s for s ∈ {½, 1/√2, 1, √2, 2}: fitted log-log exponent |Δ3/2| ≤ 1e-7 and T̂/s^{3/2} collapse ≤ 1e-8; (E) force-law control p = 2.5: zero recurrences, closest approach ≥ 0.2; (F) stability dichotomy: fig-8 λ(100) ≤ 0.06 AND decaying vs t (the ln t/t signature), every resolved free-fall scalene triple (module's own chaos distribution, encounter-adaptive h = min(5e-4, 0.003·r_min^{3/2}), validity cut |ΔE/E| < 1e-4 discards 1 of 8 — conservatively, the MOST chaotic one) λ ≥ 0.25 with ≥ 6 e-folds and ≥ 4× contrast; (G) dt-halving mesh, T̂ shift ≤ 1e-10; (H) energy/momentum/angular momentum at roundoff; (I) verbatim module mirror (softened kick–drift–kick, dt = 1/2000, ε² = 0.0016, module Benettin params, fixed chaos seed 1234567): pins λ = 0.0773 → 'stable choreography' verdict, λ = 0.8495 → 'CHAOTIC' verdict, 0 ionisations, drift 0.0000%, and the softened period T_soft = 6.354386 (+0.45%, the module's disclosed systematic). Tamper self-test by hand: known_value → 6.5 ⇒ gates A/B/C FAIL, exit 1, recovered value unchanged, restored by hand-edit. HONEST-MODULE CERT (gates J–N): the derisk executes the shipped ThreeBodyModule.ts headless (sha pin, 33 strip pairs, stubbed Babylon/DOM) — executed init statics + 12-D ICs pinned bit-for-bit to the oracle's reference inputs, 14640-call lockstep per preset bit-exact vs an independent replica at every call including displays, shown λ(60) === gate I's mirror bit-for-bit in both presets, chaos RNG stream + ionisation restart certified as a trajectory.
Three equal masses from the standard figure-8 ICs trace one figure-eight, T/3 out of phase, with period T = 6.32591398 (Chenciner & Montgomery 2000; refined ICs and period by Simó 2000); the orbit is linearly stable — a rare λ ≈ 0 island — while generic three-body motion is chaotic; mechanical similarity gives T → s^{3/2}T under r → s·r, v → v/√s (Landau–Lifshitz §10).
4.6800e-8
9/9 oracle gates + 5/5 honest-module cert gates (J–N) = 14/14 in ~78 s; tamper: known_value → 6.5 ⇒ only A/B/C FAIL exit 1 with recovery unchanged at 6.3259139869, restored by hand; THREEBODY_TAMPER=sha ⇒ only J; THREEBODY_TAMPER=ulp (1-ulp SOFT2) ⇒ K+N only — the fig-8 lockstep absorbs a 1-ulp softening shift for all 243999 substeps (below the rounding grain of O(1) pair separations)
Chenciner & Montgomery's figure-8 periodic solution of the equal-mass three-body problem (Ann. of Math. 152, 2000) with Simó's refined period T = 6.32591398, plus its numerically-established linear stability (Simó 2000; Galán et al. 2002), plus mechanical similarity T ∝ L^{3/2} (Landau–Lifshitz §10 — Kepler's third law generalized). Non-circular because the generator integrates ONLY a = Σ Gm(r_j−r_i)/r³ by RK4: no period value, no choreography or periodicity assumption, no orbit theory anywhere in generation or detection — the recurrence finder records EVERY deep minimum of the blind phase-space distance with no prior on where they fall, the integer-multiple structure t_k = k·T̂ (residual 1.3e-10) is discovered, not imposed, and the tamper self-test proves the scorer is the only consumer of the known value (tampered known ⇒ FAIL with the recovered 6.3259139869 unchanged). The recovery is sharper than 'the orbit looks periodic' in four ways: it returns T to 1.1e-9 relative with seed-level uncertainty; it recovers the mechanical-similarity exponent 3/2 to 5e-10 by fit across 16× in period; it MEASURES the stability dichotomy that makes the eight famous (λ decaying at the choreography, λ ≥ 0.46 at every resolved generic free-fall triple — 8.5× contrast through the identical Benettin machinery); and the force-law control kills 'any smooth attractive force would close these ICs' by 5 orders of magnitude in recurrence depth. Limits, disclosed: the 8-decimal published ICs are ~1e-8 off the exactly-periodic orbit, flooring the noiseless recovery at rel 4.95e-9 (dt-independent — gate G shows the integrator contributes 7e-14); the rival ensemble's validity cut (energy drift < 1e-4) discards 1 of 8 free-fall triples whose near-collision RK4 cannot resolve, in the conservative direction (the discarded instance was the MOST chaotic); and 'stability' is measured as finite-horizon λ decay to t = 100, consistent with but not a proof of the literature's linear-stability result.
ThreeBodyModule integrates the same ICs with softened gravity (ε² = 0.0016 in force and energy) by kick–drift–kick leapfrog at dt = 1/2000: the softening slows the figure-8 period to T_soft = 6.354386, +0.45% above the pure-Newton oracle value — quantified by oracle gate I's verbatim emulation. HONEST-MODULE CERT (gates J–N EXECUTE the shipped ThreeBodyModule.ts, sha-pinned + mechanically type-stripped, 33 pairs): 14640 fixedUpdate/render calls at fl(1/120) per preset hold the full state — both 12-D systems (main + Benettin twin), clock/accumulator (243999 substeps = 50/3·14640 − 1: 3·fl(1/120) === 50·fl(1/2000) as a product but the subtractive residue drops exactly ONE substep, census {17:9759, 16:4881} pinned; spiral guard and budget exhaustion proven dead by execution), Benettin sums/λ, the 600-cap divergence log (5 shifts exercised), and the 440-cap trail ring (7693 wraps) — bit-exact vs an independent replica at EVERY call, with HUD text + chart SVG bit-equal at every %6 write (2440 writes, 194 distinct — both LIVE). ZERO instrument gap: the shown λ at t = 60 is BIT-IDENTICAL to gate I's certified mirror in both presets (fig-8 0.07729330602891304, chaos 0.8494967029855782). THE CERT CAUGHT A REAL DISPLAY BUG (13th overclaim, this one live on screen): with the old 1.5 s verdict gate the HUD called the figure-8 — the canonical STABLE choreography — 'CHAOTIC · sensitive dependence (λ>0)' from t ≈ 2.6 to t ≈ 24 at every boot, because the Benettin running average's own transient peaks at λ ≈ 0.84 and only settles below the 0.15 threshold after t ≈ 24 (measured by execution; the code comment claimed 'the fig-8's transient stays below' — false). Fixed with VERDICT_T = 30 s of λ-averaging before the verdict line commits ('measuring…' until then; the λ read-out stays live): post-verdict max fig-8 λ = 0.1269 (15% below threshold, and λ(31) > λ(60) > λ(122) pins the on-screen 'λ → 0, converging' claim by execution), chaos post-verdict min λ = 0.647 (4.3× above) — neither verdict can flip wrong over the certified 122 s. The chaos preset's ionisation at t ≈ 102.3 (substep 204558) and _restartChaos (mulberry32(1234567) draws 4→8, trail reset, renorm re-phase) are certified as a trajectory; episode-1 ICs equal the 4-draw stream prediction bit-for-bit and the fig-8 preset consumes ZERO draws (all 4 _rng() sites sit in _initState's chaos branch; Math.random ×0). Tamper phenomenology: sha → only gate J; hand-edited known_value → only scoring gates A/B/C with the recovery unchanged; post-strip 1-ulp SOFT2 → caught by the statics pin (K) and the chaos close encounters (N, diverges at call 78) while the ENTIRE fig-8 lockstep absorbs it — SOFT2's ulp (~2e-19) is below the rounding grain of the O(1) pair separations it is added to, so r² = dx²+dy²+SOFT2 lands on the same double at every one of the 243999×2 substeps' force evaluations (micro-lesson #121 at whole-trajectory level: the statics pin is the mandatory ulp tripwire on softened-dynamics worlds).
three bodies chasing each other around the figure-eight with fading trails; HUD: live Lyapunov λ read-out, twin-separation renormalisation note, energy drift %, and a verdict line that commits only after 30 s of λ-averaging — 'measuring…' → 'stable choreography (λ → 0, converging)' (the pre-fix module misread its own Benettin transient as CHAOTIC for t ≈ 2.6–24) — or, with ?preset=chaos, a free-fall scalene triple that scatters, ejects a body at t ≈ 102 and restarts, verdict 'CHAOTIC · sensitive dependence (λ>0)'; every HUD and chart write over 122 s is certified bit-equal to the executed module's state, and both verdict λ's are bit-identical to oracle gate I's mirror
npm run derisk -- threebody (scripts/threebody-derisk.mjs)scripts/oracles/threebody.reference.jsonA. Chenciner & R. Montgomery, 'A remarkable periodic solution of the three-body problem in the case of equal masses', Ann. of Math. 152 (2000) 881–901. C. Simó, 'New families of solutions of the N-body problem', Proc. 3rd European Congress of Mathematics (2000) — refined ICs and T = 6.32591398. J. Galán et al., Phys. Rev. Lett. 88, 241101 (2002) — stability. L.D. Landau & E.M. Lifshitz, 'Mechanics', §10 — mechanical similarity.