The pendulum's anharmonic period recovered by timing the bare equation of motion

Galileo asserted (Discorsi 1638, First Day) that a pendulum's swings take the same time 'whether the arc is eighty degrees or two' — isochronism at all amplitudes. Huygens knew better within a generation: his 1673 Horologium Oscillatorium adds cycloidal cheeks precisely because the circular pendulum is NOT isochronous, and the true period is T(θ₀) = T₀·(2/π)·K(sin(θ₀/2)), a complete elliptic integral. Can that entire law — the 18% excess at 90°, the θ₀²/16 correction, the logarithmic divergence near 180°, and T ∝ √L — EMERGE from nothing but the bare equation of motion θ'' = −(g/L)·sinθ and a stopwatch, with the elliptic integral coded nowhere in the recovery path?

Units: dimensionless (T(90°)/T₀ = (2/π)·K(sin 45°); K(1/√2) = Γ(¼)²/(4√π))

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The finding

The pendulum's anharmonic period recovered by timing the bare equation of motion — with ONLY θ'' = −(g/L)·sinθ and a symplectic integrator coded (no elliptic integral, no AGM, no correction series anywhere in the recovery path), the swing itself returns T(90°)/T₀ = 1.1803405990165 vs the elliptic-integral law (2/π)K(sin 45°) = 1.1803405990161 — rel 3.4e-13 Richardson-extrapolated, emergent q = 1.999 — the full amplitude law tracked to ~1e-12 at 8 amplitudes up to the emergent near-180° divergence (T(179°)/T₀ = 3.90), the textbook correction series c₁ = 1/16 (rel 6.1e-7) and c₂ = 11/3072 (rel 1.8e-3) emerging from a fit to the timed data, T ∝ √L at slope 0.5000000002, a 24-seed noisy stopwatch landing 0.10 SE from known, and Galileo's isochronism (Discorsi 1638) falsified both ways: 59.1% wrong at large arc yet only 0.34% wrong in the 4°–14° chandelier regime that fooled him

Method

Velocity-Verlet integration of θ'' = −(g/L)·sinθ (g = 9.81, L = 4 m) released from rest at θ₀, so the first θ = 0 crossing is exactly T/4 by symmetry. The crossing is refined with a cubic-Hermite Newton root (θ and ω are both known at each step) so event detection matches the integrator's own O(dt²) accuracy — a plain step-quantized detection would bury the signal (the module's own on-screen timer has exactly that quantization, disclosed below). Richardson extrapolation over dt = 2e-4/1e-4/5e-5 removes the leading dt² term (emergent q = 1.999). The recovered ratio is interrogated 8 ways: the primary 90° recovery, an 8-amplitude sweep 10°–179° against the elliptic table (each an independent timing run), an LSQ fit of (T/T₀ − 1) on {θ₀², θ₀⁴} over 2°–8° whose coefficients must return the textbook series, a 5-length L-sweep for the square-root law, the two-sided Galileo falsification, a 24-seed noisy stopwatch (σ = 20 ms on 51 sim-detected turning-point times, period = regression slope on swing count — SE ∝ σ/M^1.5 beats endpoint differencing), a 50-period symplectic-stability check on the detected turning points (spread 3.6e-11), and an energy-conservation gate (5.1e-9 of the energy scale). All known ratios live in scripts/oracles/pendulum.reference.json and are loaded ONLY to score. 8 gates, ~0.1 s. ?world=pendulum.

Measurements, controls & cross-checks

Raw finest dt

Dt

5.0000e-5

Ratio

1.1803406

Rel

1.7500e-10

Note

un-extrapolated; the dt ladder has emergent convergence exponent q = 1.999 and Richardson lands 3 orders past the finest grid

Amplitude sweep

Amplitudes deg

• 10
• 30
• 60
• 90
• 120
• 150
• 170
• 179

Worst rel

1.2100e-12

Monotone

true

T179 over T0

3.901

Note

each amplitude is an independent timing run from rest; the sequence tracks (2/π)K(sin θ₀/2) to ~1e-12 and the near-180° logarithmic blowup is emergent — T(179°) runs 3.9× the naive law

Small angle series

C1

0.062499962

C1 expected

0.0625

C1 rel

6.1200e-7

C2

0.0035872

C2 expected

0.0035807

C2 rel

0.00182

Note

LSQ of (T/T₀ − 1) on {θ₀², θ₀⁴} over 2°/4°/6°/8°: the textbook expansion 1 + θ₀²/16 + 11θ₀⁴/3072 emerges from timing alone — both coefficients never coded

L scaling

Slope

0.5

Expected

0.5

L values m

• 1
• 2
• 4
• 8
• 16

Note

d log T / d log L at fixed 90° amplitude — Galileo's OTHER pendulum law, the correct one, emergent from the same timer

Rival galileo isochronism

Large arc excess

0.5914

Small swing variation

0.00344

Note

Galileo (Discorsi 1638, First Day): swings are isochronous 'whether the arc is eighty degrees or two'. The same timer that recovers the elliptic ratio finds T(140°) exceeds T(10°) by 59.1% — structural, ~11 orders above the integrator's numerics. And the oracle quantifies WHY he believed it: over the 4°→14° range a decaying chandelier actually spans, the period varies by only 0.34%, far below pulse-beat timing resolution. Huygens' cycloidal clock cheeks (Horologium Oscillatorium 1673) exist because the claim is false

Noisy stopwatch

Seeds

24

Sigma ms

20

Swings

50

T

4.73568844 ± 4.76e-5 s

Se dist

0.1

Known T s

4.7356839

Turning point spread rel

3.6000e-11

Note

51 successive turning points detected in the sim (Hermite-refined ω zero crossings), each timestamp noised per seed; period = regression slope on swing count. The noiseless detected periods spread 3.6e-11 relative over 50 swings — the symplectic integrator holds the period

Energy

Max drift over scale

5.1100e-9

Periods

50

Note

velocity-Verlet energy oscillation stays below 5.1e-9 of the energy scale gL — bounded, not secular

What it reduces to

The exact simple-pendulum period T(θ₀) = T₀·(2/π)·K(sin(θ₀/2)) (Whittaker, Analytical Dynamics §44; Abramowitz & Stegun §17.4; Landau & Lifshitz, Mechanics §11), with its small-angle series 1 + θ₀²/16 + 11θ₀⁴/3072 + … and the T ∝ √L law. It VALIDATES, not derives: the generator is the bare equation of motion plus a generic symplectic integrator and a stopwatch — the elliptic integral, the AGM, and every correction coefficient are measured OUTPUTS, present only in the reference file's scoring targets. Non-circularity: K, AGM, and the series appear nowhere in the derisk's recovery path (tamper test: falsifying known_value flips exit to 1 with the recovered 1.18034059902 unchanged). The decisive control is historical: Galileo's isochronism claim fails by 59.1% at large arc under the same timer, while the 4°–14° chandelier regime he actually watched varies by only 0.34% — the rival is falsified as physics and explained as history.

Module systematics

Both systematics disclosed at validation are now CERTIFIED by the honest-module section (M1–M8, refiner 2026-07-23): (1) the ~4.2 ms step-quantized turning-point detection is the D term of the exact decomposition below — computed bit-exactly, telescoping to (δ₁₀₈ − δ₁₀₄)/4 with every offset in (0, dt] and |D| ≤ dt/4 a priori; (2) the LAW-FED on-screen T_exact line (T₀/AGM(1, cos(θ₀/2))) is pinned to agree with the reference scoring target 1.37288050061835·T₀ to 1.6e-16, and the AGM appears only in the display-reproduction path, never in any recovery — the non-circular recovery of the same number remains gate B's independently-timed 120° sweep member (rel 6.9e-13). Module untouched this run (no src/ change).

Confidence & reproduction

Confidence

high

Validation

derisk-pass

Re-run the check

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

Oracle

scripts/oracles/pendulum.reference.json

Sources

E. T. Whittaker, A Treatise on the Analytical Dynamics of Particles and Rigid Bodies (CUP, 4th ed. 1937) §44; Abramowitz & Stegun, Handbook of Mathematical Functions (1964) §17.4, K via AGM §17.6; L. D. Landau & E. M. Lifshitz, Mechanics (3rd ed.) §11 problem 1; Belendez et al., 'Exact solution for the nonlinear pendulum', Rev. Bras. Ens. Fís. 29 (2007) 645. The rival: Galileo, Discorsi (1638), First Day; falsified by C. Huygens, Horologium Oscillatorium (1673).