Syntonic Engine: the law knows when to speed up (cadence bench, simulation)

Syntonic Engine sets the time window of a correction or an estimate from statistics measured in the signal, by Theorem U' of the Syntony Principle: \(\tau^* = \kappa_H (\sigma^2/\lambda_H)^{1/(2H+1)}\). Version 0.2.7 is a bench and documentation release; the code is that of 0.2.0.

New: the cadence bench (bench/cadence_2026-10-31/), self-contained, on the servo model of examples/servo_bench.py (MAESTRO 1 baton arm, loop at 2 kHz, 20 paired runs). Each joint's servo frame is chosen live from the law: the slowest frame whose floor stays under the law's window. Every script and its predictions were pushed to GitHub before its run.

Timing (prediction T1, held). The law's own frames replayed 15 s out of step, with the same frames, shares and switches, lose 12 % of baton-tip accuracy, in 20 runs of 20. The law's window tells the machine when the motion needs a faster cadence.

Along the e-geodesic (predictions E1 to E3, all held). Carried on the rate-linear path of Dual Geodesy, \(1/\tau(t)\) affine in \(t\), on a 0.5 ms grid, the cadence becomes a slope instead of a staircase. It keeps its timing (11.3 % against its own schedule shifted 15 s, 20 of 20) and its efficiency (6.7 % below the fixed frames at the same frame rate; the staircase: 7.1 %). The geometry gives the cadence its form, not more accuracy.

Budget. Against the 3 ms frame of today's servos, the law's cadence cuts the tip error by 20 % at a mean 541 frames per second: 80 % of the gain of a fixed 1 ms frame, for 54 % of its frames.

What failed, and the limits. The first run's main prediction, parity with a fixed 1 ms frame, failed: the law's cadence stays 7.1 % above it. An exploratory diagnostic found why: 96 % of the gain of a faster frame comes from refreshing the command (plan and correction), 4 % from the window's floor, the only part the rule covers. Against one fixed frame per joint chosen from the law's own average, the timing adds 2.7 % (20 of 20). Frames under 3 ms are a simulation probe: the bench's servos accept about 330 Hz. Simulation only: one servo model, two excerpts of one piece; nothing has run on hardware.

Ten predictions, each committed before its run: P0 to P5 (5 of 6 held), T1 (held), E1 to E3 (held). Details and every figure: bench/cadence_2026-10-31/NOTE.md and results/ in the repository.

Engine in Python, C++ (single header) and Rust (no_std), with the bench scripts and every result file: https://github.com/jpbronsard/syntonic-engine