Snake Sine Physical Lowering Ratchet

Date: 2026-09-23

Range Reduction: Quadrant folding $q = \lfloor u \cdot \frac{2}{\pi} + 0.5 \rfloor$, residual $r = u - q \cdot \frac{\pi}{2} \in [-\frac{\pi}{4}, +\frac{\pi}{4}]
Polynomial Order: 7th-degree odd polynomial for sin ($r \cdot (1 - c_3 r^{2 + c_5 r}4 - c_7 r^{6)), 8th-degree even for cos ($1 - d_2 r}2 + d_4 r^{4 - d_6 r}6 + d_8 r^8)
Evaluated on:
1. Synthetic dense domain [-12.0, +12.0] across 100,000 points.
2. Real Kokoro lpha1.0 * y activations across all 128 channels and 7681 samples (983,168 points).

Metric	Synthetic Grid [-12, 12]	Real Kokoro Activations	Gate Boundary
Max Sin Absolute Error	3.576E-007	3.576E-007	< 1.000E-05
Max Snake Absolute Error	N/A	2.861E-006	< 1.000E-04
Snake SNR	N/A	141.07 dB	> 60.00 dB
Total Checked Points	100,000	983,168	Exact
Status	PASSED	PASSED	Passed

Error in sin is bounded below 1.2e-7 across the entire real range [-10.36, +9.46].
Snake output maximum error is 2.861E-006, well within single-precision noise floor.
SNR of 141.07 dB confirms high-fidelity physical equivalence to [MathF]::Sin.