Does a dissipative chaotic PDE have discrete Ruelle resonances?

Kuramoto–Sivashinsky at L = 22, 3×10⁹ steps, 426,748 Poincaré section crossings. Ulam transition matrix on grids from 12×12 to 40×40.

Convergence holds and is well measured. |λ₂| = 0.5215, 0.5196, 0.5191, 0.5170, 0.5145, 0.5135 — monotone, decelerating, total drift 0.0080 over an elevenfold range in cell count, which is 1.5% of the value. A 1/n fit extrapolates to 0.5109 with an rms residual of 8.5×10⁻⁴, the scale of the last quoted digit. Correlation time about 1.5 section returns.

The prediction's second half was not tested, and the failure is in the test. The "weighting" check reported a shift of 0.0001 at every grid, and it means nothing: a weight that is a function of state is a similarity transform, which moves eigenvalues by exactly zero (verified, 3×10⁻¹⁶). A weight that is i.i.d. noise on the counts measures sampling variance. Neither probes partition dependence.

A prediction that was wrong, and instructively. I expected |λ₂| to drift upward with grid size, reading a trend off the smoke run where 171 crossings gave 0.614 → 0.815. That +0.20 swing was pure sampling noise; with three orders of magnitude more data the drift reverses and is twenty-five times smaller. A trend read off an underpowered run is not a trend.

Resolved. The prediction's second half — partition-independence — is established by run 022, whose offset test on this run's full 426,748-crossing set (ks22_full.npy) came in at a mean spread of 0.0023, scaling as sampling noise (ratio 2.6 vs 3.16 for pure 1/√N) against a floor predicted at 0.0024. Both halves of the prediction now hold. Superseded on the stability claim by 022; kept for the convergence measurement and the underpowered-trend lesson.