Precision, Round-off, and the Illusion of Local Predictability in Chaotic Hamiltonian Systems: A Million-Window Benchmark

Abstract

I perform a large-scale benchmark of whether local geometric features can predict the oscillation-amplitude variation of the energy error in conservative chaotic systems. Over 1.25 million sliding windows from 22,380 double-pendulum trajectories are computed at three precision levels (Float64+Kahan, Float64, Float32). Eight geometric features, including attention-based entropy, are correlated with the instantaneous slope of the relative energy error envelope. Across all groups, Pearson and Spearman correlations are consistently negligible (\(r \approx -0.09\), \(\rho \approx -0.15\)), and symbolic regression collapses to a constant. A lag-1 backward signal (Spearman \(\rho \approx -0.14\)) is shown to be a phase artifact via surrogate analysis. Power spectral density analysis reveals that the dominant spectral lines are precisely the nonlinear-shifted normal modes of the double pendulum (low-frequency centroid mode at 0.25 Hz and high-frequency antiphase mode at 0.9286 Hz). Kahan compensation decoheres the phase-locking of round-off error to the high-frequency mode, producing spectral shaping. Crucially, a dummy-feature benchmark demonstrates that a purely synthetic sine wave (containing only the 0.25 Hz and 0.9286 Hz frequencies with random phases) yields a median correlation \textit{higher} than the attention-based features, with the observed \(|r| \approx 0.09\) lying well within the 95\% confidence interval of the dummy distribution (\(p > 0.92\) for all groups). I conclude that the tested local geometric features are not viable diagnostics for the chosen error proxy, and I release the full dataset as a community benchmark for future studies on round-off structure in Hamiltonian chaos.