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

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 an interesting spectral-shaping effect of Kahan compensation, suppressing a distinct high-frequency resonance while elevating broadband noise. 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.