When More Data Become Less Informative: Finite-Precision Periodicization and Collapse of Forecast-Error Lyapunov Estimates

What the math gives to ML

The paper demonstrates that reduced-precision chaotic trajectories are finite-state dynamical systems: after a transient they enter a digital cycle, so extending one record eventually adds repeated futures rather than new information. In the logistic-map experiment, all 10,000 float32 trajectories recurred before 7,612 iterations, and a forecast-error Lyapunov estimate collapsed from approximately 0.69 to approximately zero as the record length crossed the recurrence scale, while independently restarted short trajectories preserved the correct exponent. The transferable asset is a precision-aware protocol for neural dynamical systems: detect digital recurrence and replace overly long single trajectories with independent restarts, or stop trusting long-horizon stability estimates after saturation. This is particularly relevant to RNNs, state-space models, neural simulators, and quantized inference, where finite precision can create artificial attractors and misleading long-horizon diagnostics.