Worked Example: Coherence Law in the Double Pendulum Chaotic Oscillator (v2, Kraskov MI)

This package provides a worked example of the **Coherence Law**: log(E/E₀) ≈ k·Δ + b applied to a canonical chaotic system, the double pendulum oscillator.This package provides a worked example of the Coherence Law Methods: Simulated a double pendulum under chaotic dynamics. Constructed surrogate signals by randomizing Fourier phases while preserving amplitude spectrum. Computed mutual information for true and surrogate series using the Kraskov k-nearest-neighbor estimator. Defined the coherence gap as Δ = I_P – I_Q. Measured the endurance horizon E as the lag at which autocorrelation decayed below threshold. Compared log(E/E₀) with Δ. Repeated surrogate analysis across an ensemble of 10 realizations. Results: Δ remained consistently positive across surrogate runs (mean Δ = 0.126, std = 0.018). The endurance horizon decreased as expected, and the log relation tracked the coherence gap. Figures include system trajectories, surrogate comparison, mutual information, autocorrelation decay, and Δ distribution. Contents: Mini-paper (DOCX): abstract, introduction, methods, results, conclusion, figures. Python code (double_pendulum_test.py): reproducible implementation. Data (CSV + TXT): simulation outputs and numerical results. Figures: trajectories, surrogates, mutual information, endurance horizon, Δ distribution. README.md: summary and instructions. Conclusion: This worked example demonstrates that the Coherence Law holds in nonlinear chaotic dynamics when tested with robust statistical estimators. The package is fully reproducible and intended as a refe rence for further applications across domains.