Periodic-Orbit Entropy Calibration

1. Integration point: initially use this mechanism as a validation metric for neural ODEs, latent Hamiltonian-like flows, or recurrent world models with recurrent data. Only use it as a training regularizer when the observed system has a positive-entropy recurrence target; otherwise it can force artificial chaos. 2. Pseudocode: choose periods T on a logarithmic grid; initialize many latent states z0; solve the shooting residual r(z0,T)=phi_T^theta(z0)-z0; optimize each seed with Newton or L-BFGS; retain roots with norm(r)<tau; remove duplicates using trajectory Hausdorff distance and phase-shift matching; count the remaining solutions N_hat(T); fit h_hat and b_hat by regression; add lambda L_orbit to the objective or use it for checkpoint selection. 3. The exp(hT)/T law and -log(T) correction are taken from the paper. Root-finding, tolerance tau, clustering, finite-horizon counts, and entropy estimates are empirical, so report seed coverage and bootstrap confidence intervals. 4. First cheap experiment: train small neural ODEs on Lorenz-63 or a synthetic multi-cycle system, comparing one-step-loss training with orbit-calibrated training. Plot log N_hat(T) versus T and log[T N_hat(T)] versus T. The quantitative prediction is that the latter plot has an approximately linear slope h_hat and lower residual than a pure exp(hT) fit; orbit-collapse models should show subexponential counts, whereas over-chaotic models should show excess counts or failure of the -log(T) correction.