Near-Return Entropy Monitor for Recurrent Latent Dynamics

Integrate the monitor after the hidden-state rollout of an RNN, SSM, neural ODE, or learned world model. Collect K short trajectories from nearby latent initial states x_j, using the same input sequence when evaluating a predictive model. Detect approximate returns by finding T_x and T_y for which ||z(T_x)-z(0)|| and ||z'(T_y)-z'(0)|| are at most delta/2, while the initial distance is at most delta/2. For each candidate pair, search t_0<T_x and test whether z(t_0) stays at least 2 eta from every sampled point on the second trajectory; reject candidates whose paths leave a chosen neighborhood B_delta(Lambda), estimated as the empirical support of latent states. Build a graph connecting two rollouts when their minimum separation over [0,T] is below eta, and greedily compute a maximal separated set M_T(eta). The paper supplies the near-return, time-scale, and separation conditions implying positive topological entropy; M_T and h_hat_T are empirical finite-horizon approximations. First test a GRU and a 2D neural ODE on Lorenz-63 prediction and damped-oscillator prediction, using teacher-forced training followed by free rollout. Compare ordinary MSE training with a penalty on detected certificate events or early stopping when h_hat_T exceeds a threshold. The quantitative prediction is that certificate events should coincide with a positive slope of log M_T(eta) versus T, namely log M_T(eta) approximately hT+C with h>0. Stable oscillator baselines should show bounded M_T and slope statistically compatible with zero, while chaotic models should show a persistent positive slope and rapidly growing long-horizon error.