Coboundary Spectral-Gap Monitor for Latent Dynamics

Integrate the monitor into an RNN, state-space model, or deterministic world model with latent update z_(t+1) = F_theta(z_t). Every K optimization steps, collect N latent states from held-out rollouts and evaluate m bounded basis functions, initially random Fourier features or normalized low-degree polynomials. Construct the N by m matrix A whose entry in row i and column j is psi_j(z_(i+1)) minus psi_j(z_i), where z_(i+1) is generated by the current transition. Compute a truncated SVD, discard the constant-function direction and any singular values below numerical precision, and define gamma_hat as the smallest retained singular value divided by the largest. If gamma_hat remains below a threshold for q consecutive evaluations, reduce the latent integration step eta, increase a Jacobian contraction penalty, or add a soft p-cycle loss ||F_theta composed p times of z minus z|| squared for a selected small p. The paper supplies the coboundary operator and the closed-range classification; singular values, thresholds, invariant directions, and the best intervention are estimated empirically. First test this on a GRU world model trained on sequential MNIST or a small pendulum dataset, comparing the monitor against no intervention and Jacobian spectral normalization. Evaluate trajectory lengths 32, 64, 128, and 256 and basis sizes 32 and 64. The predicted signature is that gamma_hat decreases with trajectory length for poorly conditioned nonperiodic latent dynamics, while an explicitly p-periodic transition gives a trajectory-length-stable nonzero gap after invariant directions are removed. Falsify the idea if the gap does not distinguish periodicized and unconstrained models, or if intervention fails to reduce long-horizon error.