Polynomial Orbit-Pattern Regularization

Integrate this at the hidden-state trajectory of an RNN, latent ODE discretization, or state-space model. Run the model for horizons up to $H$ from a batch of $B$ inputs or perturbed initial states, normalize each hidden coordinate using running quantiles, and map each normalized vector $h_t$ to a discrete cell index $q(h_t)\in\{1,\ldots,K\}$ using a fixed grid, random-hyperplane hash, or k-means centroids. For each subset size $r$, sample several time sets $F\subset\{0,\ldots,H-1\}$ with $|F|=r$, form the label string $(q(h_j^{(b)}))_{j\in F}$ for every trajectory, and count distinct strings after merging strings within a small Hamming tolerance. Use the hinge penalty in the formula, or use it only as a controller that reduces the recurrent gain, increases teacher forcing, or clips the recurrent update when the penalty is positive. The exact paper-derived quantities are the join-cover interpretation and the polynomial-versus-sublinear growth criterion; $C$, $d$, quantization scale, and finite-sample corrections must be estimated. The first cheap experiment is a vanilla GRU and gated state-space model on sequential MNIST or Copy Memory, compared with identical models without the penalty. Use $K=8$ random-hyperplane cells, $H=128$, $B=512$, and $r\in\{1,2,4,8,16,32\}$. Plot $\log\widehat p$ versus $\log(r+1)$ and fit the slope. The predicted signature is an approximately linear log-log curve for the regularized model, with slope $\widehat d$ stable as $H$ increases; an unstable unregularized model should show upward curvature and an effective slope increasing with $r$. Sweep recurrent gain or initialization scale and locate the point where fitted slope begins increasing with horizon. The regularizer should shift this transition to larger gains without materially increasing task loss.