Orlicz-Controlled Local Temporal Stability

1. Integration point: apply the regularizer to an RNN, state-space model, or world model that produces an observable u_{b,j}=g_theta(s_{b,j}) along each rollout. For every minibatch item, create K nearby initial states by adding perturbations with norm at most r, then roll all states forward for H steps. Add lambda times the Orlicz penalty and gamma times a local-consistency penalty to the task loss. 2. Pseudocode: initialize cumulative sums c=0; for j from 1 to H, update c=c+u_j, compute a_j=c/j, and update m=max(m,norm(a_j)); after the rollout add Phi_q(m)=m*(log(e+m))^(q+1). For each perturbation, compute a_j^(k) and penalize max_j norm(mean_k(a_j^(k))-a_j^(0)). Backpropagate through the rollout. 3. Computed from the paper: the temporal averages, maximal-envelope criterion, and logarithmic Orlicz threshold. Estimated empirically: the maximal function, perturbation-ball measure, and finite-horizon approximation to integrability. 4. First experiment: train a small GRU on logistic-map or Lorenz-63 prediction, comparing no regularizer, L2 trajectory regularization, and the q=0 Orlicz penalty. Measure long-horizon error and local-neighborhood average discrepancy while increasing H. The predicted signature is a crossover where the batch estimate of E[m log(e+m)] stabilizes and local-average error stops exhibiting persistent spikes; q=1 should require the stronger m log(e+m)^2 penalty for heavy-tailed maximal envelopes. Test whether the measured transition agrees across at least three rollout horizons and whether local discrepancy decays with H rather than remaining bounded away from zero.