Horizon-Dependent Error Tubes for Recurrent Rollouts

1. Integration point: apply this wrapper to a GRU, vanilla RNN, or diagonal/low-rank SSM used for multi-step forecasting or world-model rollouts. Keep the network's nominal hidden state and add a parallel nonnegative vector ebar; do not sample trajectories during the first implementation. At each rollout step, compute or upper-bound the hidden Jacobian $A_j$. For a GRU, use automatic differentiation followed by interval or absolute-value bounds; for a diagonal SSM, use the exact diagonal transition magnitudes. Estimate $\bar w_j$ from a held-out residual quantile, sensor-noise specification, or injected quantization noise. 2. Pseudocode: initialize h=encoder(x0), ebar=initial_error; for j in range(H), compute h_next=f_theta(h,u[j]), compute A_abs=abs(jacobian(f_theta,h)) or a conservative bound, set ebar=A_abs@ebar+wbar[j], compute output margin margin=G_abs@ebar, and either reject or tighten unsafe predictions or add lambda*relu(margin-|constraint slack|)^2 to the loss. Estimate a uniform certificate matrix $M$ by taking element-wise maxima of $|A_j|$ over training rollouts, or use a blockwise local $M$. 3. From the paper, the required mechanism is the element-wise recursion and Perron–Frobenius condition; Jacobian bounds and disturbance quantiles are empirical estimates. 4. First cheap experiment: train a small GRU on noisy sine-wave or Lorenz forecasting and compare ordinary rollout training with tube-regularized training at horizons 1--100. Scale recurrent gain by a factor $\alpha$ and measure $\rho(M)$. The quantitative prediction is a sharp transition near $\rho(M)=1$: for $\rho(M)<1$, median propagated error should plateau at approximately $(I-M)^{-1}\bar w$; for $\rho(M)>1$, it should grow exponentially with slope close to $\log\rho(M)$, and the measured long-horizon violation rate should increase sharply across this boundary.