Horizon-Adaptive Neural Tube Rollouts

1) Integration point: use a GRU, residual RNN, or neural state-space model with transition $z_{k+1}=f_\theta(z_k,u_k)$. During training, maintain the nominal hidden state $\hat z_k$ and an elementwise radius vector $r_k$; for a policy, include the policy derivative in the closed-loop Jacobian. 2) Pseudocode: initialize $\hat z=E(o_0)$ and $r=r_0$; at each step compute $u=\pi_\phi(\hat z)$, update $\hat z\leftarrow f_\theta(\hat z,u)$, obtain $J$ by automatic differentiation, update $r\leftarrow |J|r+d$, and add $\lambda_r\|r\|_1$ plus penalties for violated robust constraints $g(\hat z)+|\nabla g|r>0$ to the ordinary prediction loss. Stop gradients through $r$ initially if memory is limiting. 3) Compute the tube recursion and Jacobians exactly; estimate $d_k$ from held-out transition-residual quantiles, and estimate $r_0$ from repeated encoder evaluations or initial-state uncertainty. Use a high residual quantile such as 99%. 4) First cheap experiment: train a small GRU world model on CartPole or a nonlinear pendulum with injected observation and transition noise, comparing ordinary multi-step training with tube-regularized training at equal parameter count. Sweep rollout horizon and a scalar Jacobian-gain multiplier. The quantitative prediction is a sharp change near the measured spectral-radius boundary $\rho(J^{\mathrm{cl}})=1$: below it, median $r_k$ should plateau; above it, $\log r_k$ should grow with slope approximately $\log\rho(J^{\mathrm{cl}})$. Constraint violations should decrease when robust backoff inequalities hold.