Hard-Saturated Neural Feedback

1. Integration point: replace the final action head in an RL policy, learned controller, or neural ODE control input with a bounded sigmoid or tanh map. Do not merely clip actions outside the computational graph: the saturated action must be passed into the learned dynamics and Lyapunov calculations. 2. Pseudocode: compute logits $a=\mathrm{Policy}_{\theta}(x)$; map them to $u$ using the displayed bound-preserving formula; predict $x^{+}=\hat f_{\phi}(x,u)$; evaluate the task loss and Lyapunov decrease penalty; backpropagate through the saturation map. Initialize the final-layer scale so that most actions lie in the nonsaturated central region, then gradually impose the actual actuator bounds. At deployment, log the saturation fraction and Lyapunov margin. 3. The hard-bound guarantee is exact and analytical. The effect of saturation on stability is not assumed: estimate the decrease margin and certified radius numerically or with interval verification using the actual bounded action. 4. First cheap experiment: train identical MLP policies on a two-dimensional nonlinear stabilization task with action limits $u_{\max}\in\{0.25,0.5,1,2\}$, comparing unconstrained output plus post-hoc clipping against the hard-saturated architecture. For each limit, estimate $r_{\mathrm{cert}}$ and the fraction of states at which the Lyapunov decrease inequality holds. The predicted signature is monotone shrinkage of $r_{\mathrm{cert}}$ as $u_{\max}$ decreases, with a qualitative transition when saturation prevents the required control authority; hard saturation should have no action-bound violations, whereas post-hoc clipping can show a certificate mismatch because the verified action differs from the trained action.