Incremental-ISS Contractive Recurrent Block

1. Integration point: apply this to a GRU-like cell, vanilla RNN, or residual SSM update $h_{k+1}=F_\theta(h_k,u_k)$, before the output head. Use spectral norm for a Euclidean certificate, or estimate the matrix measure for an $\ell_1$ or $\ell_\infty$ certificate. Add $\mathcal{R}_{\mathrm{ISS}}$ to the task loss, and optionally project recurrent weights after every optimizer step so that the estimated state-Jacobian norm does not exceed $q_{\mathrm{target}}<1$. 2. Pseudocode: sample a batch of hidden states and inputs; compute $A=\mathrm{jvp}(F,h)$ and $B=\mathrm{jvp}(F,u)$ using automatic differentiation; estimate $a=\|A\|_2$ and $b=\|B\|_2$ by 1--3 power iterations; set $L=L_{task}+\alpha[\max(0,a-q)^2+\gamma\max(0,b-b_{target})^2]$; backpropagate and update; evaluate perturbation trajectories using the bound $q^k d_0+b\varepsilon(1-q^k)/(1-q)$. 3. Computed versus estimated: the ISS recursion is the paper-derived certificate; Jacobian norms are minibatch estimates, so validation must sample states from clean, noisy, and adversarial trajectories. For stronger guarantees, use interval bounds or global spectral normalization rather than sampled Jacobians. 4. First cheap experiment: train identical 1-layer tanh RNNs on sequential MNIST or a synthetic delayed-copy task, comparing unconstrained training with $q=0.95$ and $q=0.8$. Inject Gaussian or bounded input noise after training and measure hidden-state discrepancy over 100--500 steps. The falsifiable signature is geometric decay with slope approximately $\log q$ for zero input difference and a plateau no larger than $b\varepsilon/(1-q)$ under persistent bounded noise. Estimate the empirical divergence boundary by scaling recurrent weights: instability should begin near the point where the measured Jacobian norm crosses $1$, and the constrained model should remain bounded below that threshold.