ISS-Constrained Modular Recurrent Network

1. Integration point: implement a recurrent sequence model with a perception block producing z and a cognition block producing x; attach the prediction head to x or to the concatenation [z,x]. Use tanh or another 1-Lipschitz activation. Spectrally normalize every recurrent and input matrix in the perception block so its estimated state Jacobian norm is at most alpha. Implement cognition as a leaky residual update with explicit step size dt and leak gamma. 2. Pseudocode: compute z_next = Perception(z,u); compute f = CognitionMLP(x,z_next,u); compute x_next = x + dt*(f - gamma*x); compute y = Head(x_next); backpropagate task loss plus a penalty max(0,J_bound-a)^2. Estimate J_bound with one or two power iterations on Jacobian-vector products, or conservatively bound it using the product of spectral norms of the cognition layers. Reduce dt or increase gamma whenever the bound exceeds the target a. 3. Computed from the mechanism: the contraction inequalities, geometric decay rate a, and steady-state ISS bound (L_z Z_max + L_u U)/(1-a). Estimated empirically: local Jacobian norms, hidden-state norms, and perturbation decay from paired rollouts. 4. First cheap experiment: compare a vanilla GRU, a spectrally normalized GRU, and this modular model on Sequential MNIST or adding-memory MNIST with matched parameter counts and identical Adam settings. Sweep dt or gamma across the predicted boundary where the measured Jacobian norm reaches one. For two trajectories initialized with perturbation delta_0, the predicted signature is ||delta_k|| approximately proportional to a^k below the boundary and non-decaying or growing above it; fit the slope of log ||delta_k|| and test agreement with log(a) within 20%. Also verify that maximum hidden-state norm remains below the predicted ISS bound and that long-horizon loss degrades more slowly in the stable regime.