Partitioned Gain-Phase Stable Neural Feedback

1. Integration point: use the certificate as a structural constraint for a deep equilibrium model, linear state-space model, or RNN represented as a feedback interconnection. Represent each block with state-space matrices A_i, B_i, C_i, and D_i, or estimate its local input-output Jacobian along minibatch trajectories. Assign ordinary MLP or tanh blocks with gain bounds to the gain partition G. Assign blocks designed with positive-real, skew-symmetric, or damped-oscillatory dynamics to the phase partition P. 2. Pseudocode: create a logarithmic frequency grid Omega. For every block and frequency, compute H_i(exp(j omega)) = C_i (exp(j omega) I - A_i)^(-1) B_i + D_i. Compute g_i as the largest singular value. Estimate phase-sector bounds from numerical-range samples v^* H_i v divided by v^* v, or use a cheaper singular-vector phase approximation. Add a penalty L_cert equal to the sum of squared positive violations of the gain and phase inequalities, and add lambda times L_cert to the task loss. Reject equilibrium solves when the worst margin is negative. 3. The gain and phase inequalities are taken from the paper; transfer functions, frequency responses, and phase bounds are computed or estimated for the neural blocks. For nonlinear networks, estimate local Jacobians over held-out trajectories and use the maximum observed violation as a robust surrogate. 4. First experiment: compare a four-block deep equilibrium network on sequential MNIST and on synthetic long-horizon oscillatory regression against spectral-normalized and unconstrained baselines. Sweep a scalar feedback gain alpha multiplying one block. Measure the predicted critical value alpha_c at which the maximum gain product reaches one, or the maximum phase sum reaches pi. The falsifiable prediction is that exploding fixed-point iterations, increasing Jacobian spectral radius, or loss divergence begins within twenty percent of the certificate boundary, while the mixed partition remains stable in cases rejected by an all-small-gain design.