Caratheodory-kernel passivity regularizer

Apply the regularizer to a linear SSM or to a frozen-point linearization of an RNN. The exact integration point is the training loss, after constructing the transition matrix A_theta and input/output matrices B and C; the forward recurrence remains unchanged. Choose M complex sample points lambda_j = rho_j exp(i omega_j), with rho_j between 0.8 and 0.99 and omega_j uniformly covering [0, 2 pi). For each point, solve the linear system (I - lambda_j A_theta) X_j = B rather than explicitly forming an inverse, then compute Phi_j = I + C X_j and H_j = (Phi_j + Phi_j^*)/2. Penalize the negative eigenvalue margin using relu(epsilon - lambda_min(H_j)) squared. For a second test, select m at most 16 sample points, form K[i,j] = (Phi_i + Phi_j^*)/(2(1 - lambda_i conjugate(lambda_j))), symmetrize K as (K+K^*)/2, and add relu(epsilon - lambda_min(K)) squared. Pseudocode: sample lambdas; solve each resolvent system; calculate Phi and H; calculate minimum Hermitian eigenvalues; build K; add alpha times the total penalty to the task loss. The Herglotz representation and kernel formula come from the paper; the frequency grid, epsilon, penalty coefficient alpha, and identity offset are empirical design choices. Begin with a 64-state SSM on copy, adding, and ListOps tasks. Compare no regularizer, spectral-radius clipping, and Caratheodory-kernel regularization with matched optimization settings. Track the minimum sampled eigenvalue, rollout error at horizons 2x and 4x training length, gradient norms, validation loss, and wall-clock overhead. Success means improved long-horizon extrapolation and a nonnegative or near-nonnegative response margin without degrading short-context accuracy by more than a small tolerance.