Phase-Aware Jacobian Stiffness Certificate

1. Integration point: start with a GRU, diagonal-plus-low-rank state-space model, or complex-valued RNN. Represent hidden features as paired amplitude and phase channels; for a real model, designate two channel groups and apply the same block decomposition. Add the stiffness penalty to the task loss and use the score as an early-warning monitor. 2. Pseudocode: for each minibatch, run a short sequence trajectory; choose 16 logarithmically spaced frequencies; estimate the Jacobian response using forward-mode Jacobian-vector products or finite sinusoidal perturbations; use three to five power iterations to estimate each block's top singular value; compute the maximum score; then update L = L_task + lambda times max(0, log(s_JVSM divided by tau)) squared. Initially detach the score from backpropagation, then enable differentiation through one power iteration. 3. Computed from the paper's mechanism: the structured local Jacobian and separate magnitude and phase sensitivities. Estimated empirically: the frequency response, normalization matrices, threshold tau, and coupling weight gamma. Use running RMS statistics for W_u and W_y. 4. First cheap experiment: train a small GRU on permuted-MNIST, comparing vanilla training, spectral normalization, and this penalty. Inject sinusoidal hidden-state perturbations and measure amplification and long-horizon prediction error. The predicted signature is a sharp change near s_JVSM = tau. With tau = 1, models whose hidden Jacobian norm remains below 0.9 should show decay close to 0.9^t, whereas models above 1 should show non-decaying or growing perturbations. The measured transition should agree with the predicted threshold within approximately 20 percent.