Sideband-Aware Stability Monitor for Periodic Training

1. Integration point: apply this to an optimizer with a periodic learning-rate, momentum, gradient-clipping, or data-order schedule, or to an SSM/RNN whose recurrent matrix is periodically gated. Every \(S\) optimizer steps, estimate the local parameter-space Jacobian at each schedule phase. For a cheap MVP, use a two-layer MLP on MNIST with cosine learning-rate modulation of periods \(S=8,16,32\), and compare ordinary SGD or momentum SGD with the sideband-aware controller. 2. Pseudocode: for each phase \(r=0,\ldots,S-1\), compute Jacobian-vector products \(A_rv\) with autograd at the current minibatch or a fixed probe batch; apply the phase operators successively to Lanczos or power-iteration vectors using \(v\leftarrow v+\eta A_rv\); estimate \(\rho(\mathcal M)\). If the estimate exceeds \(1-\epsilon\), shrink \(\eta\leftarrow0.8\eta\) or reduce modulation amplitude. Optionally estimate Fourier coefficients \(A_m=S^{-1}\sum_r A_re^{-2\pi imr/S}\) and construct the block sideband matrix with \(K=1\) or \(2\). 3. Computed versus empirical: monodromy and lifted operators are computed from Jacobian-vector products; minibatch noise and nonlinear basin effects are measured empirically. The averaged baseline uses only \(\bar A=S^{-1}\sum_rA_r\), deleting off-diagonal sideband blocks. 4. First experiment: train the same MLP on MNIST across learning-rate sweeps and modulation periods \(S=8,16,32\), recording loss divergence, gradient norm, \(\rho(\mathcal M)\), and the averaged-Jacobian prediction. The falsifiable prediction is that the actual divergence threshold follows \(\rho(\mathcal M)=1\) within 20%, while the averaged model increasingly overestimates the safe learning rate when modulation amplitude is large or perturbation content is near the schedule's Nyquist frequency. A second signature is a sharp rise in sideband gain near \(\omega\approx\omega_s/2\) or its aliases.