Periodic Lyapunov Guard for Cyclic Training

1. Integration point: use this as a controller around a cyclic learning-rate optimizer, momentum optimizer, or recurrent inference loop. Define the phase-specific update $z_{k+1}=F_k(z_k)$; for ordinary training, $z_k$ can concatenate flattened parameters and optimizer momentum, while for an RNN it can concatenate hidden state and a local parameter subset. At the end of every schedule period, freeze a minibatch and estimate the Jacobians $J_k$ with forward-mode JVPs or finite differences on a low-dimensional random subspace. 2. Pseudocode: collect $J_0,\ldots,J_{M-1}$; solve an SDP for $P_0,\ldots,P_{M-1}\succ\epsilon I$ minimizing $\gamma$ subject to $J_k^\top P_{k+1}J_k\preceq\gamma^2P_k$ and $P_M=P_0$; if $\gamma>\gamma_{\max}$, multiply the learning rate and schedule amplitude by $\gamma_{\max}/\gamma$ and retry; otherwise perform the next period. For large models, use diagonal or block-diagonal $P_k$ and Jacobian sketches instead of a full SDP. 3. Computed from the paper's mechanism: the periodic Lyapunov inequalities, phase-dependent energy, and monodromy stability criterion. Estimated empirically: the Jacobians, local trajectory, and SDP margin. 4. First cheap experiment: train a two-layer MLP on MNIST with cosine learning rate of period $M=20$, comparing SGD, gradient clipping, and the Lyapunov guard using a diagonal $P_k$ on a 100-dimensional parameter projection. Sweep the base learning rate and measure the largest stable rate. Prediction: divergence begins near the measured boundary $\rho(\Phi)=1$; below it, squared perturbation energy should decay approximately as $\gamma^{2t}$, and the guarded run should maintain $\gamma<0.95$ while unguarded runs become unstable when $\rho(\Phi)>1$.