Transfer-Spectrum Pseudo-Transition Scheduler

Integrate this into an optimizer for a small tanh RNN, state-space model, or deep residual network. Treat a minibatch of consecutive hidden states h_t as samples from a local propagation process. Every K optimization steps, estimate a low-rank transfer operator T by fitting h_{t+1} approximately equal to T h_t for linear models, or by using Jacobian-vector products J_t v = partial h_{t+1} / partial h_t times v and randomized Arnoldi iteration for nonlinear models. Compute the two or three eigenvalues with largest magnitude, normalize by the largest magnitude, and estimate p_k from projections of the current state batch onto the corresponding eigenvectors. Pseudocode: collect hidden states; run m power or Arnoldi iterations; obtain lambda_0, lambda_1, and lambda_2; calculate G, H, and xi; if G is below its trigger and H is above its trigger, multiply the learning rate by gamma = 0.5 and optionally increase weight decay; otherwise update the learning rate using the gap controller; then perform the ordinary gradient step. The transfer-matrix interpretation is taken from the paper, whereas T, eigenvalues, and thresholds are estimated empirically. First test on a one-layer tanh RNN trained on permuted MNIST or the adding problem, comparing Adam, fixed-step SGD, and the scheduler. Sweep learning rates and locate the largest stable rate. The quantitative prediction is that gradient variance and hidden-state sensitivity peak in a narrow band where G is minimized and H rises. Damping should reduce this peak without eliminating the surrounding stable regimes. Across seeds, the measured response peak should coincide with the minimum-gap location within 20 percent, and the measured memory decay should follow C(k) proportional to exp(-k / xi) using the estimated subleading eigenvalue.