Lyapunov-gap regularization for recurrent dynamics

(1) Integration point: apply this to the recurrent transition Jacobian, not to the output head. For a linear or diagonal SSM, M_t is the discretized state transition. For a GRU or LSTM, M_t is the Jacobian of h_t with respect to h_{t-1}, evaluated on a minibatch trajectory. Use a smaller number k of leading directions if the hidden dimension is large. (2) Pseudocode: initialize Q with k orthonormal columns; set s=0; for t=1,...,T, compute M_t, form Z=M_t Q, compute reduced QR Z=Q_new R, set Q=Q_new, and accumulate s_i += log(abs(R[i,i])+1e-8); after the loop set Lhat_i=s_i/T and add lambda_gap times the displayed L_gap to the task loss. (3) The QR recursion and exponent estimator come directly from the Lyapunov definition. The target delta/W, ceiling rho, and penalty weights are empirical hyperparameters; estimate c only if desired by measuring gaps in a baseline model. Use stop-gradient through M_t initially for numerical stability, then test differentiable Jacobian products. (4) First experiment: compare a vanilla GRU, orthogonally initialized GRU, spectral-normalized GRU, and this regularizer on sequential MNIST and the adding problem. Record loss versus optimizer steps and FLOPs, gradient norms versus sequence length, hidden-state effective rank, and validation accuracy. Success is fewer exploding or vanishing gradients, larger effective rank, and improved long-context accuracy at equal parameter count and compute.