Margulis-Balanced Expanding Recurrent Layer

Integration point: apply this to a GRU-like recurrent cell, a recurrent state-space model, or an invertible residual block with update z_next = f_theta(z,u). Add lambda times L_Marg to the ordinary supervised or sequence-modeling loss. Use a small vector network to produce v_theta(z), normalize it to unit Euclidean norm, and compute J_theta(z) v_theta(z) with an automatic-differentiation Jacobian-vector product; do not construct the full Jacobian. Pseudocode: sample hidden states z and inputs u from truncated training sequences; compute z_next = f_theta(z,u); compute q = J_theta(z) v_theta(z); compute ell = log(norm(q) + epsilon); compute h as the detached minibatch mean of ell; evaluate phi(z) and phi(z_next); form r = ell - h - phi(z_next) + phi(z); add lambda times mean(r squared) to the task loss. For neighboring states z and z_prime generated from nearby prefixes, add gamma times the squared difference of their ell values. Update theta, the direction network, and phi jointly. The paper supplies the constant scaling and coboundary mechanism; the direction, entropy estimate, and neighbor pairs are empirical estimates. First run a two-dimensional linear-map unit test, then train a small tanh RNN on adding-task regression or sequential MNIST against vanilla training and spectral normalization with matched parameter counts. Estimate finite-time Lyapunov growth using repeated tangent-vector products over 100 recurrent steps. Quantitative prediction: increasing lambda should monotonically reduce the batch standard deviation of r, and should reduce the variance of finite-time log expansion across initial states. Sweep the recurrent gain g and measure the horizon at which hidden-state norms diverge. The uncontrolled model should become unstable near the point where its empirical mean log expansion crosses zero; the proposed model should show a narrower expansion distribution and a delayed divergence boundary. Report whether the measured standard deviation of corrected log expansion falls by at least 20 percent at equal task loss.