Derivative-Dispersion Forcing Regularizer

Attach the regularizer to the forcing subnetwork of an RNN, linear state-space model, or gated recurrent block, not to the recurrent matrix itself. For each minibatch, sample B distinct inputs z_b=(u_b,x_b), use automatic differentiation or one forward-mode Jacobian-vector pass to obtain J_b=partial phi_theta(z_b)/partial z_b, form M_b=J_b J_b^T, and add lambda_disp times negative log determinant of the batch average plus epsilon I to the task loss. If d is large, replace the determinant by the log of the product of four to eight Lanczos-estimated eigenvalues, or use a cheaper trace-minus-Frobenius surrogate. At every optimizer step, estimate rho(A) or the spectral norm; project or rescale A to enforce norm at most q=0.95. The paper supplies the derivative-dispersion mechanism and the contraction-versus-branch-separation interpretation; the exact regularizer and finite-batch estimator are the proposed neural adaptation. First test on a width-16 vanilla RNN trained on adding-two-numbers, copy-memory, and sequential MNIST, comparing no regularizer, output-Jacobian log-determinant regularization, and a standard weight-diversity penalty. Sweep lambda_disp. The quantitative prediction is that increasing dispersion raises the smallest eigenvalue of the forcing covariance and delays hidden-state covariance collapse until a task-dependent optimum. With norm(A)=q<1, perturbation norms should decay geometrically at rate no slower than q^n after forcing differences are removed; runs with q at least one should show a clear gradient or state-norm instability boundary.