Singular-perturbation continuation schedule

1. Integration point: apply this to an implicit recurrent layer or neural ODE whose hidden state is split into slow memory P and fast auxiliary state z. During stage zero, replace the fast state by a learned algebraic decoder z=h_theta(P), and train the reduced model with ordinary backpropagation. 2. Pseudocode: initialize epsilon=1; train the reduced loss using z=h_theta(P); construct the finite-epsilon cell; for each continuation stage, run a short warm start, estimate the largest real eigenvalue of J_perp using 5-10 power iterations or autograd Jacobian-vector products, and decrease epsilon by a factor 0.5 only if mu>mu_min; otherwise reduce learning rate, add a barrier -lambda log(mu), or stop continuation. Train to convergence at each stage. 3. Computed from the paper: the fast/reduced decomposition and the requirement that trajectories remain near a normally hyperbolic critical manifold. Estimated: J_perp, mu, reconstruction error ||z-h_theta(P)||, and task loss. 4. First cheap experiment: use a 1-2 layer implicit MLP on a synthetic long-horizon nonlinear recurrence with known stiff and nonstiff regimes, comparing direct training against continuation. Measure fixed-point failures and gradient norms as epsilon decreases. The predicted signature is a sharp rise in solver iterations and gradient variance when mu approaches zero; successful continuation should maintain mu>mu_min and reduce final training failures, while reduced-to-full trajectory discrepancy should initially scale approximately linearly with epsilon away from folds.