Compactified Burst Controller

1. Integration point: insert the controller inside every neural ODE function evaluation or recurrent hidden-state update, before advancing the solver. It is intended for long-horizon RNN and SSM rollouts, DEQ iterations, and polynomial residual blocks where rare norm bursts cause NaNs or excessive adaptive solver steps. 2. Pseudocode: compute r = norm(h) plus epsilon and u = h/r; evaluate the estimated leading field v = F_n(u); compute a = dot(u,v) and q = norm(v - a u). Maintain an exponential moving average of a. If r exceeds r_c, the averaged radial growth exceeds a margin, and q is below an angular-equilibrium threshold, replace dh/dt = F(h) by dh/dt = F(h) - kappa(r)u. Estimate angular instability every K steps using finite differences or automatic differentiation of G. 3. The paper supplies the radial and tangential infinity mechanism; r_c, the growth margin, q threshold, damping gain k, and transition width delta are calibrated on non-divergent training rollouts. 4. First cheap experiment: compare vanilla updates, global gradient clipping, and this controller on a cubic RNN trained on the copy and addition tasks, evaluating at ten times the training horizon. Sweep the cubic gain lambda. Prediction: uncontrolled bursts begin when averaged a becomes positive while q is initially small; controller interventions occur before the norm maximum, reduce solver failures, and preserve trajectories with negative a. The measured critical lambda for burst onset should agree with the infinity-atlas prediction within 20 percent.