Basin-Aware Hysteresis Guard

1. Integration point: apply the guard to tanh or gated RNNs, continuous-time RNNs, neural ODEs, and state-space models during inference or truncated-BPTT training. Define the desired hidden-state branch using the zero-input state, a running teacher-forced state, or a learned reference state. Add controllable damping c to the hidden update, for example h_{t+1}=(1-c)h_t+c f_phi(h_t,u_t), and estimate the local Jacobian spectral radius around the reference trajectory. 2. Pseudocode: compute Jv power iterations and estimate alpha or rho; if the spectral margin is unsafe, increase c or apply a contraction penalty. If the margin becomes safe, draw M perturbations h_ref+epsilon z_m, run each for T steps with the current input held fixed, and calculate B_hat as the fraction returning within delta of h_ref. Keep damping whenever B_hat<1-epsilon; release it only after K consecutive safe checks. 3. The paper supplies the distinction between threshold restoration and basin recovery, plus the saddle-node condition; alpha, B_hat, and the appropriate perturbation scale are estimated from the model. 4. First cheap experiment: use a 2-layer tanh RNN on the adding problem and sequential MNIST, deliberately initialize recurrent gain near a regime with multiple fixed points. Compare ordinary spectral normalization, threshold-only damping, and the hysteresis guard. Sweep gain and perturbation amplitude, measuring fixed-point multiplicity, recovery probability, long-horizon error, and whether trajectories remain trapped after gain restoration. The predicted signature is a hysteresis loop: the gain at which the high-error state appears should exceed the gain at which it disappears, and threshold-only control should show recovery failure below the local instability boundary. The guard should reduce post-restoration failure probability by at least 50% without reducing stable-regime throughput by more than 10%.