Critical-Slowing-Down Safety Monitor

1. Integration point: for an RNN, SSM, or neural RL controller, log the hidden state, action residual, or one-step prediction error at every control step. For an optimizer version, use the minibatch loss, gradient norm, or a low-dimensional projection of the parameter update. Detrend each channel with a local rolling mean, normalize by a robust rolling scale, and optionally project a high-dimensional hidden state onto its first one to three PCA directions. 2. Pseudocode: maintain a window of length W; compute the rolling mean, variance, and AR(1) coefficient using the displayed formulas; exponentially smooth the estimates; if the smoothed coefficient exceeds a_c and the variance slope is positive, multiply the policy gain, action scale, or learning rate by exp(-gamma*(a_hat-a_c)); optionally enter a clipped-action or frozen-optimizer fallback. 3. Computed from the paper's mechanism: the recovery-rate interpretation and the joint variance/autocorrelation warning signals. Estimated empirically: W, a_c, gamma, PCA directions, confidence intervals, and the baseline distribution under stable operation. Estimate a_c from clean stable runs, for example the 99th percentile of a_hat, and reserve a second threshold for emergency intervention. 4. First cheap experiment: train a small GRU policy on CartPole or a quadrotor attitude simulator with artificial observation/action delay, comparing no monitor, variance-only, autocorrelation-only, and joint CSD intervention. Sweep delay and controller gain. The predicted signature is a monotonic rise of a_hat toward one before divergence, with variance approximately proportional to 1/(1-a_hat^2); the detected instability boundary should occur within 20% of the independently measured boundary and provide positive warning lead time. Test generalization by applying thresholds learned on one delay or plant to another without retraining.