Gumbel escape-time controller

Attach the monitor to an optimizer that uses stale gradients, delayed target parameters, or any deliberately delayed update. At the beginning of each delayed burst, define an escape event as the first step at which normalized displacement q at time t equals the parameter distance from the burst start divided by the square root of the estimated gradient-noise trace plus epsilon, and q exceeds a chosen radius R. Run 16 to 32 inexpensive replicas with different minibatch orderings or random seeds, recording each first-passage time T. Fit the Gumbel location mu and scale beta by maximum likelihood, and estimate r from the slope of log(q) during the approximately linear amplification phase. Permit another short burst only if a goodness-of-fit test is acceptable, for example Kolmogorov-Smirnov p greater than 0.05, beta times r is stable across bursts, and escaped replicas improve validation loss. If displacement grows without successful escape, reduce tau toward 0.8 tau c, lower the learning rate, or terminate the burst. The paper provides the logarithmic amplification and Gumbel mechanism; estimate r, mu, beta, effective curvature k, and noise scale empirically. First test a two-dimensional noisy double-well system, followed by a small MNIST MLP. Sweep delays around tau c=pi divided by 2k and collect at least 32 passage times per setting. The falsifiable prediction is that below threshold most runs do not escape within the observation window, while above threshold a Gumbel fit improves over an exponential fit and beta times r remains approximately constant, within 30 percent, across delay settings.