Gumbel escape-time controller MVP

Implementation

experiment.py numerically solves the delayed characteristic equation, generates extreme-value-seeded escape times, fits Gumbel and exponential models, checks rate and boundary scaling, and demonstrates a minimal burst-gating controller. Running it writes results.json.

Command:

/home/maxwelhelp/main/bin/python3 experiment.py

Quantitative checks

1. Delay threshold: with k=1, theory predicts tau_c=pi/(2k)=1.5707963. The numerical root sweep estimated 1.5707963, relative error 2.3e-13. The dominant real root was negative below threshold, approximately zero at threshold, and positive above it.

2. Gumbel law and rate scaling: for growth rates r={0.20,0.35,0.60,0.90}, fitted beta*r was {0.996,1.005,0.964,1.049}, within about 5% of the predicted constant 1. Every KS p-value exceeded 0.05, and Gumbel AIC was substantially lower than exponential AIC in all 512-sample sweeps.

3. Logarithmic boundary scaling: at r=0.45, fitting mean passage time versus log(R) for R={20,100,500,2500} gave slope 2.248, versus predicted 1/r=2.222 (1.16% relative error).

The toy controller rejected a non-improving burst and accepted improving bursts whose escape samples passed the Gumbel check. The distributional checks use an explicit extreme-value seed, so they verify the proposed mathematical mechanism rather than establish that arbitrary minibatch noise is Gumbel.

Limitations

No stale-gradient optimizer, MNIST MLP, GPU experiment, true noisy double-well integration, or measured validation-loss improvement was tested. The controller's validation-improvement input is supplied externally in this MVP. Thus the mechanism manifested clearly in the synthetic model, but optimizer speedup, stability, and generalization remain unverified.