# 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:

```bash
/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.