# Periodic Lyapunov Guard MVP

Run:

```bash
/home/maxwelhelp/main/bin/python3 periodic_guard_experiment.py
```

The experiment is a deterministic two-dimensional, two-phase linear time-periodic system. The phase-0 Jacobian is temporarily expansive, while phase 1 damps it:
`J0=I+alpha*diag(1.2,-0.2)` and `J1=I+alpha*diag(-1.8,-0.2)`.
The controller scales a requested `alpha` down until the exact diagonal periodic certificate has `gamma <= 0.95`.

## Mechanism checks

1. **Stability boundary:** the dominant monodromy eigenvalue is
`(1+1.2 alpha)(1-1.8 alpha) = 1 - 0.6 alpha - 2.16 alpha^2`.
The predicted positive high-rate crossing of `rho(Phi)=1` is `alpha=0.833333`; a 291-point sweep observed `0.835` (grid resolution 0.005).

2. **Periodic Lyapunov margin:** for diagonal periodic `P_k`, the minimum certified phase factor is predicted to be `gamma=rho(Phi)^(1/2)`. Observed and predicted values agreed to floating-point precision at alpha values 0.2, 0.5, 0.8, and 1.0. Phase 0 was expansive in Euclidean norm (spectral norms 1.24, 1.60, 1.96, 2.20), showing why a phase-independent norm can be misleading.

3. **Energy scaling:** over a full period, the observed median energy ratios were 0.65610, 0.74364, and 3.09760 at alpha 0.5, 0.8, and 1.0. The certified upper bounds were 0.65610, 0.74373, and 3.09760; the small alpha=0.8 gap is because the initial state includes a non-dominant coordinate.

## Baseline versus guard

At requested alpha=1.0, the unguarded baseline has `gamma=1.32665` and final state norm `2.319e7` after 60 phase updates. The guarded run selects `alpha=0.809835` (predicted and observed threshold agree to numerical precision), has `gamma=0.95`, and final norm `0.0461`.

**Verdict:** the proposed mechanism manifested clearly in the toy LTP system, including temporary phase expansion, the monodromy boundary, certified energy decrease, and prevention of delayed divergence. This is evidence for the mechanism, not evidence of an MNIST training win.

## Limitations

No neural network, stochastic minibatches, Jacobian sketching, SDP solver, momentum state, or wall-clock/FLOP comparison was tested. The diagonal system makes the periodic Lyapunov certificate analytically easy; higher-dimensional non-normal systems may have a gap between spectral-radius estimates and a usable diagonal/block certificate.