# Fourier-Tumble Oscillatory Memory MVP

`experiment.py` implements the first-Fourier recurrent channel:
\[
\Pi_1=\sum_j q_j e^{i\theta_j},\quad \gamma=\alpha(1-\Re\Pi_1),\quad \Omega=\alpha\Im\Pi_1,
\]
followed by the exact transition `exp(-gamma*dt) R(Omega*dt)`.

Run:

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

The script writes `results.json` and checks three mechanism predictions:

1. Discretized circular probability distributions have `|Pi_1| <= 1`; their recurrent Jacobian norm is no larger than one.
2. A homogeneous trajectory follows `exp(-gamma*t) cos(Omega*t)`; fitted damping and frequency are compared to the values induced by `Pi_1`.
3. The envelope 1/e memory time scales as `1/gamma` while the phase is fixed, with a sweep toward `|Pi_1|=1`.

Observed results from the fixed-seed run are in `results.json`: maximum sampled `|Pi_1|=0.9352`, maximum Jacobian norm `0.9373`, and zero numerical fit error for the generated exact transition. The memory sweep increases from 1.0 steps at gamma=1 to 25.29 steps at gamma=0.0395.

The delayed-recall numbers are only a secondary, untrained sanity comparison between a fixed Fourier channel and a fixed-radius scalar retention baseline; they are not evidence of a trained task-level accuracy win. No sequential-MNIST, GRU, or optimized parameter-count-matched training experiment was run.