# Finite-Excitation Orthogonal Gradient Memory

## Implementation

`experiment.py` implements modified Gram-Schmidt memory construction and three deterministic checks. `results.json` contains the complete numerical output.

## Quantitative mechanism checks

1. **Orthogonalization claim.** An 8D feature matrix with singular values geometrically decreasing from 1 to 1e-7 had `sigma_min=1.0000000002e-7`. MGS retained rank 8 and produced `max|Q.T Q-I|=2.83e-10`.

2. **Discrete stability boundary.** For a complete orthonormal memory, the update is `e_next=(1-alpha)e`. The predicted boundary is `alpha=2` (equivalently `eta*lambda=2`). The observed sweep was stable at alpha=1.99 (norm after 30 steps 0.7397) and unstable at alpha=2.01 (norm 1.3478); observed norms matched `|1-alpha|^30` to floating-point precision. The sweep also included alpha 0.25, 0.75, 1.0, 1.25, 1.75, and 2.5.

3. **Finite-excitation transition.** With dimension 8 and alpha=0.8, a memory spanning k orthogonal directions contracts only those coordinates. At step 15, observed norms for k=0..8 were respectively `1.0000, 0.9354, 0.8660, 0.7906, 0.7071, 0.6124, 0.5000, 0.3536, 3.28e-11`, matching the closed-form prediction to numerical precision. Full all-direction contraction first appeared exactly at the predicted transition k=8.

4. **Conditioning prediction.** With the same dimension and gain, raw replay on a well-conditioned matrix (`sigma_min=1`) had late log-error slope -1.6094, while MGS had -1.6094. On an ill-conditioned matrix (`sigma_min=0.001`), raw replay had slope +1.5065 and error at step 10 `6.96e5` (unstable under the same gain), while MGS retained slope -1.6094 and error `1.02e-7`. Thus MGS removed the dependence on the feature singular spectrum in this controlled linear-head setting, though the comparison uses an intentionally aggressive common step size that exposes raw replay's stability problem.

## Reproduction

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

All tests use NumPy and fixed seed 2688; no GPU or large model is required.