# Weighted Resolvent-Equivariant Attention MVP

## Implementation

`experiment.py` implements a weighted reversal operator `G = Delta^2 R`, numerical commutator/power/resolvent checks, and a small single-head attention classifier using the normalized penalty `||GP-PG||_F^2/(||G||_F^2||P||_F^2+eps)`. It uses CUDA when available and falls back to CPU on runtime failure.

## Stage-1 math verification

All checks use fixed seed 2327.

1. **Linear perturbation scaling.** For `P(eps)=P0+eps E` with `[G,P0]=0`, the prediction is `||[G,P(eps)]||_F = eps||[G,E]||_F`. The predicted slope was 1.9490096903 and the fitted slope was 1.9490096903. Maximum relative deviation over eps from 0 to 0.3 was 1.9e-16. Observed norms at eps `[0.001, 0.01, 0.1, 0.3]` were `[0.00194901, 0.01949010, 0.19490097, 0.58470291]`.

2. **Multi-step compatibility.** For commuting `P0`, every tested power and its resolvent should commute with `G`. The maximum norm of `[G,P0^t]` for t=0..8 was 6.34e-16, and the resolvent commutator at gamma=0.37 was 5.04e-16: numerical zero.

3. **Damping behavior.** The identity `[G,R(gamma)] = (1-gamma)R(gamma)[G,P]R(gamma)` predicts zero at gamma=1 and linear behavior in `(1-gamma)` near gamma=1. The identity relative error stayed below 1.2e-13 across the sweep, and the gamma=1 norm was exactly 0 numerically. The near-gamma=1 fitted slope versus `(1-gamma)` was 0.2129. The full resolvent norm was non-monotonic at gamma=0 to 0.1 because the selected non-contractive toy matrix is close to a resolvent singularity; it then decreased strongly toward zero. This is a limitation of that sweep, not a failure of the algebraic identity.

## Attention pilot

Two fixed seeds, 180 optimization steps, identical model and data for lambda=0 and lambda=1:

- Baseline lambda=0: mean accuracy **0.6994**, mean validation loss **0.6377**, mean commutator RMS **3.3943**.
- Weighted-commutator lambda=1: mean accuracy **0.7351**, mean validation loss **0.5181**, mean commutator RMS **0.3932**.

Thus the penalty reduced the measured commutator by about **8.6x** and improved mean accuracy by 3.6 percentage points / validation loss by 0.1196 in this tiny pilot. The requested 10x equivariance target was not reached, so the empirical effect is promising but not a full success against that threshold.

## Reproduction

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

Outputs are saved to `results.json`.

## Limitations

This is not a 4-layer, 8-head transformer or a second graph benchmark. Only two seeds and one synthetic attention task were tested; FLOP matching, learned kappa, reversal augmentation, sample-complexity curves, and explicit function-level equivariance were not evaluated. The toy resolvent matrix was not constrained to be a well-conditioned Markov transition matrix, causing a non-monotonic low-gamma curve, although the exact resolvent identity and gamma-to-one prediction were verified.