# Semantic Pushforward Uncertainty Head MVP

`experiment.py` implements the finite-state semantic pushforward and the proposed calibrated head:

`p[j] = sum(response_prob[r] for phi[r] == j)`

`hat_pi = softmax(a * log(p + eps) + b)`

The experiment uses exact synthetic Bayesian posteriors `q`, a controlled response distortion `p proportional to q**gamma`, and separate fitting/test splits. This isolates the calibration mechanism while retaining the finite-state total-variation metric.

## Reproduce

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

The script writes `results.json` and prints the same JSON.

## Mechanism checks

- Pushforward aggregation error: `0.0`.
- Redistributing mass among synonymous responses changes pushed state mass by only `1.11e-16`.
- Across gamma = 0.5, 0.75, 1, 1.5, 2, fitted mean slopes gave gamma*a = 0.916, 0.983, 0.995, 0.980, 1.026 (near the predicted 1).
- At gamma=2, calibration reduced TV to exact q from 0.1420 to 0.0070, ECE from 0.1324 to 0.0175, and NLL from 0.8894 to 0.7791; 90% prediction-set coverage rose from 0.8884 to 0.9680.

This is not a transformer integration: response probabilities are supplied by a controlled finite candidate distribution rather than token log-probabilities from an LM. It therefore verifies the measure aggregation and calibration mechanism, not language-model decoding, malformed-output rates, or extra-forward-pass cost.