# Topological Reachable-Set Coverage Scheduler

## Implementation
`coverage_scheduler_experiment.py` uses the unit square as an exactly known reachable set and an exact 1-Lipschitz radial residual

`r_L(x) = margin + L ||x-(0.5,0.5)||`, with `margin=0.05`.

The scheduler starts from four corners and either samples uniformly at random or greedily selects the point maximizing the prescribed coverage score. Coverage is measured exactly over a dense grid using a KD-tree. Results are in `results.json`.

## Stage-1 mechanism checks

1. **Lipschitz bound prediction:** for every tested `L` in `{0.25, 0.5, 1, 2, 4}` and all 4 random seeds, the computed lower bound `min_sample_residual - L*delta` was no greater than the true grid minimum. Observed bound validity: **100% (80/80)**.

2. **Certification threshold prediction:** certification should occur when `delta < epsilon/L`, with `epsilon=0.04`. The observed crossing ratios `delta/(epsilon/L)` were approximately **0.2357--0.2485**, hence below 1 (the coarse grid causes a substantial discrete jump). For `L=0.25`, adaptive sampling reached certification in median **36 samples**, versus **93** for random; for `L=0.5`, adaptive reached it in **88**, while random's successful runs required median **266.5**. At `L=1`, neither method reached the threshold within 300 samples on this grid, consistent with the tighter predicted radius `0.15`.

3. **Lipschitz scaling prediction:** at fixed sample geometry, the median residual gap divided by `L` remained constant at **0.141421**, while median `L*delta` scaled from **0.08333** at `L=.25` to **1.33333** at `L=4`. This is the predicted linear scaling, and the bound remained conservative (`gap <= L*delta`).

## Secondary equal-budget comparison

At `L=1`, median coverage radius for adaptive versus random was:

- 12 samples: `0.32998` vs `0.43013`
- 25 samples: `0.18856` vs `0.32473`
- 50 samples: `0.10541` vs `0.22326`
- 80 samples: `0.10000` vs `0.17309`

Thus the adaptive/farthest scheduler gives a clear coverage win in this toy geometry.

## Reproduction

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

## Limitations

This is not a neural-policy or physical simulator experiment: the reachable set is known, two-dimensional, static, and sampled from a finite grid. The residual is analytically Lipschitz, the adaptive score's low-margin term is inactive because all true residuals exceed the chosen epsilon, and no learned policy, estimated reachable-set error, noisy residual, latent metric, or finite-sample confidence interval was tested.