# Task-Tangent Capture Pruning MVP

## Mathematical predictions and checks

The experiment uses a random tangent operator `T` and calibration output directions `U`, with columns `g_i = T^*u_i`.

1. **Relative-capture bound:** removing the lowest task-energy coordinates until the cumulative discarded energy reaches `epsilon^2` should produce `r <= epsilon`, where `r = sqrt(sum_i ||(I-P)g_i||^2 / sum_i ||g_i||^2)`. Observed `(epsilon, r)` were `(0.05, 0.04953)`, `(0.10, 0.09938)`, `(0.20, 0.19810)`, `(0.30, 0.29999)`, and `(0.50, 0.49954)`. All satisfy the bound.

2. **Squared-energy scaling:** the empirical squared ratio should equal the discarded tangent-energy fraction. Across random masks, the maximum absolute error in this identity was `1.11e-16`. For the epsilon sweep, observed squared ratios were `0.002454, 0.009876, 0.039245, 0.089997, 0.249538`, matching the squares of the observed ratios exactly.

3. **Task-aware ranking:** at equal sparsity, ranking coordinates by task tangent energy should discard less task energy than unrelated magnitude ranking. Ratios (tangent vs magnitude) were: 25%: `0.2303 vs 0.4866`; 50%: `0.4128 vs 0.7363`; 80%: `0.6882 vs 0.8798`.

## Mini experiment

A fixed-seed small tanh regression model compares tangent-energy pruning and magnitude pruning, followed by 30 masked SGD steps. At 50% pruning, initial MSE was `0.4957` tangent vs `0.4680` magnitude, and final MSE was `0.3942` vs `0.4182`. At 80% pruning, initial MSE was `0.5123` vs `0.4624`, and final MSE was `0.4111` vs `0.4624`. Thus tangent pruning had a larger immediate shock in this single run but better equal-step recovery/final loss.

## Reproduction

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

This writes `results.json` and prints the same measurements. CUDA is used when available, with CPU fallback.

## Limitations

This is a toy linear-algebra verification and a tiny regression, not CIFAR/ResNet-scale evidence. It does not measure genuinely sparse-kernel speed or memory, does not compare gradient pruning, and uses one seed and one calibration batch; held-out task-direction generalization and refresh-after-finetuning were not tested.