# Matched-Execution Metric Attribution Report

- Run directory: `outputs\rl\metric_attribution_5seed_safe10_worker_cached`
- High-noise evaluation point: `0.30`
- Seeds: `[0, 1, 2, 3, 4]`
- Timesteps per method: `3000`

## Main Matched-Execution Table

| Method | Return | RMSE | Applied Var | Diff MSE | Dense violation | B-L max | Projection trigger | Projection mag. |
|---|---:|---:|---:|---:|---:|---:|---:|---:|
| Full execution, no P-ALE | -9.600 +/- 0.662 | 0.286 +/- 0.011 | 0.0063 +/- 0.0009 | 0.0069 +/- 0.0020 | 0.000 +/- 0.000 | 6.230 +/- 3.233 | 0.146 +/- 0.088 | 0.0121 +/- 0.0063 |
| Full execution + plain Var | -9.370 +/- 0.153 | 0.284 +/- 0.004 | 0.0060 +/- 0.0006 | 0.0064 +/- 0.0018 | 0.000 +/- 0.000 | 5.260 +/- 2.365 | 0.134 +/- 0.069 | 0.0111 +/- 0.0055 |
| Full execution + identity metric | -10.056 +/- 0.857 | 0.279 +/- 0.003 | 0.0100 +/- 0.0034 | 0.0203 +/- 0.0128 | 0.000 +/- 0.000 | 6.082 +/- 0.729 | 0.321 +/- 0.150 | 0.0371 +/- 0.0222 |
| Full execution + diagonal M | -10.056 +/- 0.857 | 0.279 +/- 0.003 | 0.0100 +/- 0.0034 | 0.0203 +/- 0.0128 | 0.000 +/- 0.000 | 6.082 +/- 0.729 | 0.321 +/- 0.150 | 0.0371 +/- 0.0222 |
| Full execution + random PSD M | -10.056 +/- 0.857 | 0.279 +/- 0.003 | 0.0100 +/- 0.0034 | 0.0203 +/- 0.0128 | 0.000 +/- 0.000 | 6.082 +/- 0.729 | 0.321 +/- 0.150 | 0.0371 +/- 0.0222 |
| Full execution + stationary N | -10.056 +/- 0.857 | 0.279 +/- 0.003 | 0.0100 +/- 0.0034 | 0.0203 +/- 0.0128 | 0.000 +/- 0.000 | 6.082 +/- 0.729 | 0.321 +/- 0.150 | 0.0371 +/- 0.0222 |
| Full CG-RVC, one-step M | -10.056 +/- 0.857 | 0.279 +/- 0.003 | 0.0100 +/- 0.0034 | 0.0203 +/- 0.0128 | 0.000 +/- 0.000 | 6.082 +/- 0.729 | 0.321 +/- 0.150 | 0.0371 +/- 0.0222 |

## Relative Gaps to Full CG-RVC

| Comparator | RMSE reduction vs comparator | Applied Var reduction vs comparator | Diff MSE reduction vs comparator | Dense violation reduction vs comparator | B-L max reduction vs comparator | Projection trigger reduction vs comparator | Projection mag. reduction vs comparator |
|---|---:|---:|---:|---:|---:|---:|---:|
| Full execution, no P-ALE | 2.4% | -59.7% | -196.1% | n/a | 2.4% | -119.8% | -206.1% |
| Full execution + plain Var | 1.6% | -66.2% | -217.5% | n/a | -15.6% | -139.3% | -234.2% |
| Full execution + identity metric | 0.0% | 0.0% | 0.0% | n/a | 0.0% | 0.0% | 0.0% |
| Full execution + diagonal M | 0.0% | 0.0% | 0.0% | n/a | 0.0% | 0.0% | 0.0% |
| Full execution + random PSD M | 0.0% | 0.0% | 0.0% | n/a | 0.0% | 0.0% | 0.0% |
| Full execution + stationary N | 0.0% | 0.0% | 0.0% | n/a | 0.0% | 0.0% | 0.0% |

## Interpretation

- All rows in the main table use the same nominal controller, residual execution stack, risk-scheduled smoothing, rate limit, and projection layer; only the actor regularizer metric is changed.
- The matched 2D runner is scalar-action (`action_dim=1`, `M=[0.716217]`). Therefore identity, diagonal, random-PSD, and stationary metrics mostly collapse to scalar weight variants in this task. The 2D table is a matched execution-layer attribution test, not a multi-input directionality test.
- The scalar-action matched table does not support a strong claim that the one-step certificate metric is the independent source of the 2D gains. Plain variance and even the no-P-ALE full-execution control are competitive, and plain variance gives lower applied variance, lower action-difference energy, and lower projection burden than the one-step M row in this run.
- The defensible claim is narrower: the 2D stress test shows that the execution layer plus source-variance shaping is sufficient to remove dense violations under the matched budget, while the certificate metric gives only a small RMSE trade-off in this scalar task. True directionality must be argued from the MIMO directional experiment below, not from the scalar RL ablation alone.

## MIMO Directional Injection Supplement

The MIMO supplement fixes the source trace and closed-loop system, then injects residual covariance along the minimum- and maximum-eigenvalue directions of `M`. This directly tests whether the certificate-induced metric distinguishes source directions.

| Scenario | cond(M) | predicted max/min | measured max/min | measured min | measured max |
|---|---:|---:|---:|---:|---:|
| nominal_hover | 123.6 | 123.6x | 128.5x | 0.000305 | 0.039221 |
| heavy_payload | 89.7 | 89.7x | 94.2x | 0.000168 | 0.015838 |
| agile_attitude | 135.8 | 135.8x | 141.0x | 0.000467 | 0.065852 |
| payload_mismatch | 94.5 | 94.5x | 99.7x | 0.000202 | 0.020164 |

Key observations:
- nominal_hover: same trace source produces 128.5x measured state-energy difference.
- heavy_payload: same trace source produces 94.2x measured state-energy difference.
- agile_attitude: same trace source produces 141.0x measured state-energy difference.
- payload_mismatch: same trace source produces 99.7x measured state-energy difference.

## Reviewer-Response Draft

We agree that the original ablation did not fully isolate the certificate metric from the execution stack. We therefore added a matched-execution attribution experiment in which the nominal controller, risk-scheduled smoothing, rate limiter, and projection layer are held fixed while the actor-side covariance metric is varied among no P-ALE, plain variance, identity, diagonal, random PSD, stationary `N`, and the proposed one-step certificate metric `M=Phi_u^T P Phi_u`. This experiment led us to narrow the claim: in the 2D runner the action channel is scalar, so certificate-weighted, identity, diagonal, random, and trace-normalized stationary metrics collapse to scalar regularizer variants. Under this scalar setting, the one-step certificate metric is not the unique source of the full-method gains; the matched execution layer and generic source-variance shaping already remove dense violations, and plain variance reduces applied variance and projection burden more strongly in this run. We therefore use the scalar 2D study as an attribution check and limitation, not as proof of multi-input metric geometry. To test true directional geometry, we add a MIMO directional-injection experiment with fixed source trace. There the measured state-energy ratios between maximum- and minimum-risk source directions track the predicted `tr(M Sigma_r)` ratios, supporting the source-amplifier metric mechanism in multi-input channels.
