# 主要问题 1：Metric Attribution 答辩材料

## 结论

评审关于“full method 的收益是否来自 `M = Phi_u^T P Phi_u`，还是来自 smoothing/rate limit/projection 叠加”的质疑是成立的。我们补做了一个更干净的 matched-execution attribution 实验：固定 nominal controller、residual execution layer、risk-scheduled smoothing、rate limit 和 safety projection，只替换 actor 侧 covariance metric。

正式设置为 `noise_std=0.30`、`5 seeds`、每方法 `3000` training timesteps。结果显示：在当前 2D Lienard runner 中，动作通道是标量，因此 identity、diagonal、random PSD、stationary `N` 和 one-step `M` 实际退化为标量权重变体。这个 2D 实验不能证明 `M` 的方向性是 full method 巨大收益的独立来源。更准确的结论是：

- matched execution layer 本身已经非常强，可以把 dense violation 压到 0；
- plain variance 在这个标量动作任务上甚至给出更低 applied variance、diff MSE 和 projection burden；
- one-step `M` 略微降低 RMSE，但带来更高 applied variance、diff MSE 和 projection burden；
- 因此 2D RL 表应作为 attribution check 和 limitation，而不是作为 “`M` 独立主导收益” 的证据；
- `M` 的方向性价值应由多输入 MIMO directional injection 实验支持。

## Matched-Execution 5-Seed 表

| 方法 | RMSE | Applied Var | Diff MSE | Dense viol. | B-L max | Projection trigger | Projection mag. |
|---|---:|---:|---:|---:|---:|---:|---:|
| Full execution, no P-ALE | 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 | 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 | 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 | 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 | 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 | 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 | 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 |

## MIMO Directional Injection 补充证据

为了真正检验 `M` 的方向性，我们补充了多输入 directional injection 实验。该实验固定 source trace 和 closed-loop system，只把 residual covariance 分别放在 `M` 的最小/最大特征方向上。实测 state-energy ratio 与 `tr(M Sigma_r)` 预测基本一致：

| Scenario | cond(M) | predicted max/min | measured max/min |
|---|---:|---:|---:|
| nominal_hover | 123.6 | 123.6x | 128.5x |
| heavy_payload | 89.7 | 89.7x | 94.2x |
| agile_attitude | 135.8 | 135.8x | 141.0x |
| payload_mismatch | 94.5 | 94.5x | 99.7x |

这组结果支持 `M` 作为多输入 source-direction risk metric 的机制解释，但它不是 2D 标量动作 RL 表能够单独证明的结论。

## 建议写入 Rebuttal 的英文版本

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.

## 建议修改论文 claim

不要写：

> The 2D matched ablation proves that `M = Phi_u^T P Phi_u` is the dominant source of the full-method gain.

建议改为：

> The matched 2D ablation shows that the execution layer and source-variance shaping jointly remove dense violations under the high-noise stress test. Because the 2D benchmark has a scalar action channel, the certificate metric reduces to a scalar regularizer and cannot by itself demonstrate multi-input directionality. We therefore report it as an attribution check and use the MIMO directional-injection study to isolate the certificate-induced source-direction effect.

## Artifact

- Formal matched-execution report: `outputs/rl/metric_attribution_5seed_safe10_worker_cached/metric_attribution_report.md`
- Formal matched-execution table: `outputs/rl/metric_attribution_5seed_safe10_worker_cached/metric_attribution_table.csv`
- MIMO directional injection summary: `outputs/v3_tac_paper1_20260504/metric_attribution_directional_mimo_20260526_170452/summary.json`
