# Project Evidence Dossier

This dossier enriches the PaperSpine folder with evidence mined from the current project tree. It is intentionally broad because the current stage does not restrict length. The goal is to preserve enough evidence to write a strong 6-page paper centered on nonnormality in RL closed loops.

## 1. Evidence Thesis

The project already contains a strong evidence chain for the new short paper:

1. **Mechanism isolation:** CI-1 shows that changing nonnormal closed-loop geometry while fixing eigenvalues/source/input can increase state covariance dramatically.
2. **Source intervention:** CI-2 shows that reducing or reshaping the source under a fixed amplifier reduces downstream covariance, with a colored-input caveat.
3. **Method response:** P-ALE, smoothing, and projection only work cleanly when combined; scalar `Var(u)` and temporal action-diff penalties are insufficient.
4. **Execution repair:** Projection burden decreases when source shaping is added, which supports the operational claim that source control can reduce safety-layer repair.
5. **Residual feedback caveat:** Actor-Jacobian diagnostics show why residual feedback must sometimes be folded into the amplifier.
6. **External simulation relevance:** Full3D quadrotor simulations support the pattern in a 12D/4-motor setting, while ControlGym supplies both supportive and failure-mode evidence.

## 2. Authoritative Manuscript Evidence

### 2.0 Project-wide evidence maps

Copied internal evidence maps:

- `evidence_materials/project_indices/PROJECT_EXPERIMENT_EVIDENCE_MAP.md`
- `evidence_materials/project_indices/PROJECT_COMPLETE_SUMMARY_EVIDENCE_INDEX_20260521.md`

Why they matter:

- They connect figures to raw output folders, scripts, and caveats across the larger project.
- They are useful for provenance checking before submission because they distinguish curated result folders from duplicate paper-package copies.
- They confirm that Full3D, ControlGym, conditional-variance proxy, and planar UAV evidence should be treated as auxiliary or boundary evidence rather than the central mechanism proof.

Use in new paper:

- Use `experiment_claim_matrix.md` as the immediate writing scaffold.
- Use these copied maps when a final figure or table needs raw-data provenance.

### 2.1 Long source-amplifier manuscript

Canonical file:

`paper_spine_tac_source_amplifier_20260521_best/final_paper/main.tex`

Why it matters:

- Contains the most complete source-amplifier theory, method definition, experimental tables, and limitation language.
- Explicitly states that the evidence is local-surrogate-level and simulation-level.
- Contains the exact numbers most useful for the new 6-page paper.

High-value sections:

- `Introduction`: states the source-amplifier interpretation.
- `Nonnormal Source-Amplifier Theory`: contains covariance propagation, nonnormal amplification, colored-input augmentation, and residual-feedback caveats.
- `Simulation and Controlled-Intervention Studies`: contains CI-1, CI-2, 2D ablation, projection burden, residual-feedback proxy, Full3D, and ControlGym.
- `Discussion` and `Limitations`: provides conservative claim language.

Use in new paper:

- Treat it as internal source material, not final structure.
- Compress from a CG-RVC method paper into a nonnormal RL diagnostic paper.
- Retain exact numeric evidence but reduce algorithmic exposition.

### 2.2 Existing Chinese CG-RVC draft

Canonical file:

`variance_regularized_safe_rl/PAPER_FULL_DRAFT_CN.md`

Why it matters:

- Gives a Chinese explanation of the Koopman-Lyapunov proxy certificate and variance-aware safe residual RL.
- Clearly states that posterior Bellman-Lyapunov residual is a diagnostic, not theorem-level validation for the nonlinear plant.
- Contains a concise abstract with high-noise 2D Lienard evidence.

Useful excerpt-level claims:

- Continuous-control RL in safety-critical settings suffers from action jitter, local high-frequency interventions, and safety-margin degradation.
- Existing smoothing/projection methods often do not explicitly describe how local residual action fluctuations propagate along closed-loop dynamics.
- A lifted Koopman surrogate and projected closed-loop operator support a local proxy certificate.

Use in new paper:

- Mine for Chinese draft language and conceptual explanations.
- Keep final English claims narrower and more mechanism-centered.

### 2.3 Koopman-Lyapunov stability materials

Canonical files:

- `koopman_lyapunov_repro_variance_constraint_delivery/koopman_lyapunov_repro_variance_constraint_delivery/paper_draft.tex`
- `koopman_lyapunov_repro_variance_constraint_delivery/koopman_lyapunov_repro_variance_constraint_delivery/paper_draftV2.tex`

Why they matter:

- Provide background on stability-constrained Koopman learning and ALE well-posedness.
- Support the idea that spectral boundary constraints and Lyapunov equations are useful but must be interpreted carefully.

Use in new paper:

- Use for notation and caution around spectral projection and ALE.
- Do not import the full stability-constrained Koopman narrative; it is background, not the new paper's main story.

## 3. Controlled Interventions

### 3.1 CI-1: Amplifier isolation

Evidence:

- Fixed eigenvalues: `diag(0.93, 0.68)`.
- Fixed spectral radius, input channel, and source covariance.
- Changed only eigenvector geometry.
- When `kappa(V)` increases from about 1.0 to 20.05:
  - state covariance trace increases from about 0.37 to 15.66;
  - peak transient gain increases from about 1.0 to 5.55;
  - correlation between `kappa(V)` and state covariance trace is 0.9704;
  - correlation between `kappa(V)` and peak transient gain is 0.9965.

Primary figure:

`evidence_materials/selected_figures/fig1_ci1_amplifier_isolation.png`

Original figure source:

`paper_spine_tac_source_amplifier_20260521_best/submission_v2_From_Action_Smoothing_to_Source_Amplifier_Control_2026-05-22/latex_source/figures_v4_1/fig1_ci1_amplifier_isolation.png`

Writing use:

- Page 2 or Page 5.
- This is the cleanest evidence for the statement: spectral-radius stability alone is insufficient because nonnormal geometry can be an independent covariance amplifier.

Caveat:

- Do not claim monotonicity for all nonnormal systems. The correct claim is that nonnormal geometry can independently amplify variance under fixed source conditions.

### 3.2 CI-2: Source-side intervention

Evidence:

- Fixed the strongly nonnormal matrix from CI-1.
- Compared white input with first-order causal filtered input.
- Action variance decreases from 0.03992 to 0.00317.
- State covariance trace decreases from 15.849 to 12.393.
- Peak transient gain remains about 5.5486.

Primary figure:

`evidence_materials/selected_figures/fig2_ci2_source_intervention.png`

Original figure source:

`paper_spine_tac_source_amplifier_20260521_best/submission_v2_From_Action_Smoothing_to_Source_Amplifier_Control_2026-05-22/latex_source/figures_v4_1/fig2_ci2_source_intervention.png`

Writing use:

- Page 5 controlled-intervention panel.
- Supports the claim that lowering source-side energy can lower downstream covariance under a fixed amplifier.

Caveat:

- Filtered input is colored, not white. It must be interpreted through an augmented colored-input model, not as a direct white-noise monotonicity theorem.

### 3.3 CI-3: Direction-aware source test

Evidence status:

- The project plan identifies CI-3 as needed: fixed `tr(Sigma_r)`, rotate the principal direction of `Sigma_r`, and compare alignment `tr(M Sigma_r)/tr(Sigma_r)`.
- The scan found strong conceptual support but did not verify a completed CI-3 result table.

Writing use:

- Keep CI-3 as recommended enhancement if final paper needs a direct proof that scalar variance is insufficient.
- If not completed, describe it as a planned or supplementary diagnostic rather than existing evidence.

## 4. 2D Constrained RL Evidence

Primary table source:

`paper_spine_tac_source_amplifier_20260521_best/final_paper/main.tex`, table `tab:v3_pale`.

### 4.1 Matched method-component comparison

High-noise setting:

| Method | RMSE | Applied Var | Diff MSE | Dense Viol. | B-L Max |
|---|---:|---:|---:|---:|---:|
| Direct TD3 | 0.537 +/- 0.056 | 0.1138 +/- 0.0066 | 0.2731 +/- 0.0177 | 0.9389 +/- 0.1366 | 92.391 +/- 34.518 |
| TD3 + plain `Var(u)` | 0.612 +/- 0.063 | 0.1026 +/- 0.0022 | 0.2365 +/- 0.0130 | 0.9278 +/- 0.0800 | 161.860 +/- 47.677 |
| TD3 + action-diff penalty | 0.565 +/- 0.068 | 0.1118 +/- 0.0059 | 0.2657 +/- 0.0172 | 0.9389 +/- 0.1366 | 109.096 +/- 44.352 |
| TD3 + beta-ALE | 0.661 +/- 0.085 | 0.0984 +/- 0.0023 | 0.2212 +/- 0.0081 | 0.9167 +/- 0.1145 | 208.176 +/- 79.238 |
| TD3 + P-ALE | 0.592 +/- 0.056 | 0.1049 +/- 0.0032 | 0.2460 +/- 0.0156 | 0.9389 +/- 0.0569 | 143.114 +/- 45.627 |
| TD3 + P-ALE + smoothing | 0.527 +/- 0.048 | 0.0456 +/- 0.0008 | 0.0722 +/- 0.0042 | 0.9278 +/- 0.1204 | 57.246 +/- 8.177 |
| TD3 + P-ALE + projection | 0.554 +/- 0.033 | 0.0940 +/- 0.0026 | 0.2302 +/- 0.0097 | 0.1500 +/- 0.0724 | 128.767 +/- 27.250 |
| TD3 + projection only | 0.505 +/- 0.108 | 0.1033 +/- 0.0141 | 0.2761 +/- 0.0366 | 0.1222 +/- 0.0639 | 82.320 +/- 52.322 |
| CG-RVC / Full method | 0.284 +/- 0.004 | 0.0060 +/- 0.0006 | 0.0063 +/- 0.0018 | 0.0000 +/- 0.0000 | 5.239 +/- 2.698 |

Writing use:

- This is the strongest table for Page 3 and Page 5.
- It shows plain `Var(u)` and action-diff smoothing do not control dense violations or posterior residuals.
- It shows P-ALE alone is insufficient and projection alone is insufficient.
- It supports the claim that P-ALE, smoothing, and projection must be combined.

Primary figure candidates:

- `evidence_materials/selected_figures/caps_style_method_comparison_v4_1.png`
- `evidence_materials/selected_figures/suite_high_noise_comparison.png`

### 4.2 Projection burden

Primary table source:

`paper_spine_tac_source_amplifier_20260521_best/final_paper/main.tex`, table `tab:projection_burden`.

| Method | Trigger rate | Normalized correction |
|---|---:|---:|
| Projection only | 0.2683 +/- 0.1707 | 0.09362 +/- 0.06735 |
| P-ALE + projection | 0.1728 +/- 0.0836 | 0.05426 +/- 0.03293 |
| CG-RVC | 0.1331 +/- 0.0722 | 0.01093 +/- 0.00541 |

Writing use:

- Shows source shaping reduces how strongly a fixed safety layer must repair the action stream.
- This supports the operational "consequence" part of the source-amplifier chain.

Caveat:

- Zero projection trigger for methods without projection is not safety evidence.
- Projection burden is not a global safety proof.

### 4.3 Residual-feedback diagnostic

Primary table source:

`paper_spine_tac_source_amplifier_20260521_best/final_paper/main.tex`, table `tab:residual_feedback_proxy`.

| Method | Local Var | Cross Var | Total Var | `||J_r||_F` |
|---|---:|---:|---:|---:|
| Direct TD3 | 0.1699 +/- 0.1171 | 1.4439 +/- 0.9761 | 1.6078 +/- 1.0848 | 16.23 +/- 6.00 |
| TD3 + projection only | 0.2110 +/- 0.1226 | 1.5134 +/- 0.8561 | 1.7159 +/- 0.9688 | 17.76 +/- 5.93 |
| CG-RVC / Full method | 4.51e-5 +/- 2.29e-5 | 3.52e-4 +/- 1.82e-4 | 3.97e-4 +/- 2.06e-4 | 0.274 +/- 0.081 |

Writing use:

- Supports the residual-feedback caveat.
- Shows full method is consistent with a source-side interpretation because it has much smaller local sensitivity.

Caveat:

- This is a finite-difference diagnostic, not proof that residual feedback leaves the amplifier unchanged.

## 5. Full3D Quadrotor Evidence

Primary table source:

`paper_spine_tac_source_amplifier_20260521_best/final_paper/main.tex`, table `tab:full3d_high_noise`.

Simulation design:

- 12-dimensional state.
- 4 motor channels.
- 5 random seeds.
- 48 rollouts per scenario/seed.
- Horizon 80.
- Highest noise key: 0.20.

High-noise results:

| Scenario | White state cov. | Filtered state cov. | White motor var. | Filtered motor var. |
|---|---:|---:|---:|---:|
| Nominal Hover | 0.551549 +/- 0.003477 | 0.089192 +/- 0.005230 | 1.901316 +/- 0.019104 | 0.104565 +/- 0.006143 |
| Heavy Payload | 0.266570 +/- 0.003897 | 0.083648 +/- 0.006377 | 1.763106 +/- 0.013489 | 0.146811 +/- 0.010697 |
| Agile Attitude | 0.950270 +/- 0.009514 | 0.174127 +/- 0.016088 | 1.898488 +/- 0.016801 | 0.121955 +/- 0.009240 |
| Payload Mismatch | 0.336788 +/- 0.006714 | 0.115505 +/- 0.008461 | 1.579815 +/- 0.013373 | 0.153382 +/- 0.010421 |

Primary figure:

`evidence_materials/selected_figures/full3d_5seed_high_noise_summary.png`

Writing use:

- Supports simulation-level relevance beyond the 2D constrained task.
- Shows source-side filtering can reduce both state covariance and motor variance in all four Full3D high-noise scenarios.

Caveat:

- This is numerical simulation, not hardware evidence.
- It should not be used to claim broad quadrotor controller superiority.

## 6. Planar UAV / Boundary Case Evidence

Evidence from long manuscript:

- The planar quadrotor study has four scenarios: Nominal Hover, Heavy Payload, Agile Pitch, and Payload Mismatch.
- Filtering reduces action variance in all four scenarios.
- Payload Mismatch is a boundary case where state covariance rises by about 16%.

Writing use:

- Useful for limitations: scalar source variance can drop while state covariance rises if remaining energy aligns with high-gain directions.
- Motivates direction-aware penalties based on `Phi_u^T P Phi_u`.

Caveat:

- Do not present the planar UAV bundle as the main benchmark.

## 7. ControlGym Stress-Test Evidence

Primary table source:

`paper_spine_tac_source_amplifier_20260521_best/final_paper/main.tex`, table `tab:controlgym_stress`.

High-noise results:

| System | Method | State trace | Applied Var | B-L Max | Strict infeasible rate |
|---|---|---:|---:|---:|---:|
| AC3 | Unregularized | 1.314 +/- 0.325 | 0.00447 +/- 0.00095 | 9.243 +/- 0.848 | 0.0000 +/- 0.0000 |
| AC3 | CG-RVC / Full method | 0.521 +/- 0.120 | 0.00180 +/- 0.00039 | 6.428 +/- 0.742 | 0.0008 +/- 0.0011 |
| CM2 | Unregularized | 826.350 +/- 84.420 | 0.01200 +/- 0.00345 | 1487.758 +/- 190.915 | 0.0000 +/- 0.0000 |
| CM2 | CG-RVC / Full method | 811.577 +/- 84.270 | 0.85042 +/- 0.01757 | 1485.908 +/- 190.629 | 1.0000 +/- 0.0000 |

Primary figure:

`evidence_materials/selected_figures/controlgym_suite_high_noise_comparison.png`

Writing use:

- AC3 supports the diagnostic pattern in a larger linear system.
- CM2 should be framed as a failure/boundary case: projection-heavy execution can dominate source shaping.

Caveat:

- This is a stress test, not the main benchmark.

## 8. Code Evidence

### 8.1 Planar quadrotor generalization suite

Canonical file:

`variance_regularized_safe_rl/run_planar_quadrotor_generalization_suite.py`

Duplicate copies exist in paper packages. The file defines:

- planar quadrotor rollout suite;
- scenario-level parameters;
- projection strategy `eigenvalue_clipping`;
- nonnormal/transient suite logic;
- output tag `planar_quadrotor_generalization_suite`.

Use:

- Supports the claim that planar UAV evidence is a dedicated simulation suite, not just a plot.

### 8.2 Projection beta RL summaries

Canonical example:

`variance_regularized_safe_rl/outputs/rl/projection_beta_optimality_20260426_235755/summary.json`

Duplicate copies exist under paper packages and selected outputs.

Evidence:

- Contains `td3_projection_only`, `ale_projection_strategy`, projection costs, projection trigger rates, projection magnitudes, and method summaries.

Use:

- Backstops projection-burden and beta/gamma sensitivity discussion.

### 8.3 Paper package result selections

Canonical folder:

`paper_spine_tac_source_amplifier_20260521_best/submission_v2_From_Action_Smoothing_to_Source_Amplifier_Control_2026-05-22/results_selected/`

Includes selected outputs for:

- conditional variance proxy;
- ControlGym 5-seed results;
- Full3D 5-seed results.

Use:

- Prefer this folder when selecting figures for the new paper because it is a curated package, not an arbitrary intermediate output.

## 9. Visual Evidence Routing

Recommended figure set for a richer draft:

1. `source_amplifier_layers.png`: use as Figure 1 or redraw in a cleaner 6-page style.
2. `fig1_ci1_amplifier_isolation.png`: use in the mechanism figure.
3. `fig2_ci2_source_intervention.png`: pair with CI-1.
4. `suite_high_noise_comparison.png` or `caps_style_method_comparison_v4_1.png`: use for 2D RL ablation.
5. `full3d_5seed_high_noise_summary.png`: use only if space allows; otherwise cite as supplementary.
6. `controlgym_suite_high_noise_comparison.png`: use in limitations or supplementary, not main success story.

## 10. Gaps and Next Evidence Tasks

The project contains enough evidence for a richer 6-page paper, but the following gaps remain if the final target becomes a submission-ready manuscript:

- Verify formal literature citations from original DOI-level papers, not from generated reports.
- Confirm whether CI-3 direction-aware source scan has been executed; if not, run it or frame it as future/supplementary.
- Link every copied figure to the exact result data file that generated it.
- Decide whether to include Full3D in the main paper or move it to supplementary.
- Add one compact table that maps each experiment to manipulated variable, fixed variable, measured consequence, and claim boundary.
