# OID Analysis: Final Conclusions
## (Phase 2 completed 2026-06-28)

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## Six Questions Answered

### Q1: Steady cloak -- which correction structures determine suppression/restoration?

**Answer**: The force-OID and suppression-OID mode 1 overlap at 0.763, indicating that force-generating and fluctuation-suppressing correction structures are **highly related but not identical**. The steady cloak achieves 99.43% full-field RMS reduction and 38.5% recirculation area collapse. The dominant correction structures are concentrated in the near-body zone and strongly project onto total force (cum_corr=0.88 in one mode). The recirculation length barely changes (3.2% collapse) while area drops 38.5%, suggesting the control narrows the wake bubble without fully eliminating it.

- **POD rank**: r=10 (99.97% energy in 5 modes). **Rank sensitivity**: 1.000.
- **Metric**: RMS reduction, Lr/Ar collapse (NOT time-series R2).

### Q2: Karman cloak -- which correction structures determine incoming-street preservation?

**Answer**: Force-OID (S=[0.966, 0.724]) distributes force correlation across two modes (cum_corr 0.572, 1.0). The signature-OID for delayed sensor error performs similarly to current error (cum_corr 0.967 vs 0.965), suggesting the correction structures respond synchronously with the incoming street rather than predicting it ahead of time. The OID coordinate for force prediction achieves R2=0.750 (vs POD's 0.418), a clear OID advantage. However, OID-only captures only 22.5% of action variance (vs obs->act at 95.6%), meaning OID coordinates identify force-relevant structures but do NOT capture the full control law.

- **Delay**: tau_c=25 steps. **POD rank**: r=10. **Rank sensitivity**: 1.000.
- **Status**: Preliminary. Future signature prediction failed (R2~0) due to near-zero variance in the delayed error observable at the chosen delay.

### Q3: Illusion -- which correction structures determine target-shedding retuning?

**Answer**: Signature-OID significantly outperforms POD for signature prediction across all diameters. The strongest effect is at 0.75L (Sig-OID R2=0.661 vs POD -0.034). Performance degrades at larger diameters (1.5L: Sig-OID R2=0.315 vs POD 0.060). Force-OID also strongly outperforms POD for force prediction. The signature-PCD (whitened) does NOT outperform simple signature-OID, suggesting the multi-time-window whitening may be overkill for the current data quality.

- **POD rank**: r=10. **Rank sensitivity**: 1.000.
- **Delay**: tau_c varies by SI (0.75L:~50, 1.0L:~33, 1.5L:~25 steps).

### Q4: Force-relevant vs signature-relevant structures -- same or different?

**Answer**: **Systematically different, and their separation increases with task complexity.**

| Scene | Cosine similarity | Temperature |
|-------|-----------------|-------------|
| steady_cloak | +0.763 | Force and suppression are highly related |
| karman_re100 | -0.034 | Nearly orthogonal |
| illusion_0.75L | -0.082 | Near-orthogonal |
| illusion_1.0L | -0.495 | Moderate divergence |
| illusion_1.5L | -0.932 | Strong divergence |

This monotonic pattern from positive (steady) through zero (Karman) to increasingly negative (illusion with growing diameter) is the most striking result of this analysis. It suggests:

- **Steady cloak**: suppressing force IS the suppression mechanism (same structures).
- **Karman cloak**: preserving a vortex street requires correction structures that are orthogonal to force generation.
- **Illusion**: retuning to a different shedding frequency requires correction structures that increasingly oppose the natural force-generating modes as the target mismatch grows.

This directly supports the task-book hypothesis that force-OID and signature-OID must be reported separately, and their divergence is a mechanism result, not a failure.

### Q5: Scene commonality -- shared structures?

**Answer**: The cross-scene mode overlap analysis (from the master table) shows that force-OID and signature-OID modes behave systematically across scenes, but with different quantitative relationships per scene. The common pattern is that **correction-field POD captures 98-99.9% energy in 5 modes for ALL scenes** -- meaning the correction structures are consistently low-dimensional. However, which correction structures are task-relevant shifts systematically from suppression (steady) to preservation (Karman) to retuning (illusion).

### Q6: Is OID better than POD?

**Answer**: **Yes, for all scenes where comparison is meaningful.**

| Scene | Task | OID R2 (m=2) | POD R2 (m=2) | OID wins? |
|-------|------|-------------|-------------|-----------|
| karman | Force prediction | 0.750 | 0.418 | **YES** |
| illusion_0.75L | Force prediction | 0.435 | -2.426 | **YES** |
| illusion_0.75L | Sig prediction | 0.661 | -0.034 | **YES** |
| illusion_1.0L | Force prediction | 0.671 | -0.237 | **YES** |
| illusion_1.0L | Sig prediction | 0.586 | -0.160 | **YES** |
| illusion_1.5L | Force prediction | 0.640 | 0.264 | **YES** |
| illusion_1.5L | Sig prediction | 0.315 | 0.060 | **YES** |

OID consistently provides positive R2 where POD gives negative or near-zero values. The success criterion from the task book ("OID/PCD with m<=3 beats POD with m<=3") is satisfied for all scenes.

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## Additional Conclusions

### Control law completeness
The white-box analysis shows that OID coordinates alone capture only 22.5% of the Karman control law (vs 95.6% for raw sensor observations). This is expected and appropriate: OID identifies correction structures most relevant to force/signature, but the PPO policy uses additional information (e.g., FIFO state, target history) beyond what is captured by the correction-field POD subspace.

### Illusion q_blk importance
The requirement to separately collect illusion-position q_blk was validated: the geometry differences (front_x 19 vs 30, sensor_x 30 vs 40) would have contaminated Delta_q_ctl with position mismatches. Using the cloak-position q_blk would have produced a Delta_q_ctl dominated not by control effects but by geometric displacement -- invalidating all subsequent OID analysis.

### Future work recommendations
1. **Investigate the Karman future-sig R2~0 result**: The delayed sensor error observable may need a different delay or a different formulation (e.g., phase-error instead of direct error).
2. **Connect OID coordinates to SINDy white-box framework**: The OID coordinates could be input features for SINDy, potentially giving a lower-dimensional control law.
3. **Phase-conditioned analysis for periodic cases**: Instead of delay-embedded OID, try phase-conditioned OID where each phase of the shedding cycle is analyzed separately.
4. **Check rank sensitivity for the force-vs-sig overlap divergence**: The systematic trend from +0.763 to -0.932 is compelling but needs verification across multiple POD ranks.
