# Section 6 figure specifications — Kármán mean fields and CCD

**Status:** textual production specification. Inventory and existing drafts do not imply manuscript-panel readiness.

## Shared scientific identity

- **Physical case:** periodic Kármán cloak, parabolic-inflow/no-slip-wall channel configuration; inherited code label 100 retained in provenance only; use a neutral reference source-case label pending velocity-reference authority.
- **Mean-field roles:** target, physical zero, constant reference at the exact retained DRL physical action mean, and DRL.
- **Primary mean estimand:** coordinate-weighted vector RMS target error of retained role means on the exact four-role common solver-fluid mask, inclusive `34<=x/D<=54`, `|y/D|<=5`, 80,200 fluid points. Persisted velocity is already nondimensional.
- **Residual identity:** separately centred DRL cycle/bin field minus the separate constant-control 19-cycle ensemble phase template; 19 DRL cycles × 10 bins, `N=190`, no cross-role pairing.
- **Internal provenance only:** current corrected result and publication paths may retain internal chain identifiers; captions use the physical configuration above.

## S6-1 — Mean streamwise correction and vector-error accounting

### Argumentative job

Show the physically legible streamwise pattern first—passive deficit relative to target and controlled addition relative to physical zero—then bind it to the exact two-component scalar error-reduction estimand. Establish mean dominance without calling it a physical/causal contribution.

### Exact layout

- **Top row, four equal field panels (a–d):** `mean_target[u_x]`, `mean_zero[u_x]`, `mean_constant_mean[u_x]`, `mean_drl[u_x]`.
- **Bottom-left, two wider signed-difference panels (e–f):** `mean_zero[u_x] - mean_target[u_x]`; `mean_drl[u_x] - mean_zero[u_x]`.
- **Bottom-right, one compact ledger (g):** target-error markers/bars at zero `0.4694352273`, constant `0.1110140739`, DRL `0.0857787860`; ordered brackets `zero→constant=0.3584211534`, `constant→DRL=0.0252352879`, total `0.3836564413`; labels `93.42%` and `6.58%` explicitly as shares of the measured scalar error reduction.
- Crop every field to the declared ROI; show body/ROI context only in a small locator inset if needed, because bodies lie upstream of this ROI.

### Variables and subtraction order

- Panels (a–d): retained mean streamwise velocity, not speed or vector magnitude.
- Panel (e): **physical zero minus target**. Negative values denote lower streamwise velocity than target under the repository coordinate/sign convention.
- Panel (f): **DRL minus physical zero**. Positive values denote streamwise velocity added relative to passive zero.
- The two fields are compared visually as signed complements; do not label one as an exact negative of the other.
- Ledger uses both `u_x` and `u_y`; it is not computed from panels (e–f) alone.

### Roles, case, metric, ROI, mask and window

- Roles: target / physical zero / constant reference / DRL from one corrected four-role campaign.
- Case/configuration: periodic Kármán cloak, parabolic inflow and no-slip channel walls, inherited code label 100.
- Metric: `E_r=[sum_K w_k ||mean(q_r)-q_target||_2^2/sum_K w_k]^(1/2)`; coordinate weights duplicated across both velocity components.
- ROI/mask: inclusive `34<=x/D<=54`, `|y/D|<=5`; exact intersection of all four solver-derived fluid masks; 80,200 points.
- Window: retained means bound by the corrected role campaigns; do not substitute Section-4 complete-cycle standardized means. Exact retained role/window provenance must be copied into the final production manifest: role identities, parent source leaves and hashes, retained row/slice bounds, cycle/bin windows, mask/weight contract, array orientation/transposition, render command/revision, crop/scale/clipping, and output hashes. Until that record exists, production remains dependency-gated.

### Data and artifact sources

- Exact arrays: `/home/frank14f/optane/DynamisLab/ccd/karman-dynamic/canonical/results/roi-mean/arrays.npz`.
- Result summary/manifest: same directory `summary.json`, `manifest.json`; source manifest hash `f7f6141042678e52314a4d8f76d589108da8e844c8ce504f8f523a6f88a17e73`.
- Repository authority: `src/CCD_analysis/{RESULTS_INDEX.md,FINAL_RESULTS.md,MATHEMATICAL_DERIVATION.md,FIGURES_AND_DATA.md}` and `src/CCD_analysis/karman_dynamic/{CORRECTED_MATH_CONTRACT.md,dynamic_increment.py,publication.py}`.
- Existing drafts: external and repository reader copy `publication-review-fix-per-cycle-20260808T1736+0800/{01_roi_mean_performance,02_roi_mean_target_error_fields,03_roi_mean_corrections}.{png,pdf}`.
- Existing render source: `src/CCD_analysis/karman_dynamic/publication.py`.

### Readiness

- **Arrays/values:** ready and authority-bound.
- **Existing useful panels:** draft-ready only. `02` already contains `zero-target` for `u_x/u_y`; `01` supplies the scalar ledger; `03` supplies `constant-zero` and `DRL-constant` but not the requested `DRL-zero`.
- **Requested composite:** dependency-gated new render. `DRL-zero` must be directly calculated as `mean_drl - mean_zero` from the stored means and recorded in a future output manifest with input paths/hashes, subtraction order, units, orientation and output hash; no substitution by `DRL-constant` is allowed. No CFD or new scientific estimator is required.
- **Independent sign check:** weighted ROI mean-$u_x$ cosine of requested fields `-0.9961` and relative residual `0.1662` were computed read-only for design verification. Do not print these values unless the coordinator extends current claim authority.

### Color and sign conventions

- Four absolute mean-$u_x$ panels: one shared sequential or diverging scale chosen from all four arrays; label units/nondimensionalization exactly from authority. If a sequential scale obscures deviations, use a physically meaningful common scale but never independently autoscale roles.
- Difference panels: one shared symmetric diverging scale centred at zero, preferably `RdBu_r`; positive must mean the named first role has larger `u_x` under the printed subtraction.
- Mask non-fluid cells; transpose stored `(x,y)` arrays for plotting. Never infer sign from color without a labelled color bar.
- Optional zero contours on both difference panels should use identical styling.

### Caption claim

“Relative to the target, the passive pinball produces a downstream mean-streamwise deficit over much of the declared wake ROI, while DRL adds a largely opposite-signed streamwise correction relative to physical zero. This pattern is compatible with adding streamwise velocity where the passive configuration produces a deficit. The scalar ledger evaluates both velocity components on the exact common mask: the constant reference accounts for 93.42% of the measured zero-to-DRL scalar mean-field error reduction and the constant-to-DRL increment accounts for the remaining 6.58%.”

### Explicit non-claims

No momentum restoration; no exact cancellation; no closed momentum/energy budget; no causal actuator effect; no mechanism; no physical contribution percentage; no energy/explained variance; no SR-term share; no Section-4 replication/agreement; no cross-case universality.

### Supplement fallback

If the seven-panel composition is illegible, retain panels (e–g) in main text and move the four absolute means to Supplement S6-S0. If raw composite production fails, use the existing scalar ledger plus `zero-target` draft and state that `DRL-zero` is omitted; do not substitute `DRL-constant` under the wrong label.

## S6-2 — Non-paired residual and action-coordinate phase organization

### Argumentative job

Explain what CCD adds after the mean ledger: a low-rank representation of the separately centred residual associated with action coordinates and organized over phase. Keep the result descriptive and avoid a mode gallery.

### Exact layout

- **Panel (a), left 25%:** residual construction schematic with the equation `U_(c,b)=(q^D_(c,b)-bar q_D)-(hat q^C_b-bar q_C)`, `19×10`, and a visible “cycles not paired” annotation.
- **Panel (b), centre 25%:** compact 3×3 action-coordinate matrix or bars: rows `front`, `rear-symmetric`, `rear-antisymmetric`; columns modes 1–3. State global mode-sign arbitrariness.
- **Panels (c–d), right 50%:** rank-3 mean residual reconstructions at two registered representative phase bins nearest `0` and `pi` (use four bins nearest `0,pi/2,pi,3pi/2` only if final-size legibility passes). Show signed `u_x`; use the exact registered bins from `representative_phase_bins`.
- Do not include all individual mode fields in main text. A small rank/stability note may state `rank 3`, not singular-strength shares.

### Variables, roles and estimand

- Field residual uses both velocity components internally; displayed phase reconstruction is `u_x` and must be labelled as a component view.
- Constant phase template is the ensemble mean over its own 19 cycles. DRL and constant cycles are not paired.
- Observable is same-boundary DRL effective action in native front/upper/lower order, exact retained physical action mean removed, then empirical centering.
- Operator: `A=P_c(W^(1/2)U_c)^T/(N sqrt(3))`, `N=190`, `Q=1`; no whitening, standardization, lag, wrap or POD preprojection.

### Domain, source and readiness

- Preferred display domain: pinball-inclusive `29<=x/D<=54`, `|y/D|<=5`, exact common-fluid mask, 99,264 points, `M=198528`.
- Preferred result: `/home/frank14f/optane/DynamisLab/ccd/karman-dynamic/canonical/results/pinball-domain-20260809T1925+0800/`, manifest hash `7e107a895364ed3e34abf2be513a3ce9e12736766ec6383c5b831891802d5828`.
- Repository plot arrays: `src/CCD_analysis/data/karman-dynamic/canonical/publication-pinball-domain-20260809T1925+0800/plot_data.npz` and `plot_data.json`.
- Existing drafts: `01_domain_estimand`, `02_ccd_mode_identity`, `03_phase_reconstruction`, `04_raw_pod_vs_ccd` PNG/PDF groups.
- Render sources: `src/CCD_analysis/karman_dynamic/{domain_comparison.py,mode_diagnostics.py,pod_baseline.py}`; standalone single-mode redraw helper `src/CCD_analysis/plot_ccd.py` is insufficient for the full composite.
- **Readiness:** source arrays and existing views ready; manuscript recomposition dependency-gated. Phase panels may be extracted/redrawn without new CFD. The residual schematic is specification-ready.

### Color and sign conventions

- Phase residual `u_x`: one shared robust symmetric diverging limit across selected phases, centred at zero.
- Mode/action columns: if individual mode signs are shown, use the same sign convention as the bound publication and state that global sign is arbitrary. Physical phase sign is carried by coefficient × mode, not mode color alone.
- Stored plot arrays use `(x,y)` and require transpose for display. Overlay exact mask; body circles are geometry guides only.

### Caption claim

“After separate centring, the DRL fields are compared with an independently formed constant-control phase template; cycles are not paired. CCD associates the resulting rank-3 residual representation with centred same-boundary action coordinates and displays its phase organization. These are action-correlated correction-field modes, not causal responses or unique flow structures.”

### Explicit non-claims

No paired counterfactual; no response time/lag; no causal direction; no actuator authority; no mechanism; no wall-vorticity interpretation; no explained variance from CCD singular strengths; no individual-mode stability claim; no SR derivation; no independent triangulation.

### Supplement fallback

If action matrix plus phase fields remains crowded, main text keeps residual schematic + two phase reconstructions and moves action-coordinate matrix with all individual modes to S6-S1. Text then states the association without showing a mode gallery.

## Shared normalized readiness

Plan `specified`; repository authorities and named repository copies `read/path-confirmed`, external absolute payloads `mapped` rather than newly audited, raw-array recomputation `not-covered`; S6-1/S6-2 estimands `authority-bound` but exact retained role/window and final render-manifest provenance `dependency-gated`; render `redraw-required`; disposition `retain-with-gates`. A partial plot manifest authenticates only available diagnostics, not completeness/readiness. Caption ceiling remains descriptive/noncausal and different-estimand; fallback uses existing scalar ledger plus `zero-target`, never a substituted difference.

## S6-S1 — Raw POD boundary and CCD diagnostics (supplement)

### Argumentative job and layout

Define familiar POD as a variance/energy-optimal field organization under the declared weighted metric without action information; compare its rank-3 subspace with CCD on the same centred residual. Use a compact principal-cosine strip/table plus optional aligned POD/CCD mode pairs and stability inset. Detailed mode parity and body-local rotation proxy, if retained, belong here.

### Exact source and metric

- Same pinball-inclusive residual/domain/result as S6-2.
- Raw full-fit weighted POD Gram matrix `G=U_c^T W U_c`; no action observable selects POD.
- Principal cosines are singular values of `Phi^T W Psi`: `[0.9999616609, 0.9998971005, 0.9976439804]`.
- Minimum declared bin/cycle rank-3 stability cosine `0.9974651867`.
- Source: preferred `04_raw_pod_vs_ccd` draft and compact result arrays; math in `MATHEMATICAL_DERIVATION.md`.

### Readiness, caption and non-claims

**Readiness:** numerical and draft-view ready; publication recomposition required. Caption claim: POD and CCD span essentially the same tested rank-3 field subspace; CCD supplies action-coordinate association, not a distinct field basis. Do not claim CCD/POD superiority, exact identity, held-out prediction, independent uncertainty, unique modes, causal relevance or energy meaning for CCD singular strengths.

## S6-S2 — Optional cross-cloak signed mean-$u_x$ audit (supplement only)

### Argumentative job

Test whether the visual `zero-target` / `controlled-zero` sign pattern recurs beyond the deep case without transferring its quantitative estimand or interpretation.

### Candidate source inventory

- Linked root: `src/drl_pinball/data/reproduction` → external reproduction authority.
- Periodic `phase_fields.npz` normally stores `mean_ux`, `mean_uy` from complete cycles.
- Directly inspected available means:
  - parabolic/no-slip inherited Kármán labels `karman_re50`, `karman_re100`, `karman_re200`, `karman_re400`;
  - uniform/free-slip Reynolds cases `kar_re60`, five `kar_re100` controlled realizations, `kar_re200`, `kar_re400`;
  - uniform/free-slip disturbance-size cases `kar_d075`, `kar_d15`, `kar_d2`.
- Existing images: `src/drl_pinball/data/reproduction_plots/{legacy,v5}/*.png` show phase-0 vorticity, not mean-$u_x$ differences.
- Scripts: `plot_reproduction_summary.py` renders phase-0 vorticity; `plot_flow_fields.py` reads complete means but renders nonnegative vector target-error magnitude, not signed mean-$u_x$ pairs.

### Required layout and per-panel contract if commissioned

For each selected case, two adjacent fields: `mean_zero[u_x]-mean_target[u_x]` and `mean_controlled[u_x]-mean_zero[u_x]`. Print physical configuration, case, controlled realization/seed, complete-cycle role windows, mask policy and scale above each pair. Use shared symmetric scale within a case pair; use one global scale across cases only after unit/nondimensionalization and range audit. Never combine configurations in a common scalar score.

### Readiness and dependencies

**Source arrays:** available for the cases listed. **Existing signed plots:** absent. **Render code:** absent for this exact panel. **Panel readiness:** **S6-S2 remains blocked** pending case selection, role/window and mask audit, nondimensionalization/scale decision, and a dedicated read-only renderer. A preliminary unweighted full-grid check found broadly opposite signs in most cases but weakening at harder/high-Re/large-disturbance conditions; it is diagnostic only and must not appear as a result.

### Caption claim and non-claims

Allowed caption after audit: “Selected periodic cloak cases show qualitatively opposing passive-deficit and controlled-minus-passive mean-streamwise patterns under their own role and window contracts.” Prohibit common quantitative complementarity, exact cancellation, universal cloak mechanism, transfer of `93.42/6.58`, transfer of the exact common-mask ROI, CCD interpretation, robustness/generalization, or momentum restoration.