DynamisLab/docs/JFM_WYQ/ROUND3_BOTTOM_UP/08_STEADY_SYNTHESIS.md

Round 3 first-pass module — Section 8: steady context, physical synthesis and theory limits

Status: revised candidate for second audit; not self-declared ready. This module is a two-layer author-facing section draft and evidence dossier for the 6% steady return. It does not reverse the historical order: steady discovery is recalled briefly after the Kármán deep analysis, and the full steady/theory analysis appears here retrospectively. Coordinator second-audit disposition: PASS — ready for downstream Wave C imports, but not final manuscript prose.

Section question and bounded answer

Question. After the Kármán deep analysis, what bounded steady context and theory constraints contextualize why the learned periodic law retains a near-constant rear-rotation core, and where does that interpretation stop?

Bounded answer. The independently defined steady V5/free-slip branch supplies historical discovery context, sign and magnitude calibration, and a settling-qualified low-deficit endpoint compatible with the Kármán near-constant core; direct wall-sign checks, the failed potential-circulation sign route and incomplete finite-Re budgets prevent mechanism, stability, optimum and cross-chain validation claims.

Layer 1 — narrative draft

The steady result returns here because it was discovered first, not because it is a continuation of the Legacy Kármán analysis. The early capability section records the short discovery seed; the Kármán section then shows a dominant near-constant rear-rotation component and a smaller dynamic correction under its own Legacy contract. The present return asks the narrower retrospective question: whether an independently defined steady branch contextualizes why that recurrence is plausible. It does not make the V5 steady result evidence for the Legacy SR or CCD numbers, and the two chains are never numerically pooled. [Round-2 freeze; current steady README/RESULTS; Modules 05–06]

Under the modern V5 uniform-inlet, free-slip-channel contract at Re_D=50, the body order is (front, rear_y_plus, rear_y_minus) and the wall law is (U_w,V_w)=(-\omega r_y,\omega r_x). The accepted branch is [0,+\Omega,-\Omega], with \Omega>0 and s=\Omega R/U_\infty; the direct cardinal check gives downstream velocity on both gap-facing rear cardinals and upstream velocity on the two outer cardinals. The accepted s=3.55 endpoint is settling-qualified and has the authoritative downstream full-cross-section relative velocity-deficit E_inf_vector(q_ctl,q_in)=0.0231672176752314 at x/D=10; nearby values are approximately 0.0232525, 0.0231671 and 0.0312490 at s=3.45,3.55,3.65. The spatial profile L2 at x/D=10 is 0.0153919 and remains a diagnostic, while q_blk is non-ranking. Thus s=3.55 is a qualified endpoint, not a continuous optimum or universal law; the apparent local minimum remains provisional pending restarted bracket, continuation, convergence and held-out qualification. [V5 endpoint manifest; steady v3b report; display/profile authority]

The sign check gives the shortest physical reading available from the endpoint. Positive rotation of the upper rear cylinder drives its gap-bottom cardinal downstream and the lower rear cylinder’s negative rotation drives its gap-top cardinal downstream; the accepted actuation therefore belongs to a base-bleed/gap-jet family, not a boat-tail family. Endpoint gap-flux, near-wake, recirculation, vorticity and downstream-profile diagnostics are compatible with describing a gap-facing or near-wake reorganisation and a low-deficit downstream profile. They remain observations and bounded hypotheses: the available package does not establish the arrows from wall motion through shear or separation to gap momentum, base pressure, force or downstream deficit, and the accepted s=3.55 and reverse s=5.2 states are not amplitude-matched. [V5 wall oracle; NS ledger; steady display; Round-1 theory dossier]

The shortest validated theory reference is an outer impermeability calculation, not a finite-Re rotating-wall model. For one circle, the exact uniform-flow-plus-circulation trace is u_\theta(a,\theta)=-2U\sin\theta+\Gamma/(2\pi a): impermeability leaves \Gamma independent, and for U\ne0 no constant circulation can impose the full rotating no-slip tangential trace. The MFS/strip construction therefore supplies a reference family only after its validation gates; the current derivation records the exact-circle and independent unbounded Laurent checks as validated bounded mathematics, while dense off-grid, sector, conditioning, source-radius/order and image-layer gates are required before interpreting a new MFS field, and no publication-ready MFS field is implied here. The attractive cancellation route fails its decisive sign check: the strip outer reference requires g_opt=+6.147944, whereas the accepted finite-Re endpoint’s descriptive fit is g=-12.200554. The failed route is retained because it limits the interpretation; it cannot be silently replaced by a rotating no-slip closure. [DERIVATION.md; current strip and finite-Re authority; LITERATURE.md]

The integrated map therefore constrains, without explaining, one set of compatible, non-equivalent observations. The Legacy Kármán SR deletion test ranks its persistent rear constant above the tested rear-lift feedback under a fixed parent-relative DTW window; the corrected Legacy CCD ledger says its constant-control mean-field reference accounts for 93.42% of the measured zero-to-DRL ROI target-error reduction, leaving a 6.58% increment organized descriptively by action-correlated correction-field modes; the separate V5 steady branch supplies the accepted sign and a low-deficit endpoint at a rear-rotation scale of order s=3.55. These analyses organize separate action-space and field-role observations around a persistent/constant reference, without a common estimator or independent triangulation. They use different chains, roles, domains, masks, windows and estimands: they are compatible observations, not replication, validation, agreement, a pooled metric or a closed control skeleton. This bounded synthesis therefore contextualizes, but does not explain, the persistent/constant reference before bridging qualitatively to the planned open-loop experiment—steady operation may be inspected for straighter dye streaklines and the illusion operation for changed shedding frequency. Experiment remains qualitative, and the computational interpretation stops before mechanism, stability, optimum, momentum restoration, causal closure or cross-chain validation. A decisive next test would combine amplitude-matched branches with held-out endpoint evidence, a validated outer-reference comparison and a complete same-control-volume Navier–Stokes momentum/mechanical-energy budget, followed by synchronized quantitative experiment if the claim is to cross the CFD boundary. [Modules 05–06; steady NS/theory authorities; Round-2 experiment boundary]

Layer 2 — local provenance and claim-to-evidence ledger

Narrative claim	Exact current authority / artifact	Chain, solver, geometry, case and Re	Comparator, metric/estimand, ROI/window	Status / verification boundary
Historical steady discovery and distinct contract	Round-2 05_REVISED_STRUCTURE_FREEZE.md §§2, 3, 5; src/steady_pinball_theory/README.md, RESULTS.md; src/SR_analysis/results/runs/article2-steady-analysis-20260720/interpretation.md	V5 steady/free-slip versus LegacyCelerisLab Kármán; three-cylinder pinball, body order explicitly separate; V5 Re_D=50, Legacy Re_code=100 means Re_D=50 only within Legacy’s own convention	Steady is contextual; Legacy SR/CCD retain their own role-local estimands	Current authority read; not independent CFD recomputation or universal artifact audit
Accepted branch and direct sign	steady-contract-v5-torque-d20-20260808/accepted-s3.55/manifest.json; steady-article-publication-v3b-20260809/REPORT.md; contract.py/current steady README	V5 CelerisLab/LBM, D=20, R=10, uniform inlet, free-slip lateral walls, rotating no-slip cylinders; accepted action [0,+\Omega,-\Omega], \Omega=0.00355, x-right/y-up	Cardinal wall velocities; branch/sign oracle, not a field metric	Current manifest-bound contract; direct wall check recorded by authority; artifact leaf audit deferred
Low-deficit endpoint	Same accepted endpoint manifest and publication v3b; figures/summary/profile_Einf_vector.csv	V5 free-slip Re_D=50, accepted controlled endpoint, settling window 10, final tU_\infty/D=250	Full-cross-section two-component E_inf_vector(q_ctl,q_in) at x/D=10: 0.0231672176752314; E_L2_vector=0.0153918657 is diagnostic; nearby s={3.45,3.55,3.65} E_inf_vector≈{0.0232525,0.0231671,0.0312490}; q_blk non-ranking	Qualified endpoint observation; not optimum, uniqueness, attraction, stability or causal intervention; endpoint artifact audit deferred
Reverse boundary	steady-results-display-v2-20260809/README.md; current steady RESULTS.md; profile CSV	V5 same nominal plant family, reverse branch s=5.2, but distinct amplitude and reverse sign	E_inf_vector(x/D=10)=0.6898418 versus accepted 0.0231672; this is not an amplitude-matched controlled comparison	Current display/CSV authority; descriptive branch contrast only, no mechanism or stability inference
Gap-facing physical reading	V5 manifest wall_oracle; steady_pinball_theory/LITERATURE.md sign master map; ns-ledger-v2-oracle-bound-d20-20260809/analysis.json	V5 accepted branch, rear gap/outer cardinal samples; endpoint NS diagnostic ledger	Signed gap fluxes, near-wake/recirculation/vorticity and profile diagnostics are endpoint spatial observations; no single pooled score	Bounded observation/hypothesis; no causal arrow, force ownership or closed budget
One-circle outer reference	DERIVATION.md §§5, 11; LITERATURE.md Crowdy (2006), DOI 10.1016/j.euromechflu.2005.11.005; Ueda et al. (2003), DOI 10.1017/S002211200300627X	Irrotational Euler/Neumann outer problem; one-circle exact identity, not V5 finite-Re no-slip solver	Wall normal/tangential trace identity; no downstream V5 metric	Existing derivation; exact-circle mathematics separately checked in the derivation record; no finite-Re mechanism transfer
MFS/strip gate and failed sign	steady-strip-modal-analysis-20260804-v4/analysis.json; steady-finite-re-bridge-analysis-20260804-v4/analysis.json; DERIVATION.md §§11–13	Outer strip/unbounded potential references with finite-image MFS; finite-Re descriptive contour projection from accepted V5 endpoint	Strip cancellation g_opt=+6.147944; finite-Re descriptive fit g=-12.200554; same_sign=false; strip reference BOUNDED, confinement mechanism FAIL	Current authority; sign conflict is a genuine interpretation blocker for that route; MFS field interpretation remains gate-dependent and no new field is promoted
Partial NS/budget boundary	ns-ledger-v2-oracle-bound-d20-20260809/analysis.json; v3b report/display README	V5 accepted/reverse endpoint diagnostics, one body-enclosing rectangular CV	Endpoint spatial fluxes, dissipation, gap flux and actuator-on-fluid power; storage and body-adjacent terms unavailable, residual null	Current partial ledger; mechanical-energy closure NOT_CLOSED, causality NOT_ESTABLISHED, force ownership not established; authority-led raw-term deferral
Legacy SR magnitude context	src/SR_analysis/results/runs/article2-steady-analysis-20260720/interpretation.md, steady_constant_correspondence.json; SR current claims and Module 05	LegacyCelerisLab disturbance-free contextual roles; separate Legacy geometry/API and action convention	Steady sweep A=0…6, best listed A=5, Legacy DTW similarity 0.9872116; Kármán law deployment 0.9664944; rear mean (-3.430589,+3.431828), fitted magnitude 3.431209	Contextual calibration only; not pooled with V5 E_inf_vector, not momentum equality or mechanism
Legacy CCD compatible observation	Module 06 current corrected authority; src/CCD_analysis/RESULTS_INDEX.md, FINAL_RESULTS.md, corrected manifests	Legacy karman_re100, target/zero/constant/DRL roles; Re_D=50 under Legacy Re_code=100; exact four-role common mask	Coordinate-weighted ROI mean errors E_zero=0.4694352273, E_constant=0.1110140739, E_DRL=0.0857787860; 93.42/6.58 scalar error-difference share; separate 19×10 residual	Current descriptive/noncausal result; not steady evidence and not a paired intervention; dense arrays/leaf audit deferred

Existing derivation, independent checks and open conjectures

• Existing derivation: one-circle Euler trace, fixed-circulation outer uniqueness under stated hypotheses, rear-opposite potential/dipole algebra, MFS construction, strip finite-image relations, E_inf_vector definition and partial NS-ledger identity.
• Independently checked/validated at bounded level: current authority records exact-circle failure controls, bounded agreement of independently discretized unbounded MFS and Laurent/Fourier checks, finite-image convergence observations without an asymptotic tail claim, direct wall-cardinal sign, accepted endpoint settling and the finite-Re descriptive sign fit. These do not validate an MFS field as finite-Re physics.
• Open conjecture: accepted gap-facing motion may participate in a base-bleed/gap-jet-family near-wake organisation and may be compatible with a low-deficit endpoint; a persistent rear rotation may provide a mean compensation-like contribution with a smaller dynamic correction. Neither conjecture is a mechanism, momentum restoration, stability statement, optimum or causal law.

Figures, equations and table contracts

1. Accepted/reverse profiles and fields plus E_inf_vector(s) curve. Source from the current steady display/profile CSV and endpoint fields; show accepted s=3.55, reverse s=5.2, stationary reference and nearby accepted sweep values. Caption must state V5/free-slip Re_D=50, full-cross-section q_ctl versus q_in, x/D=10, diagnostic L2, reverse amplitude mismatch, and endpoint—not optimum/stability—status. Redraw is dependent on current manifests/field payloads; no new publication-ready figure is claimed.
2. Action-sign schematic. Redraw the body order and four rear cardinal velocities from the accepted manifest. It establishes the sign and gap-facing/outer-facing observation; it does not establish circulation, base-bleed causality or boat-tail equivalence.
3. Shortest outer-reference/sign-conflict panel. Use the validated one-circle trace and a compact g_opt=+6.147944 versus g=-12.200554 sign comparison. If an MFS field is retained, all dense off-grid, sector, conditioning, source-radius/order, image-layer, one-circle and independent-reference gates must be visibly complete; otherwise leave the field in the supplement/open ledger. This panel does not establish rotating no-slip finite-Re physics.
4. Integrated evidence map. Nodes are Legacy SR term ranking, Legacy corrected mean/CCD organization, V5 steady endpoint and qualitative experiment bridge. Solid arrows mean same-contract definition or result; dashed arrows mean compatible observation across non-equivalent estimands; broken/unresolved arrows mean mechanism, causality or closure is open. The map must print chain, metric and status and must not be rendered as a causal flow diagram.

Equation budget: reuse the wall law/action sign and s, the steady E_inf_vector definition, and either the one-circle trace or the partial NS-ledger identity. The full MFS, strip-image, multipole and CV derivations belong in the supplement unless their current validation gates are explicitly closed.

Claim ladder and interfaces

Green / supported: accepted V5/free-slip sign contract; direct wall-cardinal check; settling-qualified s=3.55 endpoint; E_inf_vector at x/D=10; profile L2 as diagnostic; reverse high-deficit endpoint as a distinct non-amplitude-matched observation; failed potential-cancellation sign; partial rather than closed NS ledger.

Amber / bounded: low-deficit endpoint is compatible with the recurrent Kármán near-constant rear-rotation scale; accepted wall motion is in a base-bleed/gap-jet family; steady, SR and corrected mean/CCD observations can be integrated as compatible, non-equivalent context; qualitative experiment is a prospective bridge.

Red / prohibited without new evidence: mechanism, causal gap-jet or momentum-restoration claim; Gamma(Omega) closure; MFS-to-no-slip transfer; stability, branch attraction, optimum or universal s law; Legacy/V5 numerical continuation/equivalence; arithmetic pooling or replication/validation/agreement/triangulation language; quantitative experiment validation.

Imports: Modules 03–06 supply chain, metric, Legacy SR, corrected mean/dynamic and CCD boundaries. The current steady authority supplies the V5 contract, accepted sign, endpoint, theory failure and incomplete budget.

Exports to Module 09/10: a bounded physical synthesis (persistent rear rotation + smaller correction as compatible organization only), the failed potential-sign route and incomplete-budget decisive tests, and a qualitative experiment bridge. No quantitative steady number, sign convention or endpoint metric may be transferred to Legacy without its own contract.

Near-collision check

The Legacy SR fitted constant magnitude (3.431209 in alpha=omega/U_0) and the V5 steady endpoint (s=3.55) are numerically similar dimensionless labels but use different action parameterizations, chains, plants and objectives; they are related contextual diagnostics only, not amplitude agreement, validation or an arithmetic comparison. CCD has no actuator-magnitude comparison here: its constant mean is a field-role reference under the corrected four-role estimand and can join the synthesis only as a non-equivalent mean-field observation. No arithmetic combination, replication, validation, agreement or triangulation wording is permitted.

Defects, gates and coverage ledger

Systematic/module-specific classification: the current Round-3 controls require a cross-module near-collision check; this module applies it explicitly. Module-specific risks are the V5/Legacy sign temptation, provisional s minimum, MFS validation status and partial NS budget. No control-plane edit is authorized in this one-file task.

Genuine blockers: none for the bounded section text. The potential sign conflict is a genuine blocker to that mechanism route, not to this bounded synthesis. A retained MFS field panel becomes blocked until its listed validation gates are complete; an exact artifact-audit claim becomes blocked if authority cannot bind the panel source.

Authority-led deferrals: raw endpoint fields, external/symlink-backed roots, leaf hash recomputation, complete MFS raw arrays and full NS terms are deferred because current authorities bind the roles, numbers, contract and status without conflict. These deferrals do not authorize stronger claims.

Coverage status	Sources and reason	Evidence status
read	Current Round-3 master/rule/quality/interface ledger and Wave-A audit; Round-2 freeze/style; current Modules 03–06; Round-1 steady, SR, CCD/OID and coverage dossiers; current steady README, RESULTS, DERIVATION, LITERATURE, evidence README/INDEX, v3b report, display README, V5 endpoint manifest, strip/finite-Re authorities, NS ledger, validation authority and SR steady correspondence	Read directly; not equivalent to independent recomputation or universal artifact audit
mapped	Current figure manifest/CSV and authority-linked external roots used to identify redraw dependencies; archive/history only for replacement navigation	Mapped/indexed, not read as independent evidence; not promoted
not-covered	New CFD, new MFS computation, full raw-field/NPZ audit, universal hashes, new literature search, synchronized experiment analysis	Outside first-pass scope; no result inferred
external-inaccessible	Symlink-backed/external endpoint and dense theory/NS payloads not fully exposed through ordinary repository reading	Authority-led artifact-audit deferred; nonblocking for bounded text, blocking for any retained unsupported panel or reproducibility claim

Anti-index / anti-AI audit

The narrative has five argumentative paragraphs with local caveats, not a package inventory. Numbers appear only where they change the inference; chain, comparator and metric distinctions are repeated only at the necessary bridge. No archive result is promoted, no V5 sign is transferred to Legacy, no spatial L2 is pooled with DTW or CCD, no potential field is presented as a mechanism, and no first-pass readiness claim is made.