GEMA Observational-Environment Covariance Gates in CHC

QAC-VP3 is a companion public-data record. It follows QAC-VP0, which established a minimal public simulation--observation proxy bridge, QAC-VP1, which expanded the public SIM/OBS proxy surfaces, and QAC-VP2, which identified a TNG cosmic-web simulation environment analysis and a partial covariance object. VP3 addresses the remaining observational-environment side: can the MaNGA projected-proxy rows be tied to an explicit MaNGA environment catalog and used to construct an observational environment covariance board?

The admissible interpretation is therefore: quote QAC-VP3 states a public MaNGA GEMA observational-environment covariance analysis on the declared public-source comparison surface. quote The forbidden readings are: finite-window QAC closure, theorem-level angular-momentum compensation, evidence for a rotating Universe, a full structure-formation model, or a replacement for a shell/web/host dynamics theory.

The observational environment source is the SDSS DR17 GEMA-VAC, described by SDSS as the Galaxy Environment for MaNGA Value Added Catalog and as a catalog of environment characterisations for the MaNGA DR17 sample [citation]. The kinematic proxy source remains the MaNGA DAP/MAPS surface: SDSS describes the DAP as providing stellar kinematics, emission-line properties, and spectral indices, and the DR17 data-access page gives the MAPS file pattern for per-object downloads [citation]. The simulation environment context imported from VP2 is the public IllustrisTNG/TNG DisPerSE cosmic-web analysis; the IllustrisTNG API documentation also records the authorized public-access pattern and subhalo endpoints used in earlier QAC companion records [citation].

The companion record reports the following final labels: center

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center

The raw GEMA file is data/raw/gema/GEMA_2.0.2.fits. The selected HDU is DR17_param_LSS, with 10,086 rows and columns including mangaid, mh, den1, den2, den3, t1, t2, t3, and major/minor-axis direction fields. The VP2 MaNGA proxy handoff contains five projected-proxy rows with five non-null mangaid entries. The GEMA--MaNGA join succeeds with five rows on the normalized mangaid key.

The observational covariance board is partial because the joined MaNGA sample is small, but it is structured and environment-resolved. The record reports four observational bins: center

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center

The interval/covariance result is therefore not a full covariance closure. It is a partial environment-resolved observational covariance construction. Single-object bins are retained as bins with undefined standard error rather than being treated as stable distributional estimates.

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The public source basis for VP3 identifies the declared gate summary, GEMA parse summary, GEMA--MaNGA join summary, observational covariance summary, public-source basis, and non-claim boundary. The route type is a public observational-environment construction and interval-board comparison. Admissible interpretation: QAC-VP3-GEMA-OBSENV-COVARIANCE-PARTIAL on the declared GEMA/MaNGA observational environment analysis. Excluded interpretation: dual-side simulation--observation harmonization, finite-window QAC closure, theorem-level compensation, rotating-Universe evidence, or full structure-formation modeling.

The source basis is public-source only. The companion source summary identifies the MaNGA and GEMA public-source surfaces, the VP2-derived MaNGA proxy handoff board, the GEMA parse and join summaries, the covariance board, and the public-source boundary check. Only public-source surfaces are used in the scientific label; private or restricted-source material is outside the manuscript claim.

The final VP3 classification is

\boxed{\texttt{QAC-VP3-GEMA-OBSENV-COVARIANCE-PARTIAL}}.

This classification is read as a bounded public-data covariance record because it supplies the observational environment covariance object that VP2 explicitly left open. It does not promote QAC to finite-window closure and does not support rereading QAC as a rotating-Universe or full structure-formation paper.

A follow-up after VP3 should not be merely another sample-size expansion. Increasing the five joined MaNGA rows to a larger number would thicken an existing analysis and should be treated as a robustness analysis on the same gate family, not as a separate harmonization result. A scientifically distinct follow-up would be a dual-side environment-harmonization record that aligns the TNG cosmic-web environment family from VP2 with the GEMA observational environment family from VP3 under a single declared coarse environment taxonomy and a joint interval/covariance object. Such a follow-up would be justified only if it performs cross-family harmonization, not if it simply downloads more MaNGA MAPS files.

The observational environment analysis evaluates whether GEMA summaries lie on the response manifold fixed by the other declared environments. After fitting only the calibration subset, the frozen Jacobian determines tangent directions and its left null space determines transverse predictions. The GEMA residual must be projected onto those directions with the predeclared covariance.

Refitting an environment coefficient to the GEMA block changes the manifold and removes the holdout. If the enlarged map has full row rank, any nearby GEMA summary becomes admissible and the analysis has no predictive content. The result should therefore be stated in terms of the number and magnitude of surviving transverse residuals, not only the goodness of a joint refit.

The closure test for the GEMA environment holdout gates is applied to a dimensionless observable vector $y\in\mathbb R^m$TeXy\in\mathbb R^m formed from fixed reference scales and the declared basket of GEMA angular-momentum and kinematic residuals. Let $a$TeXa range over the independent constitutive inputs comprising training environments, GEMA selection, transfer parameters, and covariance.

proposition: Functional saturation, finite closure, and sector admissibility. Suppose the unrestricted prediction map $F:a\mapsto y$TeXF:a\mapsto y is continuously differentiable on a Banach space of constitutive inputs. If $D_aF$TeXD_aF is surjective and has a bounded right inverse at the calibration point, the unrestricted family is locally open in observable space and supplies no nonzero local equality restriction on $y$TeXy. Suppose instead that a single microscopic closure replaces $a$TeXa by finite parameters $\theta\in\mathbb R^p$TeX\theta\in\mathbb R^p, with profiled nuisance coordinates $\eta\in\mathbb R^q$TeX\eta\in\mathbb R^q. If

J=D_\eta F_{\rm cl}D_\theta F_{\rm cl},
 \qquad \operatorname{rank}J=r<m,

then there are $m-r$TeXm-r independent first-order restrictions

w^{\mathsf T}\delta y=0,
 \qquad w\in\ker J^{\mathsf T}.

If the rank is constant locally, these restrictions are tangent to a compatibility manifold of codimension $m-r$TeXm-r. For this sector, the finite closure is admissible only if the GEMA environment is excluded from calibration and evaluated under the previously frozen bridge.

proof. Split surjectivity gives a bounded right inverse $R$TeXR with $D_aF\,R=I_m$TeXD_aF\,R=I_m. The Banach-space submersion theorem then makes $F$TeXF locally onto a neighborhood of the calibrated observable vector. Any smooth equality holding throughout that image must therefore vanish on an open set and contributes no model-specific local restriction. Under finite closure, the attainable first-order variations are exactly the column space of $J$TeXJ. Its orthogonal complement is $\ker J^{\mathsf T}$TeX\ker J^{\mathsf T}, whose dimension is $m-r$TeXm-r by rank--nullity, which proves reference. The constant-rank theorem supplies the stated local manifold. The sector condition is necessary because a GEMA-specific correction changes a held-out environment into an additional fit domain. Failure of that condition therefore rejects the proposed microscopic closure before parameter estimation can be counted as evidence for it.

QAC-VP3 turns the GEMA observational environment source into a declared public-data surface, joins it to the existing MaNGA projected-proxy handoff, and produces an environment-resolved observational covariance/interval board. The effective joined sample is five objects; the 10,086-row parent catalogue does not increase that inferential sample size. Consequently, the board establishes data interoperability only and cannot estimate an environment dependence or test angular-momentum compensation. The declared classification is QAC-VP3-GEMA-OBSENV-COVARIANCE-PARTIAL.

Data and code availability..

This companion manuscript uses public observational, simulation, mock-observable, or supplementary bridge materials as described in the text. Cited public references and companion statements, where provided, are identified by the companion source summaries cited in the text.

Funding and competing interests..

No external funding was received for this work. The author declares no competing interests.