Dual Environment Harmonization and Covariance-Partial Gates in CHC

QAC-VP4 asks whether the simulation-side environment analysis and the observation-side environment analysis, previously opened in QAC-VP2 and QAC-VP3, can be put under a common coarse environment taxonomy with a structured interval/covariance object. This is a new construction layer relative to earlier QAC companion records: VP0 opened a minimal simulation--observation proxy bridge, VP1 expanded the public SIM/OBS proxy boards, VP2 identified the TNG cosmic-web environment/covariance analysis, and VP3 identified the MaNGA GEMA observational-environment covariance analysis. VP4 is therefore not a sample-size expansion; it is a dual-side environment harmonization gate.

The admissible interpretation is narrow. QAC-VP4 is a bounded public-data companion-record result. It does not claim finite-window QAC compensation closure, a theorem-level angular-momentum compensation result, a rotating-Universe model, a total cosmic angular-momentum law, or a full structure-formation model.

The observational side is anchored by MaNGA DR17, its Data Analysis Pipeline products, and the GEMA value-added catalog. MaNGA DR17 is the final MaNGA release and contains the survey's data products for over ten thousand galaxies. The MaNGA DAP provides two-dimensional derived maps, including stellar kinematic quantities, which supply the projected observational proxy surface. GEMA-VAC supplies environment characterisations for the MaNGA DR17 sample.

The simulation side is anchored by IllustrisTNG public data. The TNG API and data specifications provide public access to subhalo-level group-catalog quantities and supplementary data surfaces. In the identified VP2 source summary, the TNG cosmic-web environment analysis had already been authorized for public access and parsed into a large cosmic-web board, allowing VP4 to use the simulation-side environment source as a public-data record rather than as an unidentified plan.

The QAC-VP4 gate is decomposed into four checks.

- Source check. The VP2 and VP3 public source summaries must be present and readable. - Dual environment taxonomy. Environment labels from the two sides must be mapped to a coarse common taxonomy. - Dual environment source board. Simulation and observation rows must be present in a single structured board with side labels and environment bins. - Covariance/interval construction. At least one matched coarse bin must yield an interval or stress object, with underpowered bins explicitly flagged rather than over-read.

The allowed final labels are QAC-VP4-SOURCE-CHECK-SATISFIED, QAC-VP4-DUAL-ENV-TAXONOMY-PARTIAL, QAC-VP4-DUAL-ENV-COVARIANCE-PARTIAL, and QAC-VP4-DUAL-ENV-HARMONIZATION-STRESS. Stronger closure labels are not allowed in this companion record.

The diagnostic summary returned the following final classification:

\boxed{\texttt{QAC-VP4-DUAL-ENV-COVARIANCE-PARTIAL}}.

center

Figure or table content is omitted from the web reader; use the canonical manuscript for the exact object.

center

The matched interval is the filament bin: center

Figure or table content is omitted from the web reader; use the canonical manuscript for the exact object.

center

The stress label is therefore not a failure of the bridge. It records that a matched dual-side environment interval exists, but the observation-side count is underpowered. The admissible interpretation is partial harmonization with explicit power limitation.

Figure or table content is omitted from the web reader; use the canonical manuscript for the exact object.

The public source basis for VP4 identifies the declared gate summary, dual-environment summary, dual-environment covariance object, public-source basis, and non-claim boundary. The route type is a public-data harmonization comparison over already identified VP2 and VP3 boards. Admissible interpretation: QAC-VP4-DUAL-ENV-COVARIANCE-PARTIAL with QAC-VP4-DUAL-ENV-HARMONIZATION-STRESS because the matched observational interval is underpowered. Excluded interpretation: finite-window QAC closure, theorem-level compensation, rotating-Universe evidence, total cosmic angular-momentum measurement, or full structure-formation modeling.

The result is satisfactory for the declared QAC-VP4 task because it supplies a distinct dual-environment harmonization object: it aligns simulation-side and observation-side environment labels under a common taxonomy and produces a structured covariance/interval object. It is not merely a thicker version of VP0 or VP1. However, it remains partial because only one coarse bin matches both sides and that bin has one observational row.

The companion record therefore supports this classification sentence:

quote QAC-VP4 is classified as QAC-VP4-DUAL-ENV-COVARIANCE-PARTIAL. The record combines the QAC-VP2 and QAC-VP3 public source summaries, builds a 2005-row dual environment source board, harmonizes the simulation and observational environment labels into a coarse taxonomy, and states one matched filament-bin interval. The matched interval is underpowered on the observation side. This is not finite-window QAC closure, not theorem-level angular-momentum compensation, not rotating-Universe evidence, and not a full structure-formation model. quote

Harmonization is theory-relevant only if one map acts on both environments with fewer adjustable directions than the combined summaries. Let $S$TeXS be the block sensitivity matrix after the harmonization parameters and declared nuisances are fixed. Every $w\in\ker S^{\mathsf T}$TeXw\in\ker S^{\mathsf T} defines a cross-environment contrast that must vanish to first order; these contrasts are the local equations of the shared compatibility manifold.

Giving each environment an independent transfer function can make $S$TeXS full row rank, at which point harmonization is always locally possible and no physical linkage is tested. The follow-up boundary therefore requires a frozen common map, a reported constrained dimension, and evaluation of at least one left-null contrast on data not used to construct that map.

QAC-VP4 remains a bounded dual-environment harmonization partial on the declared public-data surfaces. A simple expansion from 5 observational rows to a larger GEMA/MaNGA sample would thicken the same analysis and should be treated as robustness analysis within this gate family rather than as a distinct scientific status.

A distinct follow-up object remains possible: a mock-IFU observable-family bridge. That follow-up record would use public TNG stellar-particle cutouts to forward-project simulated subhalos into MaNGA-like line-of-sight velocity maps, then compare the resulting observable-family proxy against MaNGA DAP MAPS under the same projected measurement formalism. This would address the observable-family bridge layer rather than merely increasing row counts. It would be computationally heavier and would require the documented TNG public-access route, but the underlying public data surfaces exist.

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.

The closure test for the dual-environment harmonization 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 paired environment distributions and cross-environment contrasts. Let $a$TeXa range over the independent constitutive inputs comprising two environment calibrations, common response parameters, harmonization map, and selection nuisance.

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 one harmonization map preserves common physical parameters and reports all surviving transverse contrasts.

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 separate monotone remappings can align marginal distributions without demonstrating a shared response. Failure of that condition therefore rejects the proposed microscopic closure before parameter estimation can be counted as evidence for it.

Funding and competing interests..

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