CHC-TDC-VP1: Public-Posterior Residual Gates for Time-Delay Cosmography

The owned object of TDC--VP1 is deliberately narrow. It asks whether the declared CHC distance map can be evaluated on released time-delay distance posterior products and whether those posteriors require a nonzero propagation-shaped residual. The assigned result label is

\boxed{\texttt{BOUNDED-NULLOVERTURN}} .

The label is a null-admissible public-posterior residual-gate label. It is not a model preference label and not a detection label.

The associated time-delay-cosmography analysis keeps the standard thin-lens same-Fermat structure on the adopted variable-source strong-lens class. TDC--VP1 does not modify that theorem status, any lens equation, any Fermat-potential equation, or the total projected-density reading used by that analysis. It also does not replace the lens-inference controls used by the public posterior route. In particular, mass-sheet and source-position transformation control, line-of-sight and external-convergence control, nearby-perturber treatment, and stellar-kinematic information remain part of the upstream strong-lens posterior construction. VP1 reads those public posterior products as already constructed total projected-density distance information and applies only the residual-gate comparison defined below. The bounded scientific support summary is: quote The H0LiCOW public-posterior residual gate returns BOUNDED-NULLOVERTURN: the declared null propagation branch remains admissible on the public-posterior window, and no nonzero propagation correction is required by this companion gate. quote

The VP1 source summary separates public strong-lens posterior products from the CHC residual-gate classifications. It identifies the public source surfaces and the declared diagnostic summaries used for the bounded residual gate; large posterior products remain external public scientific sources and are not part of the manuscript claim.

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The H0LiCOW lens target list used by the VP1 source summary is shown in reference. These redshifts and posterior-policy conventions are used only to evaluate the public-posterior residual gate; they are not a new lens sample definition.

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The associated time-delay-cosmography analysis uses the standard time-delay distance relation once the adopted distance map has been fixed:

\Ddt=(1+z_\mathrm{d})\frac{\Dd\Ds}{\Dds} .

The VP1 residual gate evaluates the same object with the declared CHC angular-diameter distance map

D_A^{\CHC}(z_1,z_2;H_0,\Om,\eta)
=\frac{1}{1+z_2}\int_{z_1}^{z_2}
\frac{c_\mathrm{ref}\,\qprop(z;\eta)}{H_\mathrm{bg}(z;H_0,\Om)}\,dz,

with the declared one-parameter propagation factor

\qprop(z;\eta)=\sqrt{1-\eta\frac{z}{1+z}} .

The null branch is $\eta=0$TeX\eta=0, so $\qprop(z;0)=1$TeX\qprop(z;0)=1. This branch is part of the admissible CHC--TDC distance family.

For a lens $\ell$TeX\ell, the public posterior surface is evaluated in log-distance space. Let

\ell_\ell(H_0,\Om,\eta)

denote the declared smoothed log-likelihood contribution from that lens under the declared posterior-policy convention. The combined objective is

\mathcal L(H_0,\Om,\eta)=\sum_\ell \ell_\ell(H_0,\Om,\eta),

and the comparison between the null branch and the one-parameter residual branch is reported as

\Delta\chi^2 = -2\left[\max_{H_0,\Om,\eta}\mathcal L(H_0,\Om,\eta)-\max_{H_0,\Om}\mathcal L(H_0,\Om,0)\right] .

The information-criterion entries are recorded as penalty diagnostics for an added propagation parameter, not as a claim of model selection:

\Delta \mathrm{AIC} = \Delta\chi^2 + 2,
\qquad
\Delta \mathrm{BIC} = \Delta\chi^2 + \log N_\mathrm{lens},

with $N_\mathrm{lens}=6$TeXN_\mathrm{lens}=6 in the primary board. Since $\Delta\chi^2=0$TeX\Delta\chi^2=0, the nonzero residual branch does not improve the public-posterior objective on the declared grid.

The diagnostic construction is a five-step public-source comparison.

- Identify the H0LiCOW Paper XIII public distance-posterior products and the TDCOSMO-IV public hierarchy-analysis surface as declared source surfaces. - Normalize the lens metadata and posterior-policy conventions through the declared lens manifest and source catalogue. - Evaluate the declared $(H_0,\Om,\eta)$TeX(H_0,\Om,\eta) grid using Eqs. reference, reference, and reference and the public posterior-policy convention. - Assign the residual-gate classification from the primary KDE summary and model-comparison board. - Record public-source summaries, chunk records, and the non-claim boundary in the source summary and support summaries.

This construction is an official-data public-posterior route. It is not a mock route, not a protocol-only route, and not a hidden same-instance component-separation route.

The primary grid has $51$TeX51 values of $H_0$TeXH_0 from 68.0 to 78.0, $31$TeX31 values of $\Om$TeX\Om from 0.15 to 0.45, and $51$TeX51 values of $\eta$TeX\eta from 0.00 to 0.50. The bounded-domain sensitivity grid extends only the secondary $\eta$TeX\eta domain to 0.95. The primary classification is assigned from the $\eta\in[0,0.5]$TeX\eta\in[0,0.5] domain.

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The declared bounded reading is therefore: quote The CHC null propagation branch remains admissible and nonzero propagation correction is not required. quote The companion record may also be read in the expanded public-posterior form: on the declared H0LiCOW/TDCOSMO public-posterior route, the CHC null propagation branch remains admissible; a nonzero propagation correction is not required by the current data.

In diagnostic-summary notation, the primary fields are: center

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center These names are reproduced only to identify the result fields; they do not create stronger physical claims.

Best-fit and model-comparison board

The full three-dimensional residual comparison returns the same best-fit location for the null model and the one-parameter residual model:

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Because the optimum does not move away from $\eta=0$TeX\eta=0, the extra residual parameter is not required by the public posterior route.

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Lens residual board

The public posterior products are evaluated lens by lens before the combined residual comparison is assigned. The distance values in reference are the VP1 residual-board comparisons in Mpc.

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Leave-one-out stability board

The leave-one-out board is a stress diagnostic. It does not assign separate lens-level claims. Five of six leave-one-out routes retain $\widehat\eta=0$TeX\widehat\eta=0; the J1206-removal route shows a boundary value in the diagnostic board but does not override the primary all-lens result.

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Domain-sensitivity board

The primary classification uses the declared domain $\eta\in[0,0.5]$TeX\eta\in[0,0.5]. A wider $\eta\in[0,0.95]$TeX\eta\in[0,0.95] table is retained only as a domain-sensitivity comparison and does not replace the primary label.

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proposition: The reported profile quantile is domain-bound. The VP1 values $q_{0.95}=0.48$TeXq_{0.95}=0.48 on $[0,0.5]$TeX[0,0.5] and $q_{0.95}=0.91$TeXq_{0.95}=0.91 on $[0,0.95]$TeX[0,0.95] do not define a cutoff-independent upper limit on $\eta$TeX\eta. Consequently, they cannot be interpreted as a physical measurement or as a robust exclusion bound without an independently specified prior domain and a likelihood analysis demonstrating stability under its enlargement.

proof. Let $U$TeXU denote the imposed upper endpoint. A cutoff-independent upper limit would remain approximately invariant when $U$TeXU is enlarged beyond that limit. Here the reported quantile changes from $0.48=0.96U$TeX0.48=0.96U for $U=0.5$TeXU=0.5 to $0.91\simeq0.958U$TeX0.91\simeq0.958U for $U=0.95$TeXU=0.95. Thus the quantile tracks the chosen endpoint rather than approaching an endpoint-independent value. The numerical summaries therefore establish only that the maximum occurs at the null point on the sampled grid; they do not establish an intrinsic upper bound on $\eta$TeX\eta.

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The word BOUNDED-NULLOVERTURN denotes only the declared null-admissible public-posterior status. It does not become PASS, TDC-PASS, a detection label, or a posterior preference claim. In particular, the endpoint-sensitive profile quantile is not a physical upper limit on $\eta$TeX\eta. The companion record is bounded by the public objects named in reference and by the total projected-density reading of the time-delay-cosmography analysis.

The public companion source summary is: center

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center The displayed numerical values come from the declared primary diagnostic summary and diagnostic summaries. Raw public posterior products remain external public products and are not part of the manuscript claim.

Within the associated time-delay-cosmography analysis, TDC--VP1 supplies only bounded BOUNDED-NULLOVERTURN null-admissible public-posterior support on the H0LiCOW distance-posterior route. It does not require a nonzero propagation correction and does not assign an $\eta$TeX\eta measurement. No theorem follows from this support summary. The same-Fermat proposition, the standard thin-lens equation, and the total projected-density inference rule remain objects of the time-delay-cosmography analysis. VP1 supplies visibility for one declared public-posterior residual check.

The VP1 status is falsifiable within its declared public-posterior role. The label BOUNDED-NULLOVERTURN should be re-evaluated, not rhetorically preserved, if a later declared public-posterior route satisfies any of the following conditions:

- the null branch $\eta=0$TeX\eta=0 is no longer admissible under the declared posterior-policy convention; - a nonzero propagation residual improves the declared comparison by more than the stated penalty rule and remains stable under the declared lens and domain checks; - the H0LiCOW public-source route, lens manifest, or posterior-policy convention is found to mismatch the companion source summary used here; - upstream strong-lens controls, including mass-sheet/source-position transformation, line-of-sight or external-convergence treatment, nearby-perturber treatment, or stellar-kinematic input, are revised in a way that invalidates the public posterior objects used by VP1; - a future TDCOSMO or H0LiCOW-style public release supplies a materially different same-window posterior surface that must be evaluated as a new companion record rather than silently imported into this one.

These failure conditions do not weaken the present result. They define the boundary under which the present result remains a bounded null-admissible residual gate rather than a general time-delay cosmography closure.

A posterior residual is theory-discriminating only when it is transverse to the fitted response manifold. If a common lens--distance parameter vector generates an $m$TeXm-component posterior summary with sensitivity matrix $S$TeXS, the locally constrained directions are $\ker S^{\mathsf T}$TeX\ker S^{\mathsf T}. Projecting residuals onto that left-null space removes first-order shifts that can be produced by admissible refitting and isolates the $m-\operatorname{rank}S$TeXm-\operatorname{rank}S overidentifying components.

The projection, covariance, and retained posterior summaries must be fixed before the public chains are inspected. Lens-specific nuisance enlargement can raise the rank and absorb the same residuals; when it reaches full row rank, no local equality test remains. The analysis must therefore report both the residual statistic and the dimension of the constrained subspace, not only the best-fit displacement along the model manifold.

The closure test for the public-posterior time-delay 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 public-chain parameter and prediction residuals. Let $a$TeXa range over the independent constitutive inputs comprising posterior coordinates, covariance, lens-family mapping, and residual normalization.

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 fitted tangent directions are projected out once and only the fixed transverse posterior residuals are used as theory tests.

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 raw posterior overlap can be generated entirely by parameters already used in calibration. Failure of that condition therefore rejects the proposed microscopic closure before parameter estimation can be counted as evidence for it.

TDC--VP1 reaches the bounded classification

\boxed{\texttt{BOUNDED-NULLOVERTURN}}

on the declared H0LiCOW/TDCOSMO public-posterior residual-gate route. The primary full three-dimensional KDE board gives $\widehat H_0=73.8$TeX\widehat H_0=73.8, $\widehat\Omega_m=0.25$TeX\widehat\Omega_m=0.25, $\widehat\eta=0.0$TeX\widehat\eta=0.0, $q_{0.95}(\eta)=0.48$TeXq_{0.95}(\eta)=0.48, $\Delta\chi^2=0.0$TeX\Delta\chi^2=0.0, $\Delta\mathrm{AIC}=2.0$TeX\Delta\mathrm{AIC}=2.0, and $\Delta\mathrm{BIC}=1.791759469228055$TeX\Delta\mathrm{BIC}=1.791759469228055. The unchanged optimum and the positive AIC/BIC penalties give no preference for the additional parameter. The profile quantile follows the selected domain endpoint and therefore supplies no cutoff-independent upper limit. The CHC null propagation branch remains admissible and nonzero propagation correction is not required. The conclusion stops there: the result is a public-posterior residual comparison, not a propagation-signal discovery, not an $\eta$TeX\eta measurement, and not a replacement for strong-lens cosmology inference pipelines.

No new observational data are introduced by this companion paper. The public sources are the H0LiCOW Paper XIII distance-posterior products and the TDCOSMO-IV hierarchy-analysis repository. The displayed VP1 values summarize the stated residual comparison. Raw posterior products remain available from their official repositories; they are not redistributed here.

Funding and competing interests..

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