TDCOSMO-IV Public-Chain Mass-Profile Stress Gates in CHC Time-Delay Cosmography

Strong-lens time-delay cosmography constrains distance combinations through measured time delays, lens modeling, line-of-sight corrections, and assumptions about the deflector mass profile. The H0LiCOW XIII analysis combined six lensed quasars and reported $H_0=73.3^{+1.7}_{-1.8}\,\kmsmpc$TeXH_0=73.3^{+1.7}_{-1.8}\,\kmsmpc in flat $\Lambda$TeX\LambdaCDM [citation]. The TDCOSMO-IV analysis relaxed the profile assumptions by explicitly treating the mass-sheet transform and stellar-kinematics hierarchy; its TDCOSMO-only layer reported $H_0=74.5^{+5.6}_{-6.1}\,\kmsmpc$TeXH_0=74.5^{+5.6}_{-6.1}\,\kmsmpc, and its TDCOSMO+SLACS joint hierarchy reported $H_0=67.4^{+4.1}_{-3.2}\,\kmsmpc$TeXH_0=67.4^{+4.1}_{-3.2}\,\kmsmpc under the stated parent-population assumption [citation].

The preceding CHC-TDC-VP1 companion comparison tested a one-parameter propagation family on the H0LiCOW public posterior products and returned the bounded null-admissible label BOUNDED-NULLOVERTURN used in that paper. Here BOUNDED-NULLOVERTURN is read only as an internal residual-gate label, not as a detection, model-selection, or closure claim. VP2 is a separate stress record. It does not rerun the TDCOSMO-IV hierarchical sampler and does not revise the VP1 posterior gate. Its purpose is narrower: it asks whether public TDCOSMO-IV chain layers, read as relaxed mass-profile stress layers, overturn the VP1 null-admissible reading at a declared proxy level.

The comparison remains at the total projected-density inference level. The paper does not identify a separate CHC phase component in lensing data. The symbols and proxy labels below are therefore stress-language only; they are not measurements of a physical propagation parameter.

The public TDCOSMO-IV analysis materials contain posterior-chain products used in the published hierarchy analysis [citation]. VP2 uses those public-chain products for the declared post-processing diagnostic. Direct HDF5 inspection found sixteen chain files with `mcmc/chain', `mcmc/log_prob', and `mcmc/accepted' objects. No per-sample weights dataset was found in the inspected chain files; the summaries below therefore use flattened post-burn samples as unweighted posterior samples.

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The construction sequence is: public chain source summary, public-chain column-map normalization, published $H_0$TeXH_0 summary reproduction, tier-1 $H_0$TeXH_0 proxy evaluation, limited hybrid proxy evaluation, leave-one-layer-out stability check, and bounded classification assignment. The companion source summary also includes a public-source basis list, a source-boundary record, and a signal-target boundary record.

The frozen target layers are:

- TDCOSMO-only: the primary relaxed-profile hierarchy layer - TDCOSMO+SLACS$_{\rm IFU}$TeX_{\rm IFU}: an IFU anisotropy stress layer; - TDCOSMO+SLACS$_{\rm SDSS}$TeX_{\rm SDSS}: an SDSS/SLACS population stress layer; - TDCOSMO+SLACS$_{\rm SDSS+IFU}$TeX_{\rm SDSS+IFU}: the final joint SDSS+IFU stress layer.

The earlier apparent interval discrepancy in the SLACS layer is resolved by target-layer reconciliation. The SDSS-only chain corresponds to the published chain target $67.4^{+4.3}_{-4.7}\,\kmsmpc$TeX67.4^{+4.3}_{-4.7}\,\kmsmpc, while the abstract-level $67.4^{+4.1}_{-3.2}\,\kmsmpc$TeX67.4^{+4.1}_{-3.2}\,\kmsmpc value corresponds to the SDSS+IFU layer. This distinction matches the published layer structure reported in the TDCOSMO-IV analysis.

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Let layer $0$TeX0 denote the TDCOSMO-only primary layer. For a stress layer $L$TeXL, the tier-1 $H_0$TeXH_0-layer proxy records the median shift

\Delta H_0^{(L)} = H_{0,L}^{(50)}-H_{0,0}^{(50)}

and the logarithmic shift

\delta_H^{(L)} = \left|\log\frac{H_{0,0}^{(50)}}{H_{0,L}^{(50)}}\right|.

The absolute value is used only to register stress amplitude. This proxy is not an $\eta$TeX\eta posterior and is not an inference of a propagation correction.

A limited hybrid stress proxy is then evaluated on the frozen coordinate set

\mathcal C=\{H_0,\Lmst,\sigma(\Lmst),\alpha_\lambda\}.

For coordinate $j\in\mathcal C$TeXj\in\mathcal C, let $Z_{L,j}$TeXZ_{L,j} be the standardized median shift of layer $L$TeXL against the primary layer, using the declared public-chain interval scale for the coordinate. The hybrid score is

\Sscore_L = \left(\frac{1}{|\mathcal C|}\sum_{j\in\mathcal C} Z_{L,j}^{2}\right)^{1/2}.

The declared proxy-warning threshold is the unit standardized score. Scores below this threshold are recorded as stress-pass. The score is a compact proxy-level comparison, not a likelihood ratio and not a TDCOSMO sampler diagnostic.

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The IFU-only layer remains close to the primary relaxed-profile layer. The SLACS-informed layers provide the strongest median downward stress, driven primarily by $H_0$TeXH_0 and $\Lmst$TeX\Lmst, but remain below the declared warning threshold.

The stability gate asks whether the staged proxy reading depends on any single secondary layer. Let the active secondary set be

\mathcal L=\{\mathrm{IFU},\mathrm{SDSS},\mathrm{SDSS+IFU}\}.

For each $K\in\mathcal L$TeXK\in\mathcal L, the public-chain stress record removes $K$TeXK and recomputes the active-layer proxy reading. The baseline and all leave-one-layer-out cases preserve interval overlap with the primary layer and keep the staged reading at leans_NONOVERTURN.

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This stability result rules out a single-layer dependence of the declared proxy-level reading. It does not convert the stress proxy into a propagation-parameter measurement.

The VP2 gate sequence is:

- HDF5 source summary pass; - column-map pass; - published-summary reproduction pass; - tier-1 $H_0$TeXH_0-layer propagation-equivalent stress proxy; - limited hybrid population/IFU stress comparison; - leave-one-layer-out stability; - classification assignment.

The first six gates are satisfied by the declared public-chain post-processing record. The final classification is

\boxed{\texttt{TDC-VP2-RELAXED-PROXY-NONOVERTURN}}.

This means that the TDCOSMO-IV public-chain relaxed-profile stress layers do not overturn the VP1 null-admissible reading at the declared proxy level. It is not a pass label for a new propagation parameter, not an $\eta$TeX\eta measurement, and not a rerun of the TDCOSMO-IV hierarchical sampler.

The final declared diagnostic surface records the following non-claim entries: center

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center The reason is that no fixed transfer rule $S_\eta$TeXS_\eta has been supplied to convert the relaxed mass-profile hierarchy stress proxy into a reduced $\eta$TeX\eta-equivalent propagation inference.

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The classification is bounded by four exclusions:

- no nonzero CHC propagation correction is inferred; - no $\eta$TeX\eta measurement is assigned; - no separate phase component is identified in strong-lens data; - no full TDCOSMO-IV hierarchical-sampler reproduction is claimed.

A tier-2 reduced $\eta$TeX\eta-equivalent proxy would require a fixed, declared, and defensible transfer rule $S_\eta$TeXS_\eta connecting the VP1 propagation family to the relaxed hierarchy layer. No such transfer rule is assigned here. Therefore the only assigned result is the proxy-level stress classification.

The public-chain stress test should distinguish motion along an admitted inference manifold from disagreement orthogonal to it. With one frozen parameterization, the chain-derived response Jacobian spans the locally refittable directions. Left-null projections give parameterization-invariant stress residuals and count the independent restrictions as $m-\operatorname{rank}S$TeXm-\operatorname{rank}S.

Changing the mass-profile family, external-convergence prior, or distance response after examining the projected residual defines a new model and requires a new declared chain route. If the combined nuisance family has full observable rank, it can reproduce an open neighborhood of summaries and the test ceases to be overidentifying. The present test is therefore strongest when the model family is frozen on one subset and its transverse residuals are evaluated on a separate lens, chain product, or distance combination.

The closure test for the TDCOSMO chain-stress 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 rank stability and residual shifts across chain stresses. Let $a$TeXa range over the independent constitutive inputs comprising public-chain choice, sample weights, nuisance family, and stress transformation.

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 every stress preserves the original parameter family and observation map before changes in transverse residuals are assessed.

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 adding stress-specific nuisance directions can hide a failure by changing the model being tested. Failure of that condition therefore rejects the proposed microscopic closure before parameter estimation can be counted as evidence for it.

VP2 supplies a public-chain post-processing stress record for CHC time-delay cosmography. The public-chain stress record reproduces the targeted public-chain $H_0$TeXH_0 summaries, freezes the layer map, compares the TDCOSMO-only relaxed-profile hierarchy against SLACS-informed stress layers, evaluates a tier-1 $H_0$TeXH_0 stress proxy, evaluates a limited hybrid stress proxy over four hierarchy coordinates, and checks leave-one-layer-out stability. The IFU-only layer is close to the primary layer, while the SLACS-informed layers provide the largest stress. All active stress layers preserve overlap with the primary TDCOSMO-only interval, all hybrid stress scores remain below the declared warning threshold, and the staged reading is stable under any single secondary-layer omission. The resulting classification is TDC-VP2-RELAXED-PROXY-NONOVERTURN.

Together with VP1, this result preserves the null-admissible proxy-level reading. VP1 established that the H0LiCOW public-posterior record does not require a nonzero CHC propagation correction. VP2 adds that the public TDCOSMO-IV relaxed mass-profile hierarchy chains do not overturn that internal residual-gate reading at the declared proxy level. Neither companion record identifies a phase component or measures a nonzero propagation parameter.

The public companion source summary contains the chain source summary, column-map evidence, published-summary reproduction, layer comparison tables, tier-1 and hybrid stress proxy tables, leave-one-layer-out tables, protocol status addendum, and public-source list. The original H0LiCOW and TDCOSMO data products are external public products; the VP2 support summary summarizes their declared public-chain post-processing route and public-source summaries where available. The principal support summaries are the final classification summary, hybrid-stress proxy summary, leave-one-layer-out summary, published-summary reproduction, and signal-target boundary record.

Funding and competing interests..

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