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.

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.