Strong-Lens Time-Delay Cosmography in the CHC Framework: Fermat Potentials, Distance Factors, and Inference Control

Strong-lens time-delay cosmography uses multiply imaged variable sources to infer absolute distances. For a given lens system, the same arrival-time surface determines both the stationary image configuration and the relative delays between images [citation]. Current analyses close the inference problem with four primary ingredients: high-resolution imaging, measured delays, stellar kinematics, and control of line-of-sight structure [citation]. Recent TDCOSMO work has sharpened this chain through sub-percent velocity-dispersion measurements, updated SL2S dispersions, explicit projection-effect calibration, JWST-based lens modeling, resolved JWST-NIRSpec kinematics, and blinded end-to-end inference [citation].

The deflector is described by a total projected surface density; any baryonic-plus-residual split is treated only as a declared bookkeeping decomposition. The homogeneous background is described by $\Hbg(z)$TeX\Hbg(z), and the propagation sector by the distance map generated by $c(z)=\cinf\sqrt{1-\Xicos(z)}$TeXc(z)=\cinf\sqrt{1-\Xicos(z)}. Within the thin-lens regime considered here, the standard Fermat structure is retained. Any nonzero propagation correction is interpreted only after the lens-family, kinematic, and line-of-sight controls are satisfied.

Strong-lens time-delay observables therefore mix local deflector structure with background and propagation distance factors. The lens sector enters through the potential and its admissible perturbations, whereas the background and propagation sectors enter through the angular-diameter distances and the time-delay distance. The deflector surface-density bookkeeping decomposition and the distance factors therefore enter only through the adopted strong-lens inference chain. The total deflector sector, the homogeneous background branch, and the propagation-distance map are adopted here as imported inputs from the cited results; they are not rederived locally from the strong-lens data.

The analysis is restricted to variable-source strong lenses in the thin-lens, small-angle regime [citation]. Weak-lensing and growth observables are not used, no chronometric inference enters this strong-lens reading, and current data are not assumed to require a nonstandard propagation sector. Any secondary residual channel is subordinate to the primary imaging-delay-kinematic-line-of-sight closure. The standard limit $\Xicos\to 0$TeX\Xicos\to 0 remains part of the admissible space. Throughout, the adopted thin-lens strong-lens inference class consists of variable-source strong-lens systems read within that thin-lens, small-angle regime and through the imaging-delay-kinematic-line-of-sight decomposition stated below. All downstream claims are read only relative to this adopted thin-lens strong-lens inference class. The adopted observational channel set is given by the imaging, delay, stellar-kinematic, and line-of-sight channels entering the primary inference stack below, with any late residual treated as secondary on that same channel set. All downstream claims are read only relative to this adopted observational channel set. The adopted analysis is limited to one declared thin-lens variable-source strong-lens class with one declared observational channel stack. It does not define a widened observational-channel window, a widened reconstruction or tomography class, or any observational claim outside the declared variable-source strong-lens regime.

The TDC public-source checks have only bounded residual and stress readings. TDC-VP1 carries the bounded gate label $\mathrm{BOUNDED-NULLOVERTURN}$TeX\mathrm{BOUNDED-NULLOVERTURN} on the H0LiCOW public-posterior residual-gate route, meaning that the null propagation branch remains admissible on that declared public object. TDC-VP2 carries the bounded relaxed-proxy gate label $\mathrm{RELAXED\text{-}PROXY\text{-}NONOVERTURN}\;\mathrm{(proxy\text{-}level)}$TeX\mathrm{RELAXED\text{-}PROXY\text{-}NONOVERTURN}\;\mathrm{(proxy\text{-}level)} on the TDCOSMO-IV public-chain relaxed-profile stress route, meaning that those stress layers do not overturn the VP1 null-admissible reading. These checks are residual and stress gates on the adopted strong-lens channel set; they are not a nonzero-propagation detection, an $\eta$TeX\eta measurement, a phase-component separation, a full hierarchical cosmography closure, a TDCOSMO-IV sampler rerun, or a TDCOSMO-2025 cosmology analysis.

Total deflector sector

The strong-lens deflector sector is represented by the total projected lensing surface density

\Sigmalens(\thetavec)\equiv \Sigma_{\mathrm{tot}}(\thetavec).

The strong-lens data constrain only this total projected lensing object. They are not used here to identify baryonic, residual, phase-side, or other subcomponents as separately measured lens-plane surface densities. When a baryonic-plus-residual decomposition is declared for bookkeeping, it may be written as

\Sigmalens(\thetavec)\equiv \Sigmab(\thetavec)+\Sigmaph(\thetavec),

but the split in reference is not a component-separation result supplied by the strong-lens data.

A secondary residual perturbation channel may be written as

\psilens(\thetavec)=\psilens^{(0)}(\thetavec)+\delta\psi_{\mathrm{res}}(\thetavec),

with $\delta\psi_{\mathrm{res}}$TeX\delta\psi_{\mathrm{res}} a residual lens-potential correction on the adopted lens family. That channel is admitted only as a late residual check, after the primary imaging, delay, kinematic, and line-of-sight controls have been satisfied.

Background and propagation sectors

The homogeneous background is described by an expansion history $\Hbg(z)$TeX\Hbg(z) [citation]. The propagation law is taken to be

c(z)=\cinf\sqrt{1-\Xicos(z)},

with $\Xicos(z)$TeX\Xicos(z) the homogeneous propagation variable [citation]. Within the flat-background restriction used here, the corresponding angular-diameter distance between redshifts $z_1$TeXz_1 and $z_2>z_1$TeXz_2>z_1 is

D_A^{\mathrm{(CHC)}}(z_1,z_2)=
\frac{1}{1+z_2}\int_{z_1}^{z_2}
\frac{\cinf\sqrt{1-\Xicos(z)}}{\Hbg(z)}\,dz.

The standard limit is recovered when $\Xicos\to 0$TeX\Xicos\to 0. For the strong-lens application considered here, the propagation sector enters only through the admissible distance map of reference.

For a deflector at redshift $z_{\rm d}$TeXz_{\rm d} and source at $z_{\rm s}$TeXz_{\rm s}, define

\Dd \equiv D_A^{\mathrm{(CHC)}}(0,z_{\rm d}),
\qquad
\Ds \equiv D_A^{\mathrm{(CHC)}}(0,z_{\rm s}),
\qquad
\Dds \equiv D_A^{\mathrm{(CHC)}}(z_{\rm d},z_{\rm s}),

and the CHC time-delay distance

\DdtCHC=(1+z_{\rm d})\frac{\Dd\Ds}{\Dds}.

The corresponding critical surface density is

\SigmaCrit^{\mathrm{(CHC)}}=
\frac{\cinf^{2}}{4\pi G}\frac{\Ds}{\Dd\Dds},

In the adopted thin-lens class, $\cinf^2$TeX\cinf^2 is the fixed local normalization of the lens-plane weak-field law; any additional local propagation renormalization of the lens-plane kernel would define a different class and is excluded here. Thus

\kappalens(\thetavec)=\frac{\Sigmalens(\thetavec)}{\SigmaCrit^{\mathrm{(CHC)}}}.

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Thin-lens regime with imported distance factors

The adopted regime is the standard geometric-optics, weak-field, thin-lens, small-angle strong-lensing limit with the distance factors $\Dd$TeX\Dd, $\Ds$TeX\Ds, $\Dds$TeX\Dds, and $\DdtCHC$TeX\DdtCHC [citation]. No additional independent lens-plane propagation functional is introduced. If a system required an extra local lens-plane propagation term beyond the standard arrival-time surface, it would lie outside the regime considered here.

In this regime, reference, reference, and reference retain the standard weak-field thin-lens normalization once the distance factors are fixed. Any model that modifies the local deflection law or the lens-plane normalization beyond this mapping lies outside the regime considered here.

This restriction does not remove line-of-sight structure from the analysis. Line-of-sight effects enter through explicit additional deflectors when nonlinear perturbers are important, and through external-convergence statistics when weaker perturbers can be absorbed into an environment correction [citation].

The dimensionless lensing potential is defined in the standard way by the lens-plane Poisson equation

\nabla_{\thetavec}^{2}\psilens(\thetavec)=2\kappalens(\thetavec),

and the reduced deflection field is

\alphavec(\thetavec)=\nabla_{\thetavec}\psilens(\thetavec).

The arrival-time surface is then

\tau(\thetavec,\betavec)=
\frac{\DdtCHC}{\cinf}\,\phifermat(\thetavec,\betavec),

with Fermat potential

\phifermat(\thetavec,\betavec)
=
\frac{1}{2}\lvert\thetavec-\betavec\rvert^{2}
-\psilens(\thetavec).

proposition: Same-Fermat structure in the adopted thin-lens regime. Within the adopted thin-lens regime, image positions and relative time delays are two readouts of the same Fermat potential. Specifically:

- image positions are stationary points of $\tau$TeX\tau, equation \nabla_tau(,)=0 =-\nabla_(); equation - the relative delay between images $i$TeXi and $j$TeXj of the same source is equation t_ij = ()/() [ (\thetavec_i,)-(\thetavec_j,) ]. equation

proof. Since $\DdtCHC/\cinf$TeX\DdtCHC/\cinf is independent of $\thetavec$TeX\thetavec for fixed $(z_{\rm d},z_{\rm s})$TeX(z_{\rm d},z_{\rm s}), stationarity of $\tau$TeX\tau is equivalent to stationarity of $\phifermat$TeX\phifermat. Differentiating reference yields reference. Taking the difference of reference between two stationary images gives reference.

remark. reference is not by itself a closure of cosmographic inference. It fixes the structural coupling of image geometry and delays, but it does not remove internal lens-family degeneracies, external convergence, or nonlinear perturbations from nearby structures. Those controls are stated separately in reference. The proposition is read only on the adopted thin-lens strong-lens inference class and the adopted observational channel set fixed in the Introduction, and the downstream inference statements below are not promoted to universal exact identities outside that class.

The lens sector therefore cannot be tuned independently against image geometry and delay data. Any admissible model must preserve the same Fermat potential for both observables. At the same time, the same-Fermat statement does not determine the absolute distance factor by itself; that requires stellar kinematics and line-of-sight control [citation].

Imaging and delays alone do not close the inference problem. Modern time-delay cosmography requires the full observable stack summarized in Table reference. Following current practice, the primary cosmographic chain is imaging plus delays plus stellar kinematics plus line-of-sight control; any late residual check is secondary to that four-part closure [citation].

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Image geometry and relative delays

Imaging constrains the admissible family of lens potentials through reference. Relative delays then couple that same family to $\DdtCHC$TeX\DdtCHC through reference. The deflector sector, the background history, and the propagation sector meet here: the lens sector enters through $\psilens$TeX\psilens, the background history through the redshift dependence of the angular-diameter distances, and the propagation sector through $\Xicos$TeX\Xicos in reference.

The same-Fermat structure prevents a reading in which image positions are fitted by one lens potential while the delays are interpreted with another. Any distance-side interpretation begins only after a single admissible lens family has survived both observables.

Kinematics and absolute distances

Imaging and delays do not uniquely fix the cosmographic factor. Stellar kinematics are required to control internal mass-profile freedom and related mass-sheet directions [citation]. Recent analyses show that improved integrated and spatially resolved kinematics, together with explicit treatment of projection effects, materially tighten time-delay inference and the interpretation of mass-sheet-transformation-related systematics [citation]. Where the data are sufficiently constraining, kinematics can also provide information on the angular-diameter distance to the deflector in addition to the time-delay distance [citation]. The recent HE 1104-1805 analysis makes the same point in concrete form by combining time delays, lens modeling, stellar kinematics, and line-of-sight convergence as four distinct ingredients in one closed inference chain [citation]. Recent JWST-NIRSpec resolved kinematics strengthen the same control logic by tightening the deflector-side dynamical constraint without altering the underlying same-Fermat structure [citation].

Once the lens-family and kinematic controls are in place, the distance factor to be read is $\DdtCHC$TeX\DdtCHC, with the propagation sector entering only through the distance map. Accordingly, neither image geometry nor time delays alone are used here to infer a nonzero $\Xicos(z)$TeX\Xicos(z); any propagation-side interpretation begins only after the internal-lens and line-of-sight conditions are satisfied.

Secondary residuals

Secondary residuals are admitted only after the primary smooth-lens model has been tested against imaging, delays, kinematics, and line-of-sight information. They are not treated as co-equal with the primary cosmographic ingredients. To first order, a perturbation $\delta\psi_{\mathrm{res}}$TeX\delta\psi_{\mathrm{res}} shifts the relative delay by

\delta(\Delta t_{ij})
=
-\frac{\DdtCHC}{\cinf}
\left[
\delta\psi_{\mathrm{res}}(\thetavec_i)-\delta\psi_{\mathrm{res}}(\thetavec_j)
\right].

Thus the residual channel remains tied to the same lens-potential sector. It is not a separate distance estimator.

Time-delay cosmography is anchored primarily by the smooth-lens plus kinematic plus line-of-sight stack. Flux-ratio residuals in quasar lenses can also be contaminated by stellar microlensing and source-size effects [citation]. Source-side residuals are therefore treated as a late residual check rather than as a primary cosmographic input.

The sources of non-uniqueness must be controlled explicitly.

Internal lens-family control: mass-sheet and related directions

The best-known non-uniqueness is the mass-sheet transformation

\kappa_{\lambda}(\thetavec)=
\lambda\,\kappalens(\thetavec)+(1-\lambda),

with transformed lensing potential

{\psilens}_{\lambda}(\thetavec)=
\lambda\,\psilens(\thetavec)
+\frac{1-\lambda}{2}\lvert\thetavec\rvert^{2}.

Under the associated source rescaling, image positions are preserved, while the Fermat-potential difference scales as

(\Delta\phifermat)_{\lambda}=\lambda\,\Delta\phifermat.

For fixed measured delays, the corresponding model time-delay distance scales as

\Ddt^{\mathrm{model}}\rightarrow \lambda^{-1}\,\Ddt^{\mathrm{model}}.

Equivalently, the inferred Hubble scale transforms in the opposite direction, $H_0^{\mathrm{model}}\rightarrow \lambda H_0^{\mathrm{model}}$TeXH_0^{\mathrm{model}}\rightarrow \lambda H_0^{\mathrm{model}} when all other ingredients are held fixed [citation]. Imaging and delays alone therefore do not close the cosmographic inference.

The mass-sheet transformation is representative rather than exhaustive. Source-position transformations and related lens-family directions can preserve much of the imaging information while modifying the inferred Fermat structure and the recovered distance scale [citation]. The internal-lens admissibility condition is that imaging, delays, and stellar kinematics reduce the internal lens-family degeneracy class enough that the inferred $\DdtCHC$TeX\DdtCHC is stable within the adopted model family. Blinded workflows and explicit projection-effect calibration strengthen that requirement in current practice rather than replacing it [citation].

Line-of-sight control: external convergence and nearby perturbers

Line-of-sight structure must be separated into weak perturbations that can be summarized statistically and nonlinear perturbers that must be modeled explicitly [citation]. For the external-convergence component, the standard correction is

\Ddt^{\mathrm{true}}=
\frac{\Ddt^{\mathrm{model}}}{1-\kext},

Here $\Ddt^{\mathrm{model}}$TeX\Ddt^{\mathrm{model}} and $\Ddt^{\mathrm{true}}$TeX\Ddt^{\mathrm{true}} denote the standard lensing-inference bookkeeping variables for the same CHC time-delay distance under, respectively, uncorrected and line-of-sight-corrected modeling conventions; no additional distance object is introduced. Accordingly, positive $\kext$TeX\kext lowers the modeled distance and raises the uncorrected $H_0$TeXH_0 inference [citation]. This is not an optional refinement at current precision.

Nearby or sufficiently massive perturbers are different. They cannot in general be absorbed into a single external-convergence correction without bias. Instead they must be promoted to explicit additional deflectors or treated with an equivalent multi-plane model [citation]. Diagnostics such as the flexion-shift ranking identify which perturbers belong to that explicit class [citation].

The line-of-sight admissibility condition is that far line-of-sight structure be controlled through an external-convergence posterior and that all perturbers with non-negligible nonlinear impact be modeled explicitly or removed from the admissible sample.

Propagation-distance admissibility

The propagation sector is not used as a free distortion layer. Any nonzero $\Xicos(z)$TeX\Xicos(z) entering reference and reference must lie within the locally admissible propagation branch and within the restrictions required by the lensing application considered here. In particular, on the interval relevant to a given system, $1-\Xicos(z)$TeX1-\Xicos(z) must remain positive so that $c(z)$TeXc(z), $\Dd$TeX\Dd, $\Ds$TeX\Ds, $\Dds$TeX\Dds, and $\DdtCHC$TeX\DdtCHC are real and finite, and the thin-lens normalization through $\SigmaCrit^{\mathrm{(CHC)}}$TeX\SigmaCrit^{\mathrm{(CHC)}} remains well defined. The propagation-distance admissibility condition is that any nontrivial correction in strong-lens time-delay cosmography be carried by such an admissible $\Xicos(z)$TeX\Xicos(z).

If the distance correction required by the lens data cannot be realized within that admissible class, the propagation-side reading is rejected and the system must be read on the deflector and background sectors alone.

Secondary residual admissibility

The secondary residual channel is admitted only if it does work that the smooth model does not. The corresponding admissibility condition is that a late residual improve the joint imaging-plus-delay fit without spoiling the kinematic and line-of-sight controls of the same system.

If a demanded residual requires changes that violate the smooth-lens closure or the environment constraints, the residual channel fails for that system.

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

The TDC public-source comparisons are used only as bounded residual and stress gates. They do not alter the same-Fermat proposition, the thin-lens equations, the total projected-density reading, or the standard $\Xicos\to0$TeX\Xicos\to0 admissible limit.

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The only consequence for the time-delay-cosmography analysis is the bounded reading that the declared public posterior and public chain objects do not force a nonzero propagation correction under the VP1/VP2 conditions. The companion records are not data-selected CHC propagation detections and are not replacements for current strong-lens cosmology pipelines.

The admissible scope is fixed by explicit failure conditions.

F1: same-Fermat failure..

If no single $\phifermat$TeX\phifermat simultaneously fits imaging and the measured delays of a variable-source strong lens within one adopted lens family, the lens-sector reading fails for that system.

F2: internal lens-family non-closure..

If stellar kinematics and lens modeling do not constrain the internal degeneracy directions well enough to stabilize the inferred $\Ddt$TeX\Ddt, no closed cosmographic inference is available from that system.

F3: line-of-sight non-closure..

If the external-convergence posterior is not under control, or if nearby perturbers with non-negligible nonlinear impact cannot be modeled explicitly, the absolute-distance reading is not admissible.

F4: propagation-distance incompatibility..

If the value or redshift history of $\Xicos(z)$TeX\Xicos(z) needed to fit the inferred $\DdtCHC$TeX\DdtCHC fails to keep $c(z)$TeXc(z), $\Dd$TeX\Dd, $\Ds$TeX\Ds, $\Dds$TeX\Dds, and $\DdtCHC$TeX\DdtCHC real and finite on the relevant interval, or lies outside the locally admissible propagation branch, the propagation-side interpretation is rejected.

F5: late residual failure..

If a demanded residual cannot improve the joint imaging-plus-delay fit while preserving the kinematic and line-of-sight controls, the secondary residual channel fails for that system.

F6: standard-limit null..

If the preferred fit is $\Xicos\to 0$TeX\Xicos\to 0 and no late residual is required, the system reduces to a total-deflector reading with standard propagation distances. This is a null outcome, not a contradiction.

F7: companion-check overread..

A public-posterior or public-chain stress check is outside the admitted interpretation of the adopted TDC reading if it is used to claim an $\eta$TeX\eta detection, a separated phase component, a nonzero propagation discovery, a full lensing-cosmology closure, a full TDCOSMO hierarchical-sampler reproduction, or a TDCOSMO-2025 cosmology result without the corresponding declared data object and inference stack. This paper uses the companion VP1 and VP2 checks only for their stated null-admissible and proxy-level stress status on their declared public objects.

Within the adopted thin-lens inference class, strong-lens time-delay cosmography retains the standard thin-lens backbone. The deflector is carried by the total projected lensing surface density $\Sigmalens$TeX\Sigmalens, the background by $\Hbg(z)$TeX\Hbg(z), and the propagation sector by the distance map generated by $c(z)=\cinf\sqrt{1-\Xicos(z)}$TeXc(z)=\cinf\sqrt{1-\Xicos(z)}. Within the adopted regime, imaging and relative delays remain two readouts of one Fermat potential, while inference of $\DdtCHC$TeX\DdtCHC remains contingent on control of internal lens-family freedom, external convergence, and nearby perturbers.

Within the adopted regime, the deflector surface-density decomposition and the distance factors enter only as declared objects in the standard inference stack. No additional local lens-plane propagation term is introduced in this regime. The H0LiCOW public-posterior residual gate gives $\mathrm{BOUNDED-NULLOVERTURN}$TeX\mathrm{BOUNDED-NULLOVERTURN}: the declared null propagation branch remains admissible and a nonzero propagation correction is not required by the public posterior data. The TDCOSMO-IV public-chain relaxed-profile stress gate gives $\mathrm{RELAXED\text{-}PROXY\text{-}NONOVERTURN}\;\mathrm{(proxy\text{-}level)}$TeX\mathrm{RELAXED\text{-}PROXY\text{-}NONOVERTURN}\;\mathrm{(proxy\text{-}level)}: relaxed mass-profile hierarchy stress layers do not overturn the VP1 null-admissible reading at the declared proxy level. The standard limit $\Xicos\to0$TeX\Xicos\to0 remains admissible.

These results are not a nonzero propagation-signal detection, not an $\eta$TeX\eta measurement, not a separate phase-component inference, not a full lensing-cosmology closure, not a full TDCOSMO-IV hierarchical-sampler reproduction, and not a TDCOSMO-2025 cosmology analysis. All downstream inference statements above are read only on the adopted thin-lens strong-lens inference class and the adopted observational channel set fixed in the Introduction. Any future propagation-side signal reading must first survive same-Fermat, internal-lens, line-of-sight, propagation-distance, public-posterior residual-gate, and public-chain stress admissibility on one declared observational map.

Funding and competing interests..

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