Mass-Sheet Degeneracy in Lensing

• Mass-sheet degeneracy is a non-uniqueness in gravitational lensing where rescaled mass distributions yield identical image configurations while altering source properties.
• The transformation κ' = λκ + (1−λ) preserves image positions yet modifies magnification and time delays, challenging accurate mass and H0 inferences.
• MSD impacts time-delay cosmography, weak-lensing, and multi-plane models, necessitating external data and independent constraints to break the degeneracy.

Mass-sheet degeneracy (MSD), also called the mass-sheet transformation (MST) when written as an explicit map between lens models, is a non-uniqueness of gravitational-lens inference in which different projected mass distributions reproduce the same strong-lensing image configuration while changing the inferred source scale, absolute magnification, and time delays. In its standard single-plane form, the transformed convergence is

$$\kappa_{\lambda}(\boldsymbol{\theta})=\lambda \kappa(\boldsymbol{\theta})+(1-\lambda),$$

with a corresponding source-plane rescaling. MSD is therefore a structural limitation of lens inversion rather than a pathology of any one algorithm. It enters directly into time-delay cosmography, lens-profile inference, multi-source and multi-plane lens modeling, and newer applications such as lensed gravitational waves and strong-lensing tomography (Liesenborgs et al., 2012, Gorenstein, 4 Jan 2026).

1. Standard formulation

For a single deflector plane, the lens equation may be written in reduced form as

$$\boldsymbol{\beta}=\boldsymbol{\theta}-\boldsymbol{\alpha}(\boldsymbol{\theta}),$$

or, in the notation used by Liesenborgs and De Rijcke,

$$\boldsymbol\beta(\boldsymbol\theta)=\boldsymbol\theta-\frac{D_{\rm ds}}{D_{\rm s}}\,\hat{\boldsymbol\alpha}(\boldsymbol\theta),$$

with convergence

$$\kappa(\boldsymbol\theta)=\frac{\Sigma(\boldsymbol\theta)}{\Sigma_{\rm cr}},\qquad \Sigma_{\rm cr}=\frac{c^2}{4\pi G}\frac{D_{\rm s}}{D_{\rm ds}D_{\rm d}}.$$

The standard MST is

$$\kappa'(\boldsymbol\theta)=\lambda\,\kappa(\boldsymbol\theta)+(1-\lambda),$$

equivalently

$$\boldsymbol\alpha'(\boldsymbol\theta)=\lambda \boldsymbol\alpha(\boldsymbol\theta)+(1-\lambda)\boldsymbol\theta,$$

which implies

$$\boldsymbol\beta'(\boldsymbol\theta)=\lambda\,\boldsymbol\beta(\boldsymbol\theta).$$

The same image positions are therefore produced by a uniformly rescaled source plane (Liesenborgs et al., 2012).

Marc V. Gorenstein recasts this transformation in proper-distance coordinates, writing the ray-trace relation as

$$\boldsymbol{\Lambda}(\boldsymbol{b})=\frac{\boldsymbol{b_s}-\boldsymbol{b}}{\mathbb D},\qquad \mathbb D\equiv \frac{D_dD_{ds}}{D_s}.$$

In that form, the term

$-\frac{\boldsymbol b}{\mathbb D}$

is identified with an “Einstein Lens” having surface density

$\sigma_E\equiv \frac{1}{4\pi\mathbb D},$

and deflection law

$$\boldsymbol{\beta}=\boldsymbol{\theta}-\boldsymbol{\alpha}(\boldsymbol{\theta}),$$0

Subtracting this geometric focusing term defines the Image-Selection Relation,

$$\boldsymbol{\beta}=\boldsymbol{\theta}-\boldsymbol{\alpha}(\boldsymbol{\theta}),$$1

whose scaling symmetry yields the MST after the focusing term is restored: $$\boldsymbol{\beta}=\boldsymbol{\theta}-\boldsymbol{\alpha}(\boldsymbol{\theta}),$$2 This optical interpretation does not alter the standard mathematics; it relocates the origin of the degeneracy from the heuristic idea of “adding a mass sheet” to a scaling symmetry of the image-selection part of the lens equation (Gorenstein, 4 Jan 2026).

2. Invariant and non-invariant observables

The defining content of MSD is the split between image observables that remain unchanged and quantities that rescale. In the standard single-plane case, image positions are invariant because each observed image position $$\boldsymbol{\beta}=\boldsymbol{\theta}-\boldsymbol{\alpha}(\boldsymbol{\theta}),$$3 continues to satisfy the transformed lens equation for a rescaled source. Image morphology in the image plane is likewise preserved in the sense relevant to strong-lensing reconstruction, while the inferred unlensed source is isotropically shrunk or expanded (Liesenborgs et al., 2012).

The transformed potential is

$$\boldsymbol{\beta}=\boldsymbol{\theta}-\boldsymbol{\alpha}(\boldsymbol{\theta}),$$4

Magnification rescales as

$$\boldsymbol{\beta}=\boldsymbol{\theta}-\boldsymbol{\alpha}(\boldsymbol{\theta}),$$5

and time delays obey

$$\boldsymbol{\beta}=\boldsymbol{\theta}-\boldsymbol{\alpha}(\boldsymbol{\theta}),$$6

In time-delay cosmography this is often written as

$$\boldsymbol{\beta}=\boldsymbol{\theta}-\boldsymbol{\alpha}(\boldsymbol{\theta}),$$7

so that, to first approximation,

$$\boldsymbol{\beta}=\boldsymbol{\theta}-\boldsymbol{\alpha}(\boldsymbol{\theta}),$$8

These are the basic reasons imaging alone does not determine the absolute mass scale, and why an unaccounted MSD biases $$\boldsymbol{\beta}=\boldsymbol{\theta}-\boldsymbol{\alpha}(\boldsymbol{\theta}),$$9 (Liesenborgs et al., 2012, Chen et al., 2020).

The same logic appears in weak-lensing notation. Under

$$\boldsymbol\beta(\boldsymbol\theta)=\boldsymbol\theta-\frac{D_{\rm ds}}{D_{\rm s}}\,\hat{\boldsymbol\alpha}(\boldsymbol\theta),$$0

the reduced shear and reduced flexions remain invariant,

$$\boldsymbol\beta(\boldsymbol\theta)=\boldsymbol\theta-\frac{D_{\rm ds}}{D_{\rm s}}\,\hat{\boldsymbol\alpha}(\boldsymbol\theta),$$1

while

$$\boldsymbol\beta(\boldsymbol\theta)=\boldsymbol\theta-\frac{D_{\rm ds}}{D_{\rm s}}\,\hat{\boldsymbol\alpha}(\boldsymbol\theta),$$2

This is why shear- or flexion-only reconstructions inherit the same degeneracy, and why magnification information is special (Rexroth et al., 2016).

A recurring misconception is that sufficiently precise imaging, especially of extended Einstein rings, should by itself determine the lens profile. Schneider and Sluse argue the opposite: exquisite imaging still constrains only a finite radial range, and an MST-like family can remain nearly power-law over that range while implying significantly different time-delay scales. Their composite-lens example is reproduced extremely well by a power-law model with the same velocity dispersion but with an $$\boldsymbol\beta(\boldsymbol\theta)=\boldsymbol\theta-\frac{D_{\rm ds}}{D_{\rm s}}\,\hat{\boldsymbol\alpha}(\boldsymbol\theta),$$3 prediction differing by $$\boldsymbol\beta(\boldsymbol\theta)=\boldsymbol\theta-\frac{D_{\rm ds}}{D_{\rm s}}\,\hat{\boldsymbol\alpha}(\boldsymbol\theta),$$4 (Schneider et al., 2013).

3. Generalizations beyond the textbook single-source case

The standard derivation is not the end of the degeneracy. Liesenborgs and De Rijcke construct what they call the most general form of the MSD, showing that multiple sources at different redshifts do not, by themselves, break it. The transformed density at the image locations of source $$\boldsymbol\beta(\boldsymbol\theta)=\boldsymbol\theta-\frac{D_{\rm ds}}{D_{\rm s}}\,\hat{\boldsymbol\alpha}(\boldsymbol\theta),$$5 may satisfy

$$\boldsymbol\beta(\boldsymbol\theta)=\boldsymbol\theta-\frac{D_{\rm ds}}{D_{\rm s}}\,\hat{\boldsymbol\alpha}(\boldsymbol\theta),$$6

so different source planes can be rescaled by different factors. The transformed mass distribution must then change precisely at the image positions, with the amount of change linked to the source rescaling factors. This is one reason the paper ties MSD to unresolved substructure rather than to a purely global change of slope or normalization (Liesenborgs et al., 2012).

The same paper emphasizes a further distinction from the monopole degeneracy. A monopole transformation can alter the mass only in regions away from images; the generalized MSD changes the density at the image positions themselves. Taken together, the two degeneracies imply substantial freedom both between images and at the constrained image locations (Liesenborgs et al., 2012).

For double- and multiple-plane lensing, the degeneracy is recast in terms of effective distance factors. One formulation introduces

$$\boldsymbol\beta(\boldsymbol\theta)=\boldsymbol\theta-\frac{D_{\rm ds}}{D_{\rm s}}\,\hat{\boldsymbol\alpha}(\boldsymbol\theta),$$7

where $$\boldsymbol\beta(\boldsymbol\theta)=\boldsymbol\theta-\frac{D_{\rm ds}}{D_{\rm s}}\,\hat{\boldsymbol\alpha}(\boldsymbol\theta),$$8 contains both line-of-sight and internal sheet contributions. In that framework, internal and external sheets enter on the same footing, and the observable strong-lensing ratios are effective, not bare FRW, angular-diameter-distance ratios (Teodori, 2 Feb 2026). For double-plane time-delay cosmography, line-of-sight corrections also modify the “cosmological scaling factor”

$$\boldsymbol\beta(\boldsymbol\theta)=\boldsymbol\theta-\frac{D_{\rm ds}}{D_{\rm s}}\,\hat{\boldsymbol\alpha}(\boldsymbol\theta),$$9

so fixing $$\kappa(\boldsymbol\theta)=\frac{\Sigma(\boldsymbol\theta)}{\Sigma_{\rm cr}},\qquad \Sigma_{\rm cr}=\frac{c^2}{4\pi G}\frac{D_{\rm s}}{D_{\rm ds}D_{\rm d}}.$$0 directly from background cosmology while ignoring line-of-sight convergence is not generally safe. Using the unfolding relation, the uncertainty relevant for $$\kappa(\boldsymbol\theta)=\frac{\Sigma(\boldsymbol\theta)}{\Sigma_{\rm cr}},\qquad \Sigma_{\rm cr}=\frac{c^2}{4\pi G}\frac{D_{\rm s}}{D_{\rm ds}D_{\rm d}}.$$1 reduces to

$$\kappa(\boldsymbol\theta)=\frac{\Sigma(\boldsymbol\theta)}{\Sigma_{\rm cr}},\qquad \Sigma_{\rm cr}=\frac{c^2}{4\pi G}\frac{D_{\rm s}}{D_{\rm ds}D_{\rm d}}.$$2

i.e. the familiar first-plane mass-sheet freedom plus a line-of-sight contribution up to the second lens plane (Johnson et al., 2 Feb 2026).

Strong-lensing tomography with double-source-plane lenses adds another nuance. In true DSPLs, the intermediate source is itself a deflector, creating a secondary, independent MSD associated with that source plane. The proposed pseudo double-source-plane lens construction removes this intermediate-source mass problem by pairing independent single-source-plane lenses with self-similar deflectors, but it retains the ordinary deflector-galaxy MSD, which must then be inferred statistically at the population level (Sharma et al., 1 Jul 2026).

4. Central role in time-delay cosmography

The main limitation of time-delay cosmography is not the measurement of delays themselves but the conversion of those delays into a Fermat-potential difference under uncertain lens normalization. This is the sense in which several TDCOSMO analyses identify MSD as the dominant systematic in precision $$\kappa(\boldsymbol\theta)=\frac{\Sigma(\boldsymbol\theta)}{\Sigma_{\rm cr}},\qquad \Sigma_{\rm cr}=\frac{c^2}{4\pi G}\frac{D_{\rm s}}{D_{\rm ds}D_{\rm d}}.$$3 inference (Shajib et al., 2023, Chen et al., 2020).

A sharp formulation appears in the distinction between internal and external MST. External convergence $$\kappa(\boldsymbol\theta)=\frac{\Sigma(\boldsymbol\theta)}{\Sigma_{\rm cr}},\qquad \Sigma_{\rm cr}=\frac{c^2}{4\pi G}\frac{D_{\rm s}}{D_{\rm ds}D_{\rm d}}.$$4 acts as a line-of-sight sheet and rescales

$$\kappa(\boldsymbol\theta)=\frac{\Sigma(\boldsymbol\theta)}{\Sigma_{\rm cr}},\qquad \Sigma_{\rm cr}=\frac{c^2}{4\pi G}\frac{D_{\rm s}}{D_{\rm ds}D_{\rm d}}.$$5

while the angular-diameter distance to the lens, $$\kappa(\boldsymbol\theta)=\frac{\Sigma(\boldsymbol\theta)}{\Sigma_{\rm cr}},\qquad \Sigma_{\rm cr}=\frac{c^2}{4\pi G}\frac{D_{\rm s}}{D_{\rm ds}D_{\rm d}}.$$6, is invariant under pure external convergence in the cited formulation. Internal MST, by contrast, alters the lens galaxy’s own mass profile and enters stellar-dynamical modeling through the factor

$$\kappa(\boldsymbol\theta)=\frac{\Sigma(\boldsymbol\theta)}{\Sigma_{\rm cr}},\qquad \Sigma_{\rm cr}=\frac{c^2}{4\pi G}\frac{D_{\rm s}}{D_{\rm ds}D_{\rm d}}.$$7

or its internal+external MST generalization. A central conclusion is that lensing plus a single stellar velocity-dispersion measurement does not break the internal MST in a cosmology-free way; it does so only if the ratio $$\kappa(\boldsymbol\theta)=\frac{\Sigma(\boldsymbol\theta)}{\Sigma_{\rm cr}},\qquad \Sigma_{\rm cr}=\frac{c^2}{4\pi G}\frac{D_{\rm s}}{D_{\rm ds}D_{\rm d}}.$$8 is known or constrained (Chen et al., 2020).

The RXJ1131-1231 analysis of Birrer et al. makes the source-size direction of the degeneracy explicit by using the shapelet scale $$\kappa(\boldsymbol\theta)=\frac{\Sigma(\boldsymbol\theta)}{\Sigma_{\rm cr}},\qquad \Sigma_{\rm cr}=\frac{c^2}{4\pi G}\frac{D_{\rm s}}{D_{\rm ds}D_{\rm d}}.$$9 as a coordinate along the MSD-like family. The paper argues that imaging should be interpreted as conditional on $$\kappa'(\boldsymbol\theta)=\lambda\,\kappa(\boldsymbol\theta)+(1-\lambda),$$0, not as measuring $$\kappa'(\boldsymbol\theta)=\lambda\,\kappa(\boldsymbol\theta)+(1-\lambda),$$1 absolutely, and therefore renormalizes the imaging likelihood across source scales. In that system, the choice of source-size prior is subdominant at present, at roughly $$\kappa'(\boldsymbol\theta)=\lambda\,\kappa(\boldsymbol\theta)+(1-\lambda),$$2 in $$\kappa'(\boldsymbol\theta)=\lambda\,\kappa(\boldsymbol\theta)+(1-\lambda),$$3, whereas the choice of kinematic prior changes $$\kappa'(\boldsymbol\theta)=\lambda\,\kappa(\boldsymbol\theta)+(1-\lambda),$$4 by more than $$\kappa'(\boldsymbol\theta)=\lambda\,\kappa(\boldsymbol\theta)+(1-\lambda),$$5 (Birrer et al., 2015).

The critique by Schneider and Sluse remains foundational in this context. They argue that much of the formal precision in time-delay $$\kappa'(\boldsymbol\theta)=\lambda\,\kappa(\boldsymbol\theta)+(1-\lambda),$$6 estimates relies on the assumption that the mass profile is a perfect power law. Their explicit example shows that a composite baryons+dark-matter lens can be fit extremely well by a power-law model with the same velocity dispersion but with an $$\kappa'(\boldsymbol\theta)=\lambda\,\kappa(\boldsymbol\theta)+(1-\lambda),$$7 prediction differing by $$\kappa'(\boldsymbol\theta)=\lambda\,\kappa(\boldsymbol\theta)+(1-\lambda),$$8 (Schneider et al., 2013).

Resolved stellar kinematics are one route to reducing this model dependence. In RXJ1131−1231, TDCOSMO XII combines HST lens models, time delays, and KCWI spatially resolved spectroscopy to constrain the internal MST with a maximally flexible mass model, obtaining

$$\kappa'(\boldsymbol\theta)=\lambda\,\kappa(\boldsymbol\theta)+(1-\lambda),$$9

and

$$\boldsymbol\alpha'(\boldsymbol\theta)=\lambda \boldsymbol\alpha(\boldsymbol\theta)+(1-\lambda)\boldsymbol\theta,$$0

for flat $$\boldsymbol\alpha'(\boldsymbol\theta)=\lambda \boldsymbol\alpha(\boldsymbol\theta)+(1-\lambda)\boldsymbol\theta,$$1CDM (Shajib et al., 2023). Even then, projection effects matter: TDCOSMO XXI finds that a spherical JAM analysis of spatially unresolved kinematic data introduces a bias of up to $$\boldsymbol\alpha'(\boldsymbol\theta)=\lambda \boldsymbol\alpha(\boldsymbol\theta)+(1-\lambda)\boldsymbol\theta,$$2 in the inferred MSD, while axisymmetric JAM reduces the uncertainty to $$\boldsymbol\alpha'(\boldsymbol\theta)=\lambda \boldsymbol\alpha(\boldsymbol\theta)+(1-\lambda)\boldsymbol\theta,$$3 for axisymmetric galaxies and $$\boldsymbol\alpha'(\boldsymbol\theta)=\lambda \boldsymbol\alpha(\boldsymbol\theta)+(1-\lambda)\boldsymbol\theta,$$4 for triaxial galaxies modeled axisymmetrically (Huang et al., 28 Feb 2025).

5. Methods for constraining or breaking the degeneracy

No single strategy is universal, and the literature increasingly treats “breaking MSD” as the combination of lensing with external information rather than as an internal property of the imaging data.

A first route is additional distance information. One TDCOSMO analysis shows that the internal MST can be constrained in a cosmological-model-independent way by combining time-delay lenses with Type Ia supernovae and BAO, which constrain the shape of the expansion history and therefore $$\boldsymbol\alpha'(\boldsymbol\theta)=\lambda \boldsymbol\alpha(\boldsymbol\theta)+(1-\lambda)\boldsymbol\theta,$$5 without imposing a specific dark-energy model (Chen et al., 2020). A later supernova-based framework combines lensed SNe Ia, which provide absolute magnification information, with unlensed SNe Ia reconstructed using Gaussian processes. In that setup, the two key observables transform as

$$\boldsymbol\alpha'(\boldsymbol\theta)=\lambda \boldsymbol\alpha(\boldsymbol\theta)+(1-\lambda)\boldsymbol\theta,$$6

and, for one simulated system with fiducial $$\boldsymbol\alpha'(\boldsymbol\theta)=\lambda \boldsymbol\alpha(\boldsymbol\theta)+(1-\lambda)\boldsymbol\theta,$$7, the combined constraint is

$$\boldsymbol\alpha'(\boldsymbol\theta)=\lambda \boldsymbol\alpha(\boldsymbol\theta)+(1-\lambda)\boldsymbol\theta,$$8

corresponding to $$\boldsymbol\alpha'(\boldsymbol\theta)=\lambda \boldsymbol\alpha(\boldsymbol\theta)+(1-\lambda)\boldsymbol\theta,$$9 precision (Li et al., 17 Jul 2025).

A second route is absolute magnification from standardizable candles. For microlensed Type Ia supernovae, the magnification transformation

$$\boldsymbol\beta'(\boldsymbol\theta)=\lambda\,\boldsymbol\beta(\boldsymbol\theta).$$0

implies that standardization can constrain $$\boldsymbol\beta'(\boldsymbol\theta)=\lambda\,\boldsymbol\beta(\boldsymbol\theta).$$1, but only if stellar microlensing is sufficiently controlled. One forecast argues that SNe Ia remain standardizable if they do not cross microlensing caustics as they expand, and estimates that by the end of the ten-year LSST survey these systems could test MSD-related systematics in $$\boldsymbol\beta'(\boldsymbol\theta)=\lambda\,\boldsymbol\beta(\boldsymbol\theta).$$2 at the

$$\boldsymbol\beta'(\boldsymbol\theta)=\lambda\,\boldsymbol\beta(\boldsymbol\theta).$$3

level, or

$$\boldsymbol\beta'(\boldsymbol\theta)=\lambda\,\boldsymbol\beta(\boldsymbol\theta).$$4

if time delays can be extracted only from the third of systems with counter-images brighter than $$\boldsymbol\beta'(\boldsymbol\theta)=\lambda\,\boldsymbol\beta(\boldsymbol\theta).$$5 mag (Weisenbach et al., 2024).

A third route is weak lensing. One forecast proposes stacked galaxy–galaxy weak-lensing profiles as an external prior on the approximate-MST freedom of flexible strong-lens models. With current HSC-like data the achievable precision is $$\boldsymbol\beta'(\boldsymbol\theta)=\lambda\,\boldsymbol\beta(\boldsymbol\theta).$$6 on the MSD constraint, while LSST-like surveys are forecast to reach $$\boldsymbol\beta'(\boldsymbol\theta)=\lambda\,\boldsymbol\beta(\boldsymbol\theta).$$7, and $$\boldsymbol\beta'(\boldsymbol\theta)=\lambda\,\boldsymbol\beta(\boldsymbol\theta).$$8 if measurements can be pushed to $$\boldsymbol\beta'(\boldsymbol\theta)=\lambda\,\boldsymbol\beta(\boldsymbol\theta).$$9 (Khadka et al., 2024). In a different weak-lensing context, aperture moments of convergence inferred from magnification bias can be compared with shear and flexion moments; under MSD the convergence moment acquires an additive term while reduced shear and reduced flexion remain invariant, providing an estimator for $$\boldsymbol{\Lambda}(\boldsymbol{b})=\frac{\boldsymbol{b_s}-\boldsymbol{b}}{\mathbb D},\qquad \mathbb D\equiv \frac{D_dD_{ds}}{D_s}.$$0 (Rexroth et al., 2016).

Gravitational-wave lensing produces a more divided picture. For dark-siren galaxy-lens reconstruction, the effective luminosity distance

$$\boldsymbol{\Lambda}(\boldsymbol{b})=\frac{\boldsymbol{b_s}-\boldsymbol{b}}{\mathbb D},\qquad \mathbb D\equiv \frac{D_dD_{ds}}{D_s}.$$1

becomes

$$\boldsymbol{\Lambda}(\boldsymbol{b})=\frac{\boldsymbol{b_s}-\boldsymbol{b}}{\mathbb D},\qquad \mathbb D\equiv \frac{D_dD_{ds}}{D_s}.$$2

under a constant mass sheet, so the GW luminosity distance is entirely degenerate with the sheet; in that framework, $$\boldsymbol{\Lambda}(\boldsymbol{b})=\frac{\boldsymbol{b_s}-\boldsymbol{b}}{\mathbb D},\qquad \mathbb D\equiv \frac{D_dD_{ds}}{D_s}.$$3 remains degenerate with $$\boldsymbol{\Lambda}(\boldsymbol{b})=\frac{\boldsymbol{b_s}-\boldsymbol{b}}{\mathbb D},\qquad \mathbb D\equiv \frac{D_dD_{ds}}{D_s}.$$4 unless lens-galaxy velocity dispersion is added (Poon et al., 2024). By contrast, for point-mass GW lensing in the interference regime, the frequency-dependent fringe pattern shifts because

$$\boldsymbol{\Lambda}(\boldsymbol{b})=\frac{\boldsymbol{b_s}-\boldsymbol{b}}{\mathbb D},\qquad \mathbb D\equiv \frac{D_dD_{ds}}{D_s}.$$5

and a single lensed waveform can then partially or strongly break MSD. The quoted uncertainty on lens mass is $$\boldsymbol{\Lambda}(\boldsymbol{b})=\frac{\boldsymbol{b_s}-\boldsymbol{b}}{\mathbb D},\qquad \mathbb D\equiv \frac{D_dD_{ds}}{D_s}.$$6 at $$\boldsymbol{\Lambda}(\boldsymbol{b})=\frac{\boldsymbol{b_s}-\boldsymbol{b}}{\mathbb D},\qquad \mathbb D\equiv \frac{D_dD_{ds}}{D_s}.$$7 for current sensitivities and can drop to $$\boldsymbol{\Lambda}(\boldsymbol{b})=\frac{\boldsymbol{b_s}-\boldsymbol{b}}{\mathbb D},\qquad \mathbb D\equiv \frac{D_dD_{ds}}{D_s}.$$8 for higher-SNR events (Cremonese et al., 2021). These results are not equivalent statements about the same regime; they refer to different observables and modeling assumptions.

6. Modeling choices, misconceptions, and current interpretation

Several recurring claims in the literature are explicitly challenged by the papers discussed here. The statement that “multiple sources at different redshifts break the MSD” is rejected in the generalized construction of Liesenborgs and De Rijcke: multiple redshifts only invalidate the simplest literal constant-sheet construction, not the underlying degeneracy (Liesenborgs et al., 2012). The statement that environment characterization breaks the degeneracy is likewise too strong: Schneider and Sluse argue that internal mass-profile freedom is not physically identical to extrinsic convergence, so characterizing the line of sight does not in general remove the internal MST-like ambiguity (Schneider et al., 2013).

A further misconception is that the practical bias of a fit can be predicted by transforming the true profile into a power law near the image radius and reading off $$\boldsymbol{\Lambda}(\boldsymbol{b})=\frac{\boldsymbol{b_s}-\boldsymbol{b}}{\mathbb D},\qquad \mathbb D\equiv \frac{D_dD_{ds}}{D_s}.$$9. Numerical experiments with two-component mock galaxies show that the recovered bias in $-\frac{\boldsymbol b}{\mathbb D}$0 does not follow that simple picture. Fitting quads generated from two-component profiles with power-law ellipse+shear models, and then “informing” the fit with the true slope near the image radius, can introduce biases from $-\frac{\boldsymbol b}{\mathbb D}$1 to $-\frac{\boldsymbol b}{\mathbb D}$2 depending on the model (Gomer et al., 2019). This suggests that commonplace modeling pipelines select one branch of a broader degeneracy family in a way that is more complicated than the local-MST heuristic implies.

The broader methodological conclusion is that MSD is often broken artificially by assumptions built into inversion methods rather than by the lensing data themselves. Liesenborgs and De Rijcke explicitly mention parametric models, basis-function methods, regularization schemes, LensTool, light-traces-mass approaches, smooth overlapping basis functions, and PixeLens with explicit priors as examples in which smoothness assumptions restrict the allowed substructure and thereby select one among many degenerate mass maps (Liesenborgs et al., 2012). This does not make such methods unusable; it means their priors must be interpreted as part of the inference.

Taken together, the modern literature treats MSD less as a single algebraic curiosity than as a hierarchy of related non-uniquenesses. In the standard single-plane case it is the transformation

$-\frac{\boldsymbol b}{\mathbb D}$3

or its equivalent variants. In generalized multi-source and multi-plane settings it becomes a family of transformations involving image-position-local density changes, effective distance ratios, and line-of-sight dressings. In time-delay cosmography it remains a direct multiplicative uncertainty on $-\frac{\boldsymbol b}{\mathbb D}$4 and hence on $-\frac{\boldsymbol b}{\mathbb D}$5. The practical lesson is therefore consistent across otherwise different applications: image configurations alone do not determine the absolute mass scale, and robust inference requires external information that is genuinely sensitive to what the MST leaves free.