LensMC: Bayesian Shear Measurement for Euclid

• LensMC is a Bayesian forward-modeling method that represents galaxy images as PSF-convolved profiles for precise weak-lensing shear measurements.
• It employs MCMC sampling and marginalizes over nuisance parameters, ensuring robust ellipticity estimation even with blended sources.
• Validated through simulations and on-sky data, LensMC provides competitive calibration and reliability for Euclid cluster-lensing and dark-energy analyses.

LensMC is a weak-lensing shear measurement method developed for Euclid and Stage-IV surveys. It is based on forward modelling of galaxy images, explicit convolution with the point spread function (PSF), Markov Chain Monte Carlo (MCMC) sampling of galaxy-parameter posteriors, and marginalisation over nuisance parameters. In the Euclid programme, LensMC has a dual role: it is the designated galaxy-shape measurement method for DR1, and it has already been exercised in both pre-launch Euclid-like simulations and on early on-sky data, including the Abell 2390 Early Release Observations and the Euclid Quick Release 1 cluster-lensing catalogue (Collaboration et al., 2024, Schrabback et al., 10 Jul 2025, Congedo et al., 18 Jun 2026).

1. Scientific role within Euclid weak lensing

LensMC was designed for the Euclid VIS imager, where the PSF is diffraction-limited, chromatic, spatially varying across the field, and undersampled, with a size comparable to many galaxies. The method targets the accuracy and precision needed for percent-level dark-energy constraints and is formulated to operate at Euclid scale, namely on the order of $1.5$ billion galaxies. The original Euclid preparation study therefore framed LensMC as a shear-measurement method tailored to explicit PSF treatment, realistic neighbour handling, and statistically grounded uncertainty propagation (Collaboration et al., 2024).

Within Euclid’s operational weak-lensing pipeline, LensMC has moved from simulation validation to survey deployment. In the Abell 2390 Early Release Observations analysis, it was used alongside KSB+ and SourceXtractor++ as one of three independent shape-measurement algorithms, and that study identifies LensMC as Euclid’s designated galaxy-shape measurement method for DR1 (Schrabback et al., 10 Jul 2025). In Quick Release 1, LensMC was used to generate the first Euclid shear-measurement catalogue produced with the method in anticipation of DR1, covering the three Euclid Deep Field mosaics over approximately $63\,\mathrm{deg}^2$ (Congedo et al., 18 Jun 2026).

This positioning is methodologically significant. LensMC is not an internally calibrated shear-response scheme of the metacalibration type, nor a pure moment-based estimator of the KSB family. Instead, it is a Bayesian forward-modelling framework whose calibration strategy relies on dedicated image simulations and, in some data releases, empirical corrections. A plausible implication is that LensMC is intended to be especially useful where PSF convolution, blending, and heterogeneous morphology make pixel-level likelihood modelling preferable to lower-dimensional summaries.

2. Forward model, likelihood, and MCMC inference

At its core, LensMC models the observed image as a PSF-convolved galaxy profile plus noise,

$$I_{\rm obs}(\mathbf{x}) = [P(\mathbf{x}) * M(\mathbf{x};\boldsymbol{\theta})] + n(\mathbf{x}).$$

The Euclid preparation study describes the galaxy model as a linear mixture of two co-centred circular Sérsic-type components, disc plus bulge, rendered isotropically and then anisotropically distorted. In that baseline description, the disc is exponential, the bulge has fixed $n_b=1$, the size ratio is fixed to $r_h/r_e=0.15$, and profiles are truncated at $r_{\max}/r_e=4.5$. The Abell 2390 analysis also describes LensMC as fitting a two-component bulge+disc profile convolved with the PSF, but there the profile is reported as de Vaucouleurs bulge plus exponential disc (Collaboration et al., 2024, Schrabback et al., 10 Jul 2025).

The ellipticity enters through the model geometry. In the Euclid preparation paper, the isotropic template is transformed by an anisotropic distortion matrix,

$\tens{S}=\frac{\bar{r}_0}{q_\epsilon\,r_0} \begin{pmatrix} 1-\epsilon_1 & -\epsilon_2 \ -\epsilon_2 & 1+\epsilon_1 \end{pmatrix}, \qquad q_\epsilon=1-|\epsilon|.$

Rendering and convolution are performed in Fourier space. For isotropic $I(r)$, the 2D Fourier transform reduces to the 1D Hankel transform

$$\tilde{I}(k)=2\pi\!\int_0^\infty I(r)\,J_0(kr)\,r\,{\rm d}r,$$

after which shear, centroid shifts, and PSF convolution are applied efficiently before inverse transformation to real space (Collaboration et al., 2024).

Inference is Bayesian. With pixel data vector $\vec{D}$, ellipticity $63\,\mathrm{deg}^2$0, non-linear nuisance parameters $63\,\mathrm{deg}^2$1, and linear flux parameters $63\,\mathrm{deg}^2$2, LensMC defines

$63\,\mathrm{deg}^2$3

Assuming Gaussian pixel noise,

$63\,\mathrm{deg}^2$4

Because the model is linear in the bulge and disc fluxes, those fluxes can be marginalised analytically. LensMC then estimates ellipticity using the posterior mean,

$63\,\mathrm{deg}^2$5

This posterior-mean estimator is central to the method’s stated robustness against overfitting and non-Gaussian posteriors (Collaboration et al., 2024).

Sampling is performed with an improved Metropolis–Hastings scheme. The preparation study reports an initial deterministic maximisation, a burn-in phase with a cooling schedule, adaptive proposal-covariance updates every $63\,\mathrm{deg}^2$6 samples, $63\,\mathrm{deg}^2$7 burn-in samples, and $63\,\mathrm{deg}^2$8 kept samples. Joint likelihoods are used for neighbour groups, typically with $63\,\mathrm{deg}^2$9, so that overlapping objects are fitted simultaneously rather than circularised by masking or independent treatment (Collaboration et al., 2024).

3. Catalogue construction, blending control, and object-level outputs

LensMC produces per-object shape catalogues rather than only stacked shear estimates. In the Abell 2390 analysis, the method returns ellipticity components $$I_{\rm obs}(\mathbf{x}) = [P(\mathbf{x}) * M(\mathbf{x};\boldsymbol{\theta})] + n(\mathbf{x}).$$0, sizes, fluxes or magnitudes, positions, parameter uncertainties, reduced $$I_{\rm obs}(\mathbf{x}) = [P(\mathbf{x}) * M(\mathbf{x};\boldsymbol{\theta})] + n(\mathbf{x}).$$1, and a shear weight per object, together with quality flags; flagged objects can be assigned zero weight (Schrabback et al., 10 Jul 2025). In Q1, the merged catalogue includes right ascension and declination, ellipticities, the flux-averaged half-light radius $$I_{\rm obs}(\mathbf{x}) = [P(\mathbf{x}) * M(\mathbf{x};\boldsymbol{\theta})] + n(\mathbf{x}).$$2, $$I_{\rm obs}(\mathbf{x}) = [P(\mathbf{x}) * M(\mathbf{x};\boldsymbol{\theta})] + n(\mathbf{x}).$$3 magnitude, per-object weight $$I_{\rm obs}(\mathbf{x}) = [P(\mathbf{x}) * M(\mathbf{x};\boldsymbol{\theta})] + n(\mathbf{x}).$$4, quality flags, and selected MER and PHZ columns (Congedo et al., 18 Jun 2026).

Blending control is an explicit part of the LensMC workflow. The Euclid preparation paper groups neighbours with a friends-of-friends scale $$I_{\rm obs}(\mathbf{x}) = [P(\mathbf{x}) * M(\mathbf{x};\boldsymbol{\theta})] + n(\mathbf{x}).$$5, uses segmentation maps dilated by one pixel to mask objects outside the group, and jointly fits close pairs to mitigate neighbour bias (Collaboration et al., 2024). The Abell 2390 implementation likewise groups nearby objects with friends-of-friends at $$I_{\rm obs}(\mathbf{x}) = [P(\mathbf{x}) * M(\mathbf{x};\boldsymbol{\theta})] + n(\mathbf{x}).$$6 arcsec scale, jointly fits blended systems, applies robust sigma-clipping to mitigate residual cosmic rays and image features, uses an external segmentation map and mask, and subtracts local background gradients across $$I_{\rm obs}(\mathbf{x}) = [P(\mathbf{x}) * M(\mathbf{x};\boldsymbol{\theta})] + n(\mathbf{x}).$$7-pixel postage stamps (Schrabback et al., 10 Jul 2025). In Q1, deblending and spurious-detection control rely on the MER pipeline flags and a pre-trained spurious classifier, while LensMC is run on all MER detections and science analyses impose catalogue-level quality cuts (Congedo et al., 18 Jun 2026).

Star–galaxy separation and robustness cuts differ across analyses. In the preparation simulations, star–galaxy separation uses a measured size threshold $$I_{\rm obs}(\mathbf{x}) = [P(\mathbf{x}) * M(\mathbf{x};\boldsymbol{\theta})] + n(\mathbf{x}).$$8, with true-positive rate approximately $$I_{\rm obs}(\mathbf{x}) = [P(\mathbf{x}) * M(\mathbf{x};\boldsymbol{\theta})] + n(\mathbf{x}).$$9 and false-positive rate approximately $n_b=1$0 at $n_b=1$1 (Collaboration et al., 2024). In the Abell 2390 ERO analysis, objects are retained if the flux-averaged half-light radius exceeds $n_b=1$2, and faint galaxies with unrealistically large size estimates are removed using the magnitude-dependent bound

$n_b=1$3

In Q1, star–galaxy separation uses the magnitude-independent threshold $n_b=1$4, and analyses may optionally impose $n_b=1$5 to suppress a large-size tail that is mostly spurious or poorly modelled (Schrabback et al., 10 Jul 2025, Congedo et al., 18 Jun 2026).

Reported source densities illustrate the method’s survey role. In Euclid-like simulations LensMC measured objects with density approximately $n_b=1$6 for $n_b=1$7 across $n_b=1$8 (Collaboration et al., 2024). In Abell 2390, after masking and selections, the catalogue reached $n_b=1$9 in total, $r_h/r_e=0.15$0 in the weak-lensing magnitude range $r_h/r_e=0.15$1, and $r_h/r_e=0.15$2 for $r_h/r_e=0.15$3 (Schrabback et al., 10 Jul 2025). In Q1, the catalogue is reported as complete to $r_h/r_e=0.15$4, with $r_h/r_e=0.15$5 for $r_h/r_e=0.15$6 and $r_h/r_e=0.15$7 for $r_h/r_e=0.15$8 after quality cuts (Congedo et al., 18 Jun 2026).

4. Bias model, calibration strategy, and systematics

LensMC adopts the standard linear shear-bias model

$r_h/r_e=0.15$9

with multiplicative bias $r_{\max}/r_e=4.5$0, additive bias $r_{\max}/r_e=4.5$1, and noise $r_{\max}/r_e=4.5$2. PSF leakage is parameterised by

$r_{\max}/r_e=4.5$3

The Euclid preparation study explicitly separates measurement biases from detection and selection biases, rather than treating the full pipeline response as a single number (Collaboration et al., 2024).

In Euclid-like Flagship and VIS emulation, LensMC’s measurement-only biases were reported as

$r_{\max}/r_e=4.5$4

$r_{\max}/r_e=4.5$5

The same study found a large detection bias with multiplicative component $r_{\max}/r_e=4.5$6 and additive component $r_{\max}/r_e=4.5$7, together with measurement PSF leakage

$r_{\max}/r_e=4.5$8

A dominant contribution to the measurement multiplicative bias comes from undetected faint galaxies, with $r_{\max}/r_e=4.5$9. Morphology mismatch can add approximately $\tens{S}=\frac{\bar{r}_0}{q_\epsilon\,r_0} \begin{pmatrix} 1-\epsilon_1 & -\epsilon_2 \ -\epsilon_2 & 1+\epsilon_1 \end{pmatrix}, \qquad q_\epsilon=1-|\epsilon|.$0 of multiplicative bias when full bulge variability and a bulge-only subpopulation are allowed, but the study characterises this model bias as straightforward to calibrate because of weak sensitivity to the assumed distributions (Collaboration et al., 2024).

For the Abell 2390 cluster-regime application, calibration was performed with dedicated Flagship-based image simulations extending to $\tens{S}=\frac{\bar{r}_0}{q_\epsilon\,r_0} \begin{pmatrix} 1-\epsilon_1 & -\epsilon_2 \ -\epsilon_2 & 1+\epsilon_1 \end{pmatrix}, \qquad q_\epsilon=1-|\epsilon|.$1. In that analysis, LensMC used a refined linear multiplicative correction appropriate for native PSF sampling, with multiplicative bias $\tens{S}=\frac{\bar{r}_0}{q_\epsilon\,r_0} \begin{pmatrix} 1-\epsilon_1 & -\epsilon_2 \ -\epsilon_2 & 1+\epsilon_1 \end{pmatrix}, \qquad q_\epsilon=1-|\epsilon|.$2 and additive bias $\tens{S}=\frac{\bar{r}_0}{q_\epsilon\,r_0} \begin{pmatrix} 1-\epsilon_1 & -\epsilon_2 \ -\epsilon_2 & 1+\epsilon_1 \end{pmatrix}, \qquad q_\epsilon=1-|\epsilon|.$3. A conservative multiplicative-bias uncertainty of $\tens{S}=\frac{\bar{r}_0}{q_\epsilon\,r_0} \begin{pmatrix} 1-\epsilon_1 & -\epsilon_2 \ -\epsilon_2 & 1+\epsilon_1 \end{pmatrix}, \qquad q_\epsilon=1-|\epsilon|.$4 was adopted, with $\tens{S}=\frac{\bar{r}_0}{q_\epsilon\,r_0} \begin{pmatrix} 1-\epsilon_1 & -\epsilon_2 \ -\epsilon_2 & 1+\epsilon_1 \end{pmatrix}, \qquad q_\epsilon=1-|\epsilon|.$5 added for PSF SED-dependence not modelled, giving a total shear-calibration uncertainty of $\tens{S}=\frac{\bar{r}_0}{q_\epsilon\,r_0} \begin{pmatrix} 1-\epsilon_1 & -\epsilon_2 \ -\epsilon_2 & 1+\epsilon_1 \end{pmatrix}, \qquad q_\epsilon=1-|\epsilon|.$6; the paper states that this dominates the weak-lensing systematic budget for that single-target study (Schrabback et al., 10 Jul 2025).

The Q1 catalogue uses an empirical additive correction rather than a full per-object multiplicative calibration. Field-of-view averaged additive biases are reported as $\tens{S}=\frac{\bar{r}_0}{q_\epsilon\,r_0} \begin{pmatrix} 1-\epsilon_1 & -\epsilon_2 \ -\epsilon_2 & 1+\epsilon_1 \end{pmatrix}, \qquad q_\epsilon=1-|\epsilon|.$7 and $\tens{S}=\frac{\bar{r}_0}{q_\epsilon\,r_0} \begin{pmatrix} 1-\epsilon_1 & -\epsilon_2 \ -\epsilon_2 & 1+\epsilon_1 \end{pmatrix}, \qquad q_\epsilon=1-|\epsilon|.$8, with no significant spatial dependence detected over the Q1 footprint. The adopted correction is a map $\tens{S}=\frac{\bar{r}_0}{q_\epsilon\,r_0} \begin{pmatrix} 1-\epsilon_1 & -\epsilon_2 \ -\epsilon_2 & 1+\epsilon_1 \end{pmatrix}, \qquad q_\epsilon=1-|\epsilon|.$9, built by binning in $I(r)$0 and $I(r)$1; after correction, two-point statistics show suppressed excess power at $I(r)$2 arcmin, and B-modes are an order of magnitude below E-modes in the aperture-mass test. By contrast, Q1 does not derive a per-object $I(r)$3 calibration, and for cluster science the impact is described as sub-dominant (Congedo et al., 18 Jun 2026).

A recurrent methodological point is that LensMC’s calibration philosophy differs from internally calibrated methods. Relative to metacalibration and BFD-style approaches, it relies on realistic external simulations, and relative to KSB-style moment methods it absorbs PSF convolution and undersampling directly into the forward likelihood. This suggests that the method is best understood as a calibrated model-fitting estimator rather than a self-calibrating one (Collaboration et al., 2024, Congedo et al., 18 Jun 2026).

5. Tomographic weak-lensing analysis of Abell 2390

The Abell 2390 Early Release Observations paper provides the first end-to-end demonstration of LensMC in Euclid tomographic cluster weak lensing. The analysis combines Euclid VIS imaging in the broad $I(r)$4 band over $I(r)$5–$I(r)$6 nm and NISP $I(r)$7 imaging with Subaru/Suprime-Cam $I(r)$8 data and CFHT Megacam $I(r)$9-band imaging, although the $$\tilde{I}(k)=2\pi\!\int_0^\infty I(r)\,J_0(kr)\,r\,{\rm d}r,$$0-band is dropped from the photometric-redshift estimation because of unstable calibration under cirrus. The Euclid VIS data were obtained in three dithered ROS sequences for a total VIS exposure of $$\tilde{I}(k)=2\pi\!\int_0^\infty I(r)\,J_0(kr)\,r\,{\rm d}r,$$1 s; VIS stacks have $$\tilde{I}(k)=2\pi\!\int_0^\infty I(r)\,J_0(kr)\,r\,{\rm d}r,$$2 pixels and $$\tilde{I}(k)=2\pi\!\int_0^\infty I(r)\,J_0(kr)\,r\,{\rm d}r,$$3 depth $$\tilde{I}(k)=2\pi\!\int_0^\infty I(r)\,J_0(kr)\,r\,{\rm d}r,$$4, while NISP stacks have $$\tilde{I}(k)=2\pi\!\int_0^\infty I(r)\,J_0(kr)\,r\,{\rm d}r,$$5 pixels with $$\tilde{I}(k)=2\pi\!\int_0^\infty I(r)\,J_0(kr)\,r\,{\rm d}r,$$6 depths $$\tilde{I}(k)=2\pi\!\int_0^\infty I(r)\,J_0(kr)\,r\,{\rm d}r,$$7, $$\tilde{I}(k)=2\pi\!\int_0^\infty I(r)\,J_0(kr)\,r\,{\rm d}r,$$8, and $$\tilde{I}(k)=2\pi\!\int_0^\infty I(r)\,J_0(kr)\,r\,{\rm d}r,$$9. The overlap area provides complete azimuthal coverage out to approximately $\vec{D}$0 Mpc, and Galactic cirrus was removed from all stacks with DeNeb (Schrabback et al., 10 Jul 2025).

Photometric redshifts are estimated with Phosphoros using the NISP and Suprime-Cam bands, with the VIS $\vec{D}$1 band excluded because of chromatic detrending in ERO and the CFHT $\vec{D}$2 band excluded because of unstable offsets under reddening. Systematic photometric offsets are calibrated against limited spectroscopic redshifts at or near the cluster, with an added per-band fractional flux uncertainty of approximately $\vec{D}$3. The reported spectroscopic performance is $\vec{D}$4 and outlier rate $\vec{D}$5. Tomographic selection uses magnitude bins $\vec{D}$6 and $\vec{D}$7, and redshift bins $\vec{D}$8, $\vec{D}$9, $63\,\mathrm{deg}^2$00, $63\,\mathrm{deg}^2$01, $63\,\mathrm{deg}^2$02, and $63\,\mathrm{deg}^2$03; the cluster mass analysis uses the four central bins, omitting the lowest and highest (Schrabback et al., 10 Jul 2025).

The source redshift distributions are calibrated with a self-organising map approach using COSMOS2020. One SOM is trained on the Abell 2390 photometric space, COSMOS calibrators are assigned to the same cells, and per-cell mini-$63\,\mathrm{deg}^2$04 distributions are weighted by the sum of LensMC shear weights in the cell. For the bins used in the mass analysis, the resulting mean lensing efficiencies span $63\,\mathrm{deg}^2$05–$63\,\mathrm{deg}^2$06 and $63\,\mathrm{deg}^2$07–$63\,\mathrm{deg}^2$08 (Schrabback et al., 10 Jul 2025).

Cluster-member contamination, obscuration, and magnification are treated explicitly. Source-density profiles show strong central excess at low photometric redshift and central depletion at high photometric redshift. Obscuration is mapped through large-scale image injections at approximately $63\,\mathrm{deg}^2$09, and magnification is modelled by applying a reference NFW magnification to a Flagship population. The contamination boost factor is fitted to the corrected source-density profiles using

$63\,\mathrm{deg}^2$10

with joint scale radius $63\,\mathrm{deg}^2$11 kpc and bin-dependent amplitudes $63\,\mathrm{deg}^2$12. The uncertainty in the boost correction contributes approximately $63\,\mathrm{deg}^2$13 to the mass systematic (Schrabback et al., 10 Jul 2025).

The lensing analysis uses the reduced shear relation

$63\,\mathrm{deg}^2$14

the tangential ellipticity estimator

$63\,\mathrm{deg}^2$15

and the critical surface density

$63\,\mathrm{deg}^2$16

Mass modelling adopts a spherical NFW profile,

$63\,\mathrm{deg}^2$17

with fixed $63\,\mathrm{deg}^2$18. The tangential reduced shear is jointly fitted in eight bin combinations, corresponding to four photometric-redshift bins times two magnitude bins, over $63\,\mathrm{deg}^2$19 Mpc. The centre is fixed to the strong-lensing/MCMC mass centre at $63\,\mathrm{deg}^2$20, $63\,\mathrm{deg}^2$21, approximately $63\,\mathrm{deg}^2$22 south-east of the brightest cluster galaxy. A Wiener-filtered convergence reconstruction shows a high-significance E-mode detection aligned south-east to north-west and peaking at the BCG, with negligible B-mode residual; the paper emphasises, however, that quantitative constraints come from the $63\,\mathrm{deg}^2$23 profile fits (Schrabback et al., 10 Jul 2025).

The inferred masses are as follows.

Method	$63\,\mathrm{deg}^2$24	$63\,\mathrm{deg}^2$25
LensMC	$63\,\mathrm{deg}^2$26	$63\,\mathrm{deg}^2$27
KSB+	$63\,\mathrm{deg}^2$28	$63\,\mathrm{deg}^2$29
SourceXtractor++	$63\,\mathrm{deg}^2$30	$63\,\mathrm{deg}^2$31

The LensMC result is consistent with KSB+ at approximately $63\,\mathrm{deg}^2$32 once shear-calibration systematics are included, and all three methods agree within the quoted uncertainties. Individual tomographic-bin mass fits scatter around the joint best fit without significant trends versus photometric redshift or magnitude; one bin, the bright $63\,\mathrm{deg}^2$33 sample, lies approximately $63\,\mathrm{deg}^2$34 high, which the paper describes as not unexpected among eight bins. The cross-component $63\,\mathrm{deg}^2$35 is consistent with zero for all methods. For Abell 2390 at $63\,\mathrm{deg}^2$36, the LensMC mass is also in good agreement with earlier wide-field weak-lensing results from WtG, LoCuSS, and CCCP/MENeaCS, while the Euclid analysis reaches tighter mass uncertainties, approximately $63\,\mathrm{deg}^2$37 total statistical, primarily because of higher source density and tomographic selection (Schrabback et al., 10 Jul 2025).

6. Quick Release 1 catalogue, validation, and near-term outlook

The Q1 LensMC catalogue extends the method from a single-cluster demonstration to a survey-scale cluster-lensing resource. It covers the Euclid Deep Field North, South, and Fornax mosaics, amounting to approximately $63\,\mathrm{deg}^2$38 of VIS stacked images, with accompanying photometric redshifts. In Q1, the PSF is reconstructed from the released VIS stacked-image PSF grids, with grid pitch approximately $63\,\mathrm{deg}^2$39 arcmin over approximately $63\,\mathrm{deg}^2$40-arcmin stacks. The adopted strategy uses PSF cutouts, flagging around the core, $63\,\mathrm{deg}^2$41 oversampling with splines to avoid undersampling bias, and bilinear interpolation, falling back to nearest-neighbour interpolation when needed (Congedo et al., 18 Jun 2026).

For stacked cluster lensing, Q1 defines the tangential and cross ellipticity components in the field-of-view aligned frame as

$63\,\mathrm{deg}^2$42

and the annular estimators

$63\,\mathrm{deg}^2$43

For excess surface density in comoving units, the analysis uses

$63\,\mathrm{deg}^2$44

$63\,\mathrm{deg}^2$45

and

$63\,\mathrm{deg}^2$46

The recommended baseline cuts are $63\,\mathrm{deg}^2$47, $63\,\mathrm{deg}^2$48, $63\,\mathrm{deg}^2$49, optional $63\,\mathrm{deg}^2$50, and $63\,\mathrm{deg}^2$51, together with a background condition such as $63\,\mathrm{deg}^2$52 (Congedo et al., 18 Jun 2026).

Validation in Q1 proceeds through both internal and external tests. Two-point statistics on $63\,\mathrm{deg}^2$53 sources recover the expected trends, although excess $63\,\mathrm{deg}^2$54 power remains at large angular scales when using uncalibrated photo-$63\,\mathrm{deg}^2$55 values and nominal cosmology; the additive correction suppresses but does not fully eliminate that excess. In stacked cluster analyses, $63\,\mathrm{deg}^2$56 MaDCoWS2 candidates in the Q1 footprint show good consistency between raw and corrected catalogues out to $63\,\mathrm{deg}^2$57 Mpc, with $63\,\mathrm{deg}^2$58 consistent with zero. Cross-validation against DES redMaPPer clusters yields excellent agreement of LensMC and DES tangential shear over $63\,\mathrm{deg}^2$59 to several Mpc; the differential biases are consistent with zero, $63\,\mathrm{deg}^2$60 and $63\,\mathrm{deg}^2$61, and improve further when the farthest bins are removed. Random cluster stacks are null in most bins, but low-redshift large-radius bins show residuals consistent with edge effects, incomplete azimuthal coverage, or PSF residuals, so the Q1 paper recommends downweighting or omitting those bins in precision analyses (Congedo et al., 18 Jun 2026).

The scientific reach of the Q1 catalogue is already substantial. The paper states that, thanks to Euclid image resolution, depth, and overall control of systematic errors, stacked lensing profiles of clusters with masses $63\,\mathrm{deg}^2$62 can be constrained out to $63\,\mathrm{deg}^2$63, spanning nearly $63\,\mathrm{deg}^2$64 Gyr of evolution history. At the same time, the catalogue is explicitly transitional: Q1 PSF modelling is stack-based and grid-interpolated; no per-object multiplicative calibration is provided; and residual systematics remain at low redshift and large radius. The stated DR1 direction is a wavefront-based PSF forward model per exposure, full multiplicative-bias calibration from image simulations, refined magnitude–size and star–galaxy selections, and improved photo-$63\,\mathrm{deg}^2$65 calibration and $63\,\mathrm{deg}^2$66 uncertainty propagation, with survey area projected to reach approximately $63\,\mathrm{deg}^2$67 by the end of 2026 (Congedo et al., 18 Jun 2026).

Taken together, the simulation paper, the Abell 2390 ERO study, and the Q1 catalogue delineate LensMC as a Euclid weak-lensing framework with three defining properties: explicit pixel-level forward modelling, MCMC posterior sampling with neighbour-aware fitting, and a calibration strategy anchored in realistic simulations plus release-specific empirical corrections. The available evidence shows that this combination yields high-density shape catalogues, internally consistent tomographic cluster masses, and robust stacked cluster-lensing measurements, while also making clear that shear calibration, PSF modelling, and low-level selection effects remain the dominant technical levers for DR1 and subsequent releases (Collaboration et al., 2024, Schrabback et al., 10 Jul 2025, Congedo et al., 18 Jun 2026).