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It is based on forward modelling in order to deal with convolution by a point spread function (PSF) with comparable size to many galaxies, sampling the posterior distribution of galaxy parameters via Markov chain Monte Carlo, and marginalisation over nuisance parameters for each of the 1.5 billion galaxies observed by <jats:italic>Euclid<\/jats:italic>. We quantified the scientific performance through high-fidelity images based on the <jats:italic>Euclid<\/jats:italic> Flagship simulations and emulation of the <jats:italic>Euclid<\/jats:italic> VIS images, realistic clustering with a mean surface number density of 250 arcmin<jats:sup>\u22122<\/jats:sup> (<jats:italic>I<\/jats:italic><jats:sub>E<\/jats:sub> &lt; 29.5) for galaxies, and 6 arcmin<jats:sup>\u22122<\/jats:sup> (<jats:italic>I<\/jats:italic><jats:sub>E<\/jats:sub> &lt; 26) for stars, and a diffraction-limited chromatic PSF with a full width at half maximum of 0\u2032.\u20322 and spatial variation across the field of view. L<jats:sc>ENS<\/jats:sc>MC measured objects with a density of 90 arcmin<jats:sup>\u22122<\/jats:sup> (<jats:italic>I<\/jats:italic><jats:sub>E<\/jats:sub> &lt; 26.5) in 4500 deg<jats:sup>2<\/jats:sup>. The total shear bias was broken down into measurement (our main focus here) and selection effects (which will be addressed in future work). We found measurement multiplicative and additive biases of <jats:italic>m<\/jats:italic><jats:sub>1<\/jats:sub> = (\u22123.6 \u00b1 0.2) \u00d7 10<jats:sup>\u22123<\/jats:sup>, <jats:italic>m<\/jats:italic><jats:sub>2<\/jats:sub> = (\u22124.3 \u00b1 0.2) \u00d7 10<jats:sup>\u22123<\/jats:sup>, <jats:italic>c<\/jats:italic><jats:sub>1<\/jats:sub> = (\u22121.78 \u00b1 0.03) \u00d7 10\u2212<jats:sup>4<\/jats:sup>, and <jats:italic>c<\/jats:italic><jats:sub>2<\/jats:sub> = (0.09 \u00b1 0.03) \u00d7 10<jats:sup>\u22124<\/jats:sup>; a large detection bias with a multiplicative component of 1.2 \u00d7 10<jats:sup>\u22122<\/jats:sup> and an additive component of \u22123 \u00d7 10<jats:sup>\u22124<\/jats:sup>; and a measurement PSF leakage of <jats:italic>\u03b1<\/jats:italic><jats:sub>1<\/jats:sub> = (\u22129 \u00b1 3) \u00d7 10<jats:sup>\u22124<\/jats:sup> and <jats:italic>\u03b1<\/jats:italic><jats:sub>2<\/jats:sub> = (2 \u00b1 3) \u00d7 10<jats:sup>\u22124<\/jats:sup>. When model bias is suppressed, the obtained measurement biases are close to <jats:italic>Euclid<\/jats:italic> requirement and largely dominated by undetected faint galaxies (\u22125 \u00d7 10<jats:sup>\u22123<\/jats:sup>). Although significant, model bias will be straightforward to calibrate given its weak sensitivity on galaxy morphology parameters. L<jats:sc>ENS<\/jats:sc>MC is publicly available at gitlab.com\/gcongedo\/LensMC.<\/jats:p>","DOI":"10.1051\/0004-6361\/202450617","type":"journal-article","created":{"date-parts":[[2024,10,14]],"date-time":"2024-10-14T18:46:50Z","timestamp":1728931610000},"page":"A319","source":"Crossref","is-referenced-by-count":13,"title":["<i>Euclid<\/i> preparation"],"prefix":"10.1051","volume":"691","author":[{"name":"Euclid Collaboration","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2508-0046","authenticated-orcid":false,"given":"G.","family":"Congedo","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3376-6200","authenticated-orcid":false,"given":"L.","family":"Miller","sequence":"additional","affiliation":[]},{"given":"A. 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