Magneticum mass-observable relation for Euclid-like derived quantities. The left column shows scaling relations, relative fit (solid grey line), and a corridor corresponding to one standard deviation (dashed grey line). The right column shows the residual PDF and scatter of log-residuals σ_ln. We report the following properties: lensing concentration $c_{\mathrm{NFW}}^{2\mathrm{D}}$ (first row), lensing mass $M_{\mathrm{NFW}}^{2\mathrm{D}}$ (second row), projected richness n^2D (third row), and projected stellar mass M_⋆^2D (fourth row). The histogram of residuals for halos with M < 2 × 10¹⁴ M_⊙ is in blue dotted lines, for halos with M > 2 × 10¹⁴ M_⊙ is in orange dashed lines, and for the complete sample is in solid grey lines. Note: the three histograms almost overlap. Each distribution panel reports the value of the natural log scatter σ_ln for the complete sample.