UNIONS-3500 Weak Lensing: II. B-mode validation for cosmic shear

At Stage-III sensitivities, cosmic shear $B$ modes unambiguously indicate systematic contamination and are often used to inform data selection and scale cuts for cosmological inference. We validate $B$ modes for the Ultraviolet Near-Infrared Optical Northern Survey (UNIONS)-3500 (2894 deg$^2$, $n_\mathrm{eff} \approx 5.0$ arcmin$^{-2}$) using three $E$/$B$-separable statistics: pure-mode correlation functions $\xi_\pm^{\mathrm{B}}(\theta)$, Complete Orthogonal Sets of $E$/$B$-mode Integrals (COSEBI) $B$-mode amplitudes $B_n$, and harmonic-space power spectra $C_\ell^{BB}$. For each statistic, we compute probability-to-exceed (PTE) values over a two-dimensional grid of scale-cut boundaries; our adopted cuts lie in broad stable regions of acceptable PTE. $B$-mode detections and PTE failures on initial catalog versions led us to investigate galaxy size cuts and stellar halo masking. After cuts, all three statistics pass the null test (minimum PTE $= 0.18$). Before scale cuts, we measure an oscillatory COSEBI $B$-mode pattern consistent with repeating additive shear bias, a detector-level effect seen across multiple Stage-III surveys including CFHTLenS, which used the same MegaCam camera; scale cuts that exclude the charge-coupled device (CCD) angular scale suppress it. Although these statistics probe the same two-point shear field, scale cuts in one do not map exactly onto cuts in another, because their respective filter functions weight angular scales differently. The most conservative validation therefore requires scale and sample selections that pass null tests across all frameworks simultaneously, an approach that applies directly to Stage-IV surveys where systematic errors dominate.