Sensitivity of the predicted MW–M31 merger probability to the adopted M31 PM for the two-body configuration (MW and M31; top panel) and the four-body configuration (MW, M31, M33, and LMC; bottom panel), shown in the heliocentric frame. Colors indicate the merger probability (f_m) in each PM grid. The black triangle marks the reference PM of zero transverse velocity relative to the MW. White dashed curves indicate contours of constant M31 transverse velocity (20, 50, 80, and 110 km s⁻¹ from inner to outer). The outermost contour marks where the transverse and radial components are approximately equal. The fiducial PM adopted in this work (H. Wu et al. 2025) is shown as the magenta symbol, while the HST (R. P. van der Marel et al. 2012a) and Gaia EDR3 (J. B. Salomon et al. 2021) measurements are indicated in olive and cyan, respectively.