Procedure for addressing the M31 PM discrepancy measured between the blue and red samples in the heliocentric frame assuming a distance of 784 kpc taken from Stanek & Garnavich (1998). Each panel is labeled in the topleft corner to indicate the corresponding sample. In panels (a)–(c), our PM results from the blue and red samples are connected by solid black lines. For comparison, we also show two other measurements based on Gaia (E)DR3 from Salomon et al. (2021) (gray squares) and Rusterucci et al. (2024) (gray triangles), connected by dashed gray lines. In panel (d), we compare our final results with previous measurements. Here, the black inverted triangle indicates a purely radial orbit of M31 relative to the MW. The filled magenta and open blue and red points show our new results from 5p, 6p blue, and 6p red samples (G_BP − G_RP ≤ 2.6), respectively. In addition to Gaia (E)DR3, other measurements include results from HST observations (Sohn et al. 2012; van der Marel et al. 2012a) (olive diamond) and Gaia DR2 (van der Marel et al. 2019) (black triangle).