Comparison of distance and residual distributions for high-precision (J1713+0747) and low-precision (J1640+2224) pulsars. Upper panel: violin plots of ﹩{\dot{P}}_{{\rm{res}}}﹩ using Gala2022. MC propagation of asymmetric distance errors revises significance downward (2.59σ → 1.93σ), showing that Gaussian assumptions can inflate significance in low-precision systems. We note that ﹩{\dot{P}}_{{\rm{GW}}}﹩ is negligible and omitted. Lower panel: distance probability density functions. For J1640+2224, MC sampling (gray dashed) reveals the non-Gaussian long tail from d = 1/ϖ, while GEP (green) underestimates the uncertainty.