Euler Error for Brownian Motion with Dry Friction

We study the continuous Euler approximation of the scalar diffusion with inward sign drift and constant nonzero diffusion coefficient. At mesh size $h$, the error scaled by $h^{-3/4}$ converges stably in the uniform topology to a conditionally Gaussian process driven by local time at the discontinuity. The drift feeds the error back through the same local time, producing an explicitly solvable linear equation. A bounded Brownian corrector gives the forcing variance directly as $-\sqrt{2/\pi}\,\zeta(-1/2)$. A shrinking-band occupation argument accommodates the endogenous Euler error, and uniform moments justify convergence of the mean-square error. The terminal constant is expressed using the normal distribution function for every deterministic initial value. For the pure dry-friction interpretation, the result gives an asymptotic time-step prescription for a specified velocity error. Coupled simulations illustrate its finite-grid performance and retain both sampling uncertainty and reference-grid sensitivity.