We present a lattice gauge theory framework for verifying the renormalization stability of the residual topological density $\delta_\phi$ introduced in Topological Residue and the Yang--Mills Mass Gap. Our approach discretizes $\delta_\phi$ for Monte Carlo simulations, measures its value across ensembles of $SU(N)$ gauge configurations, and extrapolates to the continuum limit $a \to 0$. This analysis directly tests whether $\delta_\phi$ remains finite and positive under the renormalization group flow, thereby ensuring that the geometric mass gap mechanism is robust in the physical continuum theory.