Corollary: Weighted internal normalization criterion

Suppose $d$ is smooth near equal-time coincidence and, for compact output $K$, its required output derivatives satisfy

outside a fixed ball. Then the internal rod correction

exists, is cutoff independent, and has cutoff error $O(R^{1-q})$ in the stated output differentiability norms.

Proof. Near coincidence, change to relative spatial coordinates and use smoothness of $d$ and local integrability of $P_i$. At infinity, output differentiation is legitimate and the same radial estimate as (eq:cutoffestimate) applies.