Lemma: Cutoff-independent internal correction

Under Assumption ass:internal, ${\gamma }_{\mathrm{rod},R}$ converges with the stated output derivatives to

The limit is independent of the source cutoff profile and

is a distribution in $f$.

Proof. Local convergence uses smoothness of $d$ and local integrability of $P$. For large $|\mathbf{y}|$, the product is bounded by $C(1+|\mathbf{y}|{)}^{-3-ϵ}$. Dominated convergence applies to the output derivatives, and the same estimate makes the difference of any two admissible cutoff families tend to zero.