Proposition: Eventual cutoff independence

Let ${\chi }_1,{\chi }_2$ both equal one on a neighborhood of $\mathfrak{f}(L)$, where $L⋐U$. For a fixed compact mollifier, the two regularized vertex maps have identical values on tests supported in $L$ for every sufficiently small $\delta$.

Proof. The distribution $({\chi }_1-{\chi }_2)C$ vanishes on a relative neighborhood of the compact $\mathfrak{f}(L)$. Its distance from a smaller compact neighborhood is positive. For sufficiently small $\delta$, convolution with ${\rho }_{\delta }$, and all its derivatives, therefore vanish on that smaller neighborhood. The crossed coefficients and the test action (eq:vertex-test) coincide.