Proposition: Eventual exact cutoff independence

Let ${\chi }_1,{\chi }_2$ equal one near $\mathfrak{f}(L)$, $L⋐U$. For a fixed compact profile, their regularized values on tests in $L$ agree at every sufficiently small scale. The same holds for logarithmic scale derivatives and dyadic increments.

Proof. The distribution $({\chi }_1-{\chi }_2)C$ vanishes near the compact relative image. For small scales, convolution with ${\rho }_{\delta },{\vartheta }_{\delta },{\zeta }_{\delta }$, including the required derivatives, vanishes near that image. The vertex test action gives equality.