Proposition: Cutoff locality

If two compact cutoffs equal one near $\mathfrak{f}(L)$, their families have identical pairings on tests supported in $L$ at sufficiently small scales. Whenever expansions exist, their coefficient restrictions and finite parts agree on $L$.

Proof. Compact profile support implies that the convolution and its required derivatives at $\mathfrak{f}(L)$ sample only a small neighborhood of that set. The cutoff difference is zero there. Then use coefficient uniqueness.