Proposition: Auxiliary compact-localization independence

Replace ${\chi }_{\Gamma }$ by another compact smooth function ${\tilde{\chi }}_{\Gamma }$ equal to one near $K_{\Gamma }$. For every compactly supported graph test and every $M$, the difference of the resulting pairings in (eq:Pi) is $O({\tau }^M)$.

Thus auxiliary compact localization cannot change a prospective distributional limit. This statement does not prove that the limit exists.

Proof. On the compact support of $({\chi }_{\Gamma }-{\tilde{\chi }}_{\Gamma })w_{\Gamma }$, the original closed graph support conditions cannot all hold. Locally, either the weight is zero or at least one numerical line is evaluated on a compact set disjoint from its support.

Use a finite cover and a smooth partition of unity. On each nonzero patch, Lemma lem:smoothing makes that line smaller than every power of $\tau$, while every other line and its derivatives has at most polynomial growth in ${\tau }^{-1}$. The finite product, including the specified differentiations, therefore has the same arbitrarily small remainder.

Pairing the delta attachments and their derivatives with a fixed smooth graph test preserves this estimate. Summing the patches proves the assertion.