Proposition: Comparison of two auxiliary limits

Fix $K$, and suppose two local regularizations, possibly with different mollifiers, have distributional limits $t^{\rho },t^{\tilde{\rho }}$ on $U$. Then

so

If a common reference on a larger open set $V\subset U$ has separately been proved for both choices, the support is contained in $U\setminus V$.

No automatic conclusion that the difference is supported only in $N$, or is a permitted collision contact, follows.

Proof. Both limits agree with the ordinary product on ${\mathcal{O}}_K$ by Corollary cor:canonical. Equality of restrictions is exactly the stated support property. The same argument applies to any separately proved common reference domain.