Lemma: Difference of scaling-bounded point extensions

Suppose two extensions $t,\tilde{t}$ agree away from the origin and have scaling degrees at most $s<\infty$. Then

when $s\ge n$. If $s<n$, they agree.

Proof. Their difference is supported at the origin. A finite-order distribution supported at one point is a finite linear combination of delta derivatives. A nonzero highest derivative of order $m$ has scaling degree $n+m$. The bound $n+m\le s$ gives the result. For $s<n$, no nonzero point-supported term satisfies it.