Proposition: Supported coefficient-extension interface

In a product chart $U=S_0\times {\mathbb{R}}^k$, write $S=S_0\times \{0\}$ and

Fix $m<\infty$ and a closed set $Z\subset S_0$. Suppose a supported reference on $(S_0\setminus Z)\times {\mathbb{R}}^k$ has normal form

Within the class of distributions supported on $S$ with normal order at most $m$, a realization

exists if and only if all coefficient distributions ${b_{\alpha }}^{\circ }$ admit extensions to $S_0$. The coefficients are unique for a fixed realization. This equivalence does not itself prove extension existence or select an extension.

Proof. For every $\alpha$, choose a compactly supported normal test whose normal derivatives through order $m$ at zero are zero except for the $\alpha$ derivative, normalized to compensate the sign $(-1{)}^{|\alpha |}$. Such tests are obtained from the corresponding monomial times a cutoff equal to one near zero. Testing $t$ against its product with a tangential test extracts $b_{\alpha }$. This proves uniqueness and the necessary restriction identity. Conversely, inserting any coefficient extensions into the finite sum gives the required realization.