Theorem: Normal coefficients and supported extension

Every supported distribution of normal order at most $p$ has a unique representation

If it is specified only over $S\setminus N_S$, a supported extension of that order exists exactly when each coefficient admits an extension to $S$. The unrestricted difference of two such extensions is

Finite normal order does not make the ambiguity finite dimensional.

Proof. Choose a compact normal cutoff $\chi =1$ near zero and let

Its jets select exactly the $\alpha$-coefficient. Set $b_{\alpha }(\psi )=T({\mathsf{J}}_{\alpha }\psi )$. Taylor decomposition modulo vanishing jets proves representation; the selectors prove uniqueness. Restriction commutes with finite insertion maps, proving the extension and ambiguity statements. The tangential coefficient spaces are infinite dimensional.