Proposition: Infinite ambiguity at fixed normal order

A fixed normal-order bound does not, by itself, make the supported ambiguity space finite-dimensional, even when the attachment in $S_0$ is a single point.

Proof. Take $S_0=\mathbb{R}$, $A=\{0\}$, and any $k\ge 1$. For $n=0,1,2,\dots$, let

Every $T_n$ has normal order zero in the $z$-directions and is supported in $A\times \{0\}$.

The distributions $a_n$ are linearly independent: a finite linear combination can be tested against polynomial jets times a cutoff equal to one near zero, isolating the highest derivative and then each lower derivative. Coefficient extraction makes the $T_n$ linearly independent as well.