Corollary: What finiteness of the ambiguity requires

In the jet module, the unrestricted supported ambiguity of normal order at most $r$ is finite-dimensional only if $Z=\varnothing$. A normal-order bound alone never makes it finite. It becomes finite-dimensional, of dimension $M_r\left(\genfrac{}{}{0ex}{}{k+d}{d}\right)$ per puncture point, exactly when the puncture is finite and a tangential order bound $k$ is imposed. The same holds for ${\mathcal{A}}_{\mathrm{ext}}$ with $M_r$ replaced by $1$, and for ${\mathcal{S}}_Z(T^{\circ })$ in Proposition prop:exampleSZ, which has dimension $k+1$ under the bound $k$ and is infinite-dimensional without it.

Proof. Theorem thm:dimension(iii)–(v) and Proposition prop:exampleSZ.