Proposition: Finiteness of is a question about the weights

For each pair $(\Gamma ,I)$ in Definition def:Cpart let ${\mathcal{W}}_{\Gamma }$ be the vector space of admissible weights $w_{\Gamma }$, $c_I$ the codimension of ${\Delta }_I$, and suppose $D_{\Gamma }$, $S_{\Gamma /I}$ fixed. If finitely many $(\Gamma ,I)$ occur and each ${\mathcal{W}}_{\Gamma }$ is finite-dimensional, then ${\mathcal{C}}_{\mathrm{part}}$ is finite-dimensional with

If some ${\mathcal{W}}_{\Gamma }$ is infinite-dimensional, the bound $|\alpha |\le 6$ gives no finite-dimensionality by itself; whether ${\mathcal{C}}_{\mathrm{part}}$ is finite-dimensional then depends on the kernel of $w\mapsto D_{\Gamma }[w{{\partial }_r}^{\alpha }\delta (r_I)S_{\Gamma /I}]$, which is not determined by the record.

Proof. The map $(w,\alpha )\mapsto D_{\Gamma }[w{{\partial }_r}^{\alpha }\delta (r_I)S_{\Gamma /I}]$ is linear in $w$ for fixed $\alpha$, and ${\mathcal{C}}_{\mathrm{part}}$ is the span of its images over the finitely many $(\Gamma ,I)$ and the $\left(\genfrac{}{}{0ex}{}{6+c_I}{c_I}\right)$ multi-indices $|\alpha |\le 6$ in $c_I$ transverse variables. A finite sum of finite-dimensional images is finite-dimensional with the stated bound. The second sentence is the observation that the normal-order bound restricts only $\alpha$; compare Corollary cor:finite.