Lemma: What the bound alone fixes

The index set $\{(\alpha ,\beta )\in {\mathbb{N}}^4\times {\mathbb{N}}^4:|\alpha |+|\beta |\le 2\}$ has $\left(\genfrac{}{}{0ex}{}{10}{2}\right)=45$ elements. For each $(L,I,J)$, $Q^{L,I,J}$ is determined by $45$ smooth functions on ${\mathbb{R}}^4$, and the supported sector's normal order (Theorem thm:normalorder) and support (Corollary cor:K8supp) depend on the bound only.

Proof. Multi-indices in eight variables of total degree at most two are counted by $\left(\genfrac{}{}{0ex}{}{8+2}{2}\right)$. The rest restates the theorems' hypotheses.