Definition

${\mathsf{T}}_{\mathrm{mat}}$ is X-E's tree with the prefactor ${\kappa }^2$ removed; X-E's display of the one-tree observable carried ${\kappa }^2$ both inside the tree and outside the bracket, and (eq:all-tree) is the intended object. ${\mathsf{T}}_{\mathrm{mix}}$ is of degree $(1,1)$ in $(\hat{\varphi },\hat{h})$; with zero cross blocks its Wick ordering is the plain product. ${\mathsf{T}}_{\mathrm{self}}$ is of degree $(1,2)$. Lemma lem:affine of X-E (the affine slot) is what makes (eq:all-tree) the completed observable of X-C at the metric (eq:metric) through ${\kappa }^2$: $d_2[h,k_{\mathrm{cl}}+\hat{k}]=Q[h,h]+d_1[k_{\mathrm{cl}}]+d_1[\hat{k}]$.