Proposition: Classical values

Let ${\varphi }_0$, $h_0$ be smooth solutions of $P_m{\varphi }_0=0$, $P_{\mathrm{gr}}{h}_0=0$, and evaluate the underlying functionals at $\hat{\varphi }\to {\varphi }_0$, $\hat{h}\to h_0$. Then ${\hat{\Phi }}^{\mathrm{F}}(f)$ evaluates to X-C's completed classical observable at the metric

through ${\kappa }^2$; ${k^{\mathrm{cl}}}_1$ is the classical first-order retarded response of the gauge-fixed fluctuation to the switched sources. In particular ${\mathsf{T}}_{\mathrm{mat}}$ evaluates to X-E's classical tree, ${\mathsf{T}}_{\mathrm{mix}}$ to $A_1[{{\Delta }_{\mathrm{R}}}^{\mathrm{gr}}(-{\sigma }_{\epsilon }{\mathcal{Q}}^{(1)}[h_{\mathrm{cl}}]h_0)]$, and ${\mathsf{T}}_{\mathrm{self}}$ to $A_1[{{\Delta }_{\mathrm{R}}}^{\mathrm{gr}}(-\frac{1}{2}{\sigma }_{\epsilon }{\mathcal{G}}_2[h_0,h_0])]$, with $A_1[k]=⟨d_1[k],{\varphi }_0⟩$ X-A's first-order relational scalar.

Proof. Evaluation of a Wick-ordered element at a smooth configuration is evaluation of the underlying polynomial functional. $d_1$ is linear, so ${\hat{A}}_1[k_{\mathrm{cl}}]+{\mathsf{T}}_{\mathrm{mat}}+{\mathsf{T}}_{\mathrm{mix}}+{\mathsf{T}}_{\mathrm{self}}$ evaluates to $A_1[k_{\mathrm{cl}}+{k^{\mathrm{cl}}}_1]$, and $\hat{Q}[h,h]$ to $⟨Q[h_{\mathrm{cl}}+h_0,h_{\mathrm{cl}}+h_0],{\varphi }_0⟩$. By the affine-slot lemma this is the ${\kappa }^2$ coefficient of X-C's classical observable at the stated metric.