Proposition: Classical value and the dressing formula

Replace $\hat{\varphi }$ by a smooth free solution ${\varphi }_0$ and $\hat{h}$ by $0$. Then ${\mathsf{T}}_{\mathrm{tree}}$ becomes ${\kappa }^2{A}_1[k_1[{\varphi }_0]]$, where $A_1[k]=⟨d_1[k],{\varphi }_0⟩$ is X-A's first-order relational scalar at background $k$, and $k_1[{\varphi }_0]=-2\mathcal{P}{{\mathsf{E}}_0}^{\mathrm{R}}(gT[{\varphi }_0])$ is the classical retarded response of the metric to the switched stress tensor. Its coordinate part is

and its scalar part is ${\rho }_1=-{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{R}}{\mathsf{L}}_{1,g}[k_1]{\varphi }_0$. Hence

Proof. $\mathrm{supp}{k}_1\subset J^{+}(\mathrm{supp}g)\subset \{t>2\delta \}$, so ${\sigma }_{\epsilon }a[k_1]=a[k_1]$ and ${\mathsf{X}}_1[k_1]={{\mathsf{E}}_0}^{\mathrm{R}}(-{\partial }_{\nu }{a}^{\nu \mu }[k_1])={{\mathsf{E}}_0}^{\mathrm{R}}(C^{\mu }[k_1])$. Lemma lem:violation with $S=\frac{1}{2}gT[{\varphi }_0]$ and (eq:T-divergence) with $P_m{\varphi }_0=0$ give $C^{\mu }[k_1]=-2{{\mathsf{E}}_0}^{\mathrm{R}}(({\partial }_{\nu }g)T^{\nu \mu })$. Then $⟨d_1[k],{\varphi }_0⟩=-\int f{{\mathsf{X}}_1}^{\mu }[k]{\partial }_{\mu }{\varphi }_0-⟨{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}f,{\mathsf{L}}_{1,g}[k]{\varphi }_0⟩$ and $⟨{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}f,{\mathsf{L}}_{1,g}{\varphi }_0⟩=⟨f,{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{R}}{\mathsf{L}}_{1,g}{\varphi }_0⟩$.