Field content and the one-tree observable

${\mathsf{T}}_{\mathrm{tree}}$ is the ${\kappa }^2$ coefficient of ${\hat{A}}_1[\hat{h}+\kappa k_1]-{\hat{A}}_1[\hat{h}]$: the retarded Møller map of the metric slot at first order, at ${\hslash }^0$, in the same sense in which X-C's ${\mathsf{L}}_{1,g}{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}f$ is the retarded Møller correction of the scalar slot. The graviton line is the single contraction between $\kappa {\hat{A}}_1[\hat{h}]$ and $\kappa V_{1,g}[\hat{h},\hat{\varphi }]$; no $\hat{h}$ survives in ${\mathsf{T}}_{\mathrm{tree}}$.