The cubic-completed second coefficient

The second metric coefficient of the gauge-fixed equation with the scalar source is $u_3[\gamma ]+r_g[{\varphi }_0]$; by (eq:affine) the $r_g$ term contributes ${\mathcal{A}}_1[r_g[{\varphi }_0];{\varphi }_0]=-\mathcal{T}-{\mathcal{T}}_{\mathrm{sc}}$, with ${\mathcal{T}}_{\mathrm{sc}}:=\int f{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{R}}{\mathsf{V}}_{1,g}[r_g]{\varphi }_0$ the scalar-response companion of XI-C's coordinate tree. Both are independent of $\gamma$ and inert under every operation below; we follow the sheet and keep only $-\mathcal{T}$. Nothing in this section changes if $-{\mathcal{T}}_{\mathrm{sc}}$ is added.