Corollary: Scalar-sourced slot

The functional ${\mathcal{A}}_1[f,g](r_g[\varphi ];\varphi )$, with $r_g[\varphi ]=-\frac{1}{2}\mathcal{P}{E}_0^{\mathrm{R}}(g{T}_0[\varphi ])$, is a compact microcausal polynomial of degree $(0,3)$; its coordinate leg is $-\mathcal{T}(f,g)$ of XII-A's named sector and its scalar leg is $-\int f{E_m}^{\mathrm{R}}(V_{1,g}[r_g[\varphi ]]\varphi )$. The full gauge-fixed second-order coefficient on the sheet is

and its variation under $\gamma \mapsto \gamma +K\xi$ at fixed $\varphi$ equals ${\Delta }_{\xi }{{\Phi }^{(2)}}_{\mathrm{D}}$: the added term has no metric argument.

Proof. Both legs are tree compositions: the coordinate leg has vertices $x$ ($f$), $x^{\prime }$ (${\sigma }_{\epsilon }$), $x^{\prime \prime }$ ($g$) with lines $\partial E_0(x-x^{\prime })$, $E_0(x^{\prime }-x^{\prime \prime })$, two deltas from $x^{\prime \prime }$ to the stress slots, and $\partial \delta (x-y_3)$; the scalar leg has vertices $x$ ($f$), $x^{\prime }$ ($g,\partial g$), $x^{\prime \prime }$ ($g$) with lines $E_m(x-x^{\prime })$, $E_0(x^{\prime }-x^{\prime \prime })$, two deltas at $x^{\prime \prime }$, and $\partial \delta (x^{\prime }-y_3)$. Hypothesis (S) holds as in Theorem thm:admission. The identification of the coordinate leg is XI-E's XIE; the decomposition of the second slot is Proposition prop:gfcoeff with Lemma lem:affine.