Theorem: The scalar completion is a microcausal compact polynomial

The functional $(\gamma ,\varphi )\mapsto {{\mathcal{A}}_2}^{\mathrm{sc}}[f,g](\gamma ,0;\varphi )$ has a compact kernel ${t^{\mathrm{sc}}}_{f,g}\in {\mathcal{E}}^{\prime }(M^3)$, symmetric in its two metric slots, with $\mathrm{WF}({t^{\mathrm{sc}}}_{f,g})\subset Z_3$. Hence ${{\mathcal{A}}_2}^{\mathrm{sc}}(\cdot ,0;\cdot )\in {\mathcal{F}}_{\mathrm{mc},\mathrm{pol}}$, and ${{\Phi }^{(2)}}_{\mathrm{D}}(f,g)={{\mathcal{A}}_2}^{\mathrm{sc}}(\gamma ,0;\varphi )-{\mathcal{C}}_{3,\mathrm{D}}(f,g)\in {\mathcal{F}}_{\mathrm{mc},\mathrm{pol}}$.

Proof. Write $E_0={E_0}^{\mathrm{R}}$, $E_m={E_m}^{\mathrm{R}}$ for the retarded kernels and ${\Lambda }^{\mu ,ab}(x,z)={\sigma }_{\epsilon }(z){\mathfrak{a}}^{\alpha \mu ,ab}{{\partial }^z}_{\alpha }{E}_0(x-z)$, so that ${X_1}^{\mu }[\gamma ](x)=\int {\Lambda }^{\mu ,ab}(x,z){\gamma }_{ab}(z)\mathrm{d}z$ XA,XIID; this is the line $\partial E_0(x-z)$ with weight ${\sigma }_{\epsilon }$ at $z$ and a constant fiber map. Every local vertex $V_{1,g}[\gamma ]\psi$, $V_{2,g}[\gamma ,0]\psi$, $\mathcal{V}[\gamma ]\psi$, $\mathcal{W}[\gamma ,0]x^{\mu }$ is a finite sum of constant tensors times derivatives of $g$ or ${\sigma }_{\epsilon }$ at the vertex point times products of differentiated deltas joining the vertex point to each of its field slots; each such delta is a balanced line. Thus each of the five terms of (eq:Asc) at $k=0$ is a finite sum of tree compositions with external vertices $z_1,z_2$ (metric) and $y$ (scalar):

Each graph is a tree. Hypothesis (S) holds in every row: $x\in \mathrm{supp}f$ is compact; an internal vertex weighted by $g$ or $\partial g$ lies in $\mathrm{supp}g$, one weighted by ${\sigma }_{\epsilon }$ or ${\sigma }_{\epsilon }^{\prime }$ lies in $\{t\ge 0\}$, and each lies in the causal past of its retarded predecessor by the support of ${E^{\mathrm{R}}}_{\mu }$; an external metric vertex reached by $\partial E_0$ carries the weight ${\sigma }_{\epsilon }$ and lies in $J^{-}(\cdot )\cap \{t\ge 0\}$; an external vertex reached by a delta coincides with its internal vertex. Finite intersections of causal pasts of compact sets with $\{t\ge 0\}$ in the slab are compact. Lemma lem:tree applies to every row; finite sums and the symmetrization over $z_1\leftrightarrow z_2$ preserve compactness and the bound $Z_3$. Membership in ${\mathcal{F}}_{\mathrm{mc},\mathrm{pol}}$ is XII-A's definition. The last assertion adds XII-D's ${\mathcal{C}}_{3,\mathrm{D}}\in {\mathcal{F}}_{\mathrm{mc},\mathrm{pol}}$.