Theorem: Existence, normalization, continuity, and support

The map (eq:Rone) is bilinear in the quadratic insertion and the linear representative. Its output test is smooth and obeys

It is continuous in the smooth background and compact-test topologies. It satisfies

It is additive in the vertex and vanishes if $\mathrm{supp}g\cap J^{-}(\mathrm{supp}c)=\varnothing$. No new diagonal extension or subtraction is used.

Proof. The advanced Green operator maps compact smooth tests to smooth functions supported in their causal past. The coefficients of ${\mathsf{L}}_{1,g}[h]$ are smooth and supported in $\mathrm{supp}g$. Therefore $-{\mathsf{L}}_{1,g}[h]{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}c$ is a smooth compact test with (eq:retarded-support).

Formal symmetry of ${\mathsf{L}}_{1,g}[h]$ and $({{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{R}}{)}^{\mathsf{T}}={{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}$ give

Compact vertex coefficients justify the integrations by parts. Green-operator continuity on compact causal regions BGP (Chapter 3), followed by continuous smooth multiplication and differentiation, proves continuity. For fixed input support, the output has a fixed compact support. Additivity and retarded vanishing are immediate.

Only one Green distribution is transposed against a smooth test. There is no singular product requiring extension on a diagonal.