Theorem: Existence, support, normalization, and continuity

Equation (eq:Rone) defines a map

linear in the insertion test and linear in the quadratic vertex. Its output test is smooth and compact, with

For every smooth $u$,

The map is continuous in the smooth background and compact-test topologies. It is additive in the vertex and vanishes when

No diagonal extension or subtraction is used.

Proof. The advanced Green operator sends $c$ to a smooth function supported in its causal past. The differential operator $L_{1,g}[h]$ has smooth coefficients supported in $\mathrm{supp}g$, giving (eq:retardedsupport).

Since $L_{1,g}[h]u$ is compactly supported, transposition in (eq:Green) gives

The final integration by parts is legitimate because the operator coefficients are compactly supported.

On each fixed-support test stage, Green-operator continuity gives smooth seminorm estimates on the compact interaction region. Multiplication by the background coefficients and their finitely many derivatives is continuous. The output lies in the fixed space ${\mathcal{D}}_{\mathrm{supp}g}$. These estimates, and continuity of the Green operator on the LF test space, give the stated continuity and its passage to the quotient. Linearity and the causal vanishing follow directly.