Proposition: Admitted compact-test prepared coefficient

Equation (eq:response) is a continuous bilinear functional of $j,h$ on each fixed compact-support test-function space. It is real for real data and retarded: only points of $H$ in the causal past of the observation test can contribute.

Proof. The map $(j,H)\mapsto F_{j,H}$ is continuous into a fixed compact-support smooth test space. Pairing with the constructed distribution is continuous. Smooth multiplication by the fixed ramp is continuous. Reality follows from Theorem thm:retarded. The support of ${\Pi }^{\mathrm{R}}$ supplies the causal statement.