Assumption: Response and common completion domain

For every real smooth perturbation $\gamma$, the inherited response $u_{\mathrm{D}}[\gamma ]$ is a smooth zero-initial-data solution of

Smooth zero-data uniqueness holds without a spatial-decay requirement, as proved in revised XVI-D.

The diagonal completion (eq:CD) is defined on these real smooth inputs. The common retarded domain used in this definition is linear for the finite response combinations occurring in polarization. On that domain, the operators in (eq:Aone) are linear, and ${\mathcal{A}}_1$ is linear in $r$ for fixed $\varphi$ and linear in $\varphi$ for fixed $r$.