What is not identified

The retarded group and the C1 residual-wave group differ. Their intersection is zero when the retarded support requirement is imposed. The section proof for one group cannot be transplanted to the other.

Nor is C1 a linearization of the full Einstein–scalar system about a nonlinear solution $(\eta ,\Phi )$. Its metric equation is vacuum linearized gravity; the stress of $\Phi$ is not inserted into that equation. For the genuinely linearized Einstein–Klein–Gordon system on on-shell backgrounds, see Hack Hack. That system is not an additional premise here.