Proposition: A reduced response need not satisfy invariant equations

For a symmetric source $S_{\mu \nu }$, a componentwise gauge-fixed forced wave solution need not solve $G^{(1)}(j)=S$.

Proof. The invariant equation requires ${\partial }^{\mu }{S}_{\mu \nu }=0$ by Lemma lem:identities. Choose a nonzero compact source with only $S_{00}=f$ and ${\partial }_{s}f\not\equiv 0$. A componentwise wave equation for the trace reverse of $j$ has a smooth forced solution with zero initial data, by Lemma lem:evolution, but no tensor can satisfy $G^{(1)}(j)=S$, because this source is not conserved.