Lemma: Constraint compatibility

For the stated conserved sources,

Proof. Writing $j={\mathcal{R}}^{A}Q$, one has $\square \mathsf{r}j=-2Q$. The initial data give $Dj{|}_0=0$. Furthermore

so differentiating $Dj$ once in time gives zero initial data. Conservation implies $\square Dj=0$. Wave uniqueness yields $Dj=0$, and (eq:Einstein) gives $G^{(1)}j=Q$.