Theorem: Normalized graviton inverse

Its retarded and advanced inverses are

Thus the coefficient in $c\mathcal{P}{{\mathsf{E}}_0}^{\mathrm{R}}$ is $c=-1$.

Proof. Trace reversal commutes with the wave operator and satisfies ${\mathcal{P}}^2=\mathrm{id}$. Hence

and the left inverse follows on compact tests. Any coefficient $c$ instead gives $-c\mathrm{id}$, forcing $c=-1$. The support statement follows from Lemma lem:Green.