The free graviton algebra, Wick ordering, and what is not an input

The free gauge-fixed graviton algebra is generated by $\hat{h}(w)$, $w$ a compactly supported smooth symmetric tensor density, linear in $w$, with $\hat{h}(P_{\mathrm{gr}}w)=0$ and commutator $\mathrm{i}\hslash$ times the Peierls bracket of $P_{\mathrm{gr}}$, i.e.\ the pairing with ${\Delta }^{\mathrm{gr}}={{\Delta }_{\mathrm{R}}}^{\mathrm{gr}}-{{\Delta }_{\mathrm{A}}}^{\mathrm{gr}}$, in the same convention in which X-B's scalar commutator uses the Peierls bracket of $-P_m$. The multiplet Wick algebra $\mathfrak{W}$ of $(\hat{\varphi },\hat{h})$ and its microcausal admission are X-B's; the equation ideal $\mathfrak{J}$ is generated by $\hat{\varphi }(P_{m}v)$ and $\hat{h}(P_{\mathrm{gr}}w)$. Wick ordering of graviton-quadratic monomials uses a fixed Hadamard bidistribution of the form ${\mathsf{H}}_{\mathrm{gr}}=-4\mathcal{P}\otimes {\mathsf{H}}_0$, ${\mathsf{H}}_0$ Hadamard for $\square$ in $(+,-,-,-)$. It is indefinite, as $\mathcal{P}$ is; no positivity is claimed or used.

VII-C's $H_{hh}=P{\Delta }_0^{+}$ is a bisolution in the opposite signature, attached to a sharp insertion and a different algebra. It is not ${\mathsf{H}}_{\mathrm{gr}}$, not ${{\Delta }_{\mathrm{R}}}^{\mathrm{gr}}$, and not an input. Every quantity computed below is at ${\hslash }^0$ and uses ${{\Delta }_{\mathrm{R}}}^{\mathrm{gr}}$ only.