Lemma: Two facts about the equation ideal

Let $\mathfrak{J}\subset \mathfrak{W}$ be the closed two-sided ideal generated by $\hat{\varphi }(P_{m}v)$ and $\hat{h}(P_{\mathrm{gr}}w)$.

1. If $W[\hat{h}]$ is a Wick polynomial in the graviton alone and $u=P_m{u}^{\prime }$, then $:\hat{\varphi }(u)W[\hat{h}]:=\hat{\varphi }(u)\star :W[\hat{h}]:\in \mathfrak{J}$. Consequently a Wick monomial whose kernel lies in $(P_m\otimes 1\otimes \cdots \otimes 1){\mathcal{D}}^{\prime }(M^n)$, with the scalar in the first slot, belongs to $\mathfrak{J}$.

2. Every element of $\mathfrak{J}$ vanishes as a functional on every pair $({\varphi }_0,h)$ of smooth solutions $P_m{\varphi }_0=0$, $P_{\mathrm{gr}}h=0$. Hence a degree-$(1,1)$ monomial $:\hat{\varphi }\hat{h}:(B)$ with $B({\varphi }_0,h)\ne 0$ for one such pair is not in $\mathfrak{J}$.

Proof. (i) The cross blocks of the Hadamard matrix vanish, so no contraction joins $\hat{\varphi }(u)$ to $W$, and the $\star$-product equals the Wick-ordered product. $\mathfrak{J}$ is a two-sided ideal and closed. (ii) For any $F\in \mathfrak{W}$, the ${\star }_{\mathsf{H}}$-product $F\star \hat{\varphi }(P_{m}v)$ equals the pointwise product $F\cdot \hat{\varphi }(P_{m}v)$, because every term of positive order in $\hslash$ pairs a derivative of $F$ with $\mathsf{H}(\cdot ,P_{m}v)=0$; $\mathsf{H}$ is a bisolution. Likewise $P_{\mathrm{gr}}{\mathsf{H}}_{\mathrm{gr}}=0$ since ${\mathcal{P}}^2=1$ and $\square {\mathsf{H}}_0=0$. So $\mathfrak{J}$ is the closure of the pointwise ideal generated by functionals vanishing on solutions, and evaluation at a fixed smooth configuration is continuous FR,Rej.