Lemma: Graviton slot of the equation ideal

Let $v\in {\mathcal{D}}^{\prime }(M\times M)$ be compactly supported with $\mathrm{WF}(v)\subset {\Sigma }_2$, covariant in the second slot. Then the degree-$(1,1)$ monomial with kernel $(1\otimes P_{\mathrm{gr}})v$,

is an element of $\mathfrak{W}$ and lies in $\mathfrak{J}$. Every element of $\mathfrak{J}$ vanishes as a functional on all pairs $({\varphi }_0,h_0)$ of smooth free solutions.

Proof. $(1\otimes P_{\mathrm{gr}})v$ is compactly supported with wavefront set in ${\Sigma }_2$ by Lemma lem:sumzero(b), hence admitted by (eq:microcausal) and Lemma lem:sumzero(e). For a finite sum $v={\sum }_i{\psi }_i\otimes w_i$ of smooth compactly supported tensors, the monomial is ${\sum }_i\hat{\varphi }({\psi }_i)\star \hat{h}(P_{\mathrm{gr}}{w}_i)$, because the cross blocks of the Hadamard matrix vanish and no contraction joins the two factors; each $\hat{h}(P_{\mathrm{gr}}{w}_i)$ is a generator of the two-sided ideal $\mathfrak{J}$. A general $v$ is a limit of such finite sums in Hörmander's topology on distributions with wavefront set in a fixed closed cone avoiding ${\overline{V_{\pm }}}^2$ Hor (Thm. 8.2.3); $1\otimes P_{\mathrm{gr}}$ is continuous in that topology, the map from admitted kernels to Wick monomials is continuous by X-B, and $\mathfrak{J}$ is closed. The last statement is X-E's Lemma Two facts about the equation ideal, part (ii), which is proved there for both generators.