Inherited statements

The following are copied from X-A to X-E and not re-proved. Equation numbers in brackets refer to X-E.

• Density coefficients $a,b,A,B$ [(2.2)], ramp coordinates ${\mathsf{X}}_1,{\mathsf{X}}_2$ [(2.3)], Euler coefficients ${\mathsf{L}}_{1,g},{\mathsf{L}}_{2,g}$, completed tests $d_1,d_2$ [(2.5)], ${\hat{A}}_1[h]=\hat{\varphi }(d_1[h])$, ${\hat{A}}_2=\hat{\varphi }(d_2)$, $\hat{Q}[h,h]=:\hat{\varphi }(Q[h,h]):$, and the affine-slot lemma $d_2[h,k]=Q[h,h]+d_1[k]$, $d_1$ linear [Lemma 2.1]. The kernel $D_1$ of $d_1$ [(4.8)] is a compactly supported distribution on $M\times M$ with $\mathrm{supp}{D}_1\subset K_f\times (J^{-}(K_f)\cap \{t\ge 0\})$.

• With $\mathcal{E}[q]:=\sqrt{-q}{G}^{\mu \nu }[q]$,

[(4.6)], ${\mathcal{G}}_2(\cdot ,\cdot )$ its symmetric bilinear polarization; $S_{\mathrm{gr}}=-(2{\kappa }^2{)}^{-1}\int \sqrt{-q}R$, $\delta S_{\mathrm{gr}}/\delta h=\frac{1}{2\kappa }\mathcal{E}$ [Lemma 4.1]; $S_{\mathrm{gr}}=S_{\mathrm{FP}}[h]+\kappa (2S_{\mathrm{FP}}(h,k)+S_3[h])+O({\kappa }^2)$, $\delta S_{\mathrm{FP}}/\delta h=\frac{1}{2}{G}_{\mathrm{lin}}[h]$, $\delta S_3/\delta h=\frac{1}{2}{\mathcal{G}}_2[h,h]$ [Lemma 6.1, Sec. 4.6]; $G_{\mathrm{lin}}[K\xi ]=0$; the vertex identity $V_{1,g}[h,\varphi ]=-\frac{1}{2}\int g{h}_{\mu \nu }{T}^{\mu \nu }[\varphi ]$, $\delta V_{1,g}/\delta h=-\frac{1}{2}gT[\varphi ]$, $\delta V_{1,g}/\delta \varphi =-{\mathsf{L}}_{1,g}[h]\varphi$ [Lemma 4.2].

• $\mathcal{P}u=u-\frac{1}{2}\eta {\mathrm{tr}}_{\eta }u$, ${\mathcal{P}}^2=1$; $C^{\mu }[u]={\partial }_{\nu }(\mathcal{P}u{)}^{\nu \mu }$; ${S^{(0)}}_{\mathrm{gf}}[u]=\frac{1}{4}\int C_{\mu }{C}^{\mu }$;

[Lemma 6.3]. Here $(KC{)}_{\mu \nu }={\partial }_{\mu }{C}_{\nu }+{\partial }_{\nu }{C}_{\mu }$.

• The multiplet Wick algebra $\mathfrak{W}$ of $(\hat{\varphi },\hat{h})$ with Hadamard matrix of zero cross blocks, ${\mathsf{H}}_{\mathrm{gr}}=-4\mathcal{P}\otimes {\mathsf{H}}_0$, and the closed two-sided equation ideal $\mathfrak{J}$ generated by $\hat{\varphi }(P_{m}v)$, $\hat{h}(P_{\mathrm{gr}}w)$ [Sec. 6.4]. The admission criterion of X-B XB,BF: a Wick monomial of degree $n$ in the multiplet belongs to $\mathfrak{W}$ iff its kernel $u\in {\mathcal{D}}^{\prime }(M^n)$ is compactly supported and

${\overline{V}}_{\pm }$ the closed future/past covector cones including zero. The wavefront bound $\mathrm{WF}({{\mathsf{E}}_0}^{\mathrm{R}})\subset \{(x,\xi ;y,-\xi ):\xi \ne 0\}$ [Lemma 4.13, DH], and the facts that elements of $\mathfrak{J}$ vanish on all pairs of smooth free solutions [Lemma 4.17(ii)].

• Theorem 4.18 of X-E: with $h=h_{\mathrm{cl}}+\hat{h}$, under $h_{\mathrm{cl}}^{\prime }=h_{\mathrm{cl}}+K\xi$, $k_{\mathrm{cl}}^{\prime }=k_{\mathrm{cl}}+{\mathcal{L}}_{\xi }{h}_{\mathrm{cl}}+K\zeta +\frac{1}{2}{\mathcal{L}}_{\xi }K\xi$, $\hat{\varphi },\hat{h}$ fixed,

Proposition 4.21 of X-E: for $m=0$, \(=