The cubic response and the witness

XI-A's unswitched action $S_{\mathrm{EH}}=-\frac{2}{{\kappa }^2}\int \sqrt{-q}R[q]$ with $q=\eta +\kappa \gamma$ gives $\sqrt{-q}{G}^{\mu \nu }(q)=\kappa {\mathcal{H}}_1+{\kappa }^2{\mathcal{H}}_2+{\kappa }^3{\mathcal{H}}_3+O({\kappa }^4)$, ${\mathcal{H}}_1=G_{\mathrm{lin}}(\gamma )=\frac{1}{2}A\gamma$, with the finite formulas of XI-A's appendix. The cubic vertex has metric Euler derivative $J_3[\gamma ]=2{\mathcal{H}}_2[\gamma ]$, a quadratic local differential polynomial. XI-E defined

of field degree $(2,1)$, and proved, for ${\delta }_0\gamma =K\xi$, $\xi \in {\mathfrak{g}}_U$,

where $({{\mathcal{L}}^{\mathrm{dens}}}_{\xi }v{)}^{\mu \nu }={\xi }^{\rho }{\partial }_{\rho }{v}^{\mu \nu }-v^{\rho \nu }{\partial }_{\rho }{\xi }^{\mu }-v^{\mu \rho }{\partial }_{\rho }{\xi }^{\nu }+({\partial }_{\rho }{\xi }^{\rho })v^{\mu \nu }$.

The witness (XI-C, XI-D, XI-E): $m=0$,

in a causal cylinder on which $g=1$, with a compact spatial cutoff on $\xi$ whose derivatives lie outside the observation dependence region, the plateau containing $\mathrm{supp}\xi$, and $f\ge 0$, $f\not\equiv 0$ observed after $\mathrm{supp}a$. Then $\square \varphi =0$, $-\square \mathcal{P}\gamma =0$, $b=D\gamma =(-2,0,0,0)$ is constant, $A\gamma =0$, and

XI-E also proved the split cancellation $-{\mathcal{A}}_1[{\mathcal{L}}_{\xi }\gamma ]+{\mathcal{A}}_1[{\mathcal{L}}_{\xi }\gamma ]=0$ in the $(h,\gamma ,k)$ parametrization with an external $k$, and recorded that the two $I$'s are different $I$'s. The two affine numbers, $S_{\epsilon }(t)$ for the bare insertion and $\epsilon /2$ for the complete word, are not touched here.