The cubic source and the ledger, retained

$S_{\mathrm{EH}}=-\frac{2}{{\kappa }^2}\int \sqrt{-q}R[q]$, $q=\eta +\kappa \gamma$, $\sqrt{-q}{G}^{\mu \nu }(q)=\kappa {\mathcal{H}}_1+{\kappa }^2{\mathcal{H}}_2+O({\kappa }^3)$, $A\gamma =2{\mathcal{H}}_1$, $J_3=2{\mathcal{H}}_2$, with XI-E's finite prescription for ${\mathcal{H}}_2$ XIE. Two structural facts are used: $J_3$ is a bilinear differential polynomial with constant coefficients, every monomial of which carries exactly two derivatives; and the contracted Bianchi identity ${\nabla }_{\mu }{G}^{\mu \nu }=0$, written as ${\partial }_{\mu }(\sqrt{-q}{G}^{\mu \nu })+\sqrt{-q}{{\Gamma }^{\nu }}_{\mu \rho }{G}^{\mu \rho }=0$, gives at order ${\kappa }^2$

consistent with XI-E's gravitational Noether coefficient.

XI-F's classical ledger is retained and not re-proved:

and, when $A\gamma =0$ on $U$, ${{\mathcal{A}}_2}^{(3)}[\gamma +K\xi ]-{{\mathcal{A}}_2}^{(3)}[\gamma ]={\mathcal{A}}_1[K{\zeta }_{\mathrm{gf}};{\varphi }_0]$, zero under the causal plateau condition, with witness value $-I+I+0=0$. Nothing below changes a line of it.