Release: Constructed, at fixed regulator and fixed input, through κ 2 0

{Release: Constructed, at fixed regulator and fixed input, through ${\kappa }^2{\hslash }^0$}{ Frozen: X-E's sheet with the background de Donder gauge $\frac{1}{4}\int \sqrt{-\bar{q}}{\bar{q}}_{\mu \nu }{C^{\mu }}_{\bar{q}}{C^{\nu }}_{\bar{q}}$, the background-field prescription, the ramp switch on the gravitational order-$\kappa$ densities, and the Møller-defined response $\hat{k}={{\Delta }_{\mathrm{R}}}^{\mathrm{gr}}\mathsf{S}$.

Proved. F1: the exact second-order covariance identity (eq:cov2). F2: ${\mathcal{F}}^{(0)}=-\frac{1}{4}\mathcal{P}KC$, $P_{\mathrm{gr}}=\frac{1}{2}{G}_{\mathrm{lin}}+{\mathcal{F}}^{(0)}$, the commutator identity (eq:key), and the dynamical jet law (eq:jetlaw). F3: three trees, Definition def:trees; kernels, supports, admission (Lemmas lem:kernels, lem:sumzero, lem:admission), classical values. F4: Theorem thm:main, the background identity through ${\kappa }^2$ modulo $\mathfrak{J}$, quantum fields fixed. F5: the flat-gauge residual, Proposition prop:flat; its class is not decided. F6: VII-C inapplicable, not contradicted.

Not opened: ${\hslash }^1$, ghosts, BRST/BV with switched matter, positivity, the classical tadpole, gauge-parameter dependence, $\epsilon \to 0$, X-Enter, all orders in $\kappa$.}