Release: Constructed, at fixed regulator and fixed input, with a stated order boundary

: Constructed, at fixed regulator and fixed input, with a stated order boundary{ Frozen: $S_{\mathrm{gr}}=-(2{\kappa }^2{)}^{-1}\int \sqrt{-q}R$; de Donder gauge fixing $\frac{1}{4}\int C_{\mu }{C}^{\mu }$ for the quantum graviton; $P_{\mathrm{gr}}=-\frac{1}{4}\mathcal{P}\square$; ${{\Delta }_{\mathrm{R}}}^{\mathrm{gr}}=-4\mathcal{P}{{\mathsf{E}}_0}^{\mathrm{R}}$.

Proved. E1: the metric Euler equation (eq:metric-euler)–(eq:metric-euler-expanded) and the Noether identity (eq:noether) with its switch defect. E2: exact ${\mathcal{G}}_U$-invariance of the total Lagrangian, covariance of the metric Euler operator, the classical identity through ${\kappa }^2$ on the coupled constraint surface, and the Bianchi obstruction (eq:bianchi-constraint). E3: the operator, its retarded inverse, the projector, and the residual algebra. E4: one tree ${\mathsf{T}}_{\mathrm{tree}}$, Definition def:tree; kernel, support, admission, normalization. The one-tree observable is invariant on ${\mathfrak{g}}_U$ through ${\kappa }^1$. Its ${\kappa }^2$ background variation with $\hat{h}$ fixed is $-{\kappa }^2{\hat{A}}_1[{\mathcal{L}}_{\xi }\hat{h}]$ modulo the ideals, Theorem thm:ward; this class is nonzero, Proposition prop:witness. E5: no loop, no $W_g$, no subtraction.

Not opened: the self-interaction tree, the mixed loop, a second-order quantum metric, ghosts, gauge-parameter dependence, $\epsilon \to 0$, X-Enter.}