Abstract

We freeze one gravitational action, one gauge condition, and one quadratic graviton operator for the ramp relational scalar of Landmarks X-A to X-D, in signature $(+,-,-,-)$, at fixed regulator width.

The action is the Einstein–Hilbert density, $S_{\mathrm{gr}}=-(2{\kappa }^2{)}^{-1}\int \sqrt{-q}R$, with the sign fixed by positivity of the transverse-traceless kinetic term and confirmed by Newtonian attraction. Its first variation is derived from the Palatini identity. The metric Euler equation is displayed with every switch factor, and the Noether identity of the total action is proved with its explicit switch defect. On the plateau algebra the total Lagrangian is exactly invariant, so the metric Euler operator is covariant and the classical identity of X-A to X-C holds through ${\kappa }^2$ on the coupled constraint surface. That surface is generically empty: the Bianchi identity forces $({\partial }_{\mu }g)T^{\mu }{}_{\rho }[{\varphi }_0]=0$.

The gauge-fixed operator is $P_{\mathrm{gr}}=-\frac{1}{4}\mathcal{P}\square$ with retarded inverse $-4\mathcal{P}{{\mathsf{E}}_0}^{\mathrm{R}}$, $\mathcal{P}$ the trace reversal. Its residual gauge algebra meets the compactly supported plateau algebra only in zero, so the frozen algebra acts on classical backgrounds alone.

One tree is constructed: the metric variation of the switched scalar density, propagated by ${{\Delta }_{\mathrm{R}}}^{\mathrm{gr}}$ and inserted into the affine slot of the completed observable. It is a totally Wick-ordered cubic in the scalar, supported in $J^{-}(\mathrm{supp}f)$, microcausally admitted, and normalized by the classical retarded response, whose relational dressing is $-2{{\mathsf{E}}_0}^{\mathrm{R}}{{\mathsf{E}}_0}^{\mathrm{R}}(({\partial }_{\nu }g)T^{\nu \mu }[{\varphi }_0])$. The one-tree observable is invariant under the frozen algebra through ${\kappa }^1$. At ${\kappa }^2$, the background variation with the graviton held fixed equals $-{\kappa }^2{\hat{A}}_1[{\mathcal{L}}_{\xi }\hat{h}]$ modulo the equation ideals, and this class is nonzero.

The gravitational self-interaction tree, the mixed one-loop term, the second-order quantum metric, and VII-C's word $B_f\star W_g\star B_{f^{\prime }}$ are named and not constructed. The regulator is not removed.