Theorem: Fixed-regulator X-E release

For the frozen sheet of Section sec:frozen:

1. The gravitational action $S_{\mathrm{gr}}=-(2{\kappa }^2{)}^{-1}\int \sqrt{-q}R$ has metric Euler operator (eq:metric-euler), expanded in (eq:metric-euler-expanded), and the total action satisfies the Noether identity (eq:noether), whose only off-plateau terms are the switch defects. On the plateau algebra the total Lagrangian is exactly invariant, the Euler operators are covariant, and the classical completed observable is invariant through ${\kappa }^2$. The coupled constraint surface is ${\mathcal{G}}_U$-invariant and generically empty, by (eq:bianchi-constraint).

2. The de Donder gauge-fixed quadratic operator is $P_{\mathrm{gr}}=-\frac{1}{4}\mathcal{P}\square$ with retarded inverse $-4\mathcal{P}{{\mathsf{E}}_0}^{\mathrm{R}}$. Its residual algebra meets ${\mathfrak{g}}_U$ only in zero. A de Donder configuration dresses the relational scalar only through the ramp layer.

3. The tree ${\mathsf{T}}_{\mathrm{tree}}$ of Definition def:tree is a totally Wick-ordered cubic in the scalar with microcausal kernel supported in $J^{-}(K)\times J^{-}(K)$, an element of the fixed-regulator multiplet algebra of X-B. Its classical value is X-A's first-order relational scalar at the retarded backreaction $k_1=-2\mathcal{P}{{\mathsf{E}}_0}^{\mathrm{R}}(gT[{\varphi }_0])$, whose coordinate dressing is $-2{{\mathsf{E}}_0}^{\mathrm{R}}{{\mathsf{E}}_0}^{\mathrm{R}}(({\partial }_{\nu }g)T^{\nu \mu }[{\varphi }_0])$.

4. The one-tree observable is invariant under the frozen algebra, acting on classical backgrounds with the graviton fixed, through ${\kappa }^1$. At ${\kappa }^2$ its variation is $-{\kappa }^2{\hat{A}}_1[{\mathcal{L}}_{\xi }\hat{h}]\equiv 2{\kappa }^2{\hat{A}}_1[{\xi }^{\lambda }{{\Gamma }^{(1)}}_{\lambda \cdot \cdot }[\hat{h}]]$ modulo the equation ideal, and this class is nonzero for suitable fixed inputs on every nonempty plateau.

VII-C does not apply and is not contradicted. The self-interaction tree, the mixed loop, a second-order quantum metric, ghosts, gauge-parameter dependence, and the sharp limit are named and not constructed.

Proof. (i) is Lemma lem:palatini, Definition def:metric-euler, Propositions prop:noether, prop:covariance, prop:bianchi, and Theorem thm:classical. (ii) is Lemmas lem:quadratic, lem:gf, lem:layer and Proposition prop:residual. (iii) is Lemmas lem:kernel, lem:admission and Proposition prop:normalization. (iv) is Theorem thm:ward, Lemma lem:gamma, and Proposition prop:witness. Section sec:E5 audits VII-C; Section sec:named names the omitted terms.