Theorem: Fixed-regulator X-F release

For the frozen sheet of Section sec:frozen:

1. The Einstein tensor density satisfies the exact second-order covariance identity (eq:cov2). The background de Donder gauge-fixing derivative satisfies ${\mathcal{F}}^{(1)}[K\xi ]=[{\mathcal{L}}_{\xi },{\mathcal{F}}^{(0)}]$, and the kinetic operator of the fluctuation satisfies $\mathcal{Q}[\eta +\kappa h_{\mathrm{cl}}]=P_{\mathrm{gr}}+\kappa {\mathcal{Q}}^{(1)}[h_{\mathrm{cl}}]+O({\kappa }^2)$ with ${\mathcal{Q}}^{(1)}[K\xi ]=[{\mathcal{L}}_{\xi },P_{\mathrm{gr}}]$.

2. The second-order quantum metric $\hat{k}={{\Delta }_{\mathrm{R}}}^{\mathrm{gr}}\mathsf{S}$ of Definition def:content transforms under the frozen action of ${\mathfrak{g}}_U$ by (eq:jetlaw); the total second-order metric $k_{\mathrm{cl}}+\hat{k}$ obeys X-A's complete two-jet law modulo the equation ideal, with the graviton held fixed.

3. The three trees of Definition def:trees are elements of X-B's fixed-regulator multiplet algebra, with compactly supported kernels of wavefront set in ${\Sigma }_n$, and their classical values are X-A's first-order relational scalar at the classical first-order retarded response of the gauge-fixed fluctuation.

4. The all-tree observable (eq:all-tree) is invariant under the frozen algebra, acting on the classical backgrounds by the complete two-jet law with $\hat{\varphi },\hat{h}$ fixed, through ${\kappa }^2$ modulo $\mathfrak{J}$. The ${\kappa }^2$ defect of X-E's one-tree observable is the negative of the variation of the omitted mixed tree.

5. In X-E's flat gauge the corresponding variation is (eq:flat-residual); its class is not decided.

VII-C does not apply and is not contradicted. The ${\hslash }^1$ terms, ghosts, the BV master equation with switched matter, positivity, the classical self-response, the order ${\kappa }^3$, and the sharp limit are named and not constructed.

Proof. (i) is Lemmas lem:natural, lem:cov2, lem:F0, lem:fluct and Proposition prop:key. (ii) is Proposition prop:jetlaw. (iii) is Lemmas lem:kernels, lem:sumzero, lem:admission and Proposition prop:classical. (iv) is Theorem thm:main and Corollary cor:truncation. (v) is Proposition prop:flat. Section sec:F6 audits VII-C.