Declarations

. Coordinates, density coefficients, the two-jet gauge law, and the first-order relational scalar are from X-A.tex. The multiplet Wick algebra, joint admission, and the pullback non-automorphism are from X-B-v3.tex. The completed tests, recursion, affine slot, and fixed-regulator identity are from X-C-v3.tex and X-E.tex. The explicit equation-ideal representatives are from X-D-v3.tex. The Einstein–Hilbert sign, the Palatini variation, the flat gauge-fixed operator, the graviton algebra, the one-tree kernel, the background identity with its defect, and the witness are from X-E.tex. VII-C.tex is used only for the copied assumption.

proof. The exact second-order covariance identity; the background de Donder gauge and its order-$\kappa$ density; the decomposition $P_{\mathrm{gr}}=\frac{1}{2}{G}_{\mathrm{lin}}+{\mathcal{F}}^{(0)}$; the commutator identity ${\mathcal{Q}}^{(1)}[K\xi ]=[{\mathcal{L}}_{\xi },P_{\mathrm{gr}}]$; the dynamical jet law; the switched source and the Møller-defined second-order quantum metric; the three tree kernels and their supports; the sum-zero calculus and three-slot admission; the graviton-slot ideal lemma; the all-tree background identity through ${\kappa }^2$; the flat-gauge residual.

to X-E. Listed in the header. Two change the sheet: the flat de Donder gauge is replaced by the background de Donder gauge, and “one tree” is replaced by three. One changes a sentence of X-E's Section 1: a sheet that freezes the ${\kappa }^2$ quantum-metric identity on X-E's field content reaches ending (1), not (2), once all ${\kappa }^2{\hslash }^0$ trees are included in the background gauge; X-E's witness is correct for the one-tree observable and is a truncation witness. X-E's Proposition No compactly supported residual gauge and its closing remark that $\hat{h}\mapsto \hat{h}+\kappa {\mathcal{L}}_{\xi }\hat{h}$ is not an automorphism stand; the tensor law is carried by $\hat{k}$, not by an automorphism of the free algebra.

-reference corrigendum. The X-E item numbers quoted in the header and in Section sec:inherited were taken from a working copy of X-E in which E4 was numbered as Section 4. In the compiled X-E.tex the items are: completed tests (2.4), not (2.5); the $\mathcal{E}$-expansion (4.3), not (4.6); Palatini Lemma 4.2, not 4.1; vertex identity Lemma 4.3, not 4.2; $\delta S_3/\delta h$ in Section 7.5, not 4.6; gauge-fixed operator Lemma 6.4, not 6.3; one-tree observable (7.3), not (4.17); kernel $D_1$ (7.4), not (4.8); admission Lemma 7.3, not 4.13; two facts about the ideal Lemma 7.6, not 4.17; background gauge identity Theorem 7.7, not 4.18; witness Proposition 7.9, not 4.21. All other quoted numbers agree. In the body of this manuscript X-E items are cited by title.

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