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, and fixed-regulator identity are from X-C-v3.tex. The separation sheet, the layer lemma, and the explicit equation-ideal representatives are from X-D-v3.tex. VII-C.tex is used only for the copied assumption.

proof. The Palatini variation and its expansion; the Noether identity with switch defect; exact plateau invariance of the total Lagrangian and covariance of the metric Euler operator; the Bianchi obstruction; the gauge-fixed operator, its inverse, and the residual algebra; the affine-slot lemma; the tree kernel, its support, admission, and classical normalization; the background identity through ${\kappa }^1$; the explicit ${\kappa }^2$ defect, its reduction to the linearized Christoffel symbol, and the nonvanishing witness.

. Listed in the header. The two that change the sheet: the quadratic action alone cannot carry E2, so Einstein–Hilbert is frozen; and the self-interaction tree exists at ${\kappa }^2$, so “one tree” is a declaration, not a completeness statement.

status. No successful pdflatex execution, proof-assistant check, numerical verification, or independent referee review is claimed. The source requires no external figures, bibliography database, or executable code.