Structural Incompleteness of Semiclassical Gravity: No Local Equivariant Vacuum Selection

We prove that any closed propositional physical system coupling a classical spacetime metric to quantum matter through the semiclassical Einstein equations cannot determine a unique pair (gμ∗ν,|Vac⟩∗) from within its own algebra of local operators. The obstruction is structural: the vacuum bundle over the space of non-static Lorentzian metrics admits no local section that is equivariant under the action of the diffeomorphism group. We show that any attempt to restore determinism by averaging over the fibre produces a mixed state whose stress-energy tensor is not the variation of a local covariant action, violating the Wald axioms and admitting no smooth metric solution. Consequently, state selection requires a structure that is not contained in the local operator algebra. We draw a structural analogy with G ̈odel’s first incompleteness theorem and identify the analogue of the undecidable sentence. The result is independent of any particular interpretation of quantum mechanics or any proposed completion of general relativity: it is a statement about the limits of the semiclassical framework itself.Note: This document contains the technical proof of the theorem. The structural interpretation of the result, and its connection to the frontier problems of physics, are developed in: https://doi.org/10.5281/zenodo.22876515This is the English version of the work Incompletitud estructural de la gravedad semiclásica: No existencia de selección de vacío local y equivariante, also available in Spanish: https://doi.org/10.5281/zenodo.22879380