Quantum Gauge Structure of a Self-Consistent Einstein–Scalar Sector

Abstract

This paper develops the quantum gauge structure of a self-consistent Einstein–scalar sector constructed in the preceding papers of the series. Starting from the symmetric/projective BV master law of Paper 5 and the native scalar quantum theory of Papers 2–4, it constructs an operator BRST representation of the classical gauge differential on an enlarged BV/Fock carrier and develops its algebraic and analytic cohomology. The resulting BRST charge is square-zero on its declared algebraic domain, intertwines the native scalar time evolution, and admits a closed graph-Hilbert realization with ghost-number grading, Hausdorff cohomology, harmonic representatives, and a bounded BRST Hodge Laplacian. The previously established scalar Fock theory embeds isometrically into ghost-number-zero BRST cohomology as a closed, continuously retracted, dynamically invariant sector. The full degree-zero cohomology therefore decomposes into this native Einstein–scalar sector together with a precisely defined complementary BRST sector whose possible nontriviality remains an explicit structural question. The projective connection symmetry is resolved through an invertible Faddeev–Popov operator, an explicit contractible quartet, and a strong deformation retract to the reduced projective complex; on the stationary locus the zero-trace representative is the Levi–Civita connection. The linearized projective connection sequence is exact, physical scalar, stress, and Einstein-residual observables satisfy the corresponding BRST Ward identities on their maximal established domains, and native time descends unitarily to the completed cohomology. The paper also characterizes the graph realization under unitary changes of coefficient presentation, isolates the exact vertical obstruction controlling ordinary counting-space closability, proves the equality between closed core boundaries and the full analytic boundary space, and identifies the latter with the adjoint/Hodge formulation of BRST cohomology. The density-to-local-functional quotient is shown to lose density-level information through an explicit nonzero total derivative, yielding a theorem that no universal density-recovery map exists. The resulting theory provides a machine-checked analytic quantum gauge completion of the classical-metric/quantum-matter Einstein–scalar sector. Its immediate boundary is the regulator-owned perturbative BV quantum-master problem, where quantum anomaly and counterterm questions acquire their appropriate regulated meaning.