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

The first five papers in this series follow one real scalar field from singular operator selection through quantum dynamics, local quantum matter, a self-consistent Einstein–scalar sector, and the functional Batalin–Vilkovisky master law that owns the same physical action. This sixth paper completes the quantum gauge layer of that program.

The direct algebraic carrier is the tensor product of the intrinsic symmetric/projective local-functional BV space with the native scalar Fock space. The full master differential acts on the BV factor, the native physical unitary acts on the Fock factor, and the two actions commute. A typed operator realization preserves the intrinsic symmetric cotangent normalization and the independent projective covector ghost. The resulting Einstein–scalar BRST charge implements the quantum gauge-generator role, squares to zero independently, and intertwines exactly with the native physical clock.

The algebraic BRST module is completed analytically using the countable jet-monomial basis. Completion of the exact charge graph gives a closed square-zero BRST complex with intrinsic integer ghost number. Closed degree-zero cycles modulo the closure of degree-minus-one boundaries form a Hausdorff physical Hilbert cohomology, canonically equivalent to an orthogonal harmonic realization. A BRST Hodge Laplacian has kernel equal to those harmonic representatives.

The native scalar Fock space embeds faithfully into the ghost-number-zero physical cohomology. A continuous BV-vacuum retraction annihilates the closed boundary space, so every nonzero native state—including the normalized coherent source states—defines a nonzero physical BRST class. Native two-time evolution preserves the closed graph, cycles, boundaries, and harmonic representatives and descends to a unitary physical clock on the quotient.

The Palatini projective symmetry also receives a complete gauge-fixed realization. The torsion trace has an invertible Faddeev–Popov operator, a projective antighost and Lautrup field form a nonminimal doublet, and the projective quartet admits an explicit contracting homotopy. Gauge-fixed and quartet-reduced cohomologies are therefore equivalent, while the stationary zero-torsion-trace representative is the Levi–Civita connection.

Combined scalar-field, stress-tensor, and Einstein-residual Ward identities hold on their common domains, with finite-bank operator identities and controlled all-mode passage. The result closes the quantum gauge theory of the self-consistent classical-metric/quantum-matter sector at the closed BRST/Hilbert level developed here. The next boundary is a regulator-owned perturbative BV/EFT treatment; a later quantum-gravity stage requires a genuinely quantum carrier for geometric degrees of freedom.