Regulated Palatini BV Quantization of a Self-Consistent Einstein–Scalar Sector

The preceding papers construct a self-consistent Einstein–real-scalar sector, identify its symmetric/projective Palatini Batalin–Vilkovisky master action, and realize the classical gauge differential as an operator BRST complex containing the native scalar Fock theory as a closed, isometric, dynamically invariant retract. This seventh paper develops the regulator-owned quantum BV structure of the same master and classifies its finite QME, refinement, obstruction, and restricted physical-sector consequences.

Finite Palatini regulators select parity-preserving conjugate field/antifield banks with split embeddings, projections, and refinement maps. Every finite bank carries a genuine second-order odd BV Laplacian whose product defect generates the regulated antibracket and whose square vanishes. Density-dependent operators obey a separate modular-square law. The rectangular jet regulator has a nonzero total-derivative boundary defect, so regulated density representatives remain explicit.

For finite classical coefficients satisfying the finite classical master equation, the regulated quantum master equation yields exact finite quantum differentials, defect-square identities, Wess–Zumino consistency, and recursive anomaly-obstruction criteria. The actual projected physical master is evaluated independently and already has a nonzero Hamiltonian square at the first cutoff.

Regulator refinement distinguishes raw projection, flat fiber averaging, and a centered Gaussian/Berezin fiber. The normalized weighted Gaussian pushforward retracts retained observables, intertwines the density BV operator, and composes associatively across finite cutoff chains. Wick/contraction grading identifies the direct same-order shell trace at one-loop order and gives an exact counterterm recurrence.

The stronger all-observable Hamiltonian descent fails: a first-shell kernel witness has zero Gaussian pushforward but nonzero Hamiltonian residual, and the defect is already present at ℏ=0. Accordingly, the declared global Gaussian renormalized-QME completion is empty at the stated scope. The same obstruction determines a canonical positive remainder: the adjacent-cutoff comparison kernel has a unique greatest invariant square-zero refinement with genuine restricted cohomology.

The restricted physical sector contains a nonzero unit class, and spectator tensoring with the established native scalar Fock carrier preserves every nonzero native matter state as a nonzero cohomology class, including an explicit one-particle state. The local observable classification is correspondingly sharp: on the complete retained scalar polynomial superalgebra, the simultaneous source-square and Paper-VI comparison kernel reduces to constant even observables with trivial odd part. The paper therefore separates failure of global all-observable descent from survival of a nontrivial restricted quantum theory.