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

This paper develops a machine-checked finite-regulator Batalin–Vilkovisky quantization of the self-consistent Einstein–real-scalar sector constructed in the preceding papers of the series. Starting from the symmetric/projective Palatini BV master action and its operator-BRST realization, it constructs parity-preserving finite conjugate-pair regulators, a genuine second-order BV odd Laplacian, density-dependent modular-square laws, regulated quantum-master defects, Wess–Zumino consistency, and finite-CME anomaly recursion. Finite regulator refinement is analyzed directly. Raw coordinate projection has an explicit odd-Laplacian defect, flat polynomial fiber integration has a BV–Stokes moment obstruction, and a centered Gaussian/Berezin fiber supplies normalized finite pushforwards with exact density-odd-Laplacian intertwining and coherent finite-cutoff composition. The first physical shell produces an explicit same-order trace of −1/2, whose Wick-contraction grading yields the corresponding perturbative counterterm recurrence. Global all-observable quantum-differential descent is decided by an explicit cutoff-one scalar derivative–antifield witness. Its Gaussian pushforward vanishes and its Hamiltonian residual has selected coefficient −1. The obstruction is present at classical order, making the declared global renormalized-QME completion empty at the stated scope. The obstruction simultaneously selects a canonical greatest compatible square-zero subcomplex. Its restricted physical sector has nontrivial cohomology: the unit survives, and every nonzero state of the established native scalar Fock carrier determines a nonzero restricted state class, including an explicit one-particle state. The local density analysis is exhaustive on several increasingly large carriers. The scalar zero/first-jet/descendant polynomial superalgebra has simultaneous physical kernel exactly the constant even line. The complete cutoff-one generator-linear antifield-free Palatini carrier has trivial simultaneous kernel. Finally, the complete faithful 1,200-coordinate antifield-free physical polynomial algebra is classified: the Paper-6 comparison gate removes all 1,125 nonzero-shell variables, the finite-master square removes every nonconstant dependence on the remaining 75 zero-shell variables, and the common kernel is exactly the rational constants. The development is formalized in Lean 4 with mathlib. Named publication declarations, focused theorem/axiom audits, proof-integrity checks, and a frozen repository snapshot provide a reproducible formal verification boundary. Geometry remains classical in this paper; genuine quantum geometry requires a subsequent construction with a new QME ownership structure.