Master-Action Ownership of a Self-Consistent Einstein–Scalar Quantum Sector

Description This fifth paper in the series develops the functional gauge and Batalin–Vilkovisky structure underlying the self-consistent Einstein–real-scalar quantum sector constructed in Papers 1–4. Using Lean 4, it builds an untruncated spacetime-jet local-functional framework, proves the classical master equation and square-zero master differential with full antifield feedback, and connects the resulting classical gauge structure to the previously constructed quantum observables, stress tensor, physical clock, and Einstein source equation. A central result is the direct derivation and exact solution of the independent Palatini connection Euler equation. Its stationary solutions form a projective Levi–Civita class, Γρμν=Γ[g]ρμν+δνρAμ, with torsion trace τμ=−3Aμ. The projective direction is proved to be the exact kernel of the fixed-metric linearized connection Euler operator, while zero torsion trace selects the unique Levi–Civita representative. The formalization then distinguishes the original ambient free jet carrier from the intrinsic symmetric inverse-metric-density carrier required by the physical Palatini theory. On this symmetric carrier, the projective transformation is an exact action symmetry. A projective covector ghost and antifield sector are constructed, the combined diffeomorphism–projective BRST differential is proved nilpotent, and the enlarged real master satisfies the full functional classical master equation. Its generated Hamiltonian squares to zero on arbitrary antifield-dependent local functionals. The resulting master retains exact provenance from the ambient construction while projecting to the same Einstein–scalar density and native matter action used by the preceding quantum theory. The coherent Fock first moment, quantum jet moment, Hilbert stress response, Einstein residual covariance, and positive-time physical-clock intertwining are thereby connected to the same classical action lineage. The paper establishes the classical projective gauge architecture needed for the Einstein–scalar sector while leaving full functional BV properness, metric-only symplectic reduction, connection-antifield elimination, intrinsic quantum BRST/cohomology, and quantum master/anomaly analysis to subsequent work.