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

The first four papers in this series select a singular real-scalar operator law, construct its physical quantum evolution and conserved matter, and close a self-consistent nonlinear Einstein–scalar quantum sector. This fifth paper moves upstream and identifies the functional gauge/master structure that generates the same Einstein–scalar action and exchanges with the established quantum observables.

The construction begins with an ambient untruncated spacetime-jet local-functional master modulo total divergences. Its functional antibracket satisfies the classical master equation, its generated Hamiltonian is square-zero on arbitrary antifield-dependent real local functionals, and the bracket obeys graded Jacobi. The physical inverse metric density is then realized intrinsically on a symmetric-pair jet carrier with its own correctly normalized cotangent bracket and master.

Variation of the same Palatini action with respect to the independent connection is solved exactly: every stationary connection is a projective shift of Levi–Civita. The projective generator is injective, its image is exactly the pointwise linearized connection-Euler kernel, and each stationary projective orbit has a unique zero-torsion-trace representative equal to the Levi–Civita connection.

An odd projective covector ghost and antifield enlarge the symmetric carrier. The resulting prolonged BRST differential is nilpotent, the projective cotangent functional and enlarged real master satisfy the classical master equation, and the induced master differential squares to zero on arbitrary antifield-dependent local functionals. Classical projection returns the same Einstein–real-scalar density and the same native matter source used in Paper IV.

The exchange with the quantum sector is then explicit. For coherent source states, the Fock first moment equals the real mean field and its jet moment equals the mean-field gradient. Along finite coordinate-gauge orbits, the derivative of that observable equals the coefficient generated by the master law. The master-projected matter density has Hilbert metric response equal to the previously constructed coherent vacuum-relative quantum stress; the Einstein residual transforms covariantly; and gauge pullback intertwines with the positive-time physical clock.

Paper V therefore closes the classical master-action provenance through intrinsic symmetric-density and projective BV structure and binds its physical projection to the self-consistent Einstein–scalar quantum sector of Papers I–IV. The continuing boundary is the quantum gauge representation itself: functional properness and Koszul acyclicity, metric-only symplectic BV reduction, a direct quantum gauge differential on the Fock carrier, physical-state cohomology, and quantum master/anomaly analysis.