Complete Quantization of Layered Gravity: BRST Construction, Constraint Algebra, and Physical Degrees of Freedom

This work presents the complete BRST quantization of Layered Gravity (LG), a geometric extension of General Relativity in which spacetime carries an internal phase-like scalar field. Starting from the covariant action with a phase‑dependent coupling F(ϕ), we derive the full Hamiltonian, momentum, and scalar-layer constraints and show that they form a closed first-class algebra. This closure enables the construction of a BRST charge with the same geometric structure as in GR. We explicitly prove that the BRST charge is nilpotent, Q^2_BRST=0, ensuring the absence of negative‑norm states and establishing LG as a mathematically consistent quantum gauge theory. The BRST cohomology yields exactly three physical degrees of freedom: two tensor modes and one scalar-layer mode. No Ostrogradsky instabilities, higher‑derivative ghosts, or extra propagating fields arise. Appendices provide full derivations of the constraint algebra, stability analysis of the layered coupling, and quantization of the microscopic layered chain model. The results demonstrate that LG constitutes a rare example of a modified gravity theory that remains fully quantizable at the BRST level while preserving geometric simplicity and observational viability.