Constructive MTT Quantum Gravity II: A Conditional BRST/BV Compatibility and Physical-State Reconstruction Contract for SPT-Filtered Models

Building on the constructive existence of the SPT-filtered transverse–traceless (TT) sector, we establish gauge invariance and the physical Hilbert space structure of Modal Triplet Theory (MTT) quantum gravity. Using BRST symmetry and algebraic renormalization methods, we lift the nonperturbative TT construction to a fully gauge-consistent framework. We show that the Loop Vertex Expansion and Borel summability extend to the full BRST complex under SPT damping, including ghosts and auxiliary fields. Ward identities and Nielsen identities are proven to hold at the level of Borel-summed correlators via nonperturbative change-of-variables arguments. Restricting Osterwalder–Schrader positivity to the TT sector, we construct a positive physical Hilbert space on BRST cohomology and show that unphysical polarizations decouple. The result is a nonperturbatively defined, gauge-invariant quantum gravity sector with a well-defined physical state space, providing the structural bridge between ultraviolet existence and infrared dynamics in the constructive MTT program.