Theorem: Conditional Co-Recovery of Quantum Matter and Geometry from a Schur-Reduced Effective Fluctuation Operator

Abstract This paper presents a conditional consistency framework for the effective co-description of quantum matter and dynamical geometry. Starting from a general action functional expanded to second order around a regular background, the fluctuation space is decomposed into geometry-like, matter-like, and non-retained degrees of freedom. Under suitable domain, self-adjointness, and invertibility assumptions, the non-retained sector is eliminated by a Schur reduction, yielding an effective coupled matter-geometry operator. The main result is formulated as a conditional theorem: if the associated defect system vanishes, then the reduced operator defines a reciprocal, conserved, causal, and limit-compatible effective matter-geometry sector. The required defects encode Schur consistency, metric recovery, quantum matter recovery, stress-energy consistency, Bianchi compatibility, microcausality, and recovery of known limiting regimes such as quantum field theory on a fixed background, semiclassical gravity, general relativity, special relativity, and the Newtonian limit. The theorem is not presented as a complete microscopic theory of quantum gravity, nor as a derivation of physical constants, Standard Model representations, or interaction coefficients. Rather, it isolates a set of sufficient mathematical conditions under which an effective quantum-gravity-like sector can be consistently defined. The stronger task of deriving these conditions from a specified microscopic action remains an open proof obligation.