Spacetime as a Functional of Matter

Canonical quantisation of the spacetime metric as an independent dynamical variable retains a background-dependent ontology that general relativity's own machinery was built to displace. We take that displacement to its endpoint and propose that the metric is not an independent variable but a functional of the matter fields. Quantum mechanics and general relativity then integrate as coupled field equations (a Dirac equation on curved spacetime and a self-consistent metric functional whose classical limit recovers Einstein's equations). On the matter side, Wigner's classification of unitary irreducible representations of the Poincaré group, combined with Hamiltonian Hermiticity, positive-energy boundedness, and parity invariance, identifies the Dirac equation on curved spacetime as the unique first-order local equation on the four-component (½, 0) ⊕ (0, ½) representation. On the metric side, diffeomorphism invariance together with Lovelock's theorem constrains a self-consistent metric functional whose classical limit is Einstein's equations. The coupling is bidirectional: the tetrad and spin connection in the Dirac equation are built from the metric that is built from the matter multiplet, and the metric functional depends on the matter stress-energy evaluated in that same metric. Squaring the coupled Dirac operator produces the Schrödinger–Lichnerowicz form and makes the Klein-Gordon reduction, the R/4 curvature coupling, and the Fμνσμν gauge-curvature coupling explicit. In the terrestrial weak-field regime with a U(1) charge, the coupled system reduces to Einstein–Dirac–Maxwell and retrodicts three experimental families (Pound–Rebka, Colella–Overhauser–Werner, cold-atom interferometry) as measurements of one and the same coupled system. Semiclassical extrapolation toward the Planck scale carries corrections suppressed by (ℓP/λ)² and introduces no new phase or field; the Planck length is the natural boundary because Compton and Schwarzschild localisation constraints saturate there simultaneously. The framework is empirically distinguished from lowered-effective-gravity and foam-based quantum-gravity programmes by present-day null results at the LHC, Pierre Auger, Telescope Array, and in gamma-ray-burst dispersion and CMB smoothness measurements. The observable that would discriminate this formulation from ⟨Tμν⟩-sourced semiclassical gravity is a composite operator in the metric functional absent from the stress-energy expectation value, appearing in strong-curvature regimes, in coherent superposition states, and in cosmological regimes carrying dark-energy and dark-matter content.