Emergence of Einstein's Equations through the Coupling of the Dirac Operator and Entanglement Entropy in Besov Spaces: Towards a Spectral Unification

AbstractThis paper provides a derivation of Einstein’s field equations from the thermodynamics of quantum entanglement. By employing Besov spaces as a functional framework and the Dirac operator for spectral geometry, we show that General Relativity emerges as the equation of state for a localized Hilbert system. This spectral unification bridges the gap between quantum information and gravitational curvature.Keywords: Unification, General Relativity, Entanglement Entropy, Besov Spaces,Spectral Geometry, Dirac Operator.