Operator–Entropic Projection Framework for Emergent Gravity and Gauge Fields Foundations: Operator Construction and Thermodynamic Structure

We develop a mathematically structured Operator–Entropic Projection Framework in which gravitational and gauge dynamics arise as thermodynamic limits of a regulated operator algebra defined on a rigorously constructed Diagram Hilbert Space. A positive-definite inner product is obtained through regulated diagrammatic gluing amplitudes, yielding a Hilbert space via quotient and completion. A self-adjoint microscopic Hamiltonian is constructed using spectral functional calculus and perturbation theory. Thermal equilibrium is defined by Gibbs states in finite volume and by KMS states in the thermodynamic limit. Under local equilibrium conditions, Einstein’s field equations emerge as an equation of state, while gauge dynamics follow from symmetry-constrained free-energy extremization. Lorentz covariance is implemented through a strongly continuous unitary representation of the Poincaré group. The framework reproduces classical general relativity and Maxwell theory at leading order and predicts higher-order curvature–gauge couplings suppressed by a microscopic noncommutativity parameter.