Thermodynamic Operator Geometry: A Constructive Framework for Emergent Gravitation and Gauge Dynamics

We present a constructive operator-theoretic framework in which spacetime geometry and gauge interactions arise from thermodynamic properties of a regulated microscopic algebra. Instead of quantizing classical geometry, we begin from a rigorously defined Hilbert space built from regulated combinatorial structures and equip it with self-adjoint generators whose spectral properties determine macroscopic dynamics through entropic coarse-graining. Einstein’s field equations emerge as equilibrium conditions of local entropy balance, while Maxwell-type gauge equations arise from internal operator symmetries via free-energy extremization. Lorentz covariance is implemented algebraically through a unitary representation of the Poincaré group. Cosmological acceleration and galactic rotation phenomena are interpreted as entropy contributions of hidden operator sectors. The framework preserves established low-energy physics while providing a unified operator-thermodynamic origin for geometry and gauge structure.