Rigorous Foundations and Completed Solutions within the Operator–Topological Unification of Matter, Gravity, and Gauge Fields

We establish the mathematically rigorous core of the operator–topological framework unifying matter, gravitation, and gauge interactions. A direct-integral Diagram Hilbert Space (DHS) is constructed with a well-defined measure and C*-algebra of projectors. The existence and conditional uniqueness of the entropic minimizer are proven within von Neumann algebraic thermodynamics. The microcanonical partition function of topological projector sectors yields a discrete, predictive mass spectrum without parameter degeneracy. The framework is shown to preserve unitarity, consistency, and one-loop renormalization closure with projector-dependent thresholds. These results render the theory mathematically complete at the foundational level and physically predictive in its operator–entropic regime.