Entropy-Originated Topological Framework for a Unified Theory of Matter and Gravity: The Theory

We propose a unified physical framework in which both gravitation and gauge interactions emerge from a topological–entropic structure within a diagrammatic Hilbert space. Matter fields correspond to projective states of a non-commutative operator algebra, and gauge and gravitational interactions arise as spectral actions of projected Dirac operators. Within this formulation, we perform a quantitative two-loop renormalization-group (RG) analysis across a grid of projection scales and entropic weights, yielding a convergent unification scale and coupling. The resulting proton-lifetime estimates 10^34-35 years are consistent with current experimental bounds. These findings suggest that entropic projective corrections to the standard two-loop RG flow can naturally stabilize gauge unification and provide a renormalizable embedding of gravity within an operator-based topological framework.