Entropy-Originated Topological Framework to Unify the Theory of Matter and Gravity

A unified operator-based formalism is proposed in which both matter and gravity emerge from entropic and topological principles. The theory is constructed within a Diagram Hilbert Space HD equipped with projective operators P representing minimal-entropy configurations of physical states. Entropic projection selects physically realized configurations, generating effective mass operators through a multiplicative composition of partition functions across diagrammatic sectors. The renormalization flow of projective couplings is shown to exhibit ultraviolet convergence, providing an asymptotically safe framework that unifies gauge and gravitational interactions. Gravity arises from topological invariants encoded within HD, leading to an emergent metric and curvature tensor consistent with the Einstein field equations in the macroscopic limit. The resulting framework offers a renormalizable and phenomenologically testable approach to quantum gravity and unification.