Topological Discontinuous Universe: Quantum Connectivity, Dynamic Vacuum Compensation, and Regular Black Holes Without Singularities

Abstract: We propose a conceptual framework—the Topological Discontinuous Universe—replacing smooth spacetime with a discrete relational network possessing finite Planck nodes (N) and a strictly conserved total link budget (Etot). Gravity emerges as an emergent geometric consequence of connectivity, where elementary particles are topological condensates. Using link conservation between collapsing matter and voids, the model derives dynamic vacuum expansion without dark energy. The effective equation of state w(z) captures recent DESI data, including phantom crossing. Introducing an upper connectivity limit (kmax) eliminates singularities, distinguishing local black hole evaporation from a global Big Bounce. Note: This manuscript presents a conceptual framework and phenomenological ansatz; rigorous derivation from a graph Lagrangian remains ongoing work.