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

We present a formalized phenomenological framework for the Topological DiscontinuousUniverse, wherein the smooth manifold of General Relativity (GR) emerges as the long-wavelengthlimit of a finite relational graph G(t) = (V, E(t)) with a fixed number of Planck nodes (|V| = N) and astrictly conserved total link budget (|E(t)| = Etot). We formulate the discrete link conservation lawvia the commutator of the graph Hamiltonian [Ĥgraf, Êtot] = 0, rigorously separating topological edgeinvariance from quantum Hilbert-space unitarity. By partitioning links between localized mattercondensates (Em) and the cosmic vacuum skeleton (Ev), we derive the macroscopic scale factor a(t)as a function of mean void connectivity. From a logistic relaxation equation governing gravitationallink binding, we explicitly derive the effective vacuum equation of state wvac(z), proving that therelaxation rate cancels out and the resulting profile naturally exhibits a phantom crossing (wvac = −1)at the critical saturation uc = (1 − √(1 − κ)) / κ ∈ (1/2, 1), in agreement with DESI Collaboration(2024) measurements while preserving early-universe BBN and CMB epochs. In the local regime, acombinatorial upper bound on node degree (ki ≤ kmax) yields a regular black hole metric withstandardized Schwarzschild asymptotics, a horizon threshold M ≥ (3√3 / 4)Lsat, finite curvatureinvariants, and an exact effective stress-energy tensor Tμν. Finally, we formulate a unified boundarylink-flux differential equation governing both local black hole evaporation and the globalcosmological Big Bounce.