Connectivity Geometric Unified Field Theory I: Topological Emergence of Spacetime and Rigorous Derivation of the Gravitational Field Equation

Within the framework of connectivity geometry, this paper rigorously derives the microscopic topological origin of macroscopic gravity. Based on the discrete spin network and the holographic screen hypothesis, we prove that the boundary entanglement entropy is proportional to the network topological minimum cut. Using the equipartition principle on the holographic screen, the gravitational constant G is expressed as a geometric combination of the network lattice spacing ell and the long-range hopping step k, yielding Newton's law of universal gravitation and Poisson's equation. Combined with Regge discrete calculus, tensor network holographic mapping, and local Rindler horizon thermodynamics, we derive the Einstein field equations with higher-order topological correction terms, and provide a quantitative expression for the cosmological constant Lambda ~ 8piell_P^2/ell^4. We predict that in cold-atom quantum simulations, the discrete topological correction will induce a relative shift in the sound speed of the emergent acoustic metric Delta c_s/c_s ~ c_2*(ell/L)^2 (order of 10^-4 to 10^-3). Key clarification: Here ell is the network lattice spacing, and Theorem 2 (eta = ell^2/ell_P^2) is a dimensional consistency condition under holographic truncation, ensuring that the macroscopic G value and microscopic experimental predictions are self-consistent simultaneously.