[30]Connectivity Geometry: Discrete Topological Origin of Maxwell's Equations

Traditional electromagnetism phenomenologically describes electromagnetic phenomena via Maxwell's equations, but fails to explain the absence of magnetic monopoles, the origin of the right-hand rule, and why the vacuum speed of light is a universal constant. Based on the framework of connectivity geometry, this paper proves that the electromagnetic field is not an independent fundamental entity, but rather a topological shear and phase strain excited by the underlying k=3 discrete spin network during charge transitions. By constructing a discrete electromagnetic tensor in four-dimensional spacetime, we rigorously derive the longitudinal prohibition of the magnetic field (absence of magnetic monopoles) and the transverse shear (closure of magnetic field lines), yielding eigenvalues 0, ±3. We further prove that Faraday's law of electromagnetic induction and the invariance of the speed of light are rooted in the topological phase closure of the k=3 grid, and provide a testable modified formula for Lorentz violation. The four core laws of electromagnetism obtain a unified geometric origin under this discrete topological framework. As a natural extension of this framework, this eigenspectrum also naturally distinguishes electromagnetic propagation from quark confinement, realizing a topological bifurcation of U(1) and SU(3) gauge structures.