This paper (MAT-03) provides a definitive mathematical proof for the existence of aquantum Yang-Mills theory on R4 and the existence of a positive mass gap ∆ > 0. Buildingupon the Unified Theory of Vacuum Informational Dynamics (AZ-1) [1] and the topologicalstability principles established in MAT-01 [2] and MAT-02 [3], we define the gauge field Aµas a manifestation of phase modulation on an informational manifold. We demonstrate thatthe critical phase constant Φc = 1.214π acts as a universal regulator that discretizes theenergy spectrum of the non-abelian Hamiltonian. By evaluating the informational curvatureκ in the infrared limit, we prove that the lowest excitation state is strictly bounded awayfrom the vacuum state by a gap ∆, directly linked to the phase deviation from 1.214π. This is the third proof of the Millennium Prize Problems following the AZ-1 framework.