This paper presents a formal proof outline addressing the Yang–Mills Mass Gap problem, approached through a recursive harmonic framework grounded in number theory. The method integrates analytic structures derived from the Riemann zeta function, spectral determinant logic, and a recursive eigenvalue formulation. This framework proposes a constructive bridge between number theory and quantum field theory by identifying discrete mass gaps through recursion across well-defined eigenmodes. All terms are rigorously defined, logical steps are explicit, and falsifiable predictions are embedded in the formal structure.