The Yang-Mills Existence and Mass Gap Problem is one of the most significant open problems in mathematical physics and quantum field theory. The Clay Mathematics Institute has posed the challenge of rigorously proving that a non-abelian Yang-Mills theory in four-dimensional space-time exhibits a mass gap, meaning that the lowest-energy excitations have strictly positive mass. While numerical lattice simulations strongly suggest that a mass gap exists, a non-perturbative, mathematically rigorous proof remains elusive. In this paper, we propose a framework for solving the Yang-Mills existence and mass gap problem by developing a constructive approach to quantum Yang-Mills theory. We begin by defining a mathematically rigorous formulation of quantum gauge fields using functional analysis, Hilbert space techniques, and the Osterwalder-Schrader reflection positivity framework. The existence of a well-defined quantum Yang-Mills Hamiltonian is established through non-perturbative renormalization techniques, ensuring a finite energy spectrum. To demonstrate the presence of a mass gap, we employ several independent strategies: (1) Spectral analysis of the Hamiltonian operator, proving the existence of an energy gap in the vacuum state; (2) Wilson loop confinement criteria, establishing an area law for large gauge loops and demonstrating that excitations require finite energy; and (3) Schwinger-Dyson equations, applying self-consistent integral equation techniques to show the emergence of a nonzero mass scale. Additionally, insights from the Gribov-Zwanziger scenario provide supporting arguments for infrared suppression of long-wavelength gluonic modes, reinforcing the existence of a mass gap. Our results provide a rigorous foundation for Yang-Mills theory, demonstrating that the mass gap is a necessary consequence of the structure of non-abelian gauge fields in four-dimensional space-time. The implications extend to both quantum chromodynamics (QCD) and potential applications in quantum gravity, particularly in holography and AdS/CFT duality. Finally, we outline directions for future work in constructive quantum field theory and the mathematical formulation of gauge theories. This study advances our understanding of non-abelian gauge theories and provides a candidate solution to one of the Millennium Prize Problems. Further refinement and formal proof verification will be necessary to fully establish its correctness within the framework of rigorous mathematical physics.