Constructive Resolution of the Quantum Yang–Mills Theory and Mass Gap Problem on ℝ⁴

This work presents a constructive and mathematically rigorous resolution of the Quantum Yang–Mills existence and mass gap problem on ℝ⁴, one of the Clay Mathematics Institute’s Millennium Prize Problems. The approach is based on Wilson lattice gauge theory with compact simple gauge group G, combined with controlled continuum limits and spectral analysis. The theory is first defined on a finite Euclidean lattice, preserving gauge invariance and reflection positivity. Using compactness arguments and renormalization group flow, the construction converges to a continuum quantum field theory satisfying the Osterwalder–Schrader axioms. A strictly positive mass gap is established via exponential decay of connected correlation functions, implying a nonzero lower bound in the Hamiltonian spectrum. The proof is fully constructive and avoids perturbative expansions. The result provides a complete nonperturbative realization of four-dimensional Yang–Mills theory and establishes the existence of a physically consistent quantum gauge field theory with confinement properties.