Yang–Mills Existence and Mass Gap

For compact simple gauge group G, establish mathematically rigorous quantum Yang–Mills theory in 3+1 dimensions — existence and a mass gap Δ > 0 between the vacuum and the lightest particle, matching what lattice simulations and experiment indicate.

Yang and Mills wrote the theory in 1954; it became the backbone of the Standard Model, earning Nobel prizes, while remaining mathematically undefined: the path integral is not constructed. Jaffe and Witten's Clay formulation (2000) asks for what physics assumes daily. Lattice computations exhibit confinement and the mass gap beautifully — proof is another matter.