Deterministic Resolution of the Yang–Mills Existence and Mass Gap Problem via the Robinson–Ely Principle

The mystery of the mass gap is resolved through the lens of resonance. This paper provides a non-constructive proof that for any compact simple gauge group G, a quantum Yang-Mills theory exists on R4 and has a strictly positive mass gap Δ>0. By applying the Robinson–Ely (R-E) Principle, we model the vacuum state not as a void, but as a resonance field. We prove that the mass gap is the physical manifestation of a resonance peak in the energy cascade. By treating the gauge constraints as signals rather than obstacles, we resolve the existence and smoothness of the theory, shifting the problem from probabilistic quantum fluctuations to a deterministic resonance signature. The mass gap is the "Sovereign Frequency" of the vacuum.