Structural Necessity of the Mass Gap in Non-Abelian Gauge Theories: A Conditional Proof Framework

We present a structural argument demonstrating that the mass gap in four-dimensional Yang-Mills theory is a necessary consequence of the theory's existence, provided the constructive axioms are met. By analyzing the distinction between Abelian and non-Abelian geometry, we identify that the non-vanishing commutator of the gauge field, combined with the infrared growth of the coupling, enforces a strictly positive lower bound on the energy spectrum. We formulate this as a conditional theorem: if a non-trivial continuum Yang-Mills measure exists on R4 satisfying the Osterwalder-Schrader axioms, then the theory must exhibit a mass gap Δ > 0. This framework clarifies the separation between the constructive existence problem and the spectral gap problem.