Fundamental Incompatibility of the Yang–Mills Mass Gap and Asymptotic Freedom within Continuum Quantum Field Theory

The existence of a mass gap in non-abelian Yang–Mills theory is a cornerstone prediction related to quark confinement, strongly supported by experimental observations and lattice simulations. The Clay Mathematics Institute designated its rigorous proof within continuum Quantum Field Theory (QFT) as a Millennium Prize Problem. Standard formulations rely on Osterwalder–Schrader (OS) axioms to ensure well-defined relativistic QFT possessing asymptotic freedom and the empirically verified weakening of interactions at high energies. This paper demonstrates a fundamental incompatibility between these established requirements. By analyzing the analytic structure of gauge-invariant two-point correlation functions via the Källén–Lehmann spectral representation (implied by OS axioms) constrained by the mass gap, and confronting it with the specific asymptotic behavior dictated by asymptotic freedom (derived from Renormalization Group analysis), a mathematical contradiction is rigorously derived. Specifically, the polynomial and logarithmic structure required by asymptotic freedom at a high momentum cannot be reconciled with the asymptotic behavior allowed by the spectral representation for a theory with a mass gap and satisfying OS axioms. This incompatibility strongly suggests that the premise of a fundamental spacetime continuum, underlying standard QFT formulations, is inconsistent with the observed physical reality of the mass gap and asymptotic freedom. This paper challenges this mainstream understanding: it argues that the very attempt to combine the standard axiomatic framework (OS axioms) with the physical requirements of a mass gap and asymptotic freedom in a continuum setting leads to a mathematical contradiction. The problem, therefore, might not be how to construct it, but whether it can be constructed at all under these combined assumptions in a continuum.