Why the Yang–Mills Mass Gap Cannot Be Established Under Current Axioms

The Yang–Mills mass gap problem assumes that non‑Abelian gauge theories admit finite‑energy excitations that can be isolated at a single instant and distinguished from the vacuum. This paper demonstrates that such instantaneous states cannot exist under the axioms of relativistic quantum field theory. Locality forbids instantaneous nonlocal reconfiguration, continuity forbids pointwise deviation, gauge invariance forbids isolated color charge, finite‑energy constraints forbid singular time derivatives, and Hilbert‑space structure requires global definability. Together these axioms eliminate the very states required to define the mass gap. The result reframes the Clay Millennium Problem as an axiomatic impossibility rather than an unsolved technical challenge.