We prove that the Clay Mathematics Institute formulation of the Yang-Mills mass gap problem on pure ℝ⁴ contains a fundamental topological contradiction. Using Singer's theorem (1978) on the non-existence of global gauge sections and the Gribov-Zwanziger regularization, we demonstrate that any consistent non-abelian gauge theory on ℝ⁴ inevitably introduces complex-conjugate poles in the gluon propagator, strictly violating the Osterwalder-Schrader axiom of Reflection Positivity. This renders the simultaneous requirements of (A) OS axioms on ℝ⁴ and (B) existence of a mass gap Δ > 0 mutually exclusive. We propose a geometric resolution by transitioning to a compact constructible manifold M = S⁴ × 𝕋³ over the multiquadratic field 𝕂 = ℚ(√2, √3, √5). On this manifold, the mass gap emerges from Cheeger's spectral inequality and the non-abelian topology of the braid group B₃, with topological protection provided by the prime torus knot T(103,3) of Seifert genus g = 102.