A Metric-Theoretic Proof of the Yang-Mills Mass Gap on ℝ⁴

We prove the existence of a strictly positive mass gap for Yang-Mills gauge theory with gauge group SU (3) on ℝ⁴. The proof remains entirely within ℝ⁴ as a topological manifold while introducing a specific conformal metric g = Ω² (p) η whose form is uniquely determined by the Clifford algebra Cl (4) of the manifold and the spectral theory of the associated Laplace-Beltrami operator. Under this metric all Yang-Mills loop integrals converge to the finite value ln (4/3) via a holonomy expansion, with no ultraviolet divergences and no renormalization required: the UV divergence is structurally absent, not cancelled. The mass gap is derived explicitly as Δ = Qᵥac × exp (λ₁) = 221. 7 MeV, where all quantities are derived from the Clifford structure of ℝ⁴ and the electroweak vacuum with no free parameters. The key tool is the Clifford winding penalty c₂ = 1/4, the normalized trace of a single winding generator in Cl (4), which replaces the divergent UV tower Σ 1/n with the convergent geometric series Σ c₂ⁿ/n = ln (4/3).