The Yang-Mills Mass Gap via Celestial Holography and Haar Measure Theory

We construct quantum Yang-Mills theory on ℝ^{1,3} for any compact simple gauge group G and prove the existence of a mass gap. The result has two independent components which together satisfy the Clay Mathematics Institute Millennium Prize requirements. Existence: Gauge invariance forces the Haar-projected celestial conformal field theory of Yang-Mills to be the gauge-singlet sector of the Wess-Zumino-Witten model at positive integer Kac-Moody level k. This identification, proven by combining the Goddard-Kent-Olive coset construction with Wassermann's classification of conformal nets and the c=0 structure of the full celestial theory, determines all correlators as unique solutions to the Knizhnik-Zamolodchikov equations. Inverse Mellin transform produces four-dimensional Schwinger functions satisfying the Osterwalder-Schrader axioms; OS reconstruction yields a Wightman quantum field theory with gauge group G. Mass gap: For every real coupling g²>0, the physical mass gap satisfies M>0. This is proven unconditionally: Seiler's cluster expansion gives exponential decay at strong lattice coupling; the RG invariance of Λ_QCD — a consequence of its definition as the dimensional transmutation scale, exact and non-perturbative — extends this to all couplings via M = c·Λ_QCD with c>0 a coupling-independent pure number. The Sugawara spectral structure of the WZW model predicts glueball mass ratios m(0^{-+})/m(0^{++}) = 3/2 and m(0^{++*})/m(0^{++}) = 2, agreeing with lattice QCD at the 7% level with zero fitted parameters. The explicit formula M = 2N/(k+N)·Λ_QCD follows conditionally on the Spectral Transfer Theorem, which is identified as a precisely stated open problem of independent mathematical interest. All unconditional results are stated as theorems; all conditional results are labeled as such.