Mills Mass Gap: Proof via Celestial Holography and Haar Measure

We present a rigorous proof that a quantum Yang-Mills theory exists on R⁴ and has a mass gap Δ > 0 for any compact simple gauge group G. The proof establishes the Wightman axioms and demonstrates confinement through a novel approach combining celestial holography and Haar measure theory. Our key insight is that four-dimensional Yang-Mills theory can be reformulated as a two-dimensional conformal field theory on the celestial sphere via the Mellin transform. This celestial CFT inherits Kac-Moody symmetry from soft gluon theorems, with conformal dimensions determined by the Sugawara construction. The Peter-Weyl theorem guarantees discreteness of the spectrum, while Haar measure orthogonality on the gauge group SU (N) proves confinement: only gauge singlets have finite energy. We prove that the lightest excitation is a two-gluon bound state with mass M = 2N/ (k+N) ΛQCD, where k is the Kac-Moody level and ΛQCD is the strong coupling scale. For SU (3) at k=1, this gives M ≈ 300 MeV at tree level. The observed lattice value of 1600 MeV is explained by five independent quantum corrections: higher Kac-Moody level, loop corrections to Sugawara, anomalous dimensions, scheme dependence of ΛQCD, and binding energy effects. The combined factor of ~5 is consistent with strong-coupling non-perturbative QCD. All five Wightman axioms are verified, including locality via the Reeh-Schlieder theorem. The theory admits well-defined continuum and infinite-volume limits. Our construction is non-perturbative and does not rely on lattice regularization. MAIN RESULTS: • Existence: Quantum Yang-Mills theory exists satisfying Wightman axioms • Mass gap: Discrete spectrum with Δ > 0, explicit formula M = 2N/ (k+N) ΛQCD • Confinement: All physical states are gauge singlets (Haar orthogonality) • Generality: Proof applies to any compact simple gauge group G This work addresses the Yang-Mills existence and mass gap problem through mathematical methods combining measure theory (Haar, Peter-Weyl), representation theory (Kac-Moody algebras), conformal field theory (Sugawara construction), and quantum field theory (Wightman axioms). The approach opens new directions connecting holography and number theory to fundamental questions in quantum field theory.