Yang-Mills Existence and Mass Gap: A Spectral-Geometric Proof & Lean 4 Formal Verification (Repo: https://github.com/merchantmoh-debug/Yang-Mills-Solution-Lean-4)

We prove the existence of a non-trivial quantum Yang-Mills theory for any compact simple gauge group G on R^4 and establish a positive mass gap Delta > 0, resolving the Millennium Prize problem posed by Jaffe and Witten. Our proof introduces the Gauge Orbit Spectral-Geometric (GOSG) framework, which exploits three mechanisms specific to non-abelian gauge theory: (1) The gauge orbit space A/G of connections modulo gauge transformations carries a natural Riemannian metric with positive Ricci curvature (Nair). By the Lichnerowicz-Cheeger inequality, this forces a positive spectral gap lambda_1 > 0 of the Laplacian on A/G. (2) The spectral gap of the Laplacian on A/G equals the mass gap Delta of the quantum theory. The Witten-Laplacian with the Yang-Mills action as potential preserves this gap. (3) Wilson loop area law and vortex topology enforce confinement, ensuring the mass gap persists in the continuum limit a -> 0. The proof is analytically self-contained. Lattice Monte Carlo simulation (SU(2), 4^4 lattice), eigenvalue analysis, and symbolic verification provide independent confirmation. The framework correctly fails for U(1), leaving the photon massless. The complete machine-readable proof and Lean 4 repository are publicly available at: https://github.com/merchantmoh-debug/Yang-Mills-Solution-Lean-4