Proof of the Yang–Mills Mass Gap via Spectral Compactness and Gauge-Invariant Operator Theory

This article presents a continuum-analytic proof of the Yang–Mills mass gap for compact simple gauge groups on ℝ⁴. By proving that the resolvent operator T = (1 + Δ_A)^(-1), acting on the gauge-invariant Hilbert space L²(𝒜/𝒢,μ), is compact and self-adjoint, we demonstrate that the Yang–Mills Laplacian ΔA has a discrete spectrum with a strictly positive first nonzero eigenvalue λ₁ > 0. Consequently, the quantum theory exhibits a nontrivial mass gap Δ = √λ₁ > 0. The proof leverages classical spectral methods, including Mercer’s theorem, Schatten-class compactness, and a gauge-covariant coercivity inequality. It avoids both lattice discretization and perturbative expansions. Numerical validation confirms exponential eigenvalue decay and spectral stability across truncation bases. This result addresses and fulfills the Clay Millennium Problem for Yang–Mills theory, which asks:“Prove that for any compact simple gauge group G, a non-trivial quantum Yang–Mills theory exists on ℝ⁴ and has a mass gap Δ > 0." Keywords: Yang–Mills, mass gap, spectral theory, compact operator, quantum field theory, gauge theory, operator analysis, functional analysis, harmonic confinement