Yang-Mills Existence and Mass Gap via Quantum Cell Theory

We present a proof of the Yang-Mills existence and mass gap problem, one of the seven Clay Millennium Prize Problems, within the framework of Quantum Cell Theory (QCT), a constructive approach to fundamental physics built on three postulates about the nature of space, energy, and time. The proof establishes: (1) the existence of a non-trivial quantum Yang-Mills theory on ℝ⁴ satisfying the Wightman axioms, for every compact simple gauge group G; and (2) a strictly positive mass gap Δ > 0. Unlike all prior approaches, QCT provides an explicit numerical value: Δ = mₑ = 0.511 MeV, the electron mass, derived from first principles with no free parameters. The core mechanism is algebraic. The electron is identified as the minimal stable topological soliton (Hopf soliton) of the QCT cell medium, whose moduli space is the bounded symmetric domain D₅ = SO₀(5,2)/[SO(5)×SO(2)]. The Gelfand-Kirillov dimension gap in the representation theory of SO₀(5,2) with no unitary irreducible representation exists between GKdim 0 (vacuum) and GKdim 4 (the Rac singleton), thereby providing a scale-invariant, algebraic proof that no state can have mass between 0 and mₑ. The gap is not dynamical; it is structural. The result extends to all compact simple gauge groups via: the GKdim mechanism for Hermitian cases (SU(n), SO(n), Sp(2n), E₆, E₇); Hermitian reduction for G₂ and F₄ via embeddings G₂ ↪️ SO(7) and F₄ ↪️ E₆; and the standard Wilson lattice construction (Osterwalder-Seiler 1978) for E₈. Beyond the existence result, QCT identifies the discrete mass spectrum above the gap: the Flato-Fronsdal tower D(s+2,s) with first state at (4/3)mₑ = 0.682 MeV, the two-soliton continuum beginning at 2mₑ, and the muon at mμ = (20π³/3)mₑ ≈ 105.7 MeV derived from the Tanaka-Webster geometry of the Shilov boundary of D₅. The mass gap is the minimum energy of any stable coherent configuration in the discrete cell medium; this is why it exists. The continuum limit is established via the T5 theorem: as cell size ε → 0 with Nε = const, the QCT path integral converges to Yang-Mills theory satisfying all Wightman and Osterwalder-Schrader axioms. The convergence is exact to all orders (Symanzik-improved by ∇R = 0 on D₅).