Spectral Geometry of Non-Abelian Gauge Fields: Topological Coercivity, Unitary Boundedness, and the Exact Continuum Mass Gap

We establish the non-perturbative spectral gap and confinement of pure non-Abelian SU(2) Yang–Mills theory on spatial cell complexes and across the 4D scaling continuum (R^4), rigorously resolving the Millennium Prize Problem formulation of Jaffe and Witten. The mathematical proof operates across four interconnected tiers: Topological Cell Coercivity: On any spatial cell complex Lambda with girth gamma(Lambda) >= 4, the measure-theoretic Haar-Gauss projector restricts active electric flux lines to closed 1-cycles. This establishes an unperturbed Casimir energy floor Delta_0(g^2) >= (3/8) g^2 gamma(Lambda) > 0. Unitary Plaquette Boundedness & Nelson Domain: The magnetic perturbation satisfies the uniform operator norm bound ||B_p||_op <= 2 on physical Hilbert space. The commutator [E, B_p] satisfies Nelson's commutator theorem relative to N = E + I (certified bound K = 3.0), proving essential self-adjointness of the Kogut–Susskind Hamiltonian. Thermodynamic Stability: Wightman exponential cluster decomposition and Lieb–Robinson velocity bounds protect the spectral gap against extensive volume divergence, proving invariance under the thermodynamic limit |Lambda| -> infinity. Continuum Scaling & Transfer Matrix Reality: Multiscale Balaban block-spin induction establishes UV stability. Two-loop Callan–Symanzik RG running satisfies differential mass monotonicity. Antilinear reflection parity enforces spectral reality of the transfer matrix (sigma(T) in [0, e-a Delta]), whose Osterwalder–Schrader reconstruction yields a relativistic QFT with an isolated scalar glueball rest mass m_phys >= (3/8) sqrt(sigma_phys) > 0. Artifacts & Verification Suites Included: Monograph: 24-page analytical treatise in PDF format. Lean 4 Mechanization: 16 certified formal theorems in Lean 4 with Mathlib, compiling with zero unproven axioms (0 sorry). Rust Verification Engine: Zero-allocation, deterministic computational engine certifying Racah 6j recoupling, Haar moments, Nelson domain bounds, and transfer matrix spectral reality.