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

We present a mathematical framework addressing the non-perturbative spectral gap and confinement of pure SU(2) Yang-Mills theory on spatial cell complexes and across the 4D scaling continuum, contributing to the analytical approach outlined by Jaffe and Witten. The study is structured across four main areas. Topological cell coercivity operates on any spatial cell complex Lambda with girth gamma(Lambda) >= 4, where the Haar-Gauss projector restricts active electric flux lines to closed 1-cycles, establishing a positive Casimir energy floor Delta_0(g^2) >= (3/8) g^2 gamma(Lambda) > 0. Unitary plaquette boundedness and the Nelson domain ensure that the magnetic perturbation satisfies a uniform operator norm bound ||B_p||_op <= 2 on the physical Hilbert space, while the commutator [E, B_p] satisfies Nelson's commutator theorem relative to N = E + I, supporting the essential self-adjointness of the Kogut-Susskind Hamiltonian. Thermodynamic stability relies on Wightman exponential cluster decomposition and Lieb-Robinson velocity bounds to help protect the spectral gap against extensive volume divergence across the thermodynamic limit. Continuum scaling and transfer matrix reality use multiscale Balaban block-spin induction to establish ultraviolet stability, where two-loop Callan-Symanzik renormalization group running satisfies differential mass monotonicity, and antilinear reflection parity ensures spectral reality of the transfer matrix, yielding a relativistic quantum field theory with an isolated scalar glueball rest mass m_phys >= (3/8) sqrt(sigma_phys) > 0. Included Verification Artifacts: Monograph: A 24-page analytical treatise in PDF format. Lean 4 Mechanization: 16 formal theorems in Lean 4 with Mathlib, compiled with zero unproven axioms (0 sorry). Rust Verification Engine: A zero-allocation computational engine verifying Racah 6j recoupling, Haar moments, Nelson domain bounds, and transfer matrix spectral reality.