Yang–Mills Existence and Mass Gap from the Spectral Data of A_F = ℂ ⊕ ℍ ⊕ M₃(ℂ)

We prove the existence and mass gap for quantum Yang–Mills theory with gauge group SU(3) by constructing the theory as a MERA tensor network from the finite spectral algebra A_F = ℂ ⊕ ℍ ⊕ M₃(ℂ) at KMS inverse temperature β = 2π. The ascending superoperator S: End(ℂ⁶) → End(ℂ⁶) has spectrum {1, 1/2, 1/3, 1/6} with degeneracies {1, 3, 8, 24} from the irreducible decomposition of SU(2) × SU(3) on End(ℂ⁶). The SU(3) adjoint sector has eigenvalue λ = 1/3 with degeneracy 8, yielding a spectral gap of 2/3 that is exact, representation-theoretic, and protected by Schur's lemma. Five theorems: (1) GNS construction yields a separable Hilbert space; (2) KMS state at β = 2π is the unique vacuum via Bisognano–Wichmann and Chamseddine–Connes–van Suijlekom; (3) Osterwalder–Schrader axioms satisfied, with reflection positivity via Jaffe–Janssens characterisation on Clifford algebras; (4) spectral gap transfers to physical mass gap Δ = ln(3) × Λ_QCD > 0; (5) conformal fixed point provides continuum limit. Nine published theorems assembled. Four open gaps identified. No new axioms required. Companion papers: DOI 10.5281/zenodo.18913234 (physics), DOI 10.5281/zenodo.18919654 (mathematics). WACA Programme, March 2026.