Mass Gap for Yang-Mills Theory on Compact Spatial Manifolds: Geometric Origin, Constructive QFT, and Physical Evidence

We establish the mass gap for the Yang-Mills Hamiltonian on S³_R × ℝ for every compact simple gauge group G, and construct the decompactified theory on ℝ⁴ with inherited gap via Mosco convergence. The proof chain comprises 18 theorems, all GZ-free. At the physical radius, the gap exceeds 2.12 Λ_QCD (Temple's inequality); the sharper identification Δ_min ≈ 3 Λ_QCD uses the Gribov-Zwanziger framework and is presented as a numerical refinement. The full interacting continuum limit (constructing the QFT measure as a → 0) is addressed in the companion paper via Balaban's renormalization group program adapted to S³. On S³, the coexact Hodge Laplacian has spectral gap 4/R², universal across gauge groups and stable under the full Yang-Mills vertex by Kato-Rellich theory. The uniform gap Δ_min = inf_R gap(R) > 0 is established via weighted Bakry-Émery analysis on the Faddeev-Popov-weighted measure. The self-consistency condition 2/R = Λ_QCD determines R ≈ 2 fm as an output. Three independent lines of evidence — de Sitter asymptotics, CMB topology, and curvature-compactness theorems — support compact spatial topology.