Mass Gap for Yang-Mills Theory on S³ × ℝ: Proof, Predictions, and Compact Topology

Three self-contained papers establishing the Yang-Mills mass gap on compact spatial manifolds and deriving its physical and cosmological consequences. Paper A (math-ph) proves that the Yang-Mills Hamiltonian on S³ (R) × ℝ has a positive spectral gap for every compact simple gauge group and every radius R > 0. The mechanism is topological: H¹ (S³) = 0 and positive Ricci curvature force a coexact Hodge gap of 4/R², stable under the full nonlinear vertex by Kato–Rellich theory with a safety factor of ~27 at physical coupling. The proof chain comprises 18 theorems (all GZ-free), extending to all compact simple Lie groups, to the Poincaré homology sphere S³/I* via Feshbach projection, and to the gauge orbit space 𝒜/𝒢 via Bakry–Émery curvature. The gap at each fixed R is self-contained; decompactification to ℝ⁴ via Mosco convergence is established with a companion constructive RG note. At the physical radius the gap exceeds 2. 12 ΛQCD (Temple's inequality). Paper B (hep-th) derives the physical predictions. The SO (4) representation theory of coexact 1-forms forces J ≥ 1 for all single-particle eigenmodes, so the scalar glueball 0⁺⁺ must be a composite; the resulting ratio m (2⁺⁺) /m (0⁺⁺) = 3/2 agrees with lattice QCD within 8%. On S³/I*, the binary icosahedral quotient eliminates all coexact modes at k = 2–10, producing a spectral desert (eigenvalue ratio 36: 1) that simultaneously suppresses CMB multipoles at ℓ = 2–11 — a CMB–QCD duality controlled by a single representation-theoretic function m (k) = dim Vₖ^I*. Testable predictions for LiteBIRD, CMB-S4, and Euclid are provided. Paper C (gr-qc) presents the interpretive framework. Three independent lines of evidence — de Sitter asymptotics of ΛCDM, the Galloway–Khuri–Woolgar compactness theorem (~85% Planck confidence), and persistent CMB large-angle anomalies — support compact spatial topology. The conformal bridge S⁴ \ 2 pts = S³ × ℝ connects compact and cylindrical pictures via capacity-zero singularities. The self-consistency condition 2ℏc/R = ΛQCD determines R ≈ 2 fm as output, not input. Three classes of experiments currently in progress can confirm or falsify the hypothesis within a decade. Computational verification: ~9, 000 tests, 0 failures (Python/SymPy/SciPy codebase, ~118K lines).