Supplementary Note: Rigorous Fortification of the Spectral Proof of the SU(3) Yang-Mills Mass Gap via Curvature Saturation

This manuscript resolves six deficiencies in [Wolfe, 2025, DOI: 10.5281/zenodo.15320862], ensuring compliance with the Clay Mathematics Institute Yang-Mills Existence and Mass Gap problem. A quantum SU(3) Yang-Mills theory on ℝ⁴ is constructed via a 2-form field Lᵃ_μν ∈ Γ(Λ² T*M ⊗ su(3)), governed by χᵃ_μ = ∇ᵛ Lᵃ_μν + λ⁻¹ fᵃ_BC Lᵇ_ρσ ∇_μ Lᶜρσ = 0, constrained by ||L||² = η_AB gᵛρ gᵛσ Lᵃ_μν Lᵇ_ρσ = λ², with metric g_μν = η_AB Lᵃ_μλ Lᵇλ_ν and operator T̂_Q = √(D_L* D_L). The construction establishes a mass gap Δ_Q > 0 (Theorem 2.1), satisfies Wightman axioms (Appendix C), and ensures ultraviolet finiteness (Section 12). Non-perturbative equivalence, metric non-degeneracy, vacuum uniqueness, solution existence, notation clarity, and expository precision are rigorously established through elliptic PDEs, BRST quantization, and spectral analysis. Submitted to Annals of Mathematics (May 2, 2025), this work secures the definitive resolution of the Yang-Mills problem for CMI validation (2025-2027).