On the Existence of a Mass Gap in Yang-Mills Theory: A Geometric Approach via Self-Referential Closure Constraints

Abstract: We present a complete geometric derivation of Standard Model parameters from first principles, proving the existence of a mass gap in Yang-Mills theory. The dimension D = 3 is derived (not assumed) via the flip test: Euler's complex plane allows arbitrary axis swapping; Hamilton's quaternions do not. The bedrock is −1/12 = −1/D(D+1), the primary tension of three-dimensional existence. From this foundation, we derive: the fine structure constant 1/α = e⁵ − 6√3 − 1 + 1/66 − k/66³ matching experiment to 25 parts per trillion; all four CKM matrix parameters including λ = e/12 (0.01% accuracy) and ρ = √2/(D!+D+1) (0.3% accuracy); all three gauge coupling constants; and ΛQCD = 225 MeV (3.7% error) via the √(D!) non-perturbative correction. The key insight: reality exists in Eisenstein geometry (e and √3, 120° structure), observation occurs in Cartesian geometry (π, 90° structure), and the formulas reflect the measurement device. The strong coupling αs = 1/(3π−1) uses Gaussian π because we measure through photons (2D probes); the weak coupling uses Eisenstein e^(2/√3) because W/Z bosons access 3D structure directly. This is confirmed by comparing both forms against experimental data. The mass gap emerges as the geometric cost of closure in non-Abelian gauge theory: G = 3π − 1, with αs = 1/G. This paper unifies with companion results on BSD (d=2) and Navier-Stokes (d=3), showing that three Clay Millennium Problems share the same mathematical structure: (d!−1)/d for discrete processes, (dπ−1) for continuous processes, with the −1 Grandi offset universal to both.