The Quintic Resolution of α: Deriving the Fine Structure Constant from Icosahedral Geometry

We derive the fine structure constant α from icosahedral geometry and Klein's resolution of the quintic equation. The formula α⁻¹ = e⁵ − 2(3√3 − 5) − 11 + 1/66 + (harmonics) emerges from four geometric necessities: (1) e⁵ = e^(E/D!) where E = 30 edges and D! = 6, giving the icosahedral growth exponent, (2) the kissing sphere gap 3√3 − 5 = (three-phase RMS) − (vertex degree), the difference between continuous Eisenstein closure and discrete icosahedral structure, (3) Klein's coefficient 11 = V − 1 from the icosahedral vertex invariant, and (4) the prime wheel modulus 66 = D! × (V − 1) from dimensional and vertex structure. Harmonic corrections built entirely from icosahedral invariants (V = 12, E = 30, F = 20) yield agreement with CODATA 2022 to 5 parts per trillion. The face-centred formula matches CODATA 2022 (137.035999177) to 5 ppt; the vertex-centred formula matches CODATA 2018 (137.035999084) to 475 ppt using only two correction terms, with systematic search confirming no natural icosahedral ratio closes the vertex residual at higher order. The holographic 11:5 weighted average α⁻¹ = 137.035999113 constitutes the theoretical prediction for future CODATA convergence. The fine structure constant is not arbitrary — it is the unique electromagnetic coupling permitted by icosahedral spacetime geometry.