Derivation of the Fine-Structure Constant from the Binary Icosahedral Group 2I over Q(√5): A Complete Geometric and Arithmetic Construction

We derive the inverse fine-structure constant α⁻¹ from the geometry of the binary icosahedral group 2I acting on S³, without using α⁻¹ or any electromagnetic constant as input. The golden ratio φ = (1+√5)/2 is the central object because it is the unique positive real number satisfying N(φ) = −1 in Q(√5), where N denotes the field norm. This identity N(φⁿ) = (−1)ⁿ determines the sign of every term in the series without any free choices. The result is: α⁻¹ = 360/φ² − 2/φ³ + 5/(16φ¹⁴) + 20/(111φ²⁷) + 13/(111φ⁴¹) + 52/(111φ⁶⁸) + ··· All terms lie in Q(√5) by construction. The first five terms give S₅ = 137.035 999 206 000 184..., consistent with CODATA 2022 (α⁻¹ = 137.035 999 177(21)) within 1.38σ, and with the rubidium measurement of Morel et al. (2020) (α⁻¹ = 137.035 999 206(11)) within 1.84×10⁻¹³. The same geometric objects predict mτ/mμ = 44/φ² = 16.8065 (experimental: 16.8173, error 0.064%) with no additional parameters. A falsifiable prediction: T₆ = 52/(111φ⁶⁸) ≈ 2.88×10⁻¹⁵, testable with CODATA 2026. Fourteen results are established without fitting to α. Three open problems are stated explicitly. Additional files include Spanish (ES), French (FR), and Polish (PL) translations.