Derivation of the Fine-Structure Constant α−1 from the Spectral Geometry of the 600-Cell

Abstract

A geometric-spectral hierarchy is constructed from a single axiomthe free action of the binary icosahedral group 2I ⊂ SU(2) on S3and its canonical evaluation S6 is shown to reproduceα−1 = 137.035 999 206(MorelYaoCladéGuellati-Khélifa, 2020) with residual 1.87 × 10−13, without employing α as an inputat any step of the construction. The following are determined: the coecient c1 = 360 (Gram theorem),the numerators F3+2k and 4F3+2k (Binet and the Galois-odd functional δ), the exponents pn (traces ofthe Galois pairs of the critical sector), and the denominator 16 (dim Λ∗(Ec) = π(7)). The following arenot determined and are declared as such: the coecient c2 = −2 and the use of 111 as denominator forn ≥ 4 (hypothesis CPSD-N).On the title. The term derivation designates here the chain 2I ⇒ 600-cell ⇒ ∆ ⇒ Wedderburn ⇒Ec ⇒ R ⇒ Sn, in which each object arises from the previous one without free parameters. It does notdesignate a deduction from rst principles of physics: the nal step Sn ⇒ α−1is a high-precisionnumerical evaluation under the framework, not a theorem of eld theory, and two lattice entries remainunderived. Section 1 xes this convention with precision.Keywords. ne-structure constant; binary icosahedral group; golden ratio; 600-cell; Hodge Laplacian;Wedderburn; Q(√5); Fibonacci sequence; combinatorial Dirac operator; CODATA 2026.