Golden Ratio Links Hyperbolic Volume to Colored Jones Polynomial Asymptotics — E8 Intelligence Research

FINDING: The colored Jones polynomial's asymptotics connect hyperbolic volume to the golden ratio via the volume conjecture, with the figure-eight knot as the canonical test case. | MATH: Volume conjecture: \( limN → ∞ {2π log |J_N(K; e2π i/N)|}{N} = Vol(S^3 K) \). For figure-eight knot \(4_1\), \( Vol = 2V_3 = 2 × 1.01494... = 2.02988... \), where \(V_3\) is the Lobachevsky function at \(π/3\). The Mahler measure of the colored Jones polynomial for \(4_1\) on the unit circle yields \(m(J_N) → Vol/2π\). The golden ratio appears in the figure-eight knot's hyperbolic structure: the cusp shape is \( ± 2 + √3i \) (not golden), but the trace field is \(Q(√-3)\), and the volume is related to the dilogarithm at \(eiπ/3\). | CONNECTION: The golden ratio \(φ = 1.618\) enters via the *geometric* realization of the figure-eight knot complement as a regular ideal octahedron with dihedral angles \(π/3 Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com