Rogers–Ramanujan Continued Fraction Solves General Quintic via Icosahedral Symmetry — E8 Intelligence Research

Abstract

FINDING: The Rogers–Ramanujan continued fraction (RRCF) is a modular function that solves the general quintic, linking elliptic nome, icosahedral symmetry, and the golden ratio. MATH: - RRCF: \( R(q) = \frac{q^{1/5}}{1 + \frac{q}{1 + \frac{q^2}{1 + \frac{q^3}{1 + \cdots}}}} \), with \( q = e^{2\pi i \tau} \) (elliptic nome). - Quintic solution: A root of \( x^5 + a x^2 + b x + c = 0 \) (or general Bring–Jerrard form) is expressible as an algebraic function of \( R(q) \), where \( q \) is determined by the coefficients via modular equations (arXiv:1510.00068v2). - Key identities: - \( \frac{1}{R(q)} - 1 - R(q) = \frac{\eta(q^{1/5})}{\eta(q^5)} \) (eta-quotient relation). - \( R(q) \) satisfies \( R(q) R(q^4) = \frac{1 - R(q)^5}{1 + R(q)^5} \) (modular equation of degree 5). - At \( q = e^{-\pi} \), \( R(q) = \sqrt{\frac{5+\sqrt{5}}{2}} - \frac{1+\sqrt{5}}{2} \approx 0.09017 \) (exact algebraic value). CONNECTION: - The icosahedral equation \( 1728 z^5 (z^5 - 1)^5 Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com