Golden Ratio Map's Linear Convergence Rate Tied to Coxeter Dihedral Symmetry — E8 Intelligence Research

Abstract

FINDING: Fixed-point iteration convergence rate for the golden ratio map \( g(x) = 1 + 1/x \) is linear, with asymptotic constant equal to \( 1/\phi^2 \approx 0.382 \), linking directly to the H₂ Coxeter group's dihedral symmetry. MATH: - Fixed-point iteration: \( x_{n+1} = g(x_n) \), \( g(x) = 1 + 1/x \). - Fixed point: \( \phi = (1+\sqrt{5})/2 \approx 1.618 \). - Derivative at fixed point: \( g'(\phi) = -1/\phi^2 \approx -0.382 \). - Convergence rate: linear, asymptotic error constant \( |g'(\phi)| = 1/\phi^2 \). - H₂ Coxeter group: order 10, dihedral symmetry of the pentagon, root system with angles \( 36^\circ, 72^\circ \), reflecting golden ratio. CONNECTION: - \( 1/\phi^2 = 0.382 \) is the complementary golden ratio (also \( 1 - 1/\phi \)). - This constant governs the contraction in the fixed-point iteration, mirroring the self-similar scaling in pentagonal tilings and Penrose lattices. - The H₂ root system's Cartan matrix eigenvalues involve \( \phi \) and \( 1 Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com