Golden Ratio as Optimal Convergence Rate in Krasnoselskii-Mann Iterations with H₂ Coxeter Symmetry — E8 Intelligence Research

Abstract

FINDING: The golden ratio φ = 1.618... appears as the optimal convergence rate constant for certain fixed-point iterations, specifically linked to the H₂ Coxeter group (dihedral symmetry of order 10) and Krasnoselskii-Mann iterations. MATH: - Fixed-point iteration convergence rate: \( |x_{n+1} - x^*| \leq C \cdot r^n \), where \( r = |g'(x^*)| \) for contraction mapping \( g \). - For Krasnoselskii-Mann iterations with relaxation parameter \( \lambda \), optimal rate \( r = \max\{|1 - \lambda|, |1 - \lambda \cdot \text{spectral radius}|\} \). - When the operator is related to the H₂ Coxeter group (order 10, dihedral group D₅), the spectral radius involves eigenvalues \( 2\cos(\pi k/5) \) for \( k=1,2,3,4 \). - The golden ratio emerges: \( \varphi = 2\cos(\pi/5) = 1.618... \), and its inverse \( \varphi^{-1} = 0.618... \). - Convergence factor \( r = \varphi^{-2} = 0.382... \) appears as optimal for certain symmetric projections. CONNECTION: - H₂ Coxeter group → dihedral s Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com