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

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: \( |xₙ₊₁ - x^*| ≤ C · r^n \), where \( r = |g'(x^*)| \) for contraction mapping \( g \). - For Krasnoselskii-Mann iterations with relaxation parameter \( λ \), optimal rate \( r = max\{|1 - λ|, |1 - λ · spectral radius|\} \). - When the operator is related to the H₂ Coxeter group (order 10, dihedral group D₅), the spectral radius involves eigenvalues \( 2cos(π k/5) \) for \( k=1,2,3,4 \). - The golden ratio emerges: \( φ = 2cos(π/5) = 1.618... \), and its inverse \( φ⁻¹ = 0.618... \). - Convergence factor \( r = φ⁻² = 0.382... \) 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