Fixed-Point Iteration for φ: Convergence Rate Linked to Icosahedral Symmetry — E8 Intelligence Research

FINDING: Fixed-point iteration convergence rates for the golden ratio φ (1.618) as a fixed point of g(x)=1+1/x, linking numerical analysis to the H₂ Coxeter group (icosahedral symmetry). | MATH: The golden ratio φ satisfies φ = 1 + 1/φ, with φ = (1+√5)/2 ≈ 1.618034. Fixed-point iteration xₙ₊₁ = 1 + 1/x_n converges linearly with rate |g'(φ)| = 1/φ² ≈ 0.381966. This rate equals 1 - 1/φ = φ - 1 ≈ 0.618034, and also equals 1/φ² = (3-√5)/2 ≈ 0.381966. The convergence factor is the reciprocal of φ², linking to the golden ratio conjugate Φ = 1/φ ≈ 0.618034. | CONNECTION: The convergence rate 0.381966 is exactly 1/φ², which is the square of the golden ratio conjugate. This rate appears in the H₂ Coxeter group's Cartan matrix eigenvalues: the Coxeter number h=5 for H₂, and the eigenvalues of the Cartan matrix are 2cos(π/h) = 2cos(36°) = φ ≈ 1.618 and its reciprocal 2cos(2π/h) = 2cos(72°) = 1/φ ≈ 0.618. The fixed-point iteration's convergence rate 0.381966 = (1/φ)² = 2cos(72°)² = (√5-2) ≈ 0.2 Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com