Golden Ratio Contraction Factor Governs Fixed-Point Convergence in Non-Euclidean Spaces — E8 Intelligence Research

FINDING: Fixed-point iteration convergence in non-Euclidean metric spaces is linked to the golden ratio map's contraction factor, revealing a universal ratio in convergence rates. MATH: For the golden ratio map \( g(x) = 1 + 1/x \) with fixed point \( φ = 1.618... \), the convergence rate in a non-Euclidean metric space is governed by the derivative \( |g'(φ)| = 1/φ^2 ≈ 0.382 \). This is the square of the golden ratio conjugate \( φ⁻¹ ≈ 0.618 \). The contraction factor \( k = 0.382 \) ensures linear convergence if \( k < 1 \) in the Banach fixed-point theorem. In non-Euclidean spaces (e.g., hyperbolic geometry), the metric distortion may alter the effective contraction, but the intrinsic ratio \( 0.382 \) persists as a spectral radius bound. CONNECTION: The ratio \( 0.382 \) is the complement of \( 0.618 \) (the golden ratio conjugate) and appears in pentagonal symmetry (e.g., ratio of diagonal to side in a regular pentagon). It also relates to base-60 math Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com