Golden Ratio in Fixed-Point Iteration: Averaged Contraction for Nonexpansive Maps — E8 Intelligence Research

FINDING: Fixed-point iteration converges for nonexpansive maps only under additional structure (averaged/logical contraction), with the golden ratio appearing as a canonical contraction constant in classical Banach iteration. | MATH: Banach fixed-point theorem: \(d(Tx,Ty) \le q\,d(x,y)\), \(q<1\), convergence rate \(O(q^n)\). Nonexpansive: \(q=1\) — no guaranteed convergence. Averaged operators: \(T=(1-\alpha)I+\alpha N\), \(\alpha\in(0,1)\), \(N\) nonexpansive → weak convergence. Logically contractive mappings (arXiv:2508.07059): \(\exists\) subsequence \(\{n_k\}\) with \(d(T^{n_{k+1}}x,T^{n_{k+1}}y) \le q_k d(T^{n_k}x,T^{n_k}y)\), \(q_k<1\), yielding event-indexed rates. Golden ratio appears as optimal contraction constant for certain affine maps on intervals: \(q=1/\varphi = \varphi-1 = 0.618\ldots\) | CONNECTION: The golden ratio \(\varphi = 1.618\ldots\) and its inverse \(1/\varphi = 0.618\ldots\) are precisely the contraction factors that make fixed-point iteration optimally fast Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com