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: \(d(Tx,Ty) ≤ q\,d(x,y)\), \(q<1\), convergence rate \(O(q^n)\). Nonexpansive: \(q=1\) — no guaranteed convergence. Averaged operators: \(T=(1-α)I+α N\), \(α∈(0,1)\), \(N\) nonexpansive → weak convergence. Logically contractive mappings (arXiv:2508.07059): \(∃\) subsequence \(\{n_k\}\) with \(d(T^{nₖ₊₁}x,T^{nₖ₊₁}y) ≤ q_k d(Tn_kx,Tn_ky)\), \(q_k<1\), yielding event-indexed rates. Golden ratio appears as optimal contraction constant for certain affine maps on intervals: \(q=1/φ = φ-1 = 0.618…\) | CONNECTION: The golden ratio \(φ = 1.618…\) and its inverse \(1/φ = 0.618…\) 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