Golden Ratio as Fixed Point in Algorithmic Randomness and Prefix-Free Complexity — E8 Intelligence Research

Abstract

FINDING: Algorithmic randomness and Kolmogorov complexity intersect with the golden ratio via stable self-application, revealing a deep link between prefix-free computation and irrational recurrence. | MATH: Prefix complexity \(K(x)\) satisfies \(K(x) \leq K(x|y) + K(y) + O(1)\); a priori probability \(m(x) = \sum_{p: U(p)=x} 2^{-|p|}\) over prefix-free codes. The golden ratio \(\Phi = (1+\sqrt{5})/2 \approx 1.618\) appears as the fixed point of \(x \mapsto 1 + 1/x\), with reciprocal \(\Phi^{-1} = \Phi - 1 \approx 0.618\). The arXiv paper (2510.08934v3) treats \(\Phi\) operationally: stable local recurrence via update rule \(x_{n+1} = 1 + 1/x_n\), converging to \(\Phi\) with error decaying as \(\Phi^{-2n}\) (since \(\Phi\) is a Pisot number). | CONNECTION: The golden ratio's continued fraction \([1;1,1,1,\dots]\) is the *simplest* infinite prefix-free structure — its convergents \(F_{n+1}/F_n\) (Fibonacci ratios) are the most economical rational approximations, mirroring the prefix-fre Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com