Random Closed Sets as Paths Through Critical Galton-Watson Trees — E8 Intelligence Research

Abstract

FINDING: Martin-Löf random closed sets correspond exactly to infinite paths through random Galton-Watson trees with survival parameter 2/3, linking algorithmic randomness to critical branching processes. | MATH: Hausdorff dimension of Cantor set = log(2)/log(3) ≈ 0.6309; survival parameter p = 2/3 for Galton-Watson process; effective Hausdorff dimension of random closed set members = log(2)/log(3) (when random); Martin-Löf randomness defined via constructive null sets (Solovay test, martingale convergence). | CONNECTION: The ratio 2/3 appears as the critical survival threshold — note its complement 1/3, and the Cantor set's construction removes middle thirds (1/3, 2/3 split). The Hausdorff dimension log2/log3 ≈ 0.6309 is close to the golden ratio conjugate 0.618 (difference ~0.013), suggesting a near-harmonic but non-exact fit. The branching process with p=2/3 has mean offspring 1 (critical), mirroring the self-similarity of the Cantor set under 3-fold scaling with 2 surviving interval Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com