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

Abstract

FINDING: Martin-Löf random closed sets are exactly infinite paths through ML-random Galton-Watson trees with survival parameter 2/3, linking algorithmic randomness to branching processes. | MATH: Survival parameter p = 2/3; effective Hausdorff dimension condition; Galton-Watson criticality threshold at p = 1/2 (subcritical below, supercritical above); the 2/3 value is the unique fixed point of the generating function f(s) = (1+s+s²)/3 for a Poisson(1)-like offspring distribution truncated to 0,1,2. | CONNECTION: 2/3 = 0.666… is the complement of 1/3; the golden ratio conjugate 0.618 appears in the related critical Galton-Watson extinction probability for offspring mean 1/φ; the Cantor set (measure (2/3)^n) emerges naturally — the ternary Cantor set has Hausdorff dimension log2/log3 ≈ 0.6309, close to but distinct from 0.618. | DEPTH: 8 FINDING: Kučera's theorem — for any ML-random sequence ω and effectively open set A of measure < 1, some tail of ω avoids A — generalizes to an effecti Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com