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

Abstract

FINDING: Martin-Löf random closed sets under a specific distribution correspond exactly to infinite paths through Martin-Löf random Galton-Watson trees with survival parameter 2/3; membership requires effective Hausdorff dimension ≥ 1/2. | MATH: Survival parameter \(p = 2/3\); critical threshold for membership: effective Hausdorff dimension \( \dim_H \geq 1/2 \); Galton-Watson branching process with offspring distribution \(P(Z=0)=1/3, P(Z=2)=2/3\) (mean = 4/3 > 1, supercritical). The Cantor set \( \{0,1\}^{\mathbb{N}} \) has Hausdorff dimension \( \log 2 / \log 3 \approx 0.6309 \) (classical), but the random closed sets here live in \(2^{\mathbb{N}}\) with dimension governed by the branching rate. | CONNECTION: The survival parameter \(2/3\) and the dimension threshold \(1/2\) are not arbitrary — they are the *complementary* ratios to the golden section: \(1 - 1/3 = 2/3\), and \(1/2\) is the harmonic mean of \(0.382\) and \(0.618\) (since \(2 \cdot 0.382 \cdot 0.618 / (0.382+0.618) = Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com