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

FINDING: Martin-Löf random closed sets correspond exactly to infinite paths through Martin-Löf random Galton–Watson trees with survival parameter 2/3, linking algorithmic randomness to critical branching processes. | MATH: Survival parameter \( p = 2/3 \) for Galton–Watson process; effective Hausdorff dimension of paths; Cantor set topology (uncountable, measure zero, fractal dimension \( \log 2 / \log 3 \approx 0.6309 \)); Martin-Löf randomness defined via constructive null sets (measure-theoretic, computable). | CONNECTION: The constant \( 2/3 \) is the complement of \( 1/3 \), and \( 1/3 \) is the Cantor set's construction ratio (removing middle thirds). Note \( 2/3 \approx 0.666... \), not directly in the golden ratio family, but \( 1 - 2/3 = 1/3 \) and \( 1/3 \) relates to \( 0.382 \) via \( 1/3 \approx 0.333 \) (close but not exact). However, the Cantor set's dimension \( \log 2 / \log 3 \) is a transcendental constant; no direct golden ratio or base-60 link. The Galton–Watson cr Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com