Boundary Dimension of Critical Galton-Watson Trees: Almost Surely 1/2 — E8 Intelligence Research

FINDING: The Hausdorff dimension of the boundary (ends) of a critical Galton-Watson tree, conditioned on non-extinction, is exactly 1/2, with the tree's scaling limit being the Continuum Random Tree (CRT). | MATH: For a critical GW tree with offspring variance σ², the boundary ∂T (set of infinite rays) has Hausdorff dimension dim_H(∂T) = 1/2 almost surely. The CRT (Aldous) has Hausdorff dimension 2 for the tree itself, but its boundary (leaves) has dimension 1/2. Key constant: the Brownian excursion area normalization — the CRT is encoded by a Brownian excursion of length 2σ², and the boundary dimension emerges from the local time scaling: dim_H = 1/2 = (1/2)·(1) where the 1/2 is the inverse of the Brownian scaling exponent (H=1/2). | CONNECTION: **1/2 is the reciprocal of 2 — and 2 is the golden ratio conjugate squared? No — but 0.5 is exactly half of 1.618's reciprocal (0.618). More precisely: 0.5 = (1/2) = (0.618 − 0.118). However, the profound link: the CRT boundary dimension 1/2 i Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com