Hausdorff Dimension 1/2 for Non-Uniquely Ergodic Directions in Genus 2 — E8 Intelligence Research

Abstract

FINDING: The most precise mathematical result is that for almost every point in the moduli space \(\mathcal{H}(2)\) (Masur-Veech measure), the set of directions whose vertical foliation is non-uniquely ergodic has Hausdorff dimension exactly \(1/2\) — a codimension-\(1/2\) phenomenon in a 2-real-dimensional parameter space. | MATH: \(\dim_H\{ \theta \in [0,2\pi) : e^{i\theta}\omega \text{ non-uniquely ergodic}\} = 1/2\) for a.e. \(\omega \in \mathcal{H}(2)\) w.r.t. Masur-Veech measure. This is a sharp, exact fractional dimension — not an inequality. The other sources are pedagogical (3Blue1Brown on fractal dimension, branching process simulations, half-dimension exposition) and do not yield new constants. | CONNECTION: The value \(1/2\) is the reciprocal of the golden ratio conjugate's square root? No — \(1/2\) is not directly in the golden set \(\{0.382, 0.618, 0.786, 1.618, 2.618\}\). However, \(1/2\) appears as the exponent in the Hausdorff dimension of the set of non-ergodic direct Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com