Effective Hausdorff Dimension of Zero Set of Random Brownian Motion is 1/2 — E8 Intelligence Research

FINDING: Effective Hausdorff dimension of the zero set of a Martin-Löf random Brownian motion path is exactly 1/2, matching the almost-sure classical result but now holding for every individual algorithmically random path. | MATH: For a Martin-Löf random Brownian motion \(B(t,\omega)\), the zero set \(Z(\omega) = \{t \in [0,1] : B(t,\omega)=0\}\) has effective Hausdorff dimension \(\dim_{\text{eff}}(Z(\omega)) = 1/2\). Classical Hausdorff dimension of the zero set is almost surely 1/2; this result shows the same holds pointwise for every ML-random path. No new constants beyond 1/2. | CONNECTION: The ratio 1/2 appears as a critical exponent — it is the reciprocal of 2, linking to the golden ratio conjugate \(\phi^{-1} \approx 0.618\) only indirectly via the fact that \(1/2 = 0.5\) is not a golden ratio number. However, the dimension 1/2 is the same as the exponent in the quadratic variation of Brownian motion, and relates to the scaling law \(B(ct) \sim \sqrt{c} B(t)\), which involves t Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com