Fractal Dimensions Unify Discrete and Continuous Geometry via Randomness — E8 Intelligence Research

FINDING: Fractal dimension bridges discrete and continuous geometry via Hausdorff–Minkowski measures, with algorithmic randomness formalizing "typical" fractal behavior in Cantor space. | MATH: Hausdorff dimension \( \dim_H(X) = \inf\{s \ge 0 : \mathcal{H}^s(X)=0\} \); Minkowski–Bouligand dimension \( \dim_M = \lim_{\varepsilon\to0} \frac{\log N(\varepsilon)}{\log(1/\varepsilon)} \); for \(p\)-adic fractal strings, tube formula \( V_p(\varepsilon) = \sum_{\omega \in \mathcal{D}} c_\omega \varepsilon^{1-\omega} \) with complex dimensions \(\omega\) encoding oscillations; algorithmic randomness via Martin-Löf tests: \( \dim_{ALR}(x) = \liminf_{n} \frac{K(x\upharpoonright n)}{n} \) (Kolmogorov complexity rate). | CONNECTION: The Sierpiński carpet has Hausdorff dimension \( \log_3 8 \approx 1.8928 \), not a golden-ratio value, but the *gap* between integer dimensions (1 and 2) is \(0.8928\) — near \(0.786\) (the square root of \(0.618\)) within 13.6% error. More critically, the *complement Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com