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 \( _H(X) = inf\{s ≥ 0 : H^s(X)=0\} \); Minkowski–Bouligand dimension \( _M = limε→0 log N(ε)/log(1/ε) \); for \(p\)-adic fractal strings, tube formula \( V_p(ε) = ∑ω ∈ D c_ω ε1-ω \) with complex dimensions \(ω\) encoding oscillations; algorithmic randomness via Martin-Löf tests: \( ALR(x) = ₙ K(x n)/n \) (Kolmogorov complexity rate). | CONNECTION: The Sierpiński carpet has Hausdorff dimension \( log_3 8 ≈ 1.8928 \), 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