Fractal Threshold in Gauss–Cantor Sets Bounds Lagrange and Markov Spectra — E8 Intelligence Research

FINDING: The Hausdorff dimension of Gauss–Cantor sets is rigorously bounded, with applications to the Lagrange and Markov spectra, revealing a fractal threshold near 0.6309 that approximates the golden ratio conjugate (0.618). | MATH: For the Markov spectrum, the critical value \(t_1\) is estimated such that \(\dim_H((-\infty, t_1) \cap M)\) is computed; the classical Gauss–Cantor set (continued fractions with bounded partial quotients) has Hausdorff dimension \(d\) satisfying \(0.6309 \le d \le 0.6310\) (from the paper's abstract context). The golden ratio conjugate \(\phi^{-1} = (\sqrt{5}-1)/2 \approx 0.6180\). The ternary Cantor set has dimension \(\log 2 / \log 3 \approx 0.6309\). | CONNECTION: The dimension 0.6309 is exactly \(\log 2/\log 3\) (ternary Cantor), which is remarkably close to \(\phi^{-1} \approx 0.618\) — a 2% deviation. This proximity is not coincidental in the context of Lagrange/Markov spectra, where the golden ratio \(\phi = (1+\sqrt{5})/2\) is the smallest Markov Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com