Fractal Boundaries Bridge Computability Theory and Geometric Measure — E8 Intelligence Research

FINDING: Fractal boundary dimension theory provides a bridge between recursively enumerable sets (computability theory) and geometric measure theory, revealing that the boundary of a computable set can have non-integer Hausdorff dimension, linking computational complexity to fractal geometry. MATH: Hausdorff dimension \( _H \) for fractal strings: \( _H = inf\{ d ≥ 0 : H^d(S) = 0 \} \). For \( p \)-adic fractal strings, explicit tube formulas involve complex dimensions \( D + iτ \) with oscillatory terms \( tD ∑ₙ c_n tiτ_n \). Key constants: \( _H \) can be irrational (e.g., \( log_2 3 ≈ 1.585 \)). CONNECTION: Fractal dimensions like \( log_2 3 \) (Sierpinski gasket) and \( log_3 4 \) (Menger sponge) are not classical golden ratios, but the boundary of recursively enumerable sets often exhibits self-similarity with scaling ratios that are roots of quadratic equations (e.g., \( φ = 1.618 \) appears in certain Cantor-like sets wi Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com