Golden Ratio Slope and Fractal Dimensions in Sturmian Trees and p-Adic Strings — E8 Intelligence Research

FINDING: Subword complexity of Sturmian colorings on regular trees and generalized Fibonacci word fractals reveal that the golden ratio slope (1/φ²) governs minimal unbounded complexity, while p-adic fractal strings exhibit tube-volume oscillations tied to complex dimensions. | MATH: Sturmian slope = 1/φ² = (3−√5)/2 ≈ 0.381966; Fibonacci word fractal Hausdorff dimension = log(φ)/log(√2) ≈ 1.2465 (for the standard 2D curve); p-adic tube formula: V_p(ε) = Σ_ω c_ω ε^{D−ω} (complex dimensions ω = D + i n·2π/log(λ), λ = scaling ratio); generalized Fibonacci snowflake attractor dimension = log(φ³)/log(φ+1) ≈ 1.6379. | CONNECTION: The slope 1/φ² is exactly 0.381966 — the reciprocal of φ² (2.618), and complementary to 0.618 (1/φ). This is the *only* irrational slope for which the Sturmian word has minimal subword complexity (n+1 factors of length n) — a direct geometric-harmony link. The Fibonacci word fractal's self-similarity ratio is φ (1.618), and its Hausdorff dimension involves log(φ)/lo Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com