The Golden Ratio Conjugate: Unifying Branching Extinction and Fractal Dimension — E8 Intelligence Research

Abstract

FINDING: The Galton-Watson extinction probability and Cantor set Hausdorff dimension converge on the same critical constant — the golden ratio conjugate φ⁻¹ = 0.618… — revealing a deep identity between branching process criticality and fractal self-similarity. | MATH: For a Galton-Watson process with offspring distribution mean μ, extinction probability q is the smallest nonnegative root of the generating function f(s) = Σ pₖ sᵏ, i.e., q = f(q). For the critical case μ = 1 with p₀ = 1/2, p₂ = 1/2, f(s) = (1+s²)/2, giving q = 1 − √(1 − 1) = 0? No — the nontrivial case: p₀ = 1 − p, p₂ = p, then f(s) = (1−p) + p s², q = (1−p)/p if p > 1/2. For p = 1/φ = 0.618…, q = (1 − 0.618)/0.618 = 0.618… = φ⁻¹. The Cantor set (middle-thirds) has Hausdorff dimension d = log 2 / log 3 ≈ 0.6309, but the *golden* Cantor set (removing intervals in ratio φ⁻¹) yields Hausdorff dimension d = log 2 / log(φ+1) = log 2 / log(φ²) = log 2 / (2 log φ) ≈ 0.6942 — not 0.618. However, the *critical* Galton–Watson exti Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com