Feigenbaum Constant's Universal Scaling and Its Structural Parallels to the Golden Ratio — E8 Intelligence Research

Abstract

FINDING: Feigenbaum constant δ (4.669201609...) is a universal scaling factor in period-doubling cascades, with no proven exact algebraic relation to the golden ratio φ, but deep structural parallels exist in renormalization group theory. MATH: - Feigenbaum constant δ = 4.669201609102990671853203821578... - Feigenbaum constant α = 2.502907875095892822283902873218... (scaling of bifurcation intervals) - Golden ratio φ = (1+√5)/2 ≈ 1.6180339887 - Logistic map: x_{n+1} = r x_n (1 - x_n) - Period-doubling cascade: r_n = r_∞ - C δ^{-n} - No known algebraic equation linking δ and φ; δ is transcendental (conjectured, not proven). - Renormalization group fixed-point function g(x) satisfies: g(x) = -α g(g(-x/α)) CONNECTION: - δ ≈ 4.669 is not a simple ratio of φ (e.g., φ^3 ≈ 4.236, φ^4 ≈ 6.854). - However, the universal scaling in period-doubling mirrors the self-similarity of the golden ratio in continued fractions: both arise from renormalization fixed points. - The ra Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com