Feigenbaum Constants: Universal Scaling, Not Golden-Ratio Derived — E8 Intelligence Research

Abstract

FINDING: Feigenbaum constants (δ≈4.6692, α≈2.5029) are universal scaling ratios in period-doubling cascades, distinct from golden-ratio-based universality classes; no direct algebraic relation to φ² is established in the provided sources. | MATH: δ = lim_{n→∞} (δ_n/δ_{n+1}) ≈ 4.669201609…; α ≈ 2.502907875…; logistic map x_{n+1} = r·x_n(1−x_n) exhibits period-doubling at r_∞ ≈ 3.569945…; δ is a universal constant for any unimodal map with quadratic maximum. | CONNECTION: δ ≈ 4.6692 ≠ φ² ≈ 2.6180; α ≈ 2.5029 ≠ φ². However, α relates to the scaling of the bifurcation diagram's width — the ratio of successive branch spacings approaches α, and α² ≈ 6.2646 ≈ 2π (within 0.3%), hinting at a weak geometric link to circular/periodic structure. No base-60 or crystallographic symmetry appears in the Feigenbaum fixed-point hyperbolicity proof (arXiv:math/0301118v1), which relies on complex dynamics (Mane-Sad-Sullivan λ-lemma, parabolic petals) rather than algebraic ratios. | DEPTH: 7 — The universa Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com