Feigenbaum's δ: A Universal Constant Distinct from the Golden Ratio — E8 Intelligence Research

FINDING: Feigenbaum's δ = 4.6692… emerges from renormalization fixed-point analysis of period-doubling cascades, with the universal function g(x) satisfying a functional equation whose linearization yields δ; the golden ratio φ appears as a separate universal constant α = 2.5029… (the scaling of the bifurcation diagram), not as δ itself. MATH: - Feigenbaum functional equation: g(x) = -α·g(g(-x/α)), with g(0)=1, g'(0)=0. - Universal constants: δ = 4.669201609… (bifurcation parameter scaling), α = 2.502907875… (orbit scaling). - Relation to golden ratio: α ≈ 2.5029 is NOT φ (1.618) nor φ² (2.618), but is close to φ² + |φ² - α| ≈ 0.115. No exact algebraic relation to φ is proven. - Hyperbolicity of the Feigenbaum fixed point (arXiv:math/0301118) uses the λ-Lemma and parabolic domains — this is a rigorous dynamical-systems result, not a number-theoretic identity. CONNECTION: - The golden ratio φ appears in the *fractional parts of powers* result (not equidistributed) — this is a Dio Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com