Feigenbaum Constants: Universal Scaling in Logistic Map Period-Doubling — E8 Intelligence Research

FINDING: Feigenbaum constant δ (4.669) emerges from period-doubling cascade in logistic map, linked to universality in recursive dynamical systems and fixed-point theory. MATH: - Feigenbaum constant δ = 4.669201609... (ratio of successive bifurcation intervals). - Feigenbaum constant α = 2.502907875... (scaling of period-doubling branches). - Logistic map: xₙ₊₁ = r x_n (1 - x_n). - Fixed-point condition: f(x) = x; self-referential recursion in renormalization group equation: g(x) = α g(g(x/α)). - Gödelian diagonalization: self-reference → undecidability (fixed-point lemma). CONNECTION: - δ ≈ 4.669 → ratio 1/δ ≈ 0.214, not directly in golden-ratio family (0.382, 0.618, 0.786, 1.618, 2.618). - α ≈ 2.503 → ratio 1/α ≈ 0.399, close to 0.382 (golden ratio squared inverse) but not exact. - No direct base-60 or crystallographic symmetry; however, period-doubling universality mirrors self-similarity in fractal structures (e.g., Cantor set, Mandelbrot set). - Renormaliz Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com