Feigenbaum Constants δ and α: Universal Scaling in Period-Doubling Route to Chaos — E8 Intelligence Research

FINDING: Feigenbaum constants δ=4.669... and α=2.502... govern the universal period-doubling route to chaos in unimodal maps, proven via renormalization group fixed-point analysis. MATH: - δ = limn→∞ (r_n - rₙ₋₁)/(rₙ₊₁ - r_n) ≈ 4.669201609... (bifurcation rate) - α = limn→∞ d_n / dₙ₊₁ ≈ 2.502907875... (scaling of branch widths) - Universal function g(x) = -α g(g(x/α)) satisfies renormalization fixed-point equation. - Logistic map: xₙ₊₁ = r x_n (1 - x_n) exhibits cascade at r_∞ ≈ 3.5699456. CONNECTION: - δ ≈ 4.669 is not a classical geometric ratio (0.382, 0.618, 1.618, etc.), but its reciprocal 1/δ ≈ 0.2142 is near 0.236 (1/φ^3 ≈ 0.236) and 0.214 (1/φ^4 ≈ 0.146). No exact match. - α ≈ 2.5029 is close to √(2π) ≈ 2.5066 and φ^2 ≈ 2.618, but not identical. - The period-doubling cascade exhibits self-similarity under scaling by α, analogous to fractal symmetries in the Mandelbrot set (which contains Feigenbaum points). - No direct link to base-60, crys Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com