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

FINDING: Feigenbaum constant δ = 4. 669201609. . . governs universal period-doubling route to chaos in logistic map and other one-dimensional maps, with α = 2. 502907875. . . as the scaling factor between bifurcation branches. MATH: - Logistic map: x₍+₁ = r xₙ (1 - xₙ) - Feigenbaum constant δ = lim₊→∞ (rₖ - r₊-₁) / (r₊+₁ - rₖ) ≈ 4. 669201609 - Feigenbaum constant α = scaling factor of branch widths ≈ 2. 502907875 - Renormalization fixed-point equation: g (x) = α g (g (x/α) ) with g (0) =1, g' (0) =0 - Universal function g (x) = 1 - 1. 52763 x² + 0. 104815 x⁴ + 0. 0267057 x⁶ +. . . CONNECTION: - α ≈ 2. 5029 is close to (5/2) = 2. 5, a rational ratio with geometric significance (pentagonal symmetry, golden ratio φ = (1+√5) /2 ≈ 1. 618, and 5-fold crystallographic symmetry). - δ ≈ 4. 669 is not a simple ratio but appears in base-60 sexagesimal as 4;40, 9, 42, 36,. . . (4 + 40/60 + 9/3600 +. . . ), hinting at possible ancient numerological resonance. - No direct link to 0. 382, 0. 618, 0 Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence. com