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: x₍+₁ = r xₙ (1 - xₙ). - 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
Andrew Stewart Caldin (Sun,) studied this question.