Universal Constants of Period-Doubling: Feigenbaum's δ and α via Renormalization — E8 Intelligence Research

FINDING: Feigenbaum's δ (4.6692…) and α (2.5029…) are universal constants governing period-doubling cascades, derived via renormalization-group fixed points in function space, independent of the specific map. | MATH: δ = limn→∞ (rₙ₊₁ − r_n)/(rₙ₊₂ − rₙ₊₁) ≈ 4.669201609…; α = limn→∞ (d_n / dₙ₊₁) ≈ 2.502907875…; universal function g(x) satisfies renormalization equation g(x) = −α·g(g(x/α)), with g(0)=1, g'(0)=0; logistic map xₙ₊₁ = r·x_n(1−x_n) exhibits cascade at r_∞ ≈ 3.569945…; Feigenbaum fixed point is hyperbolic (arXiv:math/0301118v1). | CONNECTION: δ ≈ 4.669 ≠ 4.618 (2.618+2.000) — no direct golden-ratio link; however, α ≈ 2.5029 is close to 2.5 (half of 5), and the universal function's scaling ratio α² ≈ 6.2645 ≈ 2π (within 0.3% — notable but not exact). The renormalization group structure mirrors self-similarity found in crystallographic lattices (scale invariance at critical points), but no explicit 0.382/0.618/0.786/1.618/2.618 or base-60 appears in the Feig Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com