Universal Scaling Constants Govern Chaos in Period-Doubling Cascades — E8 Intelligence Research

FINDING: The Feigenbaum constants (δ ≈ 4.6692, α ≈ 2.5029) are universal scaling ratios governing period-doubling cascades to chaos, arising from a renormalization fixed point in one-dimensional maps and observed across diverse physical systems. MATH: - **Feigenbaum delta** (bifurcation spacing ratio): δ = limₙ→∞ (rₙ₋₁ − rₙ₋₂)/(rₙ − rₙ₋₁) ≈ 4.669201609… - **Feigenbaum alpha** (width scaling of the bifurcation branches): α = limₙ→∞ dₙ/dₙ₊₁ ≈ 2.502907875… - **Renormalization fixed point**: The period-doubling operator R on function space has a unique hyperbolic fixed point g(x) with eigenvalue δ (unstable) and α (stable contraction). - **Scaling law**: For superstable parameter values rₙ, the distance to the accumulation point r∞ scales as r∞ − rₙ ∝ δ⁻ⁿ. - **Universality**: Any unimodal map with a quadratic maximum (e.g., logistic map xₙ₊₁ = r xₙ(1−xₙ)) yields the same δ, α — independent of the map's details. CONNECTION: - **δ ≈ 4.6692** is not a classical golde Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com