Complex Fibonacci Spirals Encode Golden Ratio as Attracting Fixed Point — E8 Intelligence Research

FINDING: Complex-valued Fibonacci extension via Binet formula generates logarithmic spirals in the complex plane whose fixed-point dynamics encode the golden ratio as an attracting fixed point of the continued fraction map. | MATH: Binet: F(n) = (φⁿ − ψⁿ)/√5, φ = (1+√5)/2 ≈ 1.6180339887, ψ = (1−√5)/2 = −1/φ ≈ −0.6180339887. Complex extension: F(z) = (φ^z − ψ^z)/√5 for z ∈ ℂ. Continued fraction: φ = 1 + 1/(1 + 1/(1 + …)) = [1;1,1,1,…], fixed point of map T(x) = 1 + 1/x, with T(φ) = φ, |T′(φ)| = 1/φ² ≈ 0.382. Logarithmic spiral: r = a·e^(bθ), pitch b = ln(φ)/(π/2) for Fibonacci phyllotaxis. | CONNECTION: φ = 1.618, 1/φ = 0.618, 1/φ² = 0.382 — all three constants appear as: (i) growth ratio, (ii) contraction ratio of continued fraction map, (iii) spiral pitch parameter. ψ = −0.618 is the conjugate fixed point, giving the alternating/oscillatory mode. The pair (φ, ψ) satisfies x² = x + 1, discriminant 5, linking to √5 and the pentagonal/icosahedral root system H₃ (icosahedral symmetry, 5-f Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com