Golden Ratio as Attractor: Fixed-Point Recursion and Spiral Eigenvalue Scaling — E8 Intelligence Research

FINDING: Fixed-point recursion x = 1 + 1/x generates the golden ratio φ as its unique attractor, with eigenvalue scaling linking to spiral self-similarity. MATH: Fixed-point equation: x = 1 + 1/x → x² - x - 1 = 0 → x = φ = (1+√5)/2 ≈ 1.618034. Recursion eigenvalue (derivative at fixed point): |f'(φ)| = | -1/φ² | = 1/φ² ≈ 0.381966. This eigenvalue governs convergence rate and scaling in golden spiral (logarithmic spiral with growth factor φ per quarter-turn). CONNECTION: Eigenvalue 0.381966 ≈ 0.382 (complement of 0.618). Ratio 1/φ² = 2 - φ ≈ 0.382. Golden spiral scaling: radius multiplies by φ every 90° → self-similarity factor φ. Also links to base-60: φ ≈ 1;37,30 in sexagesimal (1 + 37/60 + 30/3600). DEPTH: 8 — Core fixed-point mechanism of φ, directly ties recursion, eigenvalue, and spiral scaling; foundational to phyllotaxis, quasicrystals, and Penrose tilings (5-fold symmetry). No evidence of crystallographic root systems or lattice symmetries beyond 5-fold. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com