Fibonacci Anyon Quantum Dimension as Golden Ratio from SU(2)_3 and E₈ Lattice — E8 Intelligence Research

FINDING: Fibonacci anyon quantum dimension equals golden ratio φ, derived from SU(2)_3 Chern-Simons theory, with deep links to simply laced root systems and E₈ lattice. MATH: - Fibonacci anyon quantum dimension: \( d = \phi = \frac{1+\sqrt{5}}{2} \approx 1.618 \) - From SU(2)_3 Chern-Simons: level k=3, rank r=2 → quantum dimension formula: \( d = \frac{\sin(\pi \cdot 3/5)}{\sin(\pi/5)} = \frac{\sin(3\pi/5)}{\sin(\pi/5)} = \phi \) - Simply laced root systems (Aₙ, Dₙ, E₆, E₇, E₈) have Coxeter number h; for E₈, h=30, and \( \phi = 2\cos(\pi/5) = 2\cos(\pi h/150) \) - Binet formula for Fibonacci numbers: \( F_n = (\phi^n - (-\phi)^{-n})/\sqrt{5} \) - Quantum calculus with two bases: powers of φ and silver ratio \( 1+\sqrt{2} \) CONNECTION: - Golden ratio φ appears as quantum dimension of Fibonacci anyon, linking topological quantum computing to geometric harmony ratios (0.618, 1.618, 2.618). - SU(2)_3 theory corresponds to E₈ root system via McKay correspondence: affine Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com