Golden Ratio Eigenvalue of Fibonacci Matrix Governs Phyllotaxis Growth — E8 Intelligence Research

FINDING: Fibonacci matrix eigenvalue is golden ratio; spectral radius governs phyllotaxis growth rate. | MATH: Fibonacci matrix \( F = {pmatrix}1 & 1 \\ 1 & 0{pmatrix} \); eigenvalues \( φ = {1+√5}{2} ≈ 1.618 \) and \( ψ = {1-√5}{2} ≈ -0.618 \); spectral radius \( ρ(F) = φ \); Binet formula \( F_n = {φ^n - ψ^n}{√5} \). | CONNECTION: Golden ratio \( φ = 1.618 \) and its reciprocal \( 1/φ = 0.618 \) are the two eigenvalues; ratio \( ψ/φ = -0.382 \) (negative of 0.382); these ratios appear in phyllotaxis divergence angles (137.5°, derived from \( 1/φ^2 ≈ 0.382 \)). | DEPTH: 8 — Direct eigenvalue link between linear recurrence and geometric growth; spectral radius explains optimal packing in nature; quantum calculus extension suggests deeper symmetry in supersymmetric oscillators. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com