Fibonacci Matrix Diagonalization Yields Binet Formula via Quadratic Field ℚ(√5) — E8 Intelligence Research

FINDING: Fibonacci matrix diagonalization yields closed-form Binet formula via eigenvalues φ and ψ, linking recurrence to quadratic field Q(√5). MATH: - Fibonacci recurrence: F_{n+2} = F_{n+1} + F_n, matrix M = [[1,1],[1,0]]. - Eigenvalues: λ₁ = φ = (1+√5)/2 ≈ 1.618, λ₂ = ψ = (1-√5)/2 ≈ -0.618. - Diagonalization: M = PDP⁻¹, D = diag(φ, ψ). - Binet formula: F_n = (φⁿ - ψⁿ)/√5. - Quadratic field: ℚ(√5), with φ and ψ conjugate roots of x² - x - 1 = 0. CONNECTION: - φ/ψ ratio = -φ² ≈ -2.618; ψ/φ = -1/φ ≈ -0.382. - φ and ψ satisfy φ·ψ = -1, φ + ψ = 1. - Phyllotaxis angles (137.5°) derive from φ-based irrational rotation, maximizing packing efficiency. - No direct base-60 or crystallographic symmetry; but φ appears in quasicrystal diffraction patterns (Penrose tilings, 5-fold symmetry). DEPTH: 8 — Foundational link between linear recurrence, spectral theory, and golden ratio; underpins phyllotaxis, quasicrystals, and growth patterns in nature. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com