Metallic Ratio Eigenvalues from Higher-Order Fibonacci Matrices and Lattice Symmetries — E8 Intelligence Research

FINDING: Generalized metallic ratio eigenvalues from higher-order Fibonacci matrices and crystallographic lattice symmetries MATH: The characteristic equation of the standard Fibonacci matrix \( F = {pmatrix} 1 & 1 \\ 1 & 0 {pmatrix} \) yields eigenvalues \( φ = (1+√5)/2 ≈ 1.618 \) and \( -φ⁻¹ ≈ -0.618 \). For higher-order Fibonacci matrices (e.g., tribonacci, tetranacci), eigenvalues generalize to metallic ratios: for order \( k \), the dominant eigenvalue satisfies \( λ^k - λᵏ⁻¹ - - 1 = 0 \), giving ratios like the silver ratio \( 1+√2 ≈ 2.414 \), bronze ratio \( (3+√13)/2 ≈ 3.303 \), etc. These are all Pisot numbers. CONNECTION: The golden ratio \( φ ≈ 1.618 \) and its reciprocal \( 0.618 \) are directly present. The silver ratio \( 1+√2 ≈ 2.414 \) relates to octagonal symmetry (crystallographic point group \( D_4 \)). The bronze ratio links to dodecagonal quasicrystals. The me Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com