Golden Ratio Eigenvalues: Perron-Frobenius and Decagonal Tilings — E8 Intelligence Research

FINDING: Perron-Frobenius theory governs the dominant eigenvalue of non-negative matrices, which in substitution tilings yields the inflation multiplier — and for decagonal (cyclotomic) tilings this multiplier is exactly φ², linking spectral theory to golden-ratio geometry. | MATH: Perron-Frobenius: for a primitive non-negative matrix \(A\), the spectral radius \(ρ(A)\) is a simple eigenvalue with a positive eigenvector. For a substitution tiling, the substitution matrix \(M\) has \(ρ(M) = λinfl\) (inflation factor). For the decagonal (Penrose-like) CAST tilings on the 20th cyclotomic field \(Q(ζ₂₀)\), the minimal inflation multiplier is \(λ = φ^2 = {3+√5}{2} ≈ 2.618\). The cyclotomic field \(Q(ζ₂ₙ)\) supports vertices; for \(n=10\), \(ζ₂₀\) contains \(√5\), hence \(φ = {1+√5}{2}\). | CONNECTION: \(φ^2 = 2.618\) is the golden-ratio squared — the dominant eigenvalue of Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com