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 \(\rho(A)\) is a simple eigenvalue with a positive eigenvector. For a substitution tiling, the substitution matrix \(M\) has \(\rho(M) = \lambda_{\text{infl}}\) (inflation factor). For the decagonal (Penrose-like) CAST tilings on the 20th cyclotomic field \(\mathbb{Q}(\zeta_{20})\), the minimal inflation multiplier is \(\lambda = \varphi^2 = \frac{3+\sqrt{5}}{2} \approx 2.618\). The cyclotomic field \(\mathbb{Q}(\zeta_{2n})\) supports vertices; for \(n=10\), \(\zeta_{20}\) contains \(\sqrt{5}\), hence \(\varphi = \frac{1+\sqrt{5}}{2}\). | CONNECTION: \(\varphi^2 = 2.618\) 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