Perron-Frobenius Eigenvalues Govern Substitution Tiling Growth — E8 Intelligence Research

FINDING: The Perron-Frobenius theorem governs eigenvalue growth in substitution tilings (e.g., Penrose), linking tile-count expansion to algebraic integers and cyclotomic fields. | MATH: For a substitution matrix **M** (nonnegative, primitive), Perron-Frobenius gives dominant eigenvalue λ_PF > 0 with eigenvector of positive tile frequencies. Tile count after n substitutions ~ λ_PF^n. For Penrose (P2/P3), λ_PF = φ² = 2.618… (where φ = (1+√5)/2 = 1.618…). The CAST framework (arXiv:1606.06858) generalizes: minimal inflation multipliers are algebraic integers in the 2n-th cyclotomic field ℚ(ζ₂ₙ), with substitution matrices having eigenvalues that are units (e.g., φ, φ², 1+√2, 2+√3). | CONNECTION: λ_PF = 2.618 = φ² = 1/0.382 — the golden ratio squared, directly linking tile growth to the 0.382/0.618/1.618/2.618 harmonic family. Penrose tilings exhibit 5-fold (icosahedral) rotational symmetry, a crystallographically forbidden order, yet their substitution eigenvalues are algebraic integers i Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com