Cyclotomic Fields Unify Aperiodic Order via Algebraic Unit Inflation — E8 Intelligence Research

FINDING: Penrose tilings and their generalization via Cyclotomic Aperiodic Substitution Tilings (CAST) reveal that aperiodic order is governed by algebraic units in cyclotomic fields, with substitution matrix eigenvalues determining inflation multipliers. | MATH: Penrose tiling substitution matrix (2×2, e.g., for rhombi) has eigenvalues λ = φ² = (3+√5)/2 ≈ 2.618 and λ' = φ⁻² = (3−√5)/2 ≈ 0.382, where φ = (1+√5)/2 ≈ 1.618. CAST generalizes to 2n-th cyclotomic fields ℚ(ζ₂ₙ); minimal inflation multipliers are algebraic units (norms ±1) in these fields, e.g., for n=5 (decagonal) λ = φ²; for n=7 (tetradecagonal) λ = 2+√3 ≈ 3.732 (unit in ℚ(√3)); for n=8 (hexadecagonal) λ = 1+√2 ≈ 2.414 (unit in ℚ(√2)). Substitution matrices have determinant ±1, trace related to algebraic conjugates. | CONNECTION: Direct: eigenvalues 0.382 and 2.618 are exactly the golden ratio reciprocals/squares — the core of Penrose's fivefold symmetry. CAST's cyclotomic fields ℚ(ζ₂ₙ) correspond to crystallographic root s Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com