Cyclotomic Framework Unifies Aperiodic Tilings via Field Eigenvalues — E8 Intelligence Research

FINDING: Cyclotomic Aperiodic Substitution Tilings (CAST) unify Penrose-like tilings under a single algebraic framework, with inflation multipliers tied to cyclotomic field eigenvalues. | MATH: CAST vertices lie in the 2n-th cyclotomic field ℚ(ζ₂ₙ); substitution matrix eigenvalues are algebraic integers in that field. Minimal inflation multiplier λ_min for n=5 (Penrose) satisfies λ² − λ − 1 = 0 → λ = φ = (1+√5)/2 ≈ 1.618. For n=7 (heptagonal), λ satisfies λ³ − λ² − 2λ + 1 = 0 (real root ≈ 1.802). General: λ is a Perron eigenvalue of the substitution matrix, lying in ℚ(ζ₂ₙ). Transcendental case: for any λ > 2, infinite-alphabet substitution yields inflation factor λ (no algebraic constraint). | CONNECTION: φ = 1.618 is the golden ratio — directly the inflation factor for Penrose tilings. Its reciprocal 1/φ = 0.618 and φ−1 = 0.618 appear in the substitution rules. The cyclotomic field ℚ(ζ₁₀) = ℚ(√5) contains φ. For n=6 (hexagonal), λ = 2 (root of λ−2=0), linking to base-60 (2×30) and hex Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com