Cyclotomic Aperiodic Substitution Tilings Unify Finite Rotations via Algebraic Integers — E8 Intelligence Research

FINDING: Cyclotomic Aperiodic Substitution Tilings (CAST) unify a broad class of aperiodic tilings with finite rotations, supported on the 2n-th cyclotomic field, and link minimal inflation multipliers to algebraic integers in that field. | MATH: Vertices in ℚ(ζ₂ₙ), ζ₂ₙ=eπi/n. Substitution matrix eigenvalues are algebraic integers; minimal inflation multiplier λ_min is the Perron–Frobenius eigenvalue of the substitution matrix, often a unit in the cyclotomic field. For n=5 (10th cyclotomic field), λ_min = φ² = ( (1+√5)/2 )² = (3+√5)/2 ≈ 2.618, and its reciprocal φ⁻² ≈ 0.382. | CONNECTION: 5-fold symmetry emerges from ℚ(ζ₁₀) = ℚ(√5). The golden ratio φ = (1+√5)/2 ≈ 1.618 and its powers φ²≈2.618, φ⁻¹≈0.618, φ⁻²≈0.382 are the natural inflation/deflation multipliers. This is the same algebraic field underlying Penrose tilings, icosahedral quasicrystals, and the E8 root system's golden ratio connections. | DEPTH: 9 — This provides a rigorous algebraic framework (cyclotomic fields) Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com