Cyclotomic Aperiodic Substitution Tilings: Algebraic Integer Inflation Multipliers Encoding Rotational Symmetry — E8 Intelligence Research

FINDING: Cyclotomic Aperiodic Substitution Tilings (CAST) — substitution matrices over ℤ[ζ₂ₙ] yield algebraic integer inflation multipliers whose eigenvalues encode rotational symmetries and self-similarity. | MATH: Let ζ₂ₙ = e^{2πi/2n}. CAST vertices lie in ℤ[ζ₂ₙ]. Substitution matrix M has Perron–Frobenius eigenvalue λ (inflation multiplier). λ is an algebraic integer in ℤ[ζ₂ₙ] (or its real subfield ℤ[ζ₂ₙ + ζ₂ₙ⁻¹]). Minimal polynomial of λ divides the characteristic polynomial of M. For n=5 (decagonal), λ often satisfies λ² − φλ − 1 = 0 or λ² − (φ+1)λ + φ = 0, where φ = (1+√5)/2 = 1.618…, and φ⁻¹ = 0.618…, φ⁻² = 0.382… . For n=7 (tetradecagonal), λ may be root of λ³ − (2cos(π/7))λ² − … = 0, with 2cos(π/7) ≈ 1.8019, 2cos(2π/7) ≈ 1.24698, 2cos(3π/7) ≈ 0.44504. | CONNECTION: Direct. The real subfield of ℤ[ζ₂ₙ] contains 2cos(2πk/2n). For n=5, 2cos(2π/5) = φ⁻¹ = 0.618, 2cos(π/5) = φ = 1.618, 2cos(4π/5) = −φ⁻¹, 2cos(3π/5) = −φ. For n=6 (dodecagonal), 2cos(π/6) = √3 ≈ 1.732, 2cos(π/3) = 1, Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com