Aperiodic Tilings Link Substitution Rules to Algorithmic Randomness and Undecidability — E8 Intelligence Research

FINDING: Aperiodic tilings (Penrose, chiral monotile, CAST) encode algorithmic randomness via substitution rules that are computationally irreducible, linking tiling inflation to undecidability in the domino problem. MATH: - Penrose tiling: inflation multiplier τ = (1+√5)/2 ≈ 1.618 (golden ratio). Substitution matrix eigenvalues are τ and τ⁻¹ ≈ 0.618. - Chiral aperiodic monotile: substitution rule with 12-fold rotational symmetry; inflation factor ≈ 1.618 (golden ratio) or √(2+√3) ≈ 1.9319 (depending on tile shape). - CAST (Cyclotomic Aperiodic Substitution Tilings): vertices in ℚ(ζ₂ₙ) (2n-th cyclotomic field). Minimal inflation multiplier λ = 2 cos(π/n) for n=5 gives λ = φ ≈ 1.618; for n=12, λ = √(2+√3) ≈ 1.9319. Substitution matrix spectral radius = λ². CONNECTION: - Golden ratio φ = 1.618 and its reciprocal 0.618 appear in Penrose and chiral monotile inflation. - 12-fold symmetry in chiral monotile and CAST (n=12) relates to dodecagonal quasicrystals; inflation multipl Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com