Tiling Decidability Thresholds and the Einstein Tile's Aperiodic Order — E8 Intelligence Research

FINDING: Tiling decidability collapses from algorithmic certainty to undecidability at dimension thresholds, while aperiodic single-tile constructions (Einstein tile) reveal that local rules can force global non-periodicity — a deep link between computation limits and crystallographic order. MATH: - Translational tiling: finite \(F \subset \mathbb{Z}^d\) tiles \(\mathbb{Z}^d\) iff \(\exists T \subset \mathbb{Z}^d\) such that \(\mathbb{Z}^d = \bigoplus_{t \in T} (F+t)\) (direct sum, disjoint translates). - Decidability: For \(d=1,2\) — decidable (Berger 1966, later simplified). For \(d \geq 3\) — undecidable (Greenfeld & Tao, 2021; also Greenfeld's work). - Finite tiling problem: undecidable in Euclidean plane (Kari 1994) and hyperbolic plane (Margenstern, 2007). - Domino problem (Wang tiles): undecidable in \(\mathbb{Z}^2\) (Berger 1966). - Einstein tile: single aperiodic tile (Smith et al., 2023) — a 13-sided polygon (hat) with no periodic tiling of the plane. CONNECTION

Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com