Remark: Family 155. A counterexample to periodic tiling in dimension three

Constructs a finite translational tile in ${\mathbb{Z}}^3$ that tiles space but admits no fully periodic tiling, disproving the periodic tiling conjecture in the smallest possible lattice dimension. Its unit-cube thickening gives the same counterexample in ${\mathbb{R}}^3$, even with arbitrary real translation vectors.

The authors are OpenAI. This companion does not verify the proof. The claim is not a theorem of the Enter series and not a premise of any registered sheet. It does not close O.1. O.1 stays NOT_CLOSED. It is not a de Sitter state and not the QdS external-leg scheme. No quantum coefficient is admitted. No compilation is asserted.