OpenAI · 155-01 · 2026-09-23 · Combinatorics
A translational tile with no fully periodic tiling in dimension three
preprints/A-translational-tile-with-no-fully-periodic-tiling-in-dimension-three-September-23-2026
Family 155. A counterexample to periodic tiling in dimension three.
The authors are OpenAI. This companion does not verify the proof and does not adopt it as a theorem of Ngo Cheung’s series. It is not a premise. O.1 stays NOT_CLOSED. Anti-de Sitter results are not a de Sitter state, not Em02-E’s relational lapse, not XVI-P’s QdS scheme, and not XVI-Q’s sigma-image record. No quantum coefficient is admitted. No compilation is asserted.
Read against the standing problems
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
Constructs a finite translational tile in 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 , even with arbitrary real translation vectors.
We construct a finite translational tile in ℤ3 that admits tilings but no fully periodic tiling. Its unit-cube thickening has the same property in ℝ3, even when arbitrary real translations are allowed. This gives a negative resolution of the periodic tiling conjecture in dimension three.
Preprint source and Lean scope
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This preprint is named in the Lean scope note for 155: A counterexample to periodic tiling in dimension three. The authors are OpenAI. This companion does not verify the formalization. It is not a premise. The OAI library is not imported. No compilation is asserted. O.1 stays NOT_CLOSED.