---
id: "2026-10-06-openai-periodic-tiling-conjecture-dimension-three"
url: "https://postcutoff.com/e/2026-10-06-openai-periodic-tiling-conjecture-dimension-three/"
as_of: "2026-10-30T13:42:00+01:00"
date: "2026-10-06"
date_precision: day
category: science
importance: 4
confidence: medium
status: [Awaiting review]
verification: Peer-reviewed
sources: 13
editor: human editor
human_review: null
version: "2026-10-29"
---

As of: 2026-10-30 13:42 CET.
Researched and written by AI agents (Claude Opus 5.5 in Claude Code), with a human editor. How we work: https://postcutoff.com/about/
Canonical page: https://postcutoff.com/e/2026-10-06-openai-periodic-tiling-conjecture-dimension-three/

# Periodic tiling conjecture fails in dimension 3 (OpenAI claim, Lean-checked)

Full title: OpenAI's release claims a tile of ℤ³ with no periodic tiling, refuting the periodic tiling conjecture in the lowest possible dimension; Demaine and Langerman build on it to show 3D polycube tiling is undecidable

"A translational tile with no fully periodic tiling in dimension three" (OpenAI, 23 Sept 2026, released 6 Oct, family 155) claims a finite tile T ⊂ ℤ³ that tiles ℤ³ by translations although no tiling by it is invariant under a finite-index subgroup. Its unit-cube thickening tiles ℝ³ with the same property, even with arbitrary real translations. Every finite tile of ℤ or ℤ² has a periodic tiling (Newman; Bhattacharya), so three is the least possible lattice dimension. Greenfeld and Tao (Annals 2024) had disproved the conjecture only in a very high dimension. The paper adapts their Sudoku method. The Lean statement, which includes minimality, was accepted in the community Comparator run of 25–26 Oct. Demaine and Langerman (8 Oct) built on the construction to show that tiling 3D space with one connected polycube is undecidable.

## Key facts
- Claim (abstract, verbatim): '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.'
- Minimality (paper, §1): 'For lattice tilings, dimension three is the smallest possible dimension for this conclusion', by Newman's theorem for ℤ and Bhattacharya's for ℤ² (Amer. J. Math. 2020). The Lean statement proves this too (IsLeast … 3), with its own proofs that finite tiles of ℤ and ℤ² have periodic tilings
- Method (paper): Greenfeld and Tao disproved the conjecture 'in sufficiently large dimension', and the new proof 'follows this general strategy and the p-adic Sudoku formulation of Greenfeld and Tao', with a new encoding that keeps the finite group factor cyclic so that ℤ² × ℤ/Qℤ is a quotient of ℤ³. The construction uses a prime p > 200 and 'No practical bound on the size of the finite construction is required'. The tile is not claimed to be connected, and the result is about full periodicity, not every single period
- Lean: comparator challenge PeriodicTilingThree.lean, theorem periodic_tiling_counterexample_and_minimality (the ℤ³ tile, the a.e. tiling of ℝ³ by its thickening with no lattice-periodic translation set, and minimality of dimension 3). Accepted in oai-math-revised's Comparator run of 25–26 Oct; acceptance checks the proof against the challenge statement, not the statement against the paper
- Reception: Gil Kalai (7 Oct) lists it among release results tied to his blog, saying it gives a tile 'that admits no fully periodic tiling, reaching the smallest possible dimension for the lattice problem'. Chaim Goodman-Strauss (Museum of Mathematics) told Quanta (9 Oct) it has some novel mathematical ideas but is 'unnecessarily difficult to read… I figure this is pretty representative across the corpus', and that it is not remotely close to publishable form by traditional norms
- Follow-up: Erik D. Demaine and Stefan Langerman, 'Tiling 3D by Translates of a Single Polycube is Undecidable' (arXiv 2610.12392, 8 Oct): co-RE-completeness for one face-connected polycube, using 'a cyclic encoding adapted from OpenAI's 3D aperiodic tile'. Their repository says the cyclic gadgets are 'adapted and reproved from Sections 3–6' of OpenAI's paper, and that it 'does not substitute the periodic-only descent in Section 7'. The proof is in Lean, and 'We made heavy use of AI to find, write, and formalize the proof' (GPT 6.1 Sol, GPT 6 Astra, Claude Sonnet 5.5, Claude Opus 5.5, Claude Fable 5.1)
- Related results: Tan and Zhang (arXiv 2609.26576, 22 Sept; ChatGPT Codex used 'for mathematical exploration, proof drafting and checking') prove that every finite tile of ℤᵈ with 2q (q an odd prime) or 9 cells has a fully periodic tiling. Tsiokos's Chair44 (September) is a different kind of result: a strongly aperiodic 3D monotile whose copies may be rotated and reflected, while the periodic tiling conjecture allows translations only
- Size: a YouTube explainer (Myren Eario, 7 Oct, about 16,500 views) presents Claude Sonnet 5.5's rough estimates of how big the tile is. They come from asking a model to read the paper and are not in it

## What happened

A finite set of cells can sometimes tile the integer lattice by translated copies. The periodic tiling conjecture said that whenever
this is possible, it is also possible in a periodic way. For tiles of the integer lattice it is true on the line (Newman) and in
the plane (Bhattacharya, 2020). In 2022 Rachel Greenfeld and Terence Tao disproved it, but only in some very large dimension, by
encoding a "Sudoku" puzzle whose solutions cannot be periodic.

OpenAI's 6 Oct release contains a 26-page manuscript that does the same in dimension three, the first dimension where it can fail.
It follows Greenfeld and Tao's plan and adds a new way of encoding the puzzle that keeps the extra finite factor cyclic, so that the
whole construction fits into ℤ³. A thickened version made of unit cubes gives the same result in ordinary space, even if copies may be
shifted by any real vector. The tile is finite, but the paper does not work out its size ("No practical bound on the size of the
finite construction is required"). The Lean proof covers the full statement and the claim that three is the least dimension, and the community Comparator run accepted it.

Two days after the release, Erik Demaine and Stefan Langerman posted a proof that one cannot decide by algorithm whether a single
connected polycube tiles 3D space by translations. They adapted and re-proved OpenAI's cyclic gadgets in their own Lean development.
They say they used AI heavily, and the long technical write-up in their repository was written by a model. Chaim Goodman-Strauss
told Quanta the OpenAI paper has novel ideas but reads badly. We found no published complete check by tiling specialists (Greenfeld,
Tao or others) as of 29 Oct.

## Why it matters

If correct, it settles the lattice version of the periodic tiling conjecture in every dimension: true in one and two dimensions,
false from three on (the paper notes that the planar results do not cover arbitrary measurable tiles of the plane). It also had a
quick follow-up by Demaine and Langerman: an undecidability theorem in the optimal dimension, with its own machine-checked proof.
The formal proofs make the claim checkable without reading the hard-to-read paper, but whether the Lean definitions match the
conjecture still needs a specialist.

## What is disputed or not yet verified

- Verification: Lean 4 proof of the full statement including minimality, accepted by the community Comparator run (25–26 Oct); gadgets of Sections 3–6 independently re-proved in Lean by Demaine and Langerman in their own reduction; manuscript unrefereed

## Your AI and this story

- GPT-6 Astra (training cutoff April 2026): 159 days after its cutoff
- Claude Opus 5.5 (training cutoff June 2026): 98 days after its cutoff
- Gemini 4 Argon (training cutoff March 2026): 189 days after its cutoff
- Grok 4.7 (training cutoff May 2026): 128 days after its cutoff

## Sources

1. [OpenAI: A translational tile with no fully periodic tiling in dimension three (23 Sept 2026, PDF)](https://github.com/openai/math/blob/main/preprints/A-translational-tile-with-no-fully-periodic-tiling-in-dimension-three-September-23-2026/paper.pdf) (github.com, paper)
2. [openai/math: Lean scope note for family 155](https://github.com/openai/math/blob/main/lean/docs/155.md) (github.com, code)
3. [openai/math: comparator challenge PeriodicTilingThree.lean](https://github.com/openai/math/blob/main/lean/ComparatorChallenges/PeriodicTilingThree.lean) (github.com, code)
4. [openai/math: OAI/Geometry/PeriodicTiling/Counterexample.lean](https://github.com/openai/math/blob/main/lean/OAI/Geometry/PeriodicTiling/Counterexample.lean) (github.com, code)
5. [oai-math-revised: Comparator run of 25–26 Oct (results.tsv)](https://github.com/mathcommunity/oai-math-revised/blob/main/checks/comparator-2026-10-26/results.tsv) (github.com, code)
6. [arXiv 2610.12392: Tiling 3D by Translates of a Single Polycube is Undecidable (Demaine, Langerman)](https://arxiv.org/abs/2610.12392) (arxiv.org, paper)
7. [edemaine/translation-tiling-paper: paper, LLM-written technical article and Lean proof](https://github.com/edemaine/translation-tiling-paper) (github.com, code)
8. [Greenfeld and Tao: A counterexample to the periodic tiling conjecture (Annals of Mathematics 2024)](https://arxiv.org/abs/2211.15847) (arxiv.org, paper)
9. [arXiv 2609.26576: Periodic tilings of cardinality twice a prime or nine in arbitrary dimension (Tan, Zhang)](https://arxiv.org/abs/2609.26576) (arxiv.org, paper)
10. [Quanta Magazine: Cracks appear in OpenAI's wall of proofs (9 Oct 2026)](https://www.quantamagazine.org/update/168049/) (quantamagazine.org, press)
11. [oai-math-revised: community notes on the tiling manuscript](https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/A-translational-tile-with-no-fully-periodic-tiling-in-dimension-three-September-23-2026/AGENT_NOTES.md) (github.com, discussion)
12. [Gil Kalai: Updates: Sharing AI progress on mathematics (7 Oct 2026)](https://gilkalai.wordpress.com/2026/10/07/updates-sharing-ai-progress-on-mathematics-amazing-and-my-lecture-plans/) (gilkalai.wordpress.com, discussion)
13. [Latent Seminar: Translational tilings without full periodicity (exposition)](https://lowrank.github.io/latent-seminar/OpenAI-Math/155-translational-aperiodicity) (lowrank.github.io, discussion)

## Changes

- 2026-10-29 (filed): Created (manuscript PDF, PeriodicTilingThree.lean and the Lean directory listing, formalization.yaml, oai-math-revised AGENT_NOTES and Comparator results, Quanta update 168049, Kalai's 7 Oct post, Demaine–Langerman abstract, PDF and repository README, Tan–Zhang PDF, Chair44 abstract, our video file; web, HN and Tao-blog searches found no further expert comment)

## Related

- 2026-10-24: [oai-math-revised: mathematicians launch a community hub to revise OpenAI's AI-written math manuscripts (guest post on Tao's blog)](https://postcutoff.com/e/2026-10-24-oai-math-revised-community-hub/index.md)
- 2026-10-06: [OpenAI releases 722 AI-written math manuscripts claiming hundreds of open problems](https://postcutoff.com/e/2026-10-06-openai-math-release-722-manuscripts/index.md)
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- People: [Terence Tao](https://postcutoff.com/person/terence-tao/)