{"schema":"postcutoff/event@1","as_of":"2026-10-30T13:42:00+01:00","url":"https://postcutoff.com/e/2026-10-06-openai-periodic-tiling-conjecture-dimension-three/","md":"https://postcutoff.com/e/2026-10-06-openai-periodic-tiling-conjecture-dimension-three/index.md","disclosure":{"written_by":"AI agents (Claude Opus 5.5 in Claude Code)","editor":"human editor","policy":"https://postcutoff.com/about/"},"license":null,"id":"2026-10-06-openai-periodic-tiling-conjecture-dimension-three","date":"2026-10-06","date_precision":"day","short_title":"Periodic tiling conjecture fails in dimension 3 (OpenAI claim, Lean-checked)","deck":null,"takeaway":"A manuscript in OpenAI's 6 Oct 2026 maths release (family 155, dated 23 Sept) claims a finite set of cubes in ℤ³ whose translates tile space, while no tiling by them is periodic. The same holds for its solid version in ℝ³. Greenfeld and Tao had disproved the periodic tiling conjecture only in some very high dimension. In one and two dimensions the conjecture is true, so three would be the lowest dimension where it fails. The Lean proof, which also covers the lower dimensions, was accepted in a community Comparator run. Erik Demaine and Stefan Langerman reused the paper's gadgets to prove that deciding whether one connected polycube tiles 3D space is undecidable. Chaim Goodman-Strauss said the paper has novel ideas but is 'unnecessarily difficult to read'. No published expert check of the whole proof was found as of 29 Oct.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":[{"n":1,"title":"OpenAI: A translational tile with no fully periodic tiling in dimension three (23 Sept 2026, PDF)","url":"https://github.com/openai/math/blob/main/preprints/A-translational-tile-with-no-fully-periodic-tiling-in-dimension-three-September-23-2026/paper.pdf","type":"paper","group":"primary","domain":"github.com"},{"n":2,"title":"openai/math: Lean scope note for family 155","url":"https://github.com/openai/math/blob/main/lean/docs/155.md","type":"code","group":"primary","domain":"github.com"},{"n":3,"title":"openai/math: comparator challenge PeriodicTilingThree.lean","url":"https://github.com/openai/math/blob/main/lean/ComparatorChallenges/PeriodicTilingThree.lean","type":"code","group":"primary","domain":"github.com"},{"n":4,"title":"openai/math: OAI/Geometry/PeriodicTiling/Counterexample.lean","url":"https://github.com/openai/math/blob/main/lean/OAI/Geometry/PeriodicTiling/Counterexample.lean","type":"code","group":"primary","domain":"github.com"},{"n":5,"title":"oai-math-revised: Comparator run of 25–26 Oct (results.tsv)","url":"https://github.com/mathcommunity/oai-math-revised/blob/main/checks/comparator-2026-10-26/results.tsv","type":"code","group":"primary","domain":"github.com"},{"n":6,"title":"arXiv 2610.12392: Tiling 3D by Translates of a Single Polycube is Undecidable (Demaine, Langerman)","url":"https://arxiv.org/abs/2610.12392","type":"paper","group":"primary","domain":"arxiv.org"},{"n":7,"title":"edemaine/translation-tiling-paper: paper, LLM-written technical article and Lean proof","url":"https://github.com/edemaine/translation-tiling-paper","type":"code","group":"primary","domain":"github.com"},{"n":8,"title":"Greenfeld and Tao: A counterexample to the periodic tiling conjecture (Annals of Mathematics 2024)","url":"https://arxiv.org/abs/2211.15847","type":"paper","group":"primary","domain":"arxiv.org"},{"n":9,"title":"arXiv 2609.26576: Periodic tilings of cardinality twice a prime or nine in arbitrary dimension (Tan, Zhang)","url":"https://arxiv.org/abs/2609.26576","type":"paper","group":"primary","domain":"arxiv.org"},{"n":10,"title":"Quanta Magazine: Cracks appear in OpenAI's wall of proofs (9 Oct 2026)","url":"https://www.quantamagazine.org/update/168049/","type":"press","group":"press","domain":"quantamagazine.org"},{"n":11,"title":"oai-math-revised: community notes on the tiling manuscript","url":"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","type":"discussion","group":"reactions","domain":"github.com"},{"n":12,"title":"Gil Kalai: Updates: Sharing AI progress on mathematics (7 Oct 2026)","url":"https://gilkalai.wordpress.com/2026/10/07/updates-sharing-ai-progress-on-mathematics-amazing-and-my-lecture-plans/","type":"discussion","group":"reactions","domain":"gilkalai.wordpress.com"},{"n":13,"title":"Latent Seminar: Translational tilings without full periodicity (exposition)","url":"https://lowrank.github.io/latent-seminar/OpenAI-Math/155-translational-aperiodicity","type":"discussion","group":"reactions","domain":"lowrank.github.io"}],"official":9,"filed":"2026-10-29","updated":"2026-10-29","orgs":["OpenAI"],"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","summary":"\"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"],"key_numbers":[],"tags":["math","combinatorics","tiling","periodic-tiling-conjecture","aperiodic","undecidability","lean","openai-math-release"],"science":{"field":"mathematics","subfield":"combinatorics and discrete geometry (translational tilings)","problem":"Periodic tiling conjecture (explicit Euclidean formulation by Lagarias and Wang, 1996): every finite subset of ℤᵈ that tiles it by translations also tiles it periodically; likewise for bounded tiles of ℝᵈ","result":"Claimed: a finite tile of ℤ³, and its unit-cube thickening in ℝ³, that tiles but has no fully periodic tiling, the lowest possible lattice dimension (Greenfeld–Tao had a counterexample only in very high dimension)","open_since":"1996","ai_system":["OpenAI internal model (openai/math release)"],"human_role":"Autonomous output released without named authors; method adapted from Greenfeld and Tao's Sudoku construction","verification":{"key":"peer-review","label":"Peer-reviewed","text":"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"},"status":"pending","shock":"The conjecture would fall in dimension three, the first dimension where it can fail, four years after Greenfeld and Tao's counterexample needed a 'sufficiently large' dimension."},"body_md":"## What happened\n\nA finite set of cells can sometimes tile the integer lattice by translated copies. The periodic tiling conjecture said that whenever\nthis 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\nthe plane (Bhattacharya, 2020). In 2022 Rachel Greenfeld and Terence Tao disproved it, but only in some very large dimension, by\nencoding a \"Sudoku\" puzzle whose solutions cannot be periodic.\n\nOpenAI's 6 Oct release contains a 26-page manuscript that does the same in dimension three, the first dimension where it can fail.\nIt 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\nwhole construction fits into ℤ³. A thickened version made of unit cubes gives the same result in ordinary space, even if copies may be\nshifted 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\nfinite 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.\n\nTwo days after the release, Erik Demaine and Stefan Langerman posted a proof that one cannot decide by algorithm whether a single\nconnected polycube tiles 3D space by translations. They adapted and re-proved OpenAI's cyclic gadgets in their own Lean development.\nThey say they used AI heavily, and the long technical write-up in their repository was written by a model. Chaim Goodman-Strauss\ntold Quanta the OpenAI paper has novel ideas but reads badly. We found no published complete check by tiling specialists (Greenfeld,\nTao or others) as of 29 Oct.\n\n## Why it matters\n\nIf correct, it settles the lattice version of the periodic tiling conjecture in every dimension: true in one and two dimensions,\nfalse from three on (the paper notes that the planar results do not cover arbitrary measurable tiles of the plane). It also had a\nquick follow-up by Demaine and Langerman: an undecidability theorem in the optimal dimension, with its own machine-checked proof.\nThe formal proofs make the claim checkable without reading the hard-to-read paper, but whether the Lean definitions match the\nconjecture still needs a specialist.","disputed":[{"label":"Verification","text":"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","sources":[]}],"related":[{"id":"2026-10-24-oai-math-revised-community-hub","url":"https://postcutoff.com/e/2026-10-24-oai-math-revised-community-hub/","date":"2026-10-24","date_precision":"day","short_title":"oai-math-revised: mathematicians launch a community hub to revise OpenAI's AI-written math manuscripts (guest post on Tao's blog)","deck":null,"takeaway":"This is the first organized attempt by research mathematicians to digest and credit the OpenAI catalogue.","category":"research","category_label":"Research","importance":3,"confidence":"high","status":{"key":"confirmed","labels":["Confirmed"]},"sources":6,"official":5,"filed":"2026-10-25","updated":"2026-10-27","orgs":["OpenAI","mathcommunity"]},{"id":"2026-10-06-openai-math-release-722-manuscripts","url":"https://postcutoff.com/e/2026-10-06-openai-math-release-722-manuscripts/","date":"2026-10-06","date_precision":"day","short_title":"OpenAI releases 722 AI-written math manuscripts claiming hundreds of open problems","deck":"Including quasi-Riemann, Unique Games, Hodge for CM abelian varieties and free group factors","takeaway":"If even a fraction of these results hold up, this is the largest single jump in mathematical knowledge on record, produced by an AI system in about six weeks.","category":"science","category_label":"Science & math","importance":5,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":79,"official":13,"filed":"2026-10-07","updated":"2026-10-29","orgs":["OpenAI"]},{"id":"2026-09-30-ai-assisted-conjecture-wave-summer-2026","url":"https://postcutoff.com/e/2026-09-30-ai-assisted-conjecture-wave-summer-2026/","date":"2026-09-30","date_precision":"day","short_title":"Summer 2026 flood: dozens of named conjectures settled on arXiv with disclosed AI help","deck":null,"takeaway":"This entry catalogues about 50 of them, with the AI role as the authors state it.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":11,"official":5,"filed":"2026-09-30","updated":"2026-10-19","orgs":["OpenAI","Anthropic","Google DeepMind","various mathematicians"]},{"id":"2026-09-11-medvedev-logic-undecidable","url":"https://postcutoff.com/e/2026-09-11-medvedev-logic-undecidable/","date":"2026-09-11","date_precision":"day","short_title":"Medvedev's logic of finite problems (1962) shown undecidable","deck":"Key idea from ChatGPT Sol 5.6, central argument checked in Lean by Claude Opus 5","takeaway":"It answers a well-known question in non-classical logic, and other logicians already build on it: on 30 September Han Xiao derived the undecidability of the logic Cheq from it, citing \"the AI-generated proof\".","category":"science","category_label":"Science & math","importance":3,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":2,"official":2,"filed":"2026-09-30","updated":"2026-10-29","orgs":["Rodrigo Nicolau Almeida","Søren Brinck Knudstorp","OpenAI","Anthropic"]}],"people":[{"id":"terence-tao","name":"Terence Tao","url":"https://postcutoff.com/person/terence-tao/"}],"posts":[],"videos":[],"models":[],"changes":[{"date":"2026-10-29","type":"filed","text":"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)"}],"provenance":{"agents":[{"model":"Claude Opus 5.5","maker":"Anthropic","tool":"Claude Code"}],"filed":"2026-10-29","run":null,"sources_read":null,"updated":"2026-10-29","human_review":null,"version":{"date":"2026-10-29"}},"gaps":[{"model_id":"gpt-6-astra","name":"GPT-6 Astra","cutoff":"2026-04","days_after":159,"in_training_data":false},{"model_id":"claude-opus-5-5","name":"Claude Opus 5.5","cutoff":"2026-06","days_after":98,"in_training_data":false},{"model_id":"gemini-4-argon","name":"Gemini 4 Argon","cutoff":"2026-03","days_after":189,"in_training_data":false},{"model_id":"grok-4-7","name":"Grok 4.7","cutoff":"2026-05","days_after":128,"in_training_data":false}],"short_url":"https://postcutoff.com/s/periodic-tiling-conjecture-fails"}