A counterexample to periodic tiling in dimension three
OpenAI groups related papers into a “family”. OpenAI’s catalogue describes this one:
Constructs a finite translational tile in ℤ3 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 ℝ3, even with arbitrary real translation vectors. (OpenAI’s Lean note)
Lean is a language for mathematical statements and proofs that a computer can verify. The assessments below ask whether those statements express the result OpenAI describes here.
Both independent assessments find that the Lean statements express or imply the whole result described above.Read the assessments
Possible match to OpenAI’s description, selected from the papers’ main statements because it shares the most wording. No review confirming the selection is recorded.
There is a finite nonempty set T\subset\Z^3 such that:
(i)
T tiles \Z^3 by translations, but no translation set A with A\oplus T=\Z^3 is invariant under a subgroup of finite index in \Z^3;
(ii)
the set \Omega=T+[0,1]^3 tiles \R^3 by translations up to null sets, but no such tiling has a translation set invariant under a lattice of full rank in \R^3.
Reported proof verification
One Lean statement in the release; two independent groups report that Comparator accepted runs for this file at the version shown here.
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.
One Lean statement in this family
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There is a finite nonempty set T\subset\Z^3 such that:
(i)
T tiles \Z^3 by translations, but no translation set A with A\oplus T=\Z^3 is invariant under a subgroup of finite index in \Z^3;
(ii)
the set \Omega=T+[0,1]^3 tiles \R^3 by translations up to null sets, but no such tiling has a translation set invariant under a lattice of full rank in \R^3.
The words they share: finite, index, invariant, lattice, subgroup, tiles, tiling, translation.
Possible Lean counterpart
Selected from shared wording. No review of whether the statements mean the same thing is recorded.
PeriodicTilingThree.lean, the release lists it as a main result.
theoremperiodic_tiling_counterexample_and_minimality : (∃ T : Finset (Lattice 3), T.Nonempty ∧ (∃ A : Set (Lattice 3), Tiles T A) ∧ (∀ A : Set (Lattice 3), Tiles T A → ¬ FullyPeriodic A) ∧ (∃ A : Set (Space 3), AETiles (Thickening T) A) ∧ (∀ A : Set (Space 3), AETiles (Thickening T) A → ¬ EuclideanFullyPeriodic A)) ∧ IsLeast {d : ℕ | HasTileWithoutPeriodicComplement d} 3 := bysorry
This file states the theorem with sorry, a placeholder, in place of its proof. OpenAI supplies a separate proof file. Comparator tests whether that proof proves the statement here.
Other mathematical statements in OpenAI’s Lean library
Atlas found these propositions by scanning the text of the project’s Model.lean file. Each name links to its definition, with any associated statement file beside it. This list does not record running Lean to interpret or verify those definitions.
Independent comparisons and proof-verification reports
Does the Lean express the advertised result?
MathVet and cert-machine agree on how much of the advertised result the Lean statements express.
MathVet and Carlos Toledo’s cert-machine independently assessed whether this family’s Lean statements express the whole result OpenAI describes in its catalogue. These are assessments of the family as a whole, not of each paper theorem separately.
MathVet finds that the Lean statements express or imply the whole advertised result (their term: “full”).
Their note:One conjunction states all three parts: a nonempty finite T ⊆ ℤ³ with Tiles T A for some A but no FullyPeriodic (finite-index-subgroup invariant) tiling complement; the unit-cube thickening T + [0,1]³ a.e.-tiles ℝ³ by some arbitrary real translation set A but by none invariant under a full-rank lattice; and IsLeast {d | HasTileWithoutPeriodicComplement d} 3 (minimality, which absorbs the d = 1, 2 periodicity theorems). Tiles is exact (bijective F × A → G) and the ℝ³ version uses closed cubes with AETiles (unique representation a.e.), matching the summary and docs.
Definitions the group asks readers to compare with the paper:abbrev Lattice (d : ℕ) := Fin d → ℤ shadows Mathlib’s Lattice class name inside the namespace (harmless; means ℤ^d)
“full”, in their words: The challenge statements imply the headline claim after unpacking definitions or one routine step.
Their method: An independent, model-assisted, pre-referee reading of each family’s challenge statements against its headline claim under a published rubric (PROTOCOL.md), with its own Comparator runs per challenge.
The periodic-tiling question asks whether a finite tile that tiles a lattice must admit a periodic tiling. The formalized counterexample is a finite tile in $\mathbb Z^3$ that tiles but has no complement invariant under a finite-index subgroup. Its unit-cube thickening also tiles $\mathbb R^3$ almost everywhere but admits no fully periodic tiling, even with arbitrary real translation vectors. The result also establishes that dimension three is the least lattice dimension where this failure occurs.
AI-drafted comparisons, with no recorded human review
Aperiodic translational tile in dimension threePeriodicTilingThree
Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem: a finite tile in Z³ that tiles but admits no periodic tiling, with its cube thickening in R³, and minimality of dimension three.
Who ran the proof-verification program?
These groups report running Comparator with the release’s Lean statement files. For a configured theorem, it verifies OpenAI’s separate proof against the exact statement; some files instead configure definitions for comparison by name and type. Each row is the named group’s report. Atlas has not repeated the runs.
These reports do not list the software versions, proof-verification programs, axioms, time or memory in separate fields. Any such details in a report’s prose are reproduced below.
Aperiodic translational tile in dimension threePeriodicTilingThree
Two reports, all at the release’s recorded version.
Reported by
Their word
Their account
Date
Original report
cert-machine
“CERTIFIED”
Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution.
Other people’s accounts of running proof-verification software, with links to their reports.
ma4ypic4y, haskiindahouse, [submission] annals_periodic_tiling_conjecture, max gp (GPT-6) (issue 2010), 9 October 2026. Source
Lean Eval hosted Comparator run passed for annals_periodic_tiling_conjecture using a package adapted from the release’s PeriodicTiling Lean library.
Missing or incomplete source records
The program that collects Atlas’s source records, Statement Map, could not obtain or interpret some source material. These notes describe those limits; they do not identify mathematical errors.
PeriodicTilingThree: Tracked source lean/OAI/Geometry/PeriodicTiling/Main.lean unavailable: bytes were not captured at fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb
What these papers cite in the release, and what cites them
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Papers elsewhere in the release that this family’s papers cite
This family’s papers cite no paper elsewhere in the release.
Papers elsewhere in the release that cite this family’s papers
No paper elsewhere in the release cites this family’s papers.
Follow-up work by others
Erik D. Demaine, Stefan Langerman, Tiling 3D by Translates of a Single Polycube is Undecidable, 8 October 2026. Source
Demaine and Langerman adapt OpenAI’s cyclic tiling encoding from its 3D aperiodic-tile paper to show that translational monotiling by one connected polycube in 3D is undecidable.
Erik Demaine, Stefan Langerman, edemaine/translation-tiling-paper: paper and Lean proof that tiling 3D by a single polycube is undecidable, 7 October 2026. Source
Lean proof of co-RE-completeness for single-polycube tiling in Z^3 and R^3, building on OpenAI’s 3D aperiodic tile.
References outside the release
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Explain a result, compare statements or propose a correction
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Each button opens a GitHub form with this page and its source version filled in. Opening the form does not reserve the work.
Commentary on this family by others, dated and credited. It is commentary, not a finding.
An unsigned public tier list, the OpenAI Math Tier List, classes it as a counterexample and rates its significance, if it holds up, as “Career-defining publication”. The page says that nothing on it is verified.
Konstantin Kakaes, Quanta: Cracks appear in OpenAI’s wall of proofs (and the 6 October piece), 9 October 2026. Source
A tiling expert quoted by Quanta says the three-dimensional aperiodic tile paper has novel ideas but is hard to read and would not be publishable by traditional standards.
Joseph Myers, Comments on the translational tile manuscript, 8 October 2026. Source
An expert notes that the result is weak aperiodicity rather than strong, that the paper does not mention that distinction or the work on connected tiles, and that a connected tile in three dimensions follows without extra dimensions.
Lean community, OpenAI releases solutions to hundreds of problems (117 messages, 6-10 Oct 2026), 6 October 2026. Source
Joseph Myers: the periodic tiling result is weak aperiodicity only, the paper does not use the standard terms or the Greenfeld-Kolountzakis connectedness method, and a connected tile in dimension three is not hard to obtain.
Questions awaiting review
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Read the Lean statement against the paper
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Identify the theorem the catalogue describes
Family 155The theorem shown was selected from shared wording with OpenAI’s description. Atlas records no review confirming that it is the theorem the description means.Open the form
Address a question raised in an independent assessment
The challenge statements imply the headline claim after unpacking definitions or one routine step.
Their method
An independent, model-assisted, pre-referee reading of each family’s challenge statements against its headline claim under a published rubric (PROTOCOL.md), with its own Comparator runs per challenge. Its protocol describes a planned error-rate study; this import does not establish that the study has been completed.
The challenge statements state the headline as OpenAI wrote it.
Their method
Every Comparator challenge re-run on Linux with nanoda switched on, decided by a preregistered rule; every family’s challenge statements read against OpenAI’s headline, clause by clause. Decider: Comparator (leanprover/comparator d03acab1 = tag v4.34.0, built with the release’s toolchain v4.34.1), lean4export 076e8e57, landrun 811cfff5, nanoda_lib 3a240721 — on GitHub’s Linux runners…
Re-run through Comparator in a sandboxed virtual machine: accepted.
Date
8 October 2026
Version
at the release’s recorded version
Their method
An independent, model-assisted, pre-referee reading of each family’s challenge statements against its headline claim under a published rubric (PROTOCOL.md), with its own Comparator runs per challenge.