Lean statement

Aperiodic translational tile in dimension three

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Theorem selected for verification: OAI.PeriodicTilingThree.periodic_tiling_counterexample_and_minimality

Lean statement file: PeriodicTilingThree.leanShow on the mapFamily 155. OpenAI lists this as a main result in its Lean catalogue.

See the candidate theorem Who ran the proof-verification program Report a comparison with the paper (GitHub issue)

The Lean statement

The lines of the Lean statement that state the theorem, then the definitions in the file that it uses.

theorem periodic_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 := by  sorry

A Lean statement states the theorem and leaves its proof as sorry on purpose. OpenAI’s proof is in a separate solution module, and a program called Comparator checks that proof against this statement.

Definitions needed to read it

Local definitions the statement names, followed through the definitions they name in turn. The match is by name in the source text, so a name reached only through notation can be missed; the full file is below.

abbreviation OAI.PeriodicTilingThree.Lattice line 11
```
abbrev Lattice (d : ℕ) := Fin d → ℤ
```

definition OAI.PeriodicTilingThree.Tiles line 15

def Tiles (F : Finset G) (A : Set G) : Prop :=  Function.Bijective (fun p : ↥F × A => (p.1 : G) + (p.2 : G))

definition OAI.PeriodicTilingThree.Period line 18

def Period (A : Set G) (v : G) : Prop :=  ∀ x : G, x + v ∈ A ↔ x ∈ A

definition OAI.PeriodicTilingThree.FullyPeriodic line 21

def FullyPeriodic (A : Set G) : Prop :=  ∃ P : AddSubgroup G, P.FiniteIndex ∧ ∀ v ∈ P, Period A v

abbreviation OAI.PeriodicTilingThree.Space line 26
```
abbrev Space (d : ℕ) := Fin d → ℝ
```

definition OAI.PeriodicTilingThree.castLattice line 28

def castLattice {d : ℕ} : Lattice d →+ Space d where  toFun z i := (z i : ℝ)  map_zero' := by ext i; simp  map_add' z w := by ext i; simp

definition OAI.PeriodicTilingThree.unitCube line 33

def unitCube (d : ℕ) : Set (Space d) := Set.Icc 0 1

definition OAI.PeriodicTilingThree.Thickening line 35

def Thickening {d : ℕ} (F : Finset (Lattice d)) : Set (Space d) :=  {x | ∃ z ∈ F, x - castLattice z ∈ unitCube d}

definition OAI.PeriodicTilingThree.AETiles line 38

def AETiles {d : ℕ} (Ω A : Set (Space d)) : Prop :=  ∀ᵐ x ∂volume, ∃! a : A, x - (a : Space d) ∈ Ω

definition OAI.PeriodicTilingThree.latticeOfBasis line 41

def latticeOfBasis {d : ℕ} (b : Module.Basis (Fin d) ℝ (Space d)) :    AddSubgroup (Space d) := (Submodule.span ℤ (Set.range b)).toAddSubgroup

definition OAI.PeriodicTilingThree.EuclideanFullyPeriodic line 44

def EuclideanFullyPeriodic {d : ℕ} (A : Set (Space d)) : Prop :=  ∃ b : Module.Basis (Fin d) ℝ (Space d), ∀ v ∈ latticeOfBasis b, Period A v

definition OAI.PeriodicTilingThree.HasTileWithoutPeriodicComplement line 47

def HasTileWithoutPeriodicComplement (d : ℕ) : Prop :=  ∃ T : Finset (Lattice d), T.Nonempty ∧    (∃ A : Set (Lattice d), Tiles T A) ∧    ∀ A : Set (Lattice d), Tiles T A → ¬ FullyPeriodic A

Possible paper match, with no recorded review

No paper statement clearly matched the Lean wording. Statement Map, the program that collects Atlas’s source records, selected this possible match using words in OpenAI’s catalogue and Lean note. No review of whether the statements mean the same thing is recorded.

Shared with the catalogue’s wording or OpenAI’s note, none of them matched in the Lean text: finite, index, invariant, lattice, subgroup, tiles, tiling and translation.

Theorem 1.1 of A translational tile with no fully periodic tiling in dimension three

Theorem

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.

Section 1, Introduction Read it in the paper TeX source, line 79

Show the TeX

There is a finite nonempty set $T\subset\Z^3$ such that:
\begin{enumerate}[label=\textup{(\roman*)}]
\item $T$ 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 $\Z^3$;
\item 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 $\R^3$.
\end{enumerate}

Other possible paper matches, closest wording first

Proposition 7.5 of A translational tile with no fully periodic tiling in dimension three. Words of the Lean statement, its names split into words, that also appear in the theorem: euclidean, lattice, tile and tiling. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: almost, everywhere, translational, translation and invariant.
Corollary 6.3 of A translational tile with no fully periodic tiling in dimension three. Words of the Lean statement, its names split into words, that also appear in the theorem: integer, fully, periodic, tile, nonempty, finite and one more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: admits.
Proposition 6.2 of A translational tile with no fully periodic tiling in dimension three. Words of the Lean statement, its names split into words, that also appear in the theorem: fully, periodic, nonempty, finite and tiles.
Proposition 2.4 of A translational tile with no fully periodic tiling in dimension three. Words of the Lean statement, its names split into words, that also appear in the theorem: period.
Theorem 5.2 of A translational tile with no fully periodic tiling in dimension three. Words of the Lean statement, its names split into words, that also appear in the theorem: subgroup, index, nonempty, finite and tiling. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: invariant.
Proposition 7.2 of A translational tile with no fully periodic tiling in dimension three. Words of the Lean statement, its names split into words, that also appear in the theorem: tile and tiles.

How Atlas selects possible matches: it compares words in the Lean statement with every statement in the family. It also splits Lean names into words. The release’s label and note receive less weight; a main theorem wins a close match. If the Lean wording yields no clear match, Atlas uses the catalogue description. This selects passages to compare; it does not establish that they mean the same thing.

Reported Comparator runs

two independent groups report that Comparator accepted runs for this Lean statement file: Carlos Toledo’s cert-machine; MathVet. Each group reports its own Comparator run for this statement file. Atlas has not repeated the runs.

For a configured theorem, Comparator tests whether OpenAI’s proof proves the exact Lean statement. Some files instead configure definitions for comparison by name and type. Acceptance does not establish that the statement means the same thing as the paper’s theorem.

Carlos Toledo’s cert-machine

How they ran it, in their words

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 (.github/workflows/openai-math-kernel.yml), one j…

Accepted Their term: “CERTIFIED”.
Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution.
Reported 10 October 2026, at the release version shown here. Original verification report

MathVet

How they ran it, in their words

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.

Accepted Their term: “comparator-sandboxed-pass”.
Re-run through Comparator in a sandboxed virtual machine: accepted.
Reported 8 October 2026, at the release version shown here. Original verification report

What an accepted Comparator run would establish

If Comparator is run under this configuration and accepts, the solution proves the configured theorem using only the allowed axioms, and the Lean kernel accepts every step.

This configuration lists no definition holes, so every definition the statement uses is compared as it is written in this file.

Allowed axioms: propext, Quot.sound, Classical.choice

Acceptance leaves open whether the Lean statement means the same thing as the paper’s theorem, whether the paper’s other claims are correct, and whether the result is new. Those questions require comparing the definitions and statements, reviewing the paper’s arguments, and reading the earlier literature.

Do family 155’s Lean statements express the advertised result?

These assessments compare all the family’s Lean statements with the whole result OpenAI describes in its catalogue. They do not assess this statement alone. Each group uses its own published rules.

MathVet and cert-machine agree on how much of the advertised result the Lean statements express.

MathVet
Finds the whole advertised result stated or implied Their term: “full”.
The challenge statements imply the headline claim after unpacking definitions or one routine step. Their record
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)
Carlos Toledo’s cert-machine
Finds the whole advertised result stated or implied Their term: “MATCHES”.
The challenge statements state the headline as OpenAI wrote it. Their record

Read the complete assessments on the family page.

OpenAI’s own note on what this Lean statement covers

“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.”
Quoted from OpenAI’s Lean note for this family

An AI agent drafted an interpretation of that note, classifying this statement as “states a paper’s theorem as the paper does”. No human review of the draft is recorded.

Supporting material

The complete Lean statement file

66 lines, copied from release version fd4aeeb. A name defined in this file links to its line; pointing at the name shows its definition.

Declared in this fileOpenAI’s library at fd4aeebOther linked names open the current documentation for Mathlib, Lean’s mathematical library.

Links in the file: 36 to its own declarations, 32 to Mathlib’s documentation. 13 declared names also link to the same declaration in OpenAI’s library. OpenAI uses Mathlib version d13f23b. The linked documentation describes the current version, so names that have since changed may be missing there.

Definitions and lemmas named by the statement

If Comparator accepts a proof, it proves periodic_tiling_counterexample_and_minimality exactly as written here. To compare that statement with the paper, also read the definitions and lemmas listed below. Atlas found them by following names in the source text and lists each after the names it uses. For x.foo, Atlas includes every foo defined here for a type, even when the statement may use only one of them.

It reaches all 12 other declarations of the file; with the theorem itself they span 34 lines. 8 of them are named in the statement itself.

Names in the namespace OAI.PeriodicTilingThree are listed without it.

abbrev Latticeline 11in the statementlibrary
def Tilesline 15in the statementlibrary
def Periodline 18library
def FullyPeriodicline 21in the statementlibrary
abbrev Spaceline 26in the statementlibrary
def castLatticeline 28library
def unitCubeline 33library
def Thickeningline 35in the statementlibrary
def AETilesline 38in the statementlibrary
def latticeOfBasisline 41library
def EuclideanFullyPeriodicline 44in the statementlibrary
def HasTileWithoutPeriodicComplementline 47in the statementlibrary
theorem periodic_tiling_counterexample_and_minimalityline 52selected theoremlibrary

import Mathlibnamespace OAInoncomputable sectionuniverse unamespace PeriodicTilingThreeabbrev Lattice (d : ℕ) := Fin d → ℤvariable {G : Type u} [AddCommGroup G]def Tiles (F : Finset G) (A : Set G) : Prop :=  Function.Bijective (fun p : ↥F × A => (p.1 : G) + (p.2 : G))def Period (A : Set G) (v : G) : Prop :=  ∀ x : G, x + v ∈ A ↔ x ∈ Adef FullyPeriodic (A : Set G) : Prop :=  ∃ P : AddSubgroup G, P.FiniteIndex ∧ ∀ v ∈ P, Period A vopen MeasureTheoryabbrev Space (d : ℕ) := Fin d → ℝdef castLattice {d : ℕ} : Lattice d →+ Space d where  toFun z i := (z i : ℝ)  map_zero' := by ext i; simp  map_add' z w := by ext i; simpdef unitCube (d : ℕ) : Set (Space d) := Set.Icc 0 1def Thickening {d : ℕ} (F : Finset (Lattice d)) : Set (Space d) :=  {x | ∃ z ∈ F, x - castLattice z ∈ unitCube d}def AETiles {d : ℕ} (Ω A : Set (Space d)) : Prop :=  ∀ᵐ x ∂volume, ∃! a : A, x - (a : Space d) ∈ Ωdef latticeOfBasis {d : ℕ} (b : Module.Basis (Fin d) ℝ (Space d)) :    AddSubgroup (Space d) := (Submodule.span ℤ (Set.range b)).toAddSubgroupdef EuclideanFullyPeriodic {d : ℕ} (A : Set (Space d)) : Prop :=  ∃ b : Module.Basis (Fin d) ℝ (Space d), ∀ v ∈ latticeOfBasis b, Period A vdef HasTileWithoutPeriodicComplement (d : ℕ) : Prop :=  ∃ T : Finset (Lattice d), T.Nonempty ∧    (∃ A : Set (Lattice d), Tiles T A) ∧    ∀ A : Set (Lattice d), Tiles T A → ¬ FullyPeriodic Atheorem periodic_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 := by  sorryend PeriodicTilingThreeendend OAI

Every extracted statement of the accompanying papers

The statements the atlas extracted automatically from each accompanying paper’s TeX source, with links to the full sources.

A translational tile with no fully periodic tiling in dimension three

Theorem

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.

Section 1, Introduction Read it in the paper TeX source, line 79

Show the TeX

There is a finite nonempty set $T\subset\Z^3$ such that:
\begin{enumerate}[label=\textup{(\roman*)}]
\item $T$ 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 $\Z^3$;
\item 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 $\R^3$.
\end{enumerate}

Configuration and file facts

File: lean/ComparatorChallenges/PeriodicTilingThree.lean
OpenAI’s Lean catalogue: OpenAI lists this as a main result in its Lean catalogue.
Solution module: OAI/Geometry/PeriodicTiling/Counterexample.lean
Allowed axioms: propext, Quot.sound, Classical.choiceThe axioms this configuration allows a proof to use. A replay shows which of them a proof uses.
Imports: Mathlib
Imports from outside Mathlib: None
Local vocabulary: 12 local vocabulary declarations
Lines of Lean, without comments or blank lines: 34
Second proof-verification program (nanoda): Off in OpenAI's configuration.

A shareable summary of this page: the proposed paper match, what the Lean statement expresses, and which reviews are recorded.

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