Lean statement

Optimal order for translative and lattice covering-density suprema

Lean is a language for mathematical statements and proofs that a computer can verify. This page shows one statement file from OpenAI’s release, possible paper matches and independent reports. The code-like names below identify the definitions or theorems selected by the proof-verification settings.

Theorem selected for verification: OAI.CoveringOrder.optimal_order

Lean statement file: CoveringDensity.leanShow on the mapOpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 092 link it.

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.

def OptimalOrder : Prop :=  ∃ c C : ℝ, 0 < c ∧ 0 < C ∧ ∃ n₀ : ℕ, ∀ n : ℕ, n₀ ≤ n →    orderBounds c C n (translativeSup n) ∧    orderBounds c C n (symmetricTranslativeSup n) ∧    orderBounds c C n (latticeSup n) ∧    orderBounds c C n (symmetricLatticeSup n)⋮theorem CoveringOrder.optimal_order : CoveringOrder.OptimalOrder := 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.CoveringOrder.Space line 9

abbrev Space (n : ℕ) := EuclideanSpace ℝ (Fin n)

structure OAI.CoveringOrder.LatticeCover line 12

structure LatticeCover {n : ℕ} (K : Set (Space n)) where  lattice : Submodule ℤ (Space n)  discrete : DiscreteTopology lattice  fullrank : IsZLattice ℝ lattice  covers : ∀ y, ∃ l : lattice, y - l.val ∈ K

definition OAI.CoveringOrder.euclideanVolume line 18

noncomputable def euclideanVolume (n : ℕ) :    @Measure (EuclideanSpace ℝ (Fin n))      (@WithLp.measurableSpace 2 (Fin n → ℝ)        (@MeasurableSpace.pi (Fin n) (fun _ => ℝ) (fun _ => Real.measurableSpace))) :=  @Measure.map (Fin n → ℝ) (EuclideanSpace ℝ (Fin n))    (@MeasurableSpace.pi (Fin n) (fun _ => ℝ) (fun _ => Real.measurableSpace))    (@WithLp.measurableSpace 2 (Fin n → ℝ)      (@MeasurableSpace.pi (Fin n) (fun _ => ℝ) (fun _ => Real.measurableSpace)))    (@WithLp.toLp 2 (Fin n → ℝ))    (Measure.pi (fun _ : Fin n => @volume ℝ Real.measureSpace))

definition OAI.CoveringOrder.euclideanCovolume line 29

noncomputable def euclideanCovolume {n : ℕ}    (lattice : Submodule ℤ (EuclideanSpace ℝ (Fin n))) : ℝ≥0∞ :=  addCovolume lattice (EuclideanSpace ℝ (Fin n)) (euclideanVolume n)

definition OAI.CoveringOrder.latticeDensity line 33

noncomputable def latticeDensity {n : ℕ} {K : Set (Space n)}    (cover : LatticeCover K) : ℝ≥0∞ :=  euclideanVolume n K / euclideanCovolume cover.lattice

definition OAI.CoveringOrder.thetaL line 37

noncomputable def thetaL {n : ℕ} (K : Set (EuclideanSpace ℝ (Fin n))) : ℝ≥0∞ :=  ⨅ cover : LatticeCover K, latticeDensity cover

definition OAI.TranslativeCovering.cube line 49

def cube (n : ℕ) (R : ℝ) : Set (Space n) :=  {x | ∀ i, -R ≤ x i ∧ x i ≤ R}

definition OAI.TranslativeCovering.LocallyFiniteCenters line 52

def LocallyFiniteCenters {n : ℕ} (X : Set (Space n)) : Prop :=  ∀ B : Set (Space n), Bornology.IsBounded B → (X ∩ B).Finite

definition OAI.TranslativeCovering.Covers line 55

def Covers {n : ℕ} (K X : Set (Space n)) : Prop :=  ∀ y, ∃ x ∈ X, y - x ∈ K

definition OAI.TranslativeCovering.ConvexBody line 58

def ConvexBody {n : ℕ} (K : Set (Space n)) : Prop :=  IsCompact K ∧ Convex ℝ K ∧ (interior K).Nonempty

definition OAI.TranslativeCovering.CentrallySymmetric line 61

def CentrallySymmetric {n : ℕ} (K : Set (Space n)) : Prop :=  ∃ z, ∀ x, x ∈ K ↔ (2 : ℝ) • z - x ∈ K

definition OAI.TranslativeCovering.upperCenterIntensity line 64

noncomputable def upperCenterIntensity {n : ℕ} (X : Set (Space n)) : ℝ≥0∞ :=  Filter.limsup (fun R : ℝ =>    (↑(X ∩ cube n R).ncard : ℝ≥0∞) / ENNReal.ofReal ((2 * R) ^ n)) atTop

definition OAI.TranslativeCovering.thetaT line 68

noncomputable def thetaT {n : ℕ} (K : Set (Space n)) : ℝ≥0∞ :=  ⨅ X : {X : Set (Space n) // LocallyFiniteCenters X ∧ Covers K X},    volume K * upperCenterIntensity X.val

abbreviation OAI.CoveringOrder.Body line 78

abbrev Body (n : ℕ) := {K : Set (Space n) // TranslativeCovering.ConvexBody K}

abbreviation OAI.CoveringOrder.SymmetricBody line 79

abbrev SymmetricBody (n : ℕ) :=  {K : Body n // TranslativeCovering.CentrallySymmetric K.val}

definition OAI.CoveringOrder.translativeSup line 82

noncomputable def translativeSup (n : ℕ) : ℝ≥0∞ :=  ⨆ K : Body n, TranslativeCovering.thetaT K.val

definition OAI.CoveringOrder.symmetricTranslativeSup line 85

noncomputable def symmetricTranslativeSup (n : ℕ) : ℝ≥0∞ :=  ⨆ K : SymmetricBody n, TranslativeCovering.thetaT K.val.val

definition OAI.CoveringOrder.latticeSup line 88

noncomputable def latticeSup (n : ℕ) : ℝ≥0∞ :=  ⨆ K : Body n, thetaL K.val

definition OAI.CoveringOrder.symmetricLatticeSup line 91

noncomputable def symmetricLatticeSup (n : ℕ) : ℝ≥0∞ :=  ⨆ K : SymmetricBody n, thetaL K.val.val

definition OAI.CoveringOrder.orderBounds line 95

def orderBounds (c C : ℝ) (n : ℕ) (a : ℝ≥0∞) : Prop :=  ENNReal.ofReal (c * (n : ℝ) * Real.log (n : ℝ)) ≤ a ∧    a ≤ ENNReal.ofReal (C * (n : ℝ) * Real.log (n : ℝ))

Possible paper match, with no recorded review

Atlas selected this possible match by comparing words in the Lean statement with every statement in the family. No review of whether the statements mean the same thing is recorded.

Words of the Lean statement, its names split into words, that also appear in the theorem: symmetric, centrally, convex, integer, real, theta, body, absolute and two more. Found in the paper’s title, which its main theorem states: translative, densities, covering and order.

Statements in Translative covering densities of order n log n come first because the release’s name for this Lean statement shares words with that paper’s title.

Theorem 1.1 of Translative covering densities of order n log n

Theorem

There are absolute constants c>0 and n_0 such that, for every integer n\ge n_0, some centrally symmetric convex body K_n\subset\mathbb R^n satisfies

\theta_T(K_n)>c n\log n.

Section 1, Introduction Read it in the paper TeX source, line 49

Show the TeX

There are absolute constants $c>0$ and $n_0$ such that, for every integer
$n\ge n_0$, some centrally symmetric convex body $K_n\subset\mathbb R^n$
satisfies
\[
 \theta_T(K_n)>c n\log n.
\]

Other possible paper matches, closest wording first

Proposition A.1 of Translative covering densities of order n log n. Words of the Lean statement, its names split into words, that also appear in the theorem: centered, upper, cube, euclidean, density, convex and four more.
Corollary 8.1 of Translative covering densities of order n log n. Words of the Lean statement, its names split into words, that also appear in the theorem: quantities, four, comparison, order, theta, absolute and two more.
Theorem 1.1 of A single-lattice covering bound of order n log n. Words of the Lean statement, its names split into words, that also appear in the theorem: assumed, rank, lattice, convex, integer, full and four more. Found in the paper’s title, which its main theorem states: bound, covering and order.
Corollary 9.1 of Translative covering densities of order n log n. Words of the Lean statement, its names split into words, that also appear in the theorem: symmetric, centrally, real, theta and log.
Proposition 3.1 of Translative covering densities of order n log n. Words of the Lean statement, its names split into words, that also appear in the theorem: rank, lattice, full, real, covers, finite and two more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: sufficiently and large.
Proposition 6.2 of Translative covering densities of order n log n. Words of the Lean statement, its names split into words, that also appear in the theorem: intensity, positive, pi, space, measurable, full and two more.

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 092’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: None; the lower bound for centrally symmetric bodies is stated for the translative supremum too (hence for the lattice one).
Definitions the group asks readers to compare with the paper: Hand-rolled density notions: euclideanVolume (pushforward of product Lebesgue measure), euclideanCovolume, thetaL, thetaT with upperCenterIntensity := limsup #(X ∩ cube R) / (2R)^n; standard in form
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 formalization determines the order of the largest covering density in dimension $n$. For all sufficiently large $n$, the suprema of translative covering density and lattice covering density are each bounded above and below by absolute positive multiples of $n\log n$, both over all convex bodies and over centrally symmetric convex bodies.”
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

116 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: 43 to its own declarations, 66 to Mathlib’s documentation. 23 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 optimal_order 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 22 other declarations of the file; with the theorem itself they span 63 lines. 1 of them is named in the statement itself.

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

abbrev Spaceline 9library
structure LatticeCoverline 12library
def euclideanVolumeline 18library
def euclideanCovolumeline 29library
def latticeDensityline 33library
def thetaLline 37library
abbrev OAI.TranslativeCovering.Spaceline 47library
def OAI.TranslativeCovering.cubeline 49library
def OAI.TranslativeCovering.LocallyFiniteCentersline 52library
def OAI.TranslativeCovering.Coversline 55library
def OAI.TranslativeCovering.ConvexBodyline 58library
def OAI.TranslativeCovering.CentrallySymmetricline 61library
def OAI.TranslativeCovering.upperCenterIntensityline 64library
def OAI.TranslativeCovering.thetaTline 68library
abbrev Bodyline 78library
abbrev SymmetricBodyline 79library
def translativeSupline 82library
def symmetricTranslativeSupline 85library
def latticeSupline 88library
def symmetricLatticeSupline 91library
def orderBoundsline 95library
def OptimalOrderline 102in the statementlibrary
theorem optimal_orderline 113selected theoremlibrary

import Mathlibnamespace OAInamespace CoveringOrderopen Set MeasureTheoryopen scoped ENNRealabbrev Space (n : ℕ) := EuclideanSpace ℝ (Fin n)/-- Actual discrete full-rank lattice coverings; no assumed density theorem. -/structure LatticeCover {n : ℕ} (K : Set (Space n)) where  lattice : Submodule ℤ (Space n)  discrete : DiscreteTopology lattice  fullrank : IsZLattice ℝ lattice  covers : ∀ y, ∃ l : lattice, y - l.val ∈ Knoncomputable def euclideanVolume (n : ℕ) :    @Measure (EuclideanSpace ℝ (Fin n))      (@WithLp.measurableSpace 2 (Fin n → ℝ)        (@MeasurableSpace.pi (Fin n) (fun _ => ℝ) (fun _ => Real.measurableSpace))) :=  @Measure.map (Fin n → ℝ) (EuclideanSpace ℝ (Fin n))    (@MeasurableSpace.pi (Fin n) (fun _ => ℝ) (fun _ => Real.measurableSpace))    (@WithLp.measurableSpace 2 (Fin n → ℝ)      (@MeasurableSpace.pi (Fin n) (fun _ => ℝ) (fun _ => Real.measurableSpace)))    (@WithLp.toLp 2 (Fin n → ℝ))    (Measure.pi (fun _ : Fin n => @volume ℝ Real.measureSpace))noncomputable def euclideanCovolume {n : ℕ}    (lattice : Submodule ℤ (EuclideanSpace ℝ (Fin n))) : ℝ≥0∞ :=  addCovolume lattice (EuclideanSpace ℝ (Fin n)) (euclideanVolume n)noncomputable def latticeDensity {n : ℕ} {K : Set (Space n)}    (cover : LatticeCover K) : ℝ≥0∞ :=  euclideanVolume n K / euclideanCovolume cover.latticenoncomputable def thetaL {n : ℕ} (K : Set (EuclideanSpace ℝ (Fin n))) : ℝ≥0∞ :=  ⨅ cover : LatticeCover K, latticeDensity coverend CoveringOrdernamespace TranslativeCoveringopen Set Filter MeasureTheoryopen scoped ENNRealabbrev Space (n : ℕ) := EuclideanSpace ℝ (Fin n)def cube (n : ℕ) (R : ℝ) : Set (Space n) :=  {x | ∀ i, -R ≤ x i ∧ x i ≤ R}def LocallyFiniteCenters {n : ℕ} (X : Set (Space n)) : Prop :=  ∀ B : Set (Space n), Bornology.IsBounded B → (X ∩ B).Finitedef Covers {n : ℕ} (K X : Set (Space n)) : Prop :=  ∀ y, ∃ x ∈ X, y - x ∈ Kdef ConvexBody {n : ℕ} (K : Set (Space n)) : Prop :=  IsCompact K ∧ Convex ℝ K ∧ (interior K).Nonemptydef CentrallySymmetric {n : ℕ} (K : Set (Space n)) : Prop :=  ∃ z, ∀ x, x ∈ K ↔ (2 : ℝ) • z - x ∈ Knoncomputable def upperCenterIntensity {n : ℕ} (X : Set (Space n)) : ℝ≥0∞ :=  Filter.limsup (fun R : ℝ =>    (↑(X ∩ cube n R).ncard : ℝ≥0∞) / ENNReal.ofReal ((2 * R) ^ n)) atTopnoncomputable def thetaT {n : ℕ} (K : Set (Space n)) : ℝ≥0∞ :=  ⨅ X : {X : Set (Space n) // LocallyFiniteCenters X ∧ Covers K X},    volume K * upperCenterIntensity X.valend TranslativeCoveringnamespace CoveringOrderopen Set MeasureTheoryopen scoped ENNRealabbrev Body (n : ℕ) := {K : Set (Space n) // TranslativeCovering.ConvexBody K}abbrev SymmetricBody (n : ℕ) :=  {K : Body n // TranslativeCovering.CentrallySymmetric K.val}noncomputable def translativeSup (n : ℕ) : ℝ≥0∞ :=  ⨆ K : Body n, TranslativeCovering.thetaT K.valnoncomputable def symmetricTranslativeSup (n : ℕ) : ℝ≥0∞ :=  ⨆ K : SymmetricBody n, TranslativeCovering.thetaT K.val.valnoncomputable def latticeSup (n : ℕ) : ℝ≥0∞ :=  ⨆ K : Body n, thetaL K.valnoncomputable def symmetricLatticeSup (n : ℕ) : ℝ≥0∞ :=  ⨆ K : SymmetricBody n, thetaL K.val.val/-- The finite real comparison scale, viewed in extended nonnegative reals. -/def orderBounds (c C : ℝ) (n : ℕ) (a : ℝ≥0∞) : Prop :=  ENNReal.ofReal (c * (n : ℝ) * Real.log (n : ℝ)) ≤ a ∧    a ≤ ENNReal.ofReal (C * (n : ℝ) * Real.log (n : ℝ))/-- All four extremal quantities have commonabsolute positive comparison constants. ENNReal bounds imply finiteness ratherthan mapping a potentially infinite density to zero through `toReal`. -/def OptimalOrder : Prop :=  ∃ c C : ℝ, 0 < c ∧ 0 < C ∧ ∃ n₀ : ℕ, ∀ n : ℕ, n₀ ≤ n →    orderBounds c C n (translativeSup n) ∧    orderBounds c C n (symmetricTranslativeSup n) ∧    orderBounds c C n (latticeSup n) ∧    orderBounds c C n (symmetricLatticeSup n)end CoveringOrder/-- Translative and lattice covering densities over all convex bodies and overcentrally symmetric convex bodies have the same asymptotic order. -/theorem CoveringOrder.optimal_order : CoveringOrder.OptimalOrder := by  sorryend 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 single-lattice covering bound of order n log n

Theorem

There is an absolute constant C>0 such that, for every integer n\ge2 and every convex body K\subset\R^n, one full-rank lattice \Lambda satisfies

K+\Lambda=\R^n,\qquad \frac{\vol_n(K)}{\det\Lambda}\le Cn\log n.

No symmetry or boundary regularity is assumed.

Section 1, Introduction Read it in the paper TeX source, line 16

Show the TeX

There is an absolute constant $C>0$ such that, for every integer
$n\ge2$ and every convex body $K\subset\R^n$, one full-rank lattice
$\Lambda$ satisfies
\[
 K+\Lambda=\R^n,\qquad
 \frac{\vol_n(K)}{\det\Lambda}\le Cn\log n.
\]
No symmetry or boundary regularity is assumed.

Translative covering densities of order n log n

Theorem

There are absolute constants c>0 and n_0 such that, for every integer n\ge n_0, some centrally symmetric convex body K_n\subset\mathbb R^n satisfies

\theta_T(K_n)>c n\log n.

Section 1, Introduction Read it in the paper TeX source, line 49

Show the TeX

There are absolute constants $c>0$ and $n_0$ such that, for every integer
$n\ge n_0$, some centrally symmetric convex body $K_n\subset\mathbb R^n$
satisfies
\[
 \theta_T(K_n)>c n\log n.
\]

Configuration and file facts

File: lean/ComparatorChallenges/CoveringDensity.lean
OpenAI’s Lean catalogue: OpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 092 link it.
Solution module: OAI.Geometry.CoveringDensity.Main
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: 22 local vocabulary declarations
Lines of Lean, without comments or blank lines: 63
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.

Plectis Atlas