Lean statement

Bounded-degree coboundary expanders in dimensions at least three

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.CoboundaryExpanders.main

Lean statement file: CoboundaryExpanders.leanShow on the mapOpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 177 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 MainStatement : Prop :=  ∀ d : ℕ, 3 ≤ d →    ∃ (D : ℕ) (ε : ℝ), 0 < ε ∧      ∃ X : ℕ → Complex,        (∀ m, (X m).Pure d ∧ (X m).Connected) ∧        Tendsto (fun m => ((X m).facesOf 0).card) atTop atTop ∧        (∀ m (v : Fin (X m).vertexCount), (X m).topDegree d v ≤ D) ∧        ∀ m i, i < d → ∀ f : (X m).Cochain i,          ε * (X m).distance d i f ((X m).coboundaries i) ≤            (X m).norm d (i + 1) ((X m).coboundary i f)⋮theorem main : MainStatement := 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.

structure OAI.CoboundaryExpanders.Complex line 19

structure Complex where  vertexCount : ℕ  faces : Finset (Finset (Fin vertexCount))  empty_mem : ∅ ∈ faces  downward : ∀ {σ τ}, σ ∈ faces → τ ⊆ σ → τ ∈ faces  vertex_mem : ∀ v, {v} ∈ faces

definition OAI.CoboundaryExpanders.Complex.facesOf line 29

def facesOf (X : Complex) (i : ℕ) : Finset (Finset (Fin X.vertexCount)) :=  X.faces.filter (fun σ => σ.card = i + 1)

definition OAI.CoboundaryExpanders.Complex.Face line 32

def Face (X : Complex) (i : ℕ) := {σ // σ ∈ X.facesOf i}

definition OAI.CoboundaryExpanders.Complex.Pure line 38

def Pure (X : Complex) (s : ℕ) : Prop :=  (X.facesOf s).Nonempty ∧  ∀ σ ∈ X.faces, ∃ τ ∈ X.facesOf s, σ ⊆ τ

definition OAI.CoboundaryExpanders.Complex.Connected line 43

def Connected (X : Complex) : Prop :=  Nonempty (Fin X.vertexCount) ∧  ∀ v w : Fin X.vertexCount,    Relation.ReflTransGen (fun a b => a ≠ b ∧ {a, b} ∈ X.faces) v w

abbreviation OAI.CoboundaryExpanders.Complex.Cochain line 48

abbrev Cochain (X : Complex) (i : ℕ) := X.Face i → ZMod 2

definition OAI.CoboundaryExpanders.Complex.weight line 51

def weight (X : Complex) (s i : ℕ) (σ : X.Face i) : ℝ :=  (((X.facesOf s).filter (fun τ => σ.val ⊆ τ)).card : ℝ) /    ((Nat.choose (s + 1) (i + 1) : ℝ) * (X.facesOf s).card)

definition OAI.CoboundaryExpanders.Complex.norm line 56

def norm (X : Complex) (s i : ℕ) (f : X.Cochain i) : ℝ :=  ∑ σ : X.Face i, if f σ ≠ 0 then X.weight s i σ else 0

definition OAI.CoboundaryExpanders.Complex.coboundary line 60

def coboundary (X : Complex) (i : ℕ) (f : X.Cochain i) : X.Cochain (i + 1) :=  fun τ => ∑ σ : X.Face i, if σ.val ⊆ τ.val then f σ else 0

definition OAI.CoboundaryExpanders.Complex.coboundaries line 64

def coboundaries (X : Complex) : (i : ℕ) → Set (X.Cochain i)  | 0 => {f | ∃ c : ZMod 2, f = fun _ => c}  | i + 1 => Set.range (X.coboundary i)

definition OAI.CoboundaryExpanders.Complex.distance line 69

def distance (X : Complex) (s i : ℕ) (f : X.Cochain i)    (A : Set (X.Cochain i)) : ℝ :=  sInf ((fun a => X.norm s i (f - a)) '' A)

definition OAI.CoboundaryExpanders.Complex.topDegree line 74

def topDegree (X : Complex) (s : ℕ) (v : Fin X.vertexCount) : ℕ :=  ((X.facesOf s).filter (fun τ => v ∈ τ)).card

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: vertex, finite, simplicial, dimensional, constants, pure, norm, integer and four more. Found in the paper’s title, which its main theorem states: degree and coboundary.

Theorem 1.1 of Bounded-degree coboundary expanders in every dimension

Theorem

For every integer d\ge3, there exist constants D<\infty, \eps>0, and finite connected pure d-dimensional simplicial complexes X_m, with |X_m(0)|\longrightarrow\infty, such that every vertex belongs to at most D faces of dimension d and, for every f\in C^i(X_m;\F),

\norm{\delta_i f}_{X_m,i+1} \ge \eps\,\dist_{X_m,i}(f,B^i(X_m)) \qquad(0\le i<d).

The constants are independent of m, i, and f.

Section 1, Introduction Read it in the paper TeX source, line 40

Show the TeX

For every integer $d\ge3$, there exist constants $D<\infty$, $\eps>0$,
and finite connected pure $d$-dimensional simplicial complexes $X_m$,
with $|X_m(0)|\longrightarrow\infty$, such that every vertex belongs to
at most $D$ faces of dimension $d$ and, for every
$f\in C^i(X_m;\F)$,
\begin{equation}
 \norm{\delta_i f}_{X_m,i+1}
 \ge \eps\,\dist_{X_m,i}(f,B^i(X_m))
 \qquad(0\le i<d).
\end{equation}
The constants are independent of $m$, $i$, and $f$.

Other possible paper matches, closest wording first

Proposition 1 of Bounded-degree coboundary expanders in every dimension. Words of the Lean statement, its names split into words, that also appear in the theorem: eq and complex.
Proposition 2 of Bounded-degree coboundary expanders in every dimension. Words of the Lean statement, its names split into words, that also appear in the theorem: connected.

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

three independent groups report that Comparator accepted runs for this Lean statement file: Carlos Toledo’s cert-machine; MathVet; Neal McBurnett’s openai-math-lean-tooling. 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

Neal McBurnett’s openai-math-lean-tooling

How they ran it, in their words

Each challenge’s solution is built and checked by Comparator at the pinned release, on Lean leanprover/lean4:v4.34.1, recording the verdict, the check time and the peak memory; the run was still in progress at this snapshot (192 accepted, 224 pending).

Accepted Their term: “accepted”.
Re-run through Comparator with Lean’s kernel: accepted (the check took 183 s; peak memory 6.5 GiB).
Reported 11 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 177’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 disagree 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: MainStatement : ∀ d ≥ 3, ∃ D ε, 0 < ε ∧ ∃ X : ℕ → Complex, (∀ m, Pure d ∧ Connected) ∧ #vertices → ∞ ∧ (∀ m v, topDegree d v ≤ D) ∧ ∀ m, ∀ i < d, ∀ f, ε * distance f (coboundaries i) ≤ norm (coboundary f) over ZMod 2 with top-face-weighted norms (weight = normalized incident top-face count). Standard weighted coboundary expansion; the graph and 2-dim cases are not part of the claim, as the summary says.
Carlos Toledo’s cert-machine
Finds part of it stated, or a narrower or weaker statement Their term: “NARROWER — DECLARED”.
The challenge statements state less than the headline, and OpenAI’s Lean note says so. Their record
Their note: the 'every positive dimension' conclusion (d = 1, 2 cases, which the headline attributes to prior work)

Read the complete assessments on the family page.

OpenAI’s own note on what this Lean statement covers

“The formalization constructs bounded-degree $\mathbb F_2$ coboundary expanders in every dimension $d\ge3$. For each such $d$, it gives finite pure connected $d$-dimensional simplicial complexes with vertex counts tending to infinity, one uniform bound on top-dimensional degree at each vertex, and one positive coboundary-expansion constant in every degree below $d$. The constants may depend on $d$. The graph and two-dimensional cases used for the paper's all-positive-dimensions conclusion are separate.”
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

98 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: 15 to its own declarations, 25 to Mathlib’s documentation. 14 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 main 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 13 of the file’s 15 other declarations; with the theorem itself they span 44 lines. 1 of them is named in the statement itself.

Names in the namespace OAI.CoboundaryExpanders.Complex are listed without it.

structure OAI.CoboundaryExpanders.Complexline 19library
def facesOfline 29library
def Faceline 32library
def Pureline 38library
def Connectedline 43library
abbrev Cochainline 48library
def weightline 51library
def normline 56library
def coboundaryline 60library
def coboundariesline 64library
def distanceline 69library
def topDegreeline 74library
def OAI.CoboundaryExpanders.MainStatementline 80in the statementlibrary
theorem OAI.CoboundaryExpanders.mainline 91selected theoremlibrary

Other Lean rules that may affect the statement 2

Lean can use instances, notation and attributes without naming them in the statement. These are rules for interpreting or finding mathematical objects. Atlas’s scan of names cannot follow all such uses; read the listed source as well.

instance (X : Complex) (i : ℕ) : Fintype (X.Face i) line 34
instance (X : Complex) (i : ℕ) : DecidableEq (X.Face i) line 35

import Mathlibnamespace OAI/-!# Bounded-degree coboundary expanders in every dimensionFinite connected pure complexes with uniformly bounded top-face incidence anduniform weighted coboundary expansion over F₂ in all degrees below the dimension.-/noncomputable sectionopen scoped BigOperatorsopen Filternamespace CoboundaryExpanders/-- A finite abstract simplicial complex with every labelled vertex present. -/structure Complex where  vertexCount : ℕ  faces : Finset (Finset (Fin vertexCount))  empty_mem : ∅ ∈ faces  downward : ∀ {σ τ}, σ ∈ faces → τ ⊆ σ → τ ∈ faces  vertex_mem : ∀ v, {v} ∈ facesnamespace Complex/-- Faces of dimension `i` have exactly `i + 1` vertices. -/def facesOf (X : Complex) (i : ℕ) : Finset (Finset (Fin X.vertexCount)) :=  X.faces.filter (fun σ => σ.card = i + 1)def Face (X : Complex) (i : ℕ) := {σ // σ ∈ X.facesOf i}instance (X : Complex) (i : ℕ) : Fintype (X.Face i) := Subtype.fintype _instance (X : Complex) (i : ℕ) : DecidableEq (X.Face i) := Classical.decEq _/-- Exact purity: top faces exist and every face lies in a top face. -/def Pure (X : Complex) (s : ℕ) : Prop :=  (X.facesOf s).Nonempty ∧  ∀ σ ∈ X.faces, ∃ τ ∈ X.facesOf s, σ ⊆ τ/-- Connectedness of the nonempty one-skeleton. -/def Connected (X : Complex) : Prop :=  Nonempty (Fin X.vertexCount) ∧  ∀ v w : Fin X.vertexCount,    Relation.ReflTransGen (fun a b => a ≠ b ∧ {a, b} ∈ X.faces) v wabbrev Cochain (X : Complex) (i : ℕ) := X.Face i → ZMod 2/-- Weights are normalized counts of incident top-dimensional faces. -/def weight (X : Complex) (s i : ℕ) (σ : X.Face i) : ℝ :=  (((X.facesOf s).filter (fun τ => σ.val ⊆ τ)).card : ℝ) /    ((Nat.choose (s + 1) (i + 1) : ℝ) * (X.facesOf s).card)/-- Weighted support norm. -/def norm (X : Complex) (s i : ℕ) (f : X.Cochain i) : ℝ :=  ∑ σ : X.Face i, if f σ ≠ 0 then X.weight s i σ else 0/-- Simplicial coboundary over F₂. -/def coboundary (X : Complex) (i : ℕ) (f : X.Cochain i) : X.Cochain (i + 1) :=  fun τ => ∑ σ : X.Face i, if σ.val ⊆ τ.val then f σ else 0/-- Actual coboundaries, with constant cochains in degree zero. -/def coboundaries (X : Complex) : (i : ℕ) → Set (X.Cochain i)  | 0 => {f | ∃ c : ZMod 2, f = fun _ => c}  | i + 1 => Set.range (X.coboundary i)/-- Distance in the weighted support norm; on a nonempty finite set it is a minimum. -/def distance (X : Complex) (s i : ℕ) (f : X.Cochain i)    (A : Set (X.Cochain i)) : ℝ :=  sInf ((fun a => X.norm s i (f - a)) '' A)/-- Number of top-dimensional faces incident to a vertex. -/def topDegree (X : Complex) (s : ℕ) (v : Fin X.vertexCount) : ℕ :=  ((X.facesOf s).filter (fun τ => v ∈ τ)).cardend Complex/-- Both constants are uniform over the sequence, all degrees below `d`, and all cochains. -/def MainStatement : Prop :=  ∀ d : ℕ, 3 ≤ d →    ∃ (D : ℕ) (ε : ℝ), 0 < ε ∧      ∃ X : ℕ → Complex,        (∀ m, (X m).Pure d ∧ (X m).Connected) ∧        Tendsto (fun m => ((X m).facesOf 0).card) atTop atTop ∧        (∀ m (v : Fin (X m).vertexCount), (X m).topDegree d v ≤ D) ∧        ∀ m i, i < d → ∀ f : (X m).Cochain i,          ε * (X m).distance d i f ((X m).coboundaries i) ≤            (X m).norm d (i + 1) ((X m).coboundary i f)theorem main : MainStatement := by  sorryend CoboundaryExpandersendend 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.

Bounded-degree coboundary expanders in every dimension

Theorem

\norm{\delta_i f}_{X_m,i+1} \ge \eps\,\dist_{X_m,i}(f,B^i(X_m)) \qquad(0\le i<d).

The constants are independent of m, i, and f.

Section 1, Introduction Read it in the paper TeX source, line 40

Show the TeX

For every integer $d\ge3$, there exist constants $D<\infty$, $\eps>0$,
and finite connected pure $d$-dimensional simplicial complexes $X_m$,
with $|X_m(0)|\longrightarrow\infty$, such that every vertex belongs to
at most $D$ faces of dimension $d$ and, for every
$f\in C^i(X_m;\F)$,
\begin{equation}
 \norm{\delta_i f}_{X_m,i+1}
 \ge \eps\,\dist_{X_m,i}(f,B^i(X_m))
 \qquad(0\le i<d).
\end{equation}
The constants are independent of $m$, $i$, and $f$.

Configuration and file facts

File: lean/ComparatorChallenges/CoboundaryExpanders.lean
OpenAI’s Lean catalogue: OpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 177 link it.
Solution module: OAI.Combinatorics.Coboundary.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: 15 local vocabulary declarations
Lines of Lean, without comments or blank lines: 46
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