Bounded-degree coboundary expanders

OpenAI groups related papers into a “family”. OpenAI’s catalogue describes this one:

Constructs arbitrarily large finite d-dimensional simplicial complexes, for every d ≥ 3, with uniformly bounded vertex degrees and uniform 𝔽₂ coboundary expansion in every degree below d. Together with the known graph and two-dimensional cases, this establishes the existence of such expanders in every positive dimension. (OpenAI’s Lean note)

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The two independent assessments disagree on how much of the result described above is expressed by the Lean statements. Read the assessments

The paper states

Theorem 1.1

In Bounded-degree coboundary expanders in every dimension, 24 September 2026

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.

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.

Reported proof verification: One Lean statement in the release; three independent groups report that Comparator accepted runs for this file at the version shown here.
Papers and explanations: One current paper, readable here as a page beside its PDF.

Papers

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Bounded-degree coboundary expanders in every dimension

24 September 2026, current edition.

Read it here PDF

For every integer d ≥ 3, we construct arbitrarily large finite d-dimensional simplicial complexes with uniformly bounded vertex degrees and uniform 𝔽₂ coboundary expansion in every degree below d. Together with the known graph and two-dimensional cases, this establishes the existence of bounded-degree 𝔽₂ coboundary expanders in every positive dimension.

One Lean statement in this family

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`CoboundaryExpanders`

The paper states

Theorem 1.1

In Bounded-degree coboundary expanders in every dimension, 24 September 2026

\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.

The words they share: vertex, finite, simplicial, dimensional, constants, pure, norm, integer, faces, dimension, connected, complexes, degree, coboundary.

Possible Lean counterpart

Selected from shared wording. No review of whether the statements mean the same thing is recorded.

Bounded-degree coboundary expanders in dimensions at least three

CoboundaryExpanders.lean, the release lists it as a supporting statement.

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

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.

three independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet; Neal McBurnett, openai-math-lean-tooling. Who ran the proof-verification program

cert-machine records it as “CERTIFIED”.
MathVet records it as “comparator-sandboxed-pass”.
Neal McBurnett, openai-math-lean-tooling records it as “accepted”.

Read the complete Lean statement

Independent comparisons and proof-verification reports

Does the Lean express the advertised result?

MathVet and cert-machine disagree on how much of the advertised result the Lean statements express. MathVet finds that the Lean statements express or imply the whole advertised result (their term: “full”); cert-machine finds that the Lean statements express a narrower statement than the advertised result (their term: “NARROWER — DECLARED”).

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: 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.

“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.

Read the original assessment Read MathVet’s method

cert-machine finds that the Lean statements express a narrower statement than the advertised result (their term: “NARROWER — DECLARED”).

The gap they name: the ‘every positive dimension’ conclusion (d = 1, 2 cases, which the headline attributes to prior work)

“NARROWER — DECLARED”, in their words: The challenge statements state less than the headline, and OpenAI’s Lean note says so.

Their method: Every Comparator challenge re-run on Linux with nanoda switched on, decided by a preregistered rule.

Read the original assessment Read cert-machine’s method

OpenAI’s own note on what its Lean 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.

AI-drafted comparisons, with no recorded human review

Bounded-degree coboundary expanders in dimensions at least three CoboundaryExpanders
Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem: bounded-degree F₂ coboundary expanders in every dimension d ≥ 3. The graph and two-dimensional cases behind the all-dimensions statement are separate.

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.

Bounded-degree coboundary expanders in dimensions at least three `CoboundaryExpanders`

Three 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.	10 October 2026	Original report
MathVet	“comparator-sandboxed-pass”	Re-run through Comparator in a sandboxed virtual machine: accepted.	8 October 2026	Original report
Neal McBurnett, openai-math-lean-tooling	“accepted”	Re-run through Comparator with Lean’s kernel: accepted (the check took 183 s; peak memory 6.5 GiB).	11 October 2026	Original report

What these papers cite in the release, and what cites them

These links come from the papers’ bibliographies. A citation records that one paper cites another; how each cited result is used has not been examined here.

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.

References outside the release

Each paper’s outside references are listed with it on its reading page, where each citation is placed. Every outside work cited in the release: References outside the release

25 references in Bounded-degree coboundary expanders in every dimension

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What people have said

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 construction and rates its significance, if it holds up, as “Career-defining publication”. The page says that nothing on it is verified.
Author not stated. Source

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 177 The 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

Compare the proposed paper and Lean matches

Family 177 Atlas records no review of whether each linked Lean statement and paper theorem mean the same thing. Open the form