OpenAI groups related papers into a “family”. OpenAI’s catalogue describes this one:
Proves the Grothendieck homotopy hypothesis for ∞-groupoids associated with every Grothendieck coherator in the Ara–Henry convention: these algebraic objects recover the homotopy theory of spaces. (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.
The independent assessments find that the Lean statements express only part of the result described above, or a narrower statement.Read the assessments
Possible match to OpenAI’s description, selected from shared wording. No review confirming the selection is recorded.
Fix any Grothendieck coherator \mathcal C. Let X be a cellular \mathcal C-infinity-groupoid, let n\ge0, and let a:D_n\to X be any cell. In the pushout square
where s_n is the source inclusion, the map i:X\to X^+ is a weak equivalence.
The release does not say which paper theorem its catalogue describes. The statement above shares the most wording with the description; the atlas has not reviewed that choice. Identifying it is listed under Questions awaiting review.
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 prove the Grothendieck homotopy hypothesis for every Grothendieck coherator in the Ara–Henry convention: its weak globular infinity-groupoids recover the homotopy theory of spaces. We also resolve Henry’s pushout conjecture, showing that elementary expansions preserve components and all homotopy groups of cellular infinity-groupoids.
One Lean statement in this family
Lean is a language for mathematical statements and proofs that a computer can verify. The Lean statement below is shown beside its possible paper counterpart, with the reason they were linked.
For a configured theorem, the Comparator program tests whether OpenAI’s separate proof proves the exact Lean statement using only the allowed axioms. Some statement files instead configure definitions for comparison by name and type. Acceptance does not establish that a formal statement means the same thing as the paper’s theorem.
Fix any Grothendieck coherator \mathcal C. Let X be a cellular \mathcal C-infinity-groupoid, let n\ge0, and let a:D_n\to X be any cell. In the pushout square
GrothendieckElementaryExpansion.lean, the release lists it as a main result.
theoremelementary_expansion (C : GlobularTheory.{v}) (hC : C.IsCoherator) (X : C.Model) (hX : X.IsCellular) (n : ℕ) (a : GlobularTheory.Model.disk C n ⟶ X) (Y : C.Model) (i : X ⟶ Y) (b : GlobularTheory.Model.disk C (n + 1) ⟶ Y) (square : IsPushout a (GlobularTheory.Model.J C n) i b) : GlobularTheory.Model.WeakEquivalence hC i := 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?
Neither group finds that the Lean statements express or imply the whole advertised result. They describe what the statements express in different terms.
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 a supporting result (their term: “supporting-only”).
Their note:Only elementary_expansion: for a coherator C (Ara-Henry convention, encoded by hand), a cellular model X and a pushout of J n : disk n → disk (n+1) along any disk n → X, the inclusion X → Y is a WeakEquivalence (bijection on π₀ and on every π_n at every basepoint). The homotopy hypothesis itself (comparison of these ∞-groupoid models with spaces) is not stated; docs: 'the later semi-model structure and full comparison with the homotopy theory of spaces are not included'.
Definitions the group asks readers to compare with the paper:GlobularTheory, CellularPresentation, IsCoherator C := Nonempty C.CellularPresentation ∧ ∀ p : AdmissiblePair, ∃ z, Fills (Morphism.id C) p z and Model are hand-built encodings of Grothendieck coherators and their models; noncomputable def unit C hC n := Classical.choose (hC.2 (C.unitPair n)): the degenerate-cell operation used to define baseCell, Loops, HomotopyGroup and hence WeakEquivalence is chosen by Classical.choose (any choice is valid, but the notion is not canonical)
“supporting-only”, in their words: The challenge statements state a lemma or auxiliary result, not the headline.
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.
cert-machine finds that the Lean statements express a narrower statement than the advertised result (their term: “NARROWER — DECLARED”).
The gap they name:The Grothendieck homotopy hypothesis (the comparison of the homotopy theory of coherator ∞-groupoids with spaces) is not stated; only the elementary-expansion theorem (Henry’s pushout conjecture) is.
“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.
The Grothendieck homotopy hypothesis asks whether algebraic $\infty$-groupoids recover the homotopy theory of spaces. The formalized result is the elementary-expansion theorem used in this approach: for every Grothendieck coherator in the Ara–Henry convention and every boundary-cellular model, attaching an $(n+1)$-disk along its source $n$-face induces a weak equivalence.
The later semi-model structure and full comparison with the homotopy theory of spaces are not included.
AI-drafted comparisons, with no recorded human review
Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem, the elementary-expansion weak equivalence for every Grothendieck coherator. The semi-model structure and the homotopy-hypothesis comparison with spaces, corollaries in Section 5, are not included.
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.
Reports of errors and corrections from outside readers, as they state them. The atlas has not judged them.
Lean community, OpenAI releases solutions to hundreds of problems (117 messages, 6-10 Oct 2026), 6 October 2026. Source
Riehl first judged the homotopy-hypothesis Lean to be a statement without a proof; Chris Henson showed the proof is in ElementaryExpansion.lean and Riehl added a correction.
Explanations by others
Explanations of this family written elsewhere, by the people named. Each is their own reading of the mathematics.
Emily Riehl, Lean coverage by subject and a note on the globular expansion statement, 8 October 2026. Source
A subject-level audit of claimed formalisations found topology lowest, and a first reading of this family’s Comparator file saw only a sorry; a Lean maintainer replied that it is only the statement file and the proof sits in the solution module.
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.
GrothendieckElementaryExpansion: Tracked source lean/OAI/CategoryTheory/Globular/Main.lean unavailable: bytes were not captured at fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb
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
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
Explain a result, compare statements or propose a correction
Atlas has no contributed explanation for this family yet. You can offer one with a GitHub account. You receive named credit and disclose any AI assistance.
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 proof and rates its significance, if it holds up, as “Career-defining publication”. The page says that nothing on it is verified.
David Michael Roberts, Comments on the Grothendieck homotopy hypothesis manuscript, 7 October 2026. Source
A category theorist says the result closes a problem but regards the version proved, coherators after Henry, Ara and Maltsiniotis, as a historical curiosity rather than something people use.
John Baez, Mathstodon: paper 312 proves a version of the homotopy hypothesis, 7 October 2026. Source
The originator of the name says this family proves one version of the homotopy hypothesis, that the hypothesis has versions ranging from easy to hard depending on how its terms are made precise, and that it does not affect their own work.
Questions awaiting review
Each item identifies a statement, definition or report that needs further review. Its link opens a GitHub form with the source version already recorded.
Read the Lean statement against the paper
An AI agent drafted an assessment of which part of the paper this Lean statement expresses. No named human review is recorded.Open the form
Identify the theorem the catalogue describes
Family 312No single paper statement clearly matches OpenAI’s description by wording, and no review identifying the intended theorem is recorded.Open the form
Address a question raised in an independent assessment
Family 312Atlas records no review of whether each linked Lean statement and paper theorem mean the same thing.Open the form
Why this passage is shown
Theorem 1.1
In The Grothendieck homotopy hypothesis via elementary expansions
How it was found
Its words were compared with the words of OpenAI’s catalogue description.
Words they share
coherator, grothendieck
Review
Atlas records no review confirming that this is the theorem OpenAI’s description refers to.
The record’s reason
No unique candidate with at least three shared catalogue terms.
Independent assessment
MathVet
Their term
“supporting-only”
What their term means
The challenge statements state a lemma or auxiliary result, not the headline.
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 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; 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.