OpenAI groups related papers into a “family”. OpenAI’s catalogue describes this one:
Constructs a compact subset of ℝ9 that cannot be covered by ten sets of strictly smaller diameter, disproving Borsuk’s covering assertion already in dimension nine. The example consists of rank-one orthogonal projectors onto lines in ℝ4, with the Frobenius metric. (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.
Both independent assessments find that the Lean statements express or imply the whole result described above.Read the assessments
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.
The compact set of rank-one orthogonal projectors on ℝ4, with the Frobenius metric, cannot be covered by ten sets of strictly smaller diameter. It therefore gives a counterexample to Borsuk’s conjecture in dimension nine.
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.
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?
MathVet and cert-machine agree on how much of the advertised result the Lean statements express.
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:main_theorem: the set {uu^T : ‖u‖ = 1} ⊆ EuclideanSpace ℝ (Fin 4 × Fin 4) (Frobenius metric) is compact, lies in the trace-one symmetric matrices, has Metric.diam = √2, and admits no cover by ten sets of diameter < √2 (pieces taken inside the set, w.l.o.g.). The routine step not stated in Lean is that the trace-one symmetric 4×4 matrices form a 9-dimensional affine subspace (10 - 1), isometric to ℝ⁹.
“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.
Borsuk’s conjecture predicts that every bounded subset of $\mathbb R^d$ can be covered by $d+1$ sets of strictly smaller diameter. The formalized counterexample is the compact set of rank-one orthogonal projectors onto lines in $\mathbb R^4$, with the Frobenius metric. It lies in the nine-dimensional affine space of trace-one symmetric matrices, has diameter $\sqrt2$, and cannot be covered by ten arbitrary sets of smaller diameter.
AI-drafted comparisons, with no recorded human review
Nine-dimensional Borsuk counterexampleBorsukNine
Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem: the rank-one projectors onto lines in R⁴ form a set of diameter √2 in a nine-dimensional space that cannot be covered by ten sets of smaller diameter.
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.
Nine-dimensional Borsuk counterexampleBorsukNine
Two 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.
Other people’s accounts of running proof-verification software, with links to their reports.
Robert Joseph George, Do the Lean statements in openai/math match their papers?, 7 October 2026. Source
The review reports that Comparator accepted OpenAI’s solution for this challenge, run end to end on Lean 4.34.1 with the default kernel and only the three standard axioms.
Explanations by others
Explanations of this family written elsewhere, by the people named. Each is their own reading of the mathematics.
Gil Kalai, Updates: Sharing AI progress on mathematics (amazing!); and my lecture plans, 7 October 2026. Source
Notes the nine-dimensional counterexample to Borsuk’s conjecture in the release and its relevance to their Borsuk lecture at Jeff Fest.
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.
BorsukNine: Tracked source lean/OAI/Geometry/Borsuk/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
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
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 counterexample and rates its significance, if it holds up, as “Career-defining publication”. The page says that nothing on it is verified.
Moritz Firsching, anxkhn, Paul Lezeau, Formal Conjectures: mark problems solved in the OpenAI math release and link formal proofs, 11 October 2026. Source
A Formal Conjectures maintainer, after source inspection, links the nine-dimensional counterexample to Borsuk’s conjecture, recording a diameter convention difference and that the cover is intersected with the set.
acprk, acprk/openai-math-followups: follow-up results on three OpenAI families, 9 October 2026. Source
Numerical, not yet interval-certified: the projector set of CP^2 in its 8-dimensional space splits into 9 pieces of smaller diameter, so it is not an 8-dimensional Borsuk counterexample, and the RP^3 projector set behind the dimension-nine result has Borsuk number 11, leaving no slack for cutting it down to dimension 8.
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 156The 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 156Atlas 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 A nine-dimensional counterexample to Borsuk’s covering assertion, 23 September 2026
How it was found
Its words were compared with the words of OpenAI’s catalogue description.
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. Its protocol describes a planned error-rate study; this import does not establish that the study has been completed.
The challenge statements state the headline as OpenAI wrote it.
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.