ResearchPapers, evidence & method
OpenAI repository claims a Lean-formalized proof of Barnette's 1969 graph conjecture from an unreleased model
An OpenAI repository says an unreleased internal model proved Barnette's 1969 Hamiltonian-cycle conjecture, with a Lean formalization. The claim is one of 372 results released together. It stirred a researcher who spent about 24 years on the problem, and it still awaits independent review.
AT Pulse research deskOctober 7, 2026 · 4 min read · AI-assisted analysisOpenAI's public math repository lists problem 180 as a Lean formalization that it says proves Barnette's conjecture. The conjecture holds that every 3-connected cubic bipartite planar graph has a Hamiltonian cycle. Earlier work proved only special cases, and the reference literature still lists the conjecture as open. [2] [5] [9]
The result came out in a single release of 722 manuscripts grouped into 372 families. OpenAI reports posing about 4,000 problems to an unnamed internal model and spending about three hours of compute per result on average. The manuscripts and Lean code are public, but the model is not. The repository itself warns that some results that were not formalized could have problems. [3] [6]
Jake Boggan worked on the conjecture on and off for about 24 years. In a Hacker News comment that Simon Willison later quoted, he compared the news to learning that an ex had died. His reaction shows how a single release can suddenly end a long-running human research effort. [1] [4]
Mathematicians are disputing the release. MIT's Andrew Sutherland says such claims should count as unverified until others can reproduce them. Terence Tao criticized the pace of the releases, and Daniel Litt argued that labs should not keep answers to math questions secret. OpenAI has also had to walk back overstated math claims in the past. [6] [12] [11]
the formal statement is public and machine-checkable. Implication: if independent audits confirm it, Lean checking could let outsiders trust AI-generated proofs without trusting the vendor.
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Until then, whether the formal statement matches the conjecture and which axioms it allows remain unchecked.
Executive brief
Barnette's conjecture is a graph-theory problem posed in 1969. A Lean formalization in OpenAI's public openai/math repository now states that it proves the conjecture. The result is listed as "problem 180", one of hundreds of manuscripts produced by an unreleased internal OpenAI model. The story became personal through Jake Boggan, who spent about 24 years on and off working on the problem. In a Hacker News comment that Simon Willison quoted, he compared the news to hearing that an ex had died. No independent mathematical review of the Barnette proof has been published so far.
What changed and event timeline
- Context
1969: Barnette poses the conjecture
David Barnette conjectured that every 3-connected cubic bipartite planar graph has a Hamiltonian cycle. Computer searches have found no counterexample with fewer than 86 vertices ().
OpenAI forms a math advisory group
Scientific American reports that OpenAI set up an independent advisory group after its earlier Navier–Stokes claim drew disputes over transparency and release practices ().
Barnette paper dated
The companion manuscript, "Paired states and Hamiltonian cycles in cubic bipartite planar graphs," carries this date ().
Mass release on GitHub
At 6 p.m. EDT, OpenAI published 372 results from an internal model, including a claimed solution of the four-dimensional Kakeya conjecture (;).
A researcher's reaction spreads
Boggan's comment appeared in a Hacker News thread with more than 1,000 points, and Willison quoted it (;).
Capabilities and access
- Model: an "unreleased internal OpenAI model". OpenAI has not named it (openai/math README).
- Scale: about 4,000 problems were posed to the model. The repository holds 722 manuscripts grouped into 372 families (README).
- Compute: OpenAI reports an average of about three hours of "ChatGPT Pro thinking compute" per result (README).
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- Model: an "unreleased internal OpenAI model". OpenAI has not named it (openai/math README).
- Scale: about 4,000 problems were posed to the model. The repository holds 722 manuscripts grouped into 372 families (README).
- Compute: OpenAI reports an average of about three hours of "ChatGPT Pro thinking compute" per result (README).
- Access: the manuscripts and Lean code are public under Apache-2.0. The model itself is not available to the public (Scientific American).
Technical analysis for researchers and developers
- What the formal statement says: the file
BarnetteHamiltonian.leanstates that every finite simple cubic bipartite planar 3-vertex-connected graph has a cycle that visits every vertex exactly once (180.md). - Checking pipeline: results can be checked with the
comparator,landrunandlean4exporttools (Comparator README). - Coverage: many manuscripts have been formalized, but not all. Reasoning summaries exist for only 10 selected results (README).
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- What the formal statement says: the file
BarnetteHamiltonian.leanstates that every finite simple cubic bipartite planar 3-vertex-connected graph has a cycle that visits every vertex exactly once (180.md). - Checking pipeline: results can be checked with the
comparator,landrunandlean4exporttools (Comparator README). The project page for 180 does not say which axioms the proof allows or whether a human reviewed the formal statement. - Coverage: many manuscripts have been formalized, but not all. Reasoning summaries exist for only 10 selected results (README).
- Method: one HN commenter described the proof as using complex-valued exponential sums (HN). This is unreviewed commentary.
Claims and evidence
- Barnette's conjecture is proved in Lean ()
- Nearly all results came from a single prompt to a single agent ()
- The matrix multiplication exponent was cut to 2.25 ()
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| Claim | Status |
| Barnette's conjecture is proved in Lean (180.md) | Vendor-reported. Machine-checkable, but no independent audit has been published. |
| Nearly all results came from a single prompt to a single agent (SciAm) | Vendor spokesperson. MIT's Andrew Sutherland says such claims should be treated as unverified until others can reproduce them. |
| The matrix multiplication exponent was cut to 2.25 (OfficeChai) | Reported through Steven Strogatz's reaction. No independent verification yet. |
| Barnette's conjecture is still open (Wikipedia) | The reference page has not been updated. This reflects lag, not a refutation. |
Context and prior work
- Earlier partial results: in 1975 Goodey proved the conjecture for graphs whose faces all have four or six edges. Research on approximating Barnette's conjecture was still active in 2025.
- OpenAI's recent math record: in May 2026 OpenAI said a model disproved a discrete-geometry conjecture. In September it made the Navier–Stokes claim (NPR). In October 2025 it had to walk back overstated math claims.
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- Earlier partial results: in 1975 Goodey proved the conjecture for graphs whose faces all have four or six edges. Kardoš (2020) proved a different, related conjecture, which some summaries mix up with this one (Wikipedia). Research on approximating Barnette's conjecture was still active in 2025.
- OpenAI's recent math record: in May 2026 OpenAI said a model disproved a discrete-geometry conjecture. In September it made the Navier–Stokes claim (NPR). In October 2025 it had to walk back overstated math claims.
Limitations, safety and contested findings
- The README warns that "some of the unformalized results could have issues" (openai/math).
- Terence Tao criticized the "insane" pace of results. Daniel Litt argued that labs should not keep answers to math questions secret (SciAm).
- HN commenters described the work as "strip mining" open problems with proprietary models that academics cannot use (HN).
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- The README warns that "some of the unformalized results could have issues" (openai/math).
- Terence Tao criticized the "insane" pace of results. Daniel Litt argued that labs should not keep answers to math questions secret (SciAm).
- HN commenters described the work as "strip mining" open problems with proprietary models that academics cannot use (HN).
- Alvaro Lozano-Robledo called the release a "takeover of mathematics" (OfficeChai).
Business and practitioner implications
- Formal verification is the trust layer. Lean-checked statements let outsiders check a result's logic without trusting the vendor.
- The cost per result is modest at about three hours of compute.
- Reproducibility is a differentiator. Independent researchers are calling for public access to the model before they accept the claims.
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- Formal verification is the trust layer. Lean-checked statements let outsiders check a result's logic without trusting the vendor. The weak points are reviewing the formal statement and auditing which axioms it relies on.
- The cost per result is modest at about three hours of compute. Firms working on research-grade problems in optimization, cryptography or verification should expect fast-moving competition from frontier labs.
- Reproducibility is a differentiator. Independent researchers are calling for public access to the model before they accept the claims.
- The human cost is real. Boggan's comment shows that long-running research programs can be ended overnight. Research organizations need to plan for that.
Sources
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- Quoting Jake Boggan – Simon Willison
- Hacker News: Sharing AI progress in mathematics
- openai/math repository
- openai/math lean/docs/180.md
- Comparator README
- Scientific American: OpenAI unleashes hundreds more math results
- OfficeChai: Mathematicians react
- NPR: Mathematicians learn little from AI completing unsolved problem
- Wikipedia: Barnette's conjecture
- Dagstuhl: Approximating Barnette's Conjecture
- OpenAI: Model disproves discrete geometry conjecture
- TechCrunch: OpenAI's embarrassing math