What was already known
Barnette’s conjecture had remained open since 1969. Earlier proofs covered restricted face sizes, including the quadrilateral–hexagonal case and the faces-at-most-eight case.
AI involvement
OpenAI credits its internal frontier model with producing the mathematics release and lists OpenAI as author of this manuscript. The source does not establish a more precise account of human input for this proof.
OpenAI’s release states that its internal frontier model produced the results, and OpenAI is credited as author of this manuscript. The individual human contribution is not detailed enough to establish autonomy.
- Supporting source (openai.com) — Release provenance
Importance
Resolves Barnette’s longstanding Hamiltonian-cycle conjecture for the full stated graph class.
Limitations
The paper and Lean development are OpenAI-authored reports. The scope note says the Lean theorem matches the full graph statement, while the repository records its formalization review as unchecked. No named human mathematical explanation or independent proof review is documented here.
Technical verification
OpenAI’s pinned scope note says the Lean development proves the full selected Barnette statement, and the named main theorem and Comparator configuration are in the repository. This is source-reported machine checking; the repository review status is unchecked and no independent run is recorded here.
Formal checking
The repository scope note says its Lean development proves the full selected theorem; the matching main declaration is in the linked source.
- Checker
- OpenAI-reported Lean 4 / Comparator
- Scope
- Every finite simple cubic bipartite planar 3-vertex-connected graph has a Hamiltonian cycle.
- Basis
- Reported by the linked checker
Checking a formal statement does not establish that it matches the original problem.
Human understanding
Few is the default when evidence for a higher level is insufficient. The paper explains its proof, but OpenAI is the named author and no named person is documented as reconstructing the central paired-state argument. Few is the evidence default, not a report that people cannot understand it.
Correction and publication history
Primary manuscript, prior-work and pinned formalization review of one OpenAI-authored contribution; historical non-notifying inclusion.
Read this published revision