GPT-5.6 Sol Ultra proves the 50-year-old cycle double cover conjecture
Result confirmed
Importance: historic (5 of 5)The takeaway
In mid-July 2026 OpenAI released a preprint crediting GPT-5.6 Sol Ultra, ‘in less than an hour’, with a proof of the cycle double cover conjecture (Szekeres 1973, Seymour 1979): every bridgeless graph has a collection of cycles covering each edge exactly twice.
Status
- Claim
Result confirmed
- Our reporting
- Medium confidence
- Verification
- Expert-checked
- Importance
- Historic (5 of 5)
- Last verified
- 19 October 2026
Your AI and this story
- GPT-6 Astra78 days after its cutoff
- Claude Opus 5.517 days after its cutoff
- Gemini 4 Hydrogen108 days after its cutoff
- Grok 4.747 days after its cutoff
None of these four assistants can know about it. The closest, Claude Opus 5.5, stops 17 days before it.
Key facts
- OpenAI released the proof as a PDF (cdn.openai.com); arXiv 2607.15399 is Jim Geelen’s 3-page notes ‘OpenAI’s proof of the Cycle Double Cover Theorem’ (‘my attempt to clarify, to myself, the OpenAI proof … This write-up offers no new insights’); Oum’s exposition is arXiv 2607.16356 (17 Jul 2026)
- Proof attributed entirely to GPT-5.6 Sol Ultra; the write-up was done with Codex
- Independent checks and expositions by Sang-il Oum and Jim Geelen; a public Lean formalisation is reported but not verified here
What happened
OpenAI published a proof of the cycle double cover conjecture that it attributed wholly to its top model. Leading graph theorists independently re-expounded and checked the argument within days.
Why it matters
Along with the Jacobian counterexample the same week, it marked the point where famous named conjectures, not just Erdős-list problems, began falling to AI.
Sources
4 sources from 3 sites. Numbers match the chips in the text.
4 sources: 3 primary, 1 press
Primary
- OpenAI: cycle double cover proof (PDF)cdn.openai.com, paper
- Jim Geelen: OpenAI’s proof of the Cycle Double Cover Theorem (arXiv 2607.15399)arxiv.org, paper
- Sang-il Oum: exposition of the proof (arXiv 2607.16356)arxiv.org, paper
Press
Changes
- Fixed a misattribution: arXiv 2607.15399 is Jim Geelen’s notes on OpenAI’s proof, not an OpenAI preprint (key fact and link title corrected from the arXiv metadata); verification names the independent checkers
- Filed