GPT-5.6 Sol Ultra proves the 50-year-old cycle double cover conjecture
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. Independent expositions by graph theorists Sang-il Oum and Jim Geelen followed.
Key facts
- OpenAI preprint arXiv 2607.15399; Oum's exposition 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
Science result
- Field
- mathematics / graph theory
- Problem
- Cycle double cover conjecture (open since 1973)
- Result
- Proof that every bridgeless graph admits a cycle double cover.
- AI system
- GPT-5.6 Sol Ultra
- Human role
- Autonomous proof per OpenAI; human experts wrote independent expositions
- Verification
- Expert-checked (independent expositions); preprint; Lean formalisation reported
- Status
- confirmed
- Why surprising
- One of graph theory's best-known conjectures, open for about 50 years, was reportedly proved by a model in under an hour.
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.
Changelog
- 2026-09-29: created
Related events
- OpenAI broadly releases GPT-5.6 (Sol, Terra, Luna) after government-gated preview ★★★★
- OpenAI's unreleased 'Astra' model claims ten advances in maths and theoretical CS, with Lean proofs ★★★★★
Sources (4)
- paperOpenAI: cycle double cover proof (PDF)
- paperOpenAI preprint (arXiv 2607.15399)
- paperSang-il Oum: exposition of the proof (arXiv 2607.16356)
- pressAI Weekly: OpenAI attributes cycle double cover proof to GPT-5.6 Sol Ultra
id: 2026-07-17-cycle-double-cover-conjecture-proved · updated 2026-09-29 · open in the interactive timeline