What was already known
The cycle double cover conjecture had been stated independently by Szekeres, Seymour, Itai–Rodeh and Tutte. Prior known cases included planar graphs, 3-edge-colourable cubic graphs, and bridgeless graphs without a Petersen subdivision. The proof uses the known Jaeger–Kilpatrick nowhere-zero F₂³-flow theorem, then supplies a new linear-algebra conversion from such a flow to a double cover (OpenAI manuscript, Introduction and §2).
AI involvement
OpenAI names itself as author and explicitly states that the proof is entirely due to GPT 5.6 Sol Ultra; Codex assisted with the writeup. Human mathematicians Jim Geelen and Sang-il Oum later wrote separate explanations, not the original proof.
The published prompt supplied the precise conjecture and search protocol but no proof argument; OpenAI says the proof is entirely due to GPT 5.6 Sol Ultra. Codex helped write it up.
- Supporting source (cdn.openai.com) — Full task prompt, pp. 1–2
Importance
The claimed proof resolves a longstanding general graph-theory conjecture about cycle covers of bridgeless graphs.
Limitations
The published Lean endpoint covers finite loopless bridgeless multigraphs. Geelen and Oum wrote explanatory versions of the argument; their modifications and exposition are their own work. The Lean receipt is source-reported and was not rebuilt here.
Technical verification
OpenAI reports a complete Lean build and kernel check of the unconditional finite loopless bridgeless multigraph theorem, with no project-specific axioms. This assessment did not rerun it.
Formal checking
OpenAI’s verification record reports a complete Lean 4 build, no sorry/admit or project axioms, and a matching unconditional endpoint for finite loopless bridgeless multigraphs.
- Checker
- OpenAI-reported Lean 4 kernel
- Scope
- Every finite loopless bridgeless undirected multigraph has a multiset of cycles covering each edge exactly twice.
- Basis
- Reported by the linked checker
Checking a formal statement does not establish that it matches the original problem.
Human understanding
Jim Geelen reconstructs the flow-to-bicolouring linear-algebra mechanism in a complete explanatory proof. Sang-il Oum presents a modified self-contained exposition; their accounts credit OpenAI for the original argument.
Correction and publication history
Primary proof, attributed model work, matching Lean endpoint, and later human explanations reviewed for non-notifying local inclusion.
Read this published revision