OpenAI claims the cycle double cover conjecture
In July 2026 OpenAI put a short PDF on its server with a proof of the cycle double cover conjecture: every bridgeless graph has a list of cycles in which every edge lies in exactly two. The file says the proof is "entirely due to" GPT 5.6 Sol Ultra. It was on the server by 10 July; within a week two graph theorists wrote expositions of it. No journal refereeing or formal verification is reported.
Why it matters
The proof is a few pages long: it takes the known fact that every bridgeless graph has a nowhere-zero flow with values in a group of eight elements and turns it into a cover by an elementary linear algebra step. The conjecture was posed by Tutte, Itai and Rodeh, Szekeres and Seymour. The record states the claim and that two specialists read it; it does not judge its correctness.
State of the claim. Claimed: the PDF "A proof of the cycle double cover conjecture" on cdn.openai.com, author "OpenAI"; read in full. It reduces to cubic graphs, uses a nowhere-zero flow with values in F_2^3 (Kilpatrick, Jaeger), then linear algebra. The statement of AI use in the file: "The proof in this note is entirely due to GPT 5.6 Sol Ultra and the writeup with Codex (with GPT 5.6 Sol)". Checked by people: Jim Geelen (arXiv 2607.15399, submitted 16 July) calls his note "my attempt to clarify, to myself, the OpenAI proof" and says it offers no new insights; Sang-il Oum (arXiv 2607.16356, submitted 17 July, revised 23 July and 1 August) presents the proof with slight modifications intended to make it more accessible. Neither exposition reports an error and neither is a referee’s report. No document read reports a formalisation in Lean. No dispute was found in what was read. Date. The file is undated. The Last-Modified header on OpenAI’s server, read on 10 October, says 10 July 2026 17:32 UTC, and the earliest Internet Archive capture is 10 July 18:51 UTC. So 10 July is the earliest evidence, not necessarily the first day the file was available. The 17 July of the lead list is the arXiv submission date of Oum’s note. What the record does not claim: that the proof is accepted or refereed, or that the pass checked its correctness.