OpenAI's GPT-5.6 Sol Ultra produces a proof of the Cycle Double Cover Conjecture

OpenAI has published a short paper claiming a full proof of the Cycle Double Cover Conjecture, a graph-theory problem open since the 1970s, crediting the work "entirely" to its GPT-5.6 Sol Ultra model. The note, posted directly to OpenAI's own CDN on July 10, 2026, is unusually direct about how it was produced — a complete write-up of the proof, with Codex used for typesetting and prose.

What's new

• OpenAI's own paper states the result directly: "We prove the cycle double cover conjecture, posed by Tutte, Itai and Rodeh, Szekeres, and Seymour, which asserts that every bridgeless undirected graph has a collection of cycles which covers every edge exactly twice."
• The paper's "Statement of AI use" section is explicit about authorship: "The proof in this note is entirely due to GPT 5.6 Sol Ultra and the writeup with Codex (with GPT 5.6 Sol)."
• The proof reduces the general case to loopless cubic (3-regular) graphs, then uses the 8-flow theorem — a known result attributed to Jaeger and Tutte — to build an edge-labeling scheme, before closing the argument with what the paper describes as an elementary linear algebra step.
• The document is a short, self-contained note — a few pages of proof plus a 10-item bibliography — not a journal submission. It explicitly frames itself against prior partial results: the conjecture was already known to hold for planar graphs, for 3-edge-colourable cubic graphs, and for bridgeless graphs with no Petersen subdivision.
• The paper surfaced publicly via Hacker News on July 10 and drew over 120 points and two dozen comments within hours, alongside a companion GPT-5.6 system card and ARC-AGI results published in the same release cycle.

Context

The Cycle Double Cover Conjecture was posed independently by W.T. Tutte, Itai and Rodeh, George Szekeres, and Paul Seymour between the 1970s and early 1980s, and remains, per the paper's own citations, one of graph theory's best-known open problems despite decades of partial results and prior proof attempts that did not hold up. This isn't OpenAI's first public math showcase: the company ran a "First Proof" project earlier in 2026 in which its models attempted ten unpublished research-level problems submitted by mathematicians, with mixed but promising internal results. Posting a standalone, unreviewed proof note directly to its own CDN, rather than through a journal or arXiv listing, follows that same publish-first pattern.

Why it matters

A machine-produced solution to a five-decade-old open conjecture, if it holds up under expert scrutiny, would be a landmark result for AI-assisted mathematics — arguably more consequential than a benchmark score, because Cycle Double Cover has specifically resisted human mathematicians. That caveat matters: this is a preprint posted by the model's own creator, not a peer-reviewed publication, and graph theorists will need to check every step of the flow-theoretic argument before the result is broadly accepted. OpenAI's own citation list notes the conjecture has already drawn earlier "proof" attempts that did not stand up to scrutiny — exactly the terrain where independent verification is essential before this gets treated as a settled result.