OpenAI says GPT-5.6 produced a proof of the Cycle Double Cover Conjecture

OpenAI has posted a three-page note claiming a complete proof of the Cycle Double Cover Conjecture, one of graph theory’s long-standing open problems, first posed in the 1970s by Tutte, Szekeres, Seymour, and Itai & Rodeh. The conjecture asserts that every bridgeless graph admits a multiset of cycles covering each edge exactly twice. The paper’s explicit ‘statement of AI use’ credits the mathematics entirely to the model GPT-5.6 Sol Ultra, with the writeup assembled via Codex — making the authorship, as much as the result, the actual story.

Methodologically, the argument reduces the general case to cubic graphs, then leans on the classical 8-flow theorem (Kilpatrick–Jaeger) to obtain a nowhere-zero flow valued in the group of order eight. It converts that flow into an assignment of two-element edge labels satisfying a local balance condition — effectively a relaxed variant of a proper 3-edge-coloring — and closes the gap with an elementary linear-algebra duality argument. The proof is strikingly short and self-contained for a problem that has resisted resolution for roughly fifty years.

That brevity is also the reason for caution. The note is self-published on OpenAI’s CDN, dated the same day, and has not been peer-reviewed or scrutinized by the combinatorics community. The conjecture’s known difficulty is concentrated on snarks — non-3-edge-colorable cubic graphs such as the Petersen graph — a fact the paper itself acknowledges. Any short proof that appears to sidestep that hardness warrants independent verification of the central reduction before the problem can be considered settled. If it holds up, it would be a notable milestone for AI-generated mathematics; until then it is a claimed proof, not an established one.