OpenAI’s new AI model, GPT-5.6 Sol Ultra, has produced a proof of the Cycle Double Cover Conjecture using 64 subagents working in parallel. Mathematician Thomas Bloom praises the proof but criticizes the lack of citations.
OpenAI has announced that GPT-5.6 Sol Ultra has generated a complete proof of the so-called Cycle Double Cover Conjecture. The conjecture had remained unproven for about 50 years. The AI model took just under an hour to complete the task, utilizing 64 subagents working in parallel.
Put simply, the conjecture addresses a fundamental question in graph theory: Would it be possible to find a set of cycles in any network of vertices and edges that traverses each individual edge exactly twice? The problem was formulated independently by several mathematicians in the 1970s. Since then, there have been many partial solutions for special cases, but no generally accepted proof.
According to OpenAI, the proof comes entirely from GPT-5.6 Sol Ultra. The paper was written by GPT-5.6 Sol. Mathematician Thomas Bloom of the University of Manchester calls it “a very nice proof,” noting that the solution is “short, elementary, and could have been discovered in the 1980s.” It doesn’t need any new mathematical theories, but it cleverly combines known tools.
So why didn’t humans find it? Bloom suspects the key step involved a small, counterintuitive twist in the reasoning. A human mathematician would likely have tried the obvious approach, seen it fail, and moved on. AI doesn’t get discouraged; it just keeps trying small variations until one clicks.
“One can imagine trying the natural labelling first, checking the linear algebra, and when that failed shrugging and thinking ‘oh well, I was expecting to fail, guess it can’t be done this easily’ - while the AI does not get discouraged and keeps trying small variations,” writes Bloom.
Bloom’s initial assessment is the most detailed public evaluation so far; a full mathematical verification by the scientific community is still pending.
Bloom says the core mathematical ideas behind the proof trace back at least to a 1983 paper by Bermond, Jackson, and Jaeger. He criticizes that OpenAI’s paper doesn’t mention this prior work at all. “I assume that these previous works were a big influence on the OpenAI proof, and it is a shame that it does not mention them at all,” writes Bloom. “This is a frequent issue with AI-generated proofs and papers: they use ideas and proof strategies taken from the literature without proper citation.”
Bloom compares the result to the unit distance conjecture, which OpenAI also recently solved. Both were major open problems “that turned out to be much easier than expected - no big new theories were required, and one can imagine many alternate histories when these proofs were found decades ago,” he writes.
Part of the solution is the prompt written by humans. It tells the model to assume a complete proof exists, bans internet searches to check whether the conjecture has already been solved, and rejects partial results. Verification is strict: adversarial agents check each candidate proof against typical errors. The model was told to compute for at least eight hours before it could even consider giving up. It finished in one.
OpenAI researcher Ethan Knight announced the result on X on July 10, 2026, one day after GPT-5.6 Sol Ultra became generally available. OpenAI published both the three-page proof PDF and the prompt used to produce it on its CDN. The proof has not undergone formal peer review.