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In just one hour, OpenAI GPT-5.6 Sol Ultra proved a 50-year-old mathematical conjecture
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On July 10, OpenAI announced that its latest model, GPT-5.6 Sol Ultra, successfully generated a complete proof of the Cycle Double Cover Conjecture in graph theory in less than an hour. This conjecture was independently proposed by mathematician George Szekeres in 1973 and Paul Seymour in 1979, and has remained unsolved for over 50 years, recognized as one of the most important open problems in graph theory. OpenAI has released the proof and prompts as a PDF, claiming that the entire mathematical proof was completed independently by AI.
The core question of the Cycle Double Cover Conjecture is: For any bridgeless graph, does there exist a set of cycles such that every edge appears in exactly two cycles? This long-standing conjecture is a central challenge in graph theory. OpenAI researcher Ethan Knight announced the achievement on X, stating that the model used 64 sub-agents working in parallel, dynamically managing different research lines, and set up specialized "adversarial agents" to find flaws and edge cases. The system originally reserved 8 hours of computation time but ultimately took only about 1 hour.
According to the prompts released by OpenAI, GPT-5.6 Sol Ultra was instructed to prohibit online searches, reject incomplete proofs, and must pass adversarial verification. The proof steps included: reducing the original conjecture to the cubic graph problem; using the 8-flow theorem; constructing edge labels through linear algebra over GF(3) (the finite field of three elements), ultimately proving that each edge belongs to exactly two cycles. Thomas Bloom, a mathematician at the University of Manchester, was the first to publicly evaluate the proof, calling it "very beautiful," concise and elementary, suggesting that if humans had thought of it earlier, it could have been completed in the 1980s.
Bloom noted that the greatest advantage of AI is not proposing novel mathematical ideas, but possessing far greater patience than humans, able to continuously try various subtle variations. However, he also pointed out a significant issue: the entire proof does not cite any existing literature, for example, the classic 1983 paper by Bermond, Jackson, and Jaeger is not mentioned. Bloom believes this is a common flaw in current AI-generated mathematical papers.
Currently, the proof has not undergone formal peer review. The mathematical community remains cautious, as the Cycle Double Cover Conjecture has a history of multiple "proofs" that were later refuted, with some papers even withdrawn from arXiv. Additionally, the proof was not verified using formal tools like Lean, and experts note that the current formalized mathematics library for graph theory does not support such complex theorems, making it difficult to confirm correctness via automated tools in the short term.
Estimates suggest that the computational resources consumed in this inference cost between $275 and $485 (based on OpenAI's official pricing), or up to $13,000 on the Cerebras platform. If the proof is eventually validated by the mathematical community, it would mark the first time a large language model independently solved a major problem listed on Wikipedia's "Unsolved Problems in Mathematics" page. Previous breakthroughs in mathematics by AI, such as DeepMind's work on cap sets and knot theory, were human-AI collaborations rather than independent proofs.
Thomas Bloom believes that this achievement sparks new discussions about the nature of mathematical discovery. Since the proof mostly relies on classic tools developed decades ago, the advantage of AI lies more in computational patience and persistent trial. Currently, graph theory experts anticipate a rigorous review of the proof over the coming days to weeks; only if all derivation steps pass verification will this achievement gain formal recognition from the mathematical community.
