OpenAI’s GPT-5.6 Sol Ultra reportedly solves a 50-year-old math problem in under an hour

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.

Machine persistence

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.

AI still doesn’t cite its sources

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, so that anyone reading only the paper might think the AI invented the underlying strategy itself.

“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.” The mathematician doubts the AI came up with the solution on its own, “given that its first problem-solving instinct is generally to search for all related papers on a problem and read them.”

This is a recurring debate around reasoning models. Do they “merely” find existing knowledge and recombine it? Or do they actually produce something new through creative work? For this proof, Bloom seems to lean toward the former.

AI shows what humans could have solved with more patience

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.

He expects AI systems to crack more conjectures like this, “those whose solutions require only existing, well-developed, theory, plus a lot of patience and belief.” But according to Bloom, “this is likely only a small proportion of open problems, and we don’t know in advance which they are.”

“But in this strange new world where big AI companies are spending a lot of time and money attacking many open problems at once (and only reporting the successes, of course), we will soon find out more of what was within our reach all along,” he writes.

How do you prompt a complex mathematical proof?

Part of the solution is the prompt written by humans. It essentially engineers exactly the kind of persistence Bloom describes as key to finding the proof. First, the prompt tells the model to assume a complete proof exists, cutting off its most likely honest answer right away: that the conjecture is open. Then it bans the model from searching the internet to check whether the conjecture has already been solved and from answering that the conjecture is unsolved. So the model has basically nowhere to go except solving the problem.

Verification is just as strict. Partial results, reductions to other unproven conjectures, summaries of the current state of research, explanations of why the problem is hard were all rejected as insufficient. The model can’t respond until a complete proof is ready and passes an adversarial test.

The rest of the prompt reads more like directives from a research lab than a typical AI prompt. Most of the 64 agents are deliberately kept in the dark about which approach currently looks most promising to encourage independent “thinking.” Adversarial agents then check each candidate proof against a detailed list of typical errors, looking for things like closed paths incorrectly identified as cycles or reductions that accidentally create new bridges in the graph.

The model was told to compute for at least eight hours before it could even consider giving up. It finished in one.