What happened. GPT-5.6 Sol Ultra generated a proof of the Cycle Double Cover Conjecture, a problem open in graph theory since 1970, in under an hour. The model worked with 64 sub-agents in parallel. Thomas Bloom, a mathematician at the University of Manchester, verified the result and calls it “a very beautiful proof: short, elementary, and one that could have been discovered in the 1980s.” The conjecture asked whether it’s possible to find a set of cycles in any network of vertices and edges that traverses every edge exactly twice.

Why it matters to you. As we reported on July 10, GPT-5.6 Sol is OpenAI’s model for deep reasoning. This result shows where that capability reaches on a verifiable task: not synthetic benchmarks, but a mathematical proof the community can check line by line. Bloom explains that the key step required a small counterintuitive twist in reasoning. A human mathematician would try the obvious approach, see it fail, and conclude “it can’t be done so easily.” The AI doesn’t get discouraged: it keeps trying small variations until one works. It’s an example of where mechanical persistence beats intuition when the solution space is well-defined enough.

The limitation. Bloom harshly criticizes the absence of citations in OpenAI’s paper: the key mathematical ideas underlying the proof go back at least to a 1983 paper by Bermond, Jackson and Jaeger, but the paper doesn’t mention it at all. “This is a recurring problem with AI-generated proofs: they use ideas and proof strategies from the literature without citing them.” Bloom doubts the AI invented the solution on its own, “given that its first instinct is to search for and read all relevant papers.” The technical performance is real, but intellectual honesty in attribution is another matter.

In detail

The context: what came before

The Cycle Double Cover Conjecture was formulated independently by several mathematicians in the 1970s. Since then, many partial solutions for special cases have been found, but no accepted general proof. Bloom compares this result to that on the unit distance conjecture, also recently solved by OpenAI: both are “important open problems that turned out to be much easier than expected — they didn’t require new grand theories, and you can imagine many alternative stories where these proofs were found decades ago.”

How the AI worked (and what the prompt says)

The prompt written by OpenAI for this task is a lesson in engineering persistence. First rule: explicitly tell the model to assume a complete proof exists, cutting off the more likely honest answer (“the conjecture is open”). Then: forbid the model from searching the internet to check if the conjecture was already solved, and forbid it from answering that it remains unsolved. The model has no exits: it must solve the problem.

Verification is equally rigid. Partial results, reductions to other unproven conjectures, summaries of research status, explanations of why the problem is hard: all rejected as insufficient. The model can’t respond until a complete proof is ready and passes an adversarial test. Most of the 64 agents are deliberately kept in the dark about which approach seems most promising, to encourage independent “thinking.” Adversarial agents then check each candidate against a detailed list of typical errors.

How much to trust it: Bloom’s judgment

Bloom writes that the mathematical core of the proof goes back at least to the 1983 paper. He assumes that these earlier works heavily influenced OpenAI’s proof, “and it’s a pity that it doesn’t mention them at all. […] This is a frequent problem with AI-generated proofs and papers: they use ideas and proof strategies from the literature without citing them correctly.” Bloom suspects the AI didn’t invent the solution on its own, given that its first instinct when solving problems is “generally to search for and read all related papers.”

Complete verification by the mathematical community is still underway, but Bloom’s initial judgment is the most detailed available. The proof holds technically, but the problem of intellectual credit remains open.

What it means for open problems

Bloom expects AI systems to solve other problems like this: “those whose solutions require only existing and well-developed theory, plus a lot of patience and conviction.” But he adds: “This is likely only a small proportion of open problems, and we don’t know in advance which ones.” The interesting point is different: “In this strange new world where large AI companies spend a lot of time and money attacking many open problems simultaneously (and of course reporting only the successes), we’ll soon discover more about what was within our reach all along.”

We don’t know how many attempts OpenAI made before this success, or how many other problems Sol Ultra is working on without results to announce. Mechanical persistence works where the solution space is defined and verification is objective. It remains to be seen how far this extends beyond pure mathematics.