- OpenAI says GPT-5.6 Sol Ultra generated a complete proof of the Cycle Double Cover Conjecture, unproven for about 50 years, in under an hour using 64 subagents in parallel.
- Mathematician Thomas Bloom of the University of Manchester calls it “a very nice proof” that is “short, elementary, and could have been discovered in the 1980s.”
- Bloom faults the paper for not citing prior work — notably a 1983 paper by Bermond, Jackson, and Jaeger — that he believes shaped the approach.
- A full mathematical verification by the community is still pending.
What Happened
OpenAI announced that its new model GPT-5.6 Sol Ultra generated a complete proof of the Cycle Double Cover Conjecture, a graph-theory problem unproven for roughly 50 years, according to a July 11, 2026 report from The Decoder. The model took just under an hour, using 64 subagents working in parallel; the accompanying paper was written by GPT-5.6 Sol.
Why It Matters
The conjecture, formulated independently by several mathematicians in the 1970s, asks whether any network of vertices and edges admits a set of cycles that traverses each edge exactly twice. Decades of partial results existed but no generally accepted proof. An AI system producing a full proof of a long-open problem is a notable capability claim — but it lands in an active debate about whether reasoning models create new mathematics or recombine existing knowledge.
Technical Details
Bloom’s assessment is the most detailed public evaluation so far. He calls the proof “short, elementary, and could have been discovered in the 1980s,” noting it needs no new theories but cleverly combines known tools. He suspects the key step involved a small, counterintuitive twist: “One can imagine trying the natural labelling first, checking the linear algebra, and when that failed shrugging and thinking ‘oh well… guess it can’t be done this easily’ – while the AI does not get discouraged and keeps trying small variations.” The human prompt engineers that persistence deliberately — it tells the model to assume a complete proof exists, bans internet searches to check whether the conjecture is already solved, and rejects partial results, so the model can only respond once a complete proof passes an adversarial test. Most of the 64 agents are kept unaware of which approach currently looks most promising, to encourage independent reasoning.
Who’s Affected
The result matters to mathematicians and to AI labs racing to demonstrate research-grade reasoning. Bloom traces the core ideas at least to a 1983 paper by Bermond, Jackson, and Jaeger, and criticizes their omission: “they use ideas and proof strategies taken from the literature without proper citation.” He doubts the AI invented the strategy, “given that its first problem-solving instinct is generally to search for all related papers on a problem and read them.” That citation gap is the crux of who deserves credit.
What’s Next
A full community verification is still pending, and Bloom compares the result to the unit distance conjecture OpenAI recently solved — both “turned out to be much easier than expected.” He expects AI to crack more conjectures whose solutions need only existing theory “plus a lot of patience,” but cautions that “this is likely only a small proportion of open problems, and we don’t know in advance which they are.” As he puts it, labs are “attacking many open problems at once (and only reporting the successes, of course).”
