Researchers have long viewed certain mathematical conjectures as tests of human ingenuity. Yet recent weeks have shown large language models stepping into that arena with striking results. OpenAI released a document attributing a complete proof of the Cycle Double Cover Conjecture entirely to its GPT-5.6 Sol Ultra model. The proof, posted on the company’s content delivery network, resolves a graph-theory question posed in the 1970s by mathematicians including William Tutte and Paul Seymour.
The conjecture states that every finite bridgeless loopless multigraph admits a cycle double cover. In plain terms, the edges can be covered by a collection of cycles such that each edge appears in exactly two of them. Previous work had settled special cases. Jaeger showed it for planar graphs. Others handled cubic graphs with particular properties. But the general statement resisted resolution until the AI system produced a 10-page argument that reduces the problem to cubic graphs, invokes nowhere-zero flows, and finishes with an elementary linear algebra step over a carefully chosen abelian group.
“The proof in this note is entirely due to GPT 5.6 Sol Ultra and the writeup with Codex (with GPT 5.6 Sol),” the document declares. OpenAI also published the elaborate prompt used to guide the model, a multi-page instruction that assumes a proof exists and directs the system to find it while avoiding common pitfalls. The approach relied on multiple subagents working in parallel, a technique that has become common in frontier reasoning models.
News of the result spread quickly on technical forums. A detailed discussion appeared in the r/math subreddit where users examined both the proof’s validity and its implications for the field. But the story didn’t stop there.
Soon after OpenAI’s announcement, Paul Kerger, a professor of industrial engineering and operations research at UC Berkeley, adapted the same prompting strategy for a different open question. He had wrestled for about a year with a complexity gap in deterministic zeroth-order convex optimization. The problem asks how many queries to a function-value oracle are needed to find an epsilon-approximate minimizer of a convex function over the unit ball in d dimensions.
A 1996 upper bound established that O(d² / ε) queries suffice. Lower bounds had matched this in some regimes but lagged in others, leaving a linear gap in the dimension dependence at certain accuracy levels. Kerger crafted a 10-page prompt modeled directly on OpenAI’s CDC template. It laid out known approaches, specified the error tolerance required to close the quadratic gap, and instructed the model on how to proceed step by step.
After 148 minutes of computation, GPT-5.6 Sol Pro returned a proposed proof establishing a near-quadratic lower bound of Ω(d²) queries at accuracy on the order of d^{-3}. The result appears in a preprint titled “Closing the Oracle-Complexity Gap in Derivative-Free Convex Optimization: A Near-Quadratic Lower Bound from Exact Function Values,” available at arxiv.org/pdf/2607.13335. Kerger and collaborators then formalized the argument in the Lean theorem prover, providing machine-checked verification.
“After 148 minutes, GPT-5.6 Sol Pro returned a proposed proof resolving the quadratic dimension dependence at accuracy of order d^{-3},” Kerger wrote in his accompanying Medium post. He stressed that the result has not yet undergone peer review. Still, the formalization in Lean adds confidence that the mathematics holds.
The prompt itself drew on decades of optimization literature. It suggested considering families of difficult functions and strategies for an adversarial oracle that reveals minimal information. Once the right construction appeared, the remaining steps invoked existing results from convex geometry. Nothing fundamentally new, in Kerger’s assessment. Yet finding that construction had eluded him and, apparently, the broader community for 30 years.
Kerger reflected on what this says about current AI systems. “If a result is attainable with existing techniques, modern AI methods will be able to solve those problems.” He compared theorem discovery to a search problem. Brilliant mathematicians excel at pruning the search tree. AI, by contrast, lowers the cost of exploring promising branches. The combination can be potent.
These episodes arrive amid broader progress on mathematical benchmarks. Both OpenAI and Google DeepMind announced systems that reached gold-medal performance on the 2025 International Mathematical Olympiad, solving five of six problems at a level that would earn top human contestants the highest honor. Coverage in The New York Times highlighted how companies have built AI systems explicitly better suited to mathematics over the past two years.
Yet the CDC proof and the convex-optimization lower bound stand apart. They target open research problems rather than competition questions with known answers. The CDC result, if verified by the community, would mark the first time an AI system has resolved a long-standing conjecture in graph theory. Skeptics, including some mathematicians on X, have called for independent verification and deeper analysis of whether the proof introduces genuinely novel ideas or simply recombines known ones cleverly.
Fields Medalist Terence Tao has urged caution in equating such AI outputs with human mathematical insight, according to reporting in the Times of India. The systems still struggle with very long-term projects that might require thousands of hours of focused reasoning. They shine on shorter bursts. A 148-minute session or a carefully scaffolded prompt fits their current strengths.
Even so, the speed of progress surprises many insiders. Just months ago, few would have predicted an AI producing a machine-verified proof of a conjecture that had resisted experts for half a century. Now researchers talk openly about feeding the same prompting template to other open problems on lists such as those compiled by Paul Erdős.
One X user suggested running the basic CDC-style prompt across remaining Erdős problems, noting that GPT-5.6 had already cracked one of them and produced a full Lean formalization in hours using multiple subagents. The potential to clear out accessible but stubborn results looms large.
Universities and corporate labs alike are watching. If AI can systematically surface proofs for problems that sit just beyond current human bandwidth, the pace of mathematical discovery could accelerate. But it may also change who participates. Graduate students might spend less time grinding through routine cases and more time identifying the truly novel directions that still require human creativity.
Kerger’s experience illustrates the point. He had the expertise to recognize a good construction once the model proposed it and the skill to formalize the argument. The AI acted as a powerful search collaborator. The preprint lists him as author, with appropriate credit to the model in the text. This hybrid workflow feels like a preview of research norms to come.
Questions remain about cost, reliability and intellectual property. Running these elaborate prompts with dozens of subagents consumes significant compute credits. Verification still falls to humans and formal systems like Lean. And while OpenAI attributes the CDC proof wholly to its model, the prompt itself was engineered by humans who understood the conjecture’s history and failure modes.
Nevertheless, the barrier has shifted. Problems once considered medium-hanging fruit for specialists now look reachable with the right prompt and enough inference time. The convex-optimization result closed a gap that had persisted since the Clinton administration. The Cycle Double Cover proof, if it stands, resolves a question older than many working mathematicians.
Industry observers expect more such announcements. DeepMind’s advances on the IMO suggest the capability is not limited to one lab. Other frontier models will likely tackle open questions in combinatorics, number theory and beyond. The coming months will test how many of these AI-generated proofs survive community scrutiny.
For now, mathematicians find themselves in unfamiliar territory. Some express excitement at the prospect of an assistant that never tires and can explore vast swaths of the proof space. Others worry that the craft of proof, with its emphasis on elegance and insight, might lose something in the translation to machine-generated arguments.
Both reactions miss a central fact. The tools have changed. The problems have not. Graphs still need their cycles. Convex functions still demand their complexity bounds. What matters is that the work gets done, whether by human insight alone or by a collaboration that would have seemed like science fiction only a few years ago.
And the search continues. With each new conjecture attacked by these scaled-up reasoning engines, the frontier of what counts as solvable moves outward. The question is no longer whether AI can prove hard theorems. It is how far and how fast that ability will carry the entire discipline.
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