OpenAI says its internal AI models have produced proofs for three long-standing problems: one in pure mathematics, one in theoretical computer science and one in mathematical physics. The company announced the results on October 26 and handed each of them to outside researchers to check, instead of declaring them settled.
Any one of the three would be a career-defining result for a human mathematician. The computer science claim is the most startling. OpenAI says its system disproved the Exponential Time Hypothesis, an assumption that a large part of modern complexity theory rests on. None of the results has been peer reviewed, and OpenAI describes the AI-written manuscripts as raw research data that people still have to turn into understanding.
Needles, waves and every dimension at once
The mathematics result concerns the local smoothing conjecture, a problem in harmonic analysis about how waves spread out and even out over time. Christopher Sogge proposed it in 1991. OpenAI says its models proved it in every dimension, with the first output arriving on October 18 and a version checked in the Lean proof assistant following on October 19. The material is on GitHub for anyone to inspect.
What makes this one interesting is the chain of problems hanging off it. Local smoothing is known to imply the Bochner-Riesz conjecture, which implies Stein’s restriction conjecture, which in turn implies the Kakeya conjecture. Kakeya sounds like a party trick: how small can a set be if it contains a line segment pointing in every direction? It has kept analysts busy for about a century. Hong Wang and Joshua Zahl settled the three-dimensional case in 2025, a result that drew attention far beyond their field. A proof of local smoothing in all dimensions would close every link of that chain at once.
The assumption half of complexity theory leans on
The Exponential Time Hypothesis, proposed by Russell Impagliazzo and Ramamohan Paturi around the turn of the century, says that 3-SAT cannot be solved in subexponential time. In plain terms: for this classic logic puzzle, where you look for true or false values that satisfy a long list of conditions, no algorithm can avoid a running time that grows exponentially with the number of variables. Researchers use it as a working assumption to prove that a long list of other problems cannot be solved quickly either. Those results are conditional, and they all lean on the same pillar.
OpenAI says its system found a deterministic algorithm that counts the satisfying assignments of k-CNF formulas, for any fixed k, in subexponential time and polynomial memory. Counting the solutions is at least as hard as finding one, so if the algorithm works as stated, 3-SAT is no longer exponential and the hypothesis is false. According to OpenAI, the natural-language disproof dates from October 16 and the Lean formalization from October 17.
Two limits matter before anyone gets carried away. Subexponential is not the same as fast: an algorithm can beat exponential growth on paper and still be useless on real inputs, and nobody outside OpenAI has measured this one yet. It also does not settle P versus NP, and OpenAI does not claim it does. Whether it changes anything for practical security depends on details that the reviewers have only started to examine. The people checking it are about as qualified as it gets: Impagliazzo and Paturi of UC San Diego, who formulated the hypothesis, and Ryan Williams of MIT.
Physics: existence, but no million-dollar prize
The third claim reaches into the mathematics behind the strong nuclear force. Yang-Mills theory describes how quarks and gluons interact, and the Clay Mathematics Institute has offered $1 million since 2000 for a rigorous proof that the theory exists in four dimensions and has a so-called mass gap.
OpenAI says its models produced an existence proof for certain interacting quantum field theories in four dimensions, including quantum chromodynamics, with pure Yang-Mills existence as a subproblem. The company dates the natural-language proof of Yang-Mills existence to September 9 and the QCD proof to September 18, with the Lean formalization of the Yang-Mills part completed on October 16. The mass gap is not part of the claim, so the prize is not in play.
This result is also the least visible. The physicists reviewing it, a group led by Nobel laureate David Gross that includes Michael Douglas, Arthur Jaffe, Igor Klebanov and Nikita Nekrasov, chose to write up the result themselves instead of releasing the AI’s raw output. Until they publish, the rest of the field has nothing to read.
A green checkmark in Lean is only half the answer
Lean is software that checks every step of a proof mechanically. If it accepts a proof, the logic is sound, which is a big part of why these announcements are being taken seriously at all.
The catch is what exactly was proved. Lean confirms that the formal statement follows from the formal definitions; it cannot confirm that those definitions say what mathematicians or physicists mean. For a conjecture in harmonic analysis that gap is usually small. For four-dimensional quantum field theory, where even writing down the definitions is a research project, it is the first thing the reviewers will look at. A subtle mismatch would leave a perfectly checked proof of the wrong theorem.
OpenAI says it wants none of the credit
The announcement comes with a set of rules for how OpenAI works with academic groups: no non-disclosure agreements, no academic credit for OpenAI, recognition for the researchers who verify and develop the work, and independence for those groups. The company says it seeks no credit for discoveries produced by its internal models.
The approach looks like a response to how the company’s earlier math releases went down. In early October, OpenAI published a large batch of mathematical manuscripts from an unreleased internal model in one go, and some mathematicians objected to having to sift through so much material at once. This time, each result has named experts attached before the public sees it. That has its own cost: a handful of chosen researchers get first look at work the rest of the field cannot yet examine.
Who decides what this is worth
The model behind the results has no public name, and OpenAI has not said whether or when it will be available outside the company. For now, the verdict belongs to a short list of people. The complexity theorists can work directly from the public Lean files, so the Exponential Time Hypothesis claim is likely to get the fastest answer. If it survives their review, theoretical computer science will spend years rebuilding the results that depended on it. If it does not, the lesson will be about the gap between a proof a computer accepts and a theorem people actually wanted.
Editor’s note: This article was written with AI assistance from Claude. Nicolae Panait, editor and owner of Webitlio, selected the story, reviewed the text and made the editorial decisions before publication.
