OpenAI Says AI Solved the Kakeya Conjecture in One Shot, Disproved ETH, and Proved the Existence of Yang-Mills

Wang's conjecture, cracked by AI overnight Download the iFan'er app for exclusive early access October 27, 2026, 10:34:13, from Beijing

Just moments ago, OpenAI announced that its model has delivered three breakthroughs at once: it overturned the exponential time hypothesis, which the computer science community has believed in for nearly 30 years; it proved the local smoothing conjecture in all dimensions; and it constructed rigorous mathematical existence proofs for quantum chromodynamics and Yang-Mills theory.

OpenAI declares that, tracing back along the chain of mathematical derivation, this means the Kakeya conjecture (also known as the Kakeya needle conjecture) is now solved in all dimensions.

And in early 2025, the Chinese mathematician Wang Hong had just stunned the world by proving the three-dimensional Kakeya conjecture. In just a year and a half, a breakthrough humanity was proud of has become a special case of an AI result.

Even more interestingly, OpenAI brought in the very scientist who originally proposed the ETH hypothesis to review the AI's counterproof of his own hypothesis; and it also brought in a team led by a Nobel laureate in physics to digest the proof of a Millennium Prize Problem written by the AI.

The greatest reversal in the history of mathematics is now unfolding.

Among the three results, the breakthrough in mathematics is especially noteworthy, because it directly concerns the major result achieved last year by the Chinese mathematician Wang Hong.

The Kakeya conjecture originates from a seemingly simple geometric question: if a set can contain a unit line segment in every direction, what dimension must this set have? After more than a century of research, it has become an important topic in fields such as harmonic analysis and geometric measure theory.

In 2025, the Chinese mathematician Wang Hong, in collaboration with the New York University mathematician Joshua Zahl, successfully proved the three-dimensional Kakeya conjecture, making it one of the most closely watched breakthroughs in mathematics that year.

But solving the three-dimensional problem cannot be directly generalized to four dimensions, five dimensions, or any higher dimension.

Exactly this limitation has been crossed by the result OpenAI announced this time. Its AI model proved a more fundamental mathematical proposition, namely the local smoothing conjecture proposed by the mathematician Christopher Sogge in 1991.

This conjecture studies the focusing phenomenon of waves during propagation. To put it simply, waves can form extremely strong focusing at certain moments, but this extreme state can only exist briefly. If one observes the average behavior over a period of time, the degree of focusing of the wave will be noticeably lower than at the most extreme moment.

The local smoothing conjecture is important because it can be used to derive other important conclusions such as the Kakeya conjecture and the Stein restriction conjecture.

In 2019, the mathematicians Larry Guth, Wang Hong, and Zhang Ruixun successfully proved the local smoothing conjecture in two-dimensional space. Now, by introducing a novel auxiliary partial differential equation, OpenAI's model has constructed a proof of the local smoothing conjecture that applies to all dimensions.

According to the mathematical derivation relationships, if the local smoothing conjecture holds in all dimensions, that means the Kakeya conjecture and the Stein restriction conjecture are also resolved in all dimensions.

Wang Hong and Joshua Zahl previously conquered the three-dimensional Kakeya conjecture, and OpenAI this time, by proving the stronger local smoothing conjecture, has generalized the related results to all dimensions.

OpenAI disclosed that its model generated the natural-language proof on October 18, and completed the Lean formalization the next day. It should be noted that Lean formalization can only verify the syntactic compliance of the reasoning steps; it cannot automatically guarantee that the mathematical content of the proof is free of gaps, nor can it replace peer review.

At present, both proof documents have been uploaded to a public GitHub repository for mathematicians around the world to verify and study.

AI overturns a foundational hypothesis of computer science held for nearly 30 years

Compared with the Kakeya conjecture, the result OpenAI achieved in the field of computer science may have a broader impact on algorithm research as a whole.

What the AI overturned this time is the famous Exponential Time Hypothesis (ETH) in computational complexity theory.

This hypothesis was proposed around 1999 by researchers including Russell Impagliazzo, Ramamohan Paturi, and Francis Zane. Its core question is how much computational cost a computer must pay to solve certain complex problems.

A typical example is the Boolean satisfiability problem (SAT).

Suppose there are a large number of conditions composed of logical variables, and the computer needs to determine whether there exists an assignment of variables that makes all conditions hold simultaneously. As the number of variables increases, finding the answer may become extremely difficult. ETH holds that for the general 3-SAT problem, there is no algorithm with time complexity (2^{o(n)}), where (n) represents the number of variables.

In other words, the worst-case computation time for such problems cannot be reduced to sub-exponential in the number of variables.

For nearly 30 years, ETH has been an important foundational hypothesis in computational complexity theory. Hundreds of academic papers have used ETH as a premise to derive conditional computational complexity lower bounds for a large number of problems in fields such as graph theory, computational biology, and databases.

Now, OpenAI announces that its AI system has successfully proved ETH to be false, and that this has already been formally verified with Lean.

This result means that for some computational problems long believed to be unsolvable in sub-exponential time, more efficient algorithms may exist that had not previously been discovered. The many conditional complexity conclusions built on ETH will also face re-examination.

The lineup of experts OpenAI invited for this research is also quite dramatic.

Russell Impagliazzo and Ramamohan Paturi of the University of California, San Diego, and Ryan Williams of MIT will jointly lead the external research team.

Among them, Impagliazzo and Paturi are precisely the original proposers of ETH.

OpenAI invited the scientists who proposed this hypothesis back then to personally participate in reviewing the AI's counterproof of the hypothesis. OpenAI stated that the AI model generated the natural-language counterproof on October 16, completed the Lean formalization the next day, and both materials are now public.

The external research team also asked OpenAI to disclose all raw data and to fund an academic conference to discuss the impact of this result on computational complexity theory; OpenAI has accepted these requests.

If this result is recognized by the academic community, humanity may need to rethink how much room for improvement remains in what was previously regarded as the limit of computational difficulty.

Among the three studies, the result in theoretical physics may be the most complex and also the most significant for fundamental science.

OpenAI announced that its AI model has constructed rigorous mathematical existence proofs for certain interacting quantum field theories containing gauge fields and fermions in four-dimensional space.

The research involves quantum chromodynamics (QCD), a unified completion of the Standard Model of particle physics, and the existence problem of pure Yang-Mills theory.

Modern particle physics mainly relies on the Standard Model to describe fundamental particles and their interactions, involving the electromagnetic force, the weak interaction, and the strong interaction. Quantum chromodynamics specifically studies the strong interaction and explains how quarks and gluons interact.

Although the relevant theories have been verified in a large number of experiments, some unresolved questions remain regarding their rigorous mathematical foundations.

Yang-Mills theory is the most famous example among them.

In 2000, the Clay Mathematics Institute in the United States listed the existence and mass gap problem of Yang-Mills theory as one of the seven Millennium Prize Problems, requiring researchers to rigorously prove the existence of four-dimensional quantum Yang-Mills theory and to prove that it has a positive mass gap.

The result announced by OpenAI this time involves the existence part and further advances the research to quantum chromodynamics and related unified theories.

OpenAI disclosed that the model generated a natural-language proof of the existence of Yang-Mills theory on September 9, generated a proof for quantum chromodynamics on September 18, and completed the Lean formalization of the Yang-Mills theory existence proof on October 16.

To further study this result, OpenAI invited five theoretical physicists to form a team.

The lead, David Gross, is a 2004 Nobel laureate in physics; he won the prize for participating in the discovery of asymptotic freedom in the strong interaction, and his research laid an important foundation for quantum chromodynamics.

The other members include Michael Douglas, Arthur Jaffe, Igor Klebanov, and Nikita Nekrasov, all important researchers in quantum field theory and mathematical physics.

Among them, Arthur Jaffe participated in formulating the official statement of the Yang-Mills theory Millennium Prize Problem and in founding the Clay Mathematics Institute.

Considering that the proof content is very large, the expert team decided not to publicly release the AI-generated raw materials for the time being, and researchers will further organize, review, and write up formal research results. If the research proceeds smoothly, AI is expected to help physicists establish a more rigorous mathematical foundation for modern quantum field theory.

It is worth mentioning that on October 6 of this year, OpenAI publicly released 722 AI-generated mathematics research manuscripts in one go, involving 372 unsolved mathematical problems.

But afterward, some proofs were found to contain errors, and OpenAI had to withdraw 3 manuscripts and revise another 14.

Just 20 days later, OpenAI announced three major results and at the same time changed its release strategy, treating AI-generated proofs as raw research data and inviting external scientists to participate in verification, interpretation, and organization.

The mathematics results were released directly, the computer science results were organized for review by experts such as the original proposers of ETH, and the physics results were handed to a team led by a Nobel laureate in physics for study.

OpenAI also announced that it will not seek academic authorship for these discoveries and will support scientists in conducting related research through the $5 million grant it previously provided to the Simons Foundation.

In the past, scientists might have needed decades to find a proof. Today, AI can generate proofs in a few days and even complete Lean formalization verification, but scientists still need to spend a great deal of time to truly understand what these results mean.

When the speed at which AI generates scientific results begins to exceed the speed at which humans understand them, the bottleneck of scientific research may also shift from finding answers to verifying and digesting answers. Scientific problems that once troubled humanity for decades may increasingly be answered by AI in the future.

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