A world-class problem that has baffled humanity for decades was solved by AI in just a few days.
Just now, OpenAI announced that its model has achieved three breakthroughs in one go: it has overturned the exponential-time hypothesis, which the computer science community has firmly believed for nearly 30 years; it has proved the local smoothing conjecture in all dimensions; and it has constructed rigorous mathematical existence proofs for quantum chromodynamics and Yang–Mills theory.
OpenAI announced that, by tracing back along the chain of mathematical reasoning, this means the Kakeya conjecture (also known as the Besicovitch conjecture) has been proven in all dimensions.
Meanwhile, in early 2025, Chinese mathematician Wang Hong stunned the world by proving the three-dimensional Kakeya conjecture—also known as the three-dimensional Besicovitch conjecture. In just a year and a half, this breakthrough, once hailed as a triumph of human ingenuity, has become nothing more than a special case of an AI achievement.
Even more astonishing were the subsequent arrangements. OpenAI invited the very scientist who originally proposed the ETH hypothesis to review AI-generated counterexamples to his own hypothesis, and also brought in a team led by a Nobel laureate in physics to evaluate AI-written proofs of Millennium Prize Problems.
A question that took scientists half a lifetime to solve was answered by AI in just a few days. The mathematicians’ next task is no longer to find a proof, but to verify the AI’s proof.
The biggest reversal in the history of mathematics is underway.
From Wang Hong’s three-dimensional breakthrough to AI challenging all dimensions
Among the three achievements, the breakthrough in the field of mathematics is particularly noteworthy, as it directly builds on the significant result achieved last year by Chinese mathematician Wang Hong.
Kakeya’s conjecture stems from a seemingly simple geometric problem: if a set can contain unit line segments in every direction, what dimension must that set have? After more than a century of research, it has become an important topic in areas such as harmonic analysis and geometric measure theory.
In 2025, Chinese mathematician Wang Hong, in collaboration with New York University mathematician Joshua Zahl, successfully proved the three-dimensional Kakaiya conjecture, making it one of the most closely watched breakthroughs in the mathematical community that year.
However, solutions to three-dimensional problems cannot be directly generalized to four dimensions, five dimensions, or any higher dimension.
OpenAI’s latest findings have precisely overcome this limitation. Its AI model has proven a more fundamental mathematical proposition—the local smoothing conjecture, proposed by mathematician Christopher Sogge in 1991.
This conjecture investigates the focusing phenomenon of waves during propagation. Simply put, waves can momentarily form extremely strong focal points, but such extreme conditions are only transient. If one examines the average behavior over a period of time, the degree of wave focusing will be significantly lower than at the most extreme moments.
The local smoothing conjecture is important because it can be used to derive other significant results, such as the Kakeya conjecture and the Stein restriction conjecture.
In 2019, mathematicians Larry Guth, Wang Hong, and Zhang Ruixun successfully proved the local smoothing conjecture in two-dimensional space. Now, OpenAI’s model has constructed a proof of the local smoothing conjecture applicable to all dimensions by introducing a novel auxiliary partial differential equation.
According to the mathematical derivation, the local smoothing conjecture holds in all dimensions, which implies that both the Kakeya conjecture and the Stein restriction conjecture have been resolved in every dimension.
Wang Hong and Joshua Zahl previously solved the three-dimensional Kakeya conjecture, while OpenAI has now extended these results to all dimensions by proving a stronger local smoothing conjecture.
OpenAI disclosed that its model generated a natural-language proof on October 18, which was formalized in Lean the following day. Currently, both proofs have been uploaded to a public GitHub repository for mathematicians worldwide to study.
AI overturns a foundational assumption of computer science that has stood for nearly 30 years.
Compared to the Kakeya conjecture, the achievements OpenAI has made in the field of computer science may have a more far-reaching impact on algorithmic research as a whole.
This time, what has been overturned by AI is the well-known Exponential Time Hypothesis (ETH) in computational complexity theory.
This hypothesis was proposed around 1999 by researchers including Russell Impagliazzo, Ramamohan Paturi, and Francis Zane, and its central question is: how much computational cost is actually required for a computer to solve certain complex problems?
One typical example is the Boolean satisfiability problem (SAT).
Suppose there is a large set of conditions composed of logical variables; the computer must determine whether there exists an assignment of values to these variables that makes all the conditions simultaneously true. As the number of variables grows, finding such an assignment can become extremely difficult. ETH asserts that, for general 3-SAT problems, no algorithm with time complexity (2^{o(n)}) exists, where (n) denotes the number of variables.
In other words, the computational time for this type of problem in the worst case cannot be reduced to a sub-exponential function of the number of variables.
Over the past nearly 30 years, ETH has become a fundamental assumption in computational complexity theory. Hundreds of academic papers have taken ETH as a premise, deriving conditional lower bounds on the computational complexity of numerous problems in fields such as graph theory, computational biology, and databases.
Now, OpenAI has announced that its AI system has successfully proven ETH to be false and has already undergone formal verification via Lean.
A foundational assumption that has underpinned hundreds of computer science papers has now been overturned by AI.
This result implies that some computational problems long thought to be unsolvable in sub-exponential time may, in fact, admit more efficient algorithms that have yet to be discovered. Moreover, the numerous conditional complexity results built upon the ETH framework will also require reevaluation.
The lineup of experts invited by OpenAI for this study is also quite dramatic.
Russell Impagliazzo and Ramamohan Paturi of the University of California, San Diego, along with Ryan Williams of the Massachusetts Institute of Technology, will jointly lead the external research team.
Among them, Impagliazzo and Paturi are the original proposers of ETH.
OpenAI invited the scientist who originally proposed this hypothesis to personally review AI’s counterexample to the hypothesis. OpenAI stated that the AI model generated a natural-language counterexample on October 16 and completed Lean formalization the following day; both documents have now been made public.
An external research team also requested that OpenAI disclose all raw data and fund an academic conference to discuss the implications of this achievement for computational complexity theory; OpenAI has accepted these requests.
If this achievement is recognized by the academic community, humanity may need to reconsider how much room for improvement remains in what was previously considered the ultimate limit of computational difficulty.
Nobel laureate in Physics joins in; AI tackles the Millennium Prize Problem
Among the three studies, the findings in the field of theoretical physics are likely to be the most complex and of the greatest fundamental scientific significance.
OpenAI announced that its AI model has constructed rigorous mathematical existence proofs for certain interacting quantum field theories in four-dimensional spacetime, involving gauge fields and fermions.
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 primarily relies on the Standard Model to describe fundamental particles and their interactions, which encompass the electromagnetic force, the weak interaction, and the strong interaction. Quantum chromodynamics, in turn, focuses specifically on the strong interaction, explaining how quarks and gluons interact with one another.
Although the relevant theories have been validated in numerous experiments, their rigorous mathematical foundations still leave some unresolved issues.
Yang–Mills theory is one of the most famous examples.
In 2000, the Clay Mathematics Institute in the United States designated 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 demonstrate that it possesses a positive mass gap.
OpenAI’s latest findings address the existence aspect and further advance research into 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, followed by a proof of quantum chromodynamics on September 18, and completed the Lean formalization of the proof of the existence of Yang–Mills theory on October 16.
To further investigate this achievement, OpenAI invited a team of five theoretical physicists.
The head, David Gross, is a 2004 Nobel laureate in Physics, awarded for his work on the discovery of asymptotic freedom in the strong interaction, which laid an important foundation for quantum chromodynamics.
Other members include Michael Douglas, Arthur Jaffe, Igor Klebanov, and Nikita Nekrasov, all prominent researchers in the fields of quantum field theory and mathematical physics.
Among them, Arthur Jaffe helped formulate the official statement of the Yang–Mills existence and mass gap problem, one of the Millennium Prize Problems, and co-founded the Clay Mathematics Institute.
Given the extensive nature of the evidence, the expert team has decided not to release the AI-generated raw materials at this time, leaving it to researchers to further organize, review, and draft the formal research findings. If the study progresses smoothly, AI could help physicists establish a more rigorous mathematical foundation for modern quantum field theory.
AI begins tackling proof-of-work puzzles in bulk, and scientists may be overwhelmed.
Notably, on October 6 this year, OpenAI publicly released 722 AI-generated mathematical research manuscripts at once, covering 372 unsolved mathematical problems.
However, some of the proofs were later found to contain errors, prompting OpenAI to retract three manuscripts and revise another fourteen.
Just 20 days later, OpenAI announced three major breakthroughs and simultaneously changed its release strategy, treating AI-generated proofs as original research data and inviting external scientists to participate in verification, interpretation, and organization.
Mathematical results are made publicly available directly, computer science findings are reviewed by expert panels including the original proposers from ETH, and physics results are examined by teams led by Nobel laureates in physics.
OpenAI also announced that it would not seek academic authorship for these findings and, through a previously provided $5 million grant to the Simons Foundation, would support scientists in conducting related research.
In a sense, AI research is entering a rather absurd phase.
In the past, scientists might have spent decades searching for a proof. Today, AI can generate proofs in just a few days and even carry out Lean formal verification, yet scientists still need to invest considerable time to truly understand what these results actually mean.
As the pace at which AI generates scientific findings begins to outstrip humanity’s ability to comprehend them, the bottleneck in scientific research may shift from seeking answers to verifying and assimilating those answers. Scientific problems that once stumped humans for decades may increasingly be solved by AI in the future.
AI has already begun submitting its answers one after another, but humans may not even have time to grade them.
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