OpenAI says its model disproved ETH and proved the local smoothing conjecture in all dimensions

OpenAI overturns a 30-year cornerstone of computer science, and also solves the Kakeya conjecture

Machine Heart Download the client | Exclusive first look October 27, 2026 15:00:00 From Beijing Updated October 27, 2026 16:04:04

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The discussion sparked by 722 mathematical manuscripts has not yet ended, and OpenAI has already placed an even more surprising result before scientists.

This time, it claims to have overturned an assumption in theoretical computer science that has been widely believed for nearly 30 years. And the two researchers who proposed this assumption back then are now preparing to organize their peers to jointly examine the counterproof given by AI.

On October 26, OpenAI announced new results in three fields: mathematics, computer science, and physics: proving the local smoothing conjecture in all dimensions, disproving the Exponential Time Hypothesis (ETH), and establishing a rigorous mathematical existence proof for four-dimensional quantum field theories, including quantum chromodynamics. OpenAI President Greg Brockman called them three major scientific discoveries on X, and remarked that AI has begun helping humanity deepen its understanding of the universe.

Less than three weeks ago, OpenAI had just released 722 mathematical manuscripts generated by its internal models, belonging to 372 result families and 17 research directions. We went through that list at the time and picked out the 30 most closely watched classic problems. The four-dimensional Kakeya conjecture was one of them.

Now, the Kakeya problem has appeared again, but it is no longer the most unexpected news among these results. What has drawn the entire theoretical computer science community into discussion is another abbreviation: ETH.

Suppose there is a combination lock where each digit can only be 0 or 1. We do not know the correct password, but we have several clues: the first and second digits cannot both be 0; the second digit is 0, or the third digit is 0; the first, third, and fourth digits cannot all be 1.

The question is whether we can find a set of digits that satisfies all the clues at the same time. For example, 1010 is a password that meets the conditions.

In computer science, this kind of choice with only two values, 0 and 1, can be represented by Boolean variables, corresponding to "false" and "true" respectively. If each clue is written as a logical condition in which no more than three variables or their negations are connected by "or," and then we determine whether there is an assignment that makes all conditions hold simultaneously, we obtain the classic 3-SAT problem.

A four-digit password is easy to solve; at most 16 combinations need to be checked. But if the password has 100 digits, the number of possible combinations increases to 2^100. Although clever algorithms do not have to try them one by one, how fast a computer can be when facing general 3-SAT problems has always been an important question in theoretical computer science.

In the late 1990s, Russell Impagliazzo, Ramamohan Paturi, and Francis Zane proposed the Exponential Time Hypothesis (ETH).

Its core claim is: there is no algorithm that can solve all 3-SAT problems in subexponential time. In other words, no matter how algorithms improve, they cannot reduce the worst-case running time to the order of 2^{o(n)}, where n represents the number of Boolean variables.

For nearly 30 years afterward, ETH gradually became an important foundational assumption in theoretical computer science. Researchers often use it as a premise to derive computational time lower bounds in areas such as graph algorithms, database queries, and computational biology. Hundreds of research papers have relied on this assumption to establish conditional hardness results.

And this time, OpenAI's internal model has produced an algorithm that may break through this boundary.

According to the published proof, for CNF Boolean formulas of fixed width, the model constructs a deterministic algorithm that can not only determine whether an assignment satisfying the conditions exists, but also precisely count the number of satisfying assignments. Taking the earlier password problem as an example, determining whether there is a password that meets the clues only requires finding one set; precise counting, however, can calculate exactly how many sets of passwords satisfy all conditions.

More critically, there is speed. For the relevant problems in which the number of clauses is proportional to the number of variables, OpenAI gives a running-time bound of the form 2^{O(n/√log n)} (omitting polynomial factors), which belongs to subexponential time.

If the proof holds, it means ETH has been disproved. This does not mean that all hard problems can be solved quickly, but a theoretical boundary once thought impossible to cross may already have been breached.

What is even more dramatic is who will be responsible for reviewing this result next.

OpenAI invited Impagliazzo and Paturi of the University of California, San Diego, as well as Ryan Williams of MIT, to organize an external research team. The first two are precisely the original proposers of ETH, while Williams has long studied algorithms and computational complexity lower bounds.

The scientists who proposed this assumption nearly 30 years ago now have to personally take part in examining the counterproof given by AI.

According to the timeline published by OpenAI, the model generated a natural-language counterproof of ETH on October 16, completed the Lean formalization the next day, and officially announced the results on October 26. Williams and others subsequently announced the organization of an open, independent review.

An explanation of the Lean formalization proof of the refutation of the Exponential Time Hypothesis (ETH) released by OpenAI.

A hypothesis that has been widely used in theoretical computer science for nearly 30 years may now be rewritten by AI. And its proposers are joining colleagues in examining this proof.

Last time it was the four-dimensional Kakeya

This time it has been pushed directly to all dimensions

Turning our gaze from computer science back to mathematics, longtime readers will encounter a familiar name: Kakeya.

In our previous report on 722 manuscripts, we introduced this mathematical problem that originated from a single needle: if a needle is to be rotated through all directions, exactly how much space is needed? This seemingly simple geometric problem later gave rise to dimension conjectures for high-dimensional sets and became deeply connected with harmonic analysis and partial differential equations.

In 2025, Wang Hong and Joshua Zahl solved the three-dimensional Kakeya set conjecture. Previously, OpenAI's result family No. 074 claimed to further prove the three-dimensional Kakeya maximal conjecture and the four-dimensional Kakeya dimension conjecture; at that time, these two manuscripts did not yet have corresponding Lean formalized proofs.

This time, the entry point OpenAI found has changed.

The new result first targets the Local Smoothing Conjecture proposed by mathematician Christopher Sogge in 1991. It studies how waves focus during propagation: at a certain moment, a wave may be unusually concentrated in one place; but if its behavior over a period of time is observed together, this extremely concentrated state cannot be maintained over the long term.

You can think of it as ripples meeting on the surface of water. At a certain moment, wave crests may superimpose at one location to an especially high degree, but as the wave continues to propagate, that most extreme peak quickly changes. What local smoothing aims to do is use precise inequalities to explain how much additional regularity time averaging can bring.

This is not merely a mathematical game about waves. The behavior of wave equations, restriction estimates in Fourier analysis, and the geometric structure of Kakeya sets are backed by a set of interconnected theories.

In 2019, Larry Guth, Wang Hong, and Zhang Ruixiang proved the Local Smoothing Conjecture in two-dimensional space. OpenAI now claims that the model found a new auxiliary partial differential equation and used it to push the proof to all spatial dimensions. According to OpenAI's account, this result further resolves the Kakeya conjecture and the Stein restriction conjecture in all dimensions.

From the previous specific results in three and four dimensions to this claim of a unified conclusion covering all dimensions through local smoothing, the change is quite striking.

The timeline was also very tight: OpenAI said the model obtained the proof on October 18 and completed the Lean formalization on October 19. The natural-language manuscript and the formalized proof have already been made public. Unlike with ETH, OpenAI did not designate an advance review panel for the mathematics part, but instead let researchers directly access the public materials and organize explanations and verification themselves.

OpenAI's explanation of the Lean formalization proof of the Local Smoothing Conjecture states that it covers local smoothing estimates in all spaces of dimension two and above, and proves that Kakeya sets in the corresponding dimensions have full Hausdorff dimension.

A difficult problem in physics:

Can the theories we use be rigorously constructed?

If ETH is concerned with just how fast computation can be, and Kakeya and local smoothing are concerned with the properties of space and waves, then the third result directly touches the foundations of how modern physics describes the microscopic world.

Quantum Chromodynamics (QCD) describes the strong interaction between quarks and gluons. Physicists use it to study protons, neutrons, and the more fundamental particles that constitute them, and the related calculations and experiments form an important pillar of modern particle physics.

But there is an interesting gap here: a theory can be very successful in physical calculations and experiments, yet rigorously constructing it as a quantum field theory satisfying mathematical axioms is another matter.

In four-dimensional spacetime, interactions between particles, quantum fluctuations that continually appear at different scales, and how to move from approximate calculations to rigorous definitions have long confronted constructive quantum field theory with enormous mathematical difficulties.

OpenAI this time claims that an internal model, following the constructive quantum field theory research line of mathematical physicist Tadeusz Balaban, established rigorous existence proofs for certain four-dimensional interacting quantum field theories containing gauge fields and fermions, involving QCD and a unified extension that accommodates the Standard Model gauge fields and fermions.

The work also includes an existence proof for pure Yang–Mills theory. In 2000, the Clay Mathematics Institute listed the "Yang–Mills existence and mass gap" as one of the seven Millennium Prize Problems in mathematics. The problem requires not only proving that quantum Yang–Mills theory exists, but also proving that it has a positive mass gap. What OpenAI announced this time is the existence part; it did not announce a resolution of the mass gap.

The most striking part of the physics section is also the list of scientists taking over the research.

David Gross, winner of the 2004 Nobel Prize in Physics, will lead a working group whose members include Michael Douglas, Arthur Jaffe, Igor Klebanov and Nikita Nekrasov. Gross won the prize for work related to asymptotic freedom, and asymptotic freedom is precisely the key to understanding how the strong interaction behaves at extremely short distances; Jaffe, together with Edward Witten, co-authored the formal statement of the Yang–Mills Millennium Prize Problem.

The release method this time also differs from the previous two. OpenAI revealed that the model generated the Yang–Mills existence proof as early as September 9, and then generated the QCD-related proof on September 18. Because the material was complex and the length enormous, the company waited until October 16, when the Lean formalization of the Yang–Mills existence proof was completed, before moving forward with external collaboration.

The physics working group chose to first organize, understand and write its own research text, and for the time being not to publicly release the original AI-generated proofs. In other words, this result, the one closest to the foundations of modern physics, still needs the expert group to turn the long proof into research that peers can read, judge and continue to use.

Scientists begin to take over another piece of work

Looking back at these three results, the change truly worth observing is not only the weight of the problems themselves, but also the way OpenAI handled these results.

When it published 722 manuscripts on October 6, OpenAI adopted a centralized public-release model: papers, proof files, a results catalog and some model reasoning summaries were all placed on GitHub together. Mathematicians could read them directly, and also had to face a practical problem: for hundreds of manuscripts spanning different disciplines, who exactly should check them, who should explain them, and how should one judge which of them contain genuinely new ideas?

This is not purely a question of workload. After that earlier batch of manuscripts was made public, three were retracted because of symbol errors in their proofs. On the other hand, even with Lean formalization, the academic community still needs to clarify whether the formalized theorems accurately correspond to the original conjectures, and exactly which new methods in the proofs can be transferred to other problems.

OpenAI now uses a rather interesting term for these AI products: raw research data.

This positioning separates "generating proofs" from "forming scientific understanding." The former can be completed first by AI; the latter requires researchers to review the arguments, rewrite explanations, trace prior work, judge contributions, and place the new tools into the existing knowledge system of the discipline.

Therefore, the three results announced on October 26 correspond to three arrangements: the mathematics part is released directly, letting the community organize research on its own; the computer science part releases the proof while inviting the original ETH proposers and domain experts to organize an independent review; and the physics part first hands the material to an expert team, and only after they finish organizing it will it be decided how to publish.

OpenAI also proposed several principles for cooperation: external members participate voluntarily and independently, do not sign confidentiality agreements, and do not seek academic authorship for OpenAI because of model discoveries; they should receive recognition for the explanations, proof organization and follow-up research they complete. The company also promised to support independent review, and previously announced a $5 million investment in a related Simons Foundation program for workshops, academic conferences and research projects.

Releasing proofs directly allows more researchers to participate immediately, but it also means that the enormous review work must be borne by the community itself. Handing them first to an expert team can concentrate efforts on organizing complex research results, but it also lets some people access the material earlier than other researchers.

OpenAI has clearly also recognized this problem. Compared with publishing 722 mathematics manuscripts all at once in early October, this time it not only announced the research results, but also began trying to establish corresponding review and collaboration mechanisms for different disciplines.

The ETH story happens to provide a rather symbolic scene.

Nearly 30 years ago, Impagliazzo and Paturi participated in proposing this hypothesis, and since then hundreds of research papers have used it as a premise to explore the complexity boundaries of computational problems. Now, OpenAI's model claims to have found a counterexample, and these two scientists are working with peers to re-examine this theoretical foundation that they once helped establish.

For OpenAI, the model has already completed the generation and formalization of the proofs. For the scientific community, the real research work is still continuing: reviewing the algorithms, understanding the new methods in the proofs, and judging how these results will affect subsequent research.

From 722 mathematics manuscripts to the current research results spanning three disciplines, what OpenAI demonstrates is no longer just how many proofs AI can produce; it is also beginning to touch on a more practical question: as AI keeps putting forward new scientific conclusions, how can humans turn these conclusions into knowledge that can truly be understood, verified and further developed?

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