OpenAI on Oct. 26 announced three new research claims in mathematics, computer science and physics, including a result that, if it holds up, would overturn one of theoretical computer science’s most widely used assumptions of the past three decades.
The company said its internal models proved the Local Smoothing Conjecture in all dimensions, disproved the Exponential Time Hypothesis (ETH), and established rigorous mathematical existence results for four-dimensional quantum field theories, including work involving quantum chromodynamics (QCD). OpenAI President Greg Brockman described the announcement on X as three major scientific discoveries and said AI is beginning to help humans deepen their understanding of the universe.
The release comes less than three weeks after OpenAI published 722 mathematics manuscripts generated by internal models, grouped into 372 result families across 17 research directions. Among the problems that drew attention at the time was the four-dimensional Kakeya conjecture. In the latest announcement, Kakeya appears again, but the result pulling the broader theoretical computer science community into the discussion is ETH.
A nearly 30-year-old assumption faces an AI-generated challenge
ETH is easiest to approach through a simple example. Imagine a lock where each digit can only be 0 or 1, and several clues constrain the valid combinations: the first and second digits cannot both be 0; either the second digit is 0 or the third digit is 0; and the first, third and fourth digits cannot all be 1. The question is whether there exists an assignment that satisfies every condition at once.
In computer science, such binary choices are represented by Boolean variables. If each clue is written as a logical clause involving at most three variables or their negations, connected by “or,” and the task is to determine whether some assignment satisfies all clauses, the result is the classic 3-SAT problem.
With four digits, there are at most 16 combinations to check. With 100 digits, the number of possibilities rises to 2^100. Smarter algorithms do not need to test every case one by one, but how fast a computer can solve general 3-SAT instances has long been a central question in theoretical computer science.
In the late 1990s, Russell Impagliazzo, Ramamohan Paturi and Francis Zane proposed the Exponential Time Hypothesis. Its core claim is that there is no algorithm that can solve all 3-SAT instances in subexponential time. Put differently, no matter how algorithms improve, the worst-case running time cannot be pushed down to 2^{o(n)}, where n is the number of Boolean variables.
Over nearly 30 years, ETH became a basic working assumption in theoretical computer science. Researchers have used it to derive conditional lower bounds in graph algorithms, database queries and computational biology, and hundreds of papers rely on it for hardness results.
OpenAI now says its internal model produced a possible counterexample to that boundary. According to the public proof, the model constructed a deterministic algorithm for fixed-width CNF Boolean formulas that can not only decide whether a satisfying assignment exists, but also count exactly how many satisfying assignments there are.
The speed claim is the key point. For related problems where the number of clauses is proportional to the number of variables, OpenAI gives a running-time bound of 2^{O(n/√log n)}, suppressing polynomial factors. That is subexponential. If the proof is correct, ETH would be false.
That would not mean every hard problem suddenly becomes easy. It would mean a theoretical boundary long treated as out of reach may have been crossed.
The review process adds another layer to the story. OpenAI invited Impagliazzo and Paturi, both at the University of California San Diego, along with MIT’s Ryan Williams, to organize an external research team. Impagliazzo and Paturi were among the original proposers of ETH. Williams has spent years working on algorithms and lower bounds in computational complexity.
Researchers who helped formulate the assumption nearly 30 years ago are now preparing to examine the AI-generated disproof. According to OpenAI’s timeline, the model generated a natural-language disproof of ETH on Oct. 16, completed a Lean formalization the next day, and the company published the result on Oct. 26. Williams and others then announced an open, independent review effort.
A hypothesis that has underpinned a large part of theoretical computer science for almost three decades may now be rewritten by AI, with its original authors involved in testing the claim.
From specific Kakeya results to an all-dimensions claim
In mathematics, the latest announcement returns to a familiar name: Kakeya. The problem began with a geometric question about how much space is needed to rotate a needle through every direction. It later grew into high-dimensional conjectures about the dimension of sets and became deeply tied to harmonic analysis and partial differential equations.
In 2025, Wang Hong and Joshua Zahl solved the three-dimensional Kakeya set conjecture. Earlier, OpenAI’s result family No. 074 had claimed proofs of the three-dimensional Kakeya maximal conjecture and the four-dimensional Kakeya dimension conjecture, though those manuscripts did not yet have corresponding Lean formalizations.
This time, OpenAI says the model entered through a different route. The new result targets the Local Smoothing Conjecture, proposed by mathematician Christopher Sogge in 1991. The conjecture studies how waves focus as they propagate: at a given instant, a wave may concentrate unusually strongly in one place, but over a stretch of time that extreme concentration cannot persist.
One way to picture it is through ripples on water. At one moment, wave peaks may stack unusually high at a point. As the waves continue moving, that extreme peak shifts. Local smoothing seeks precise inequalities that quantify how much extra regularity appears after averaging over time.
This is not just a technical game about waves. The behavior of wave equations, restriction estimates in Fourier analysis and the geometry of Kakeya sets are linked through the same theoretical structure.
In 2019, Larry Guth, Wang Hong and Zhang Ruixiang proved the Local Smoothing Conjecture in two-dimensional space. OpenAI now claims its model found a new auxiliary partial differential equation and used it to extend the proof to all spatial dimensions. In OpenAI’s account, that result also resolves the Kakeya conjecture and the Stein restriction conjecture in all dimensions.
The shift is substantial: from earlier claims tied to specific three- and four-dimensional cases to a unified statement covering every dimension.
The timeline is tight here as well. OpenAI said the model obtained the proof on Oct. 18 and completed the Lean formalization on Oct. 19. Both the natural-language manuscript and the formal proof have been released. Unlike the ETH result, OpenAI did not appoint a designated first-pass review group for the mathematics claim, leaving researchers to organize interpretation and verification directly from the public materials.
OpenAI’s public description of the Lean formalization says it covers local smoothing estimates in all spaces of dimension two and higher, and proves that Kakeya sets in the corresponding dimensions have full Hausdorff dimension.
The physics claim reaches into QCD and Yang–Mills existence
If ETH is about the speed of computation, and Kakeya and local smoothing are about space and waves, the third result reaches into the mathematical foundations of modern physics.
Quantum chromodynamics describes the strong interaction between quarks and gluons. Physicists use it to study protons, neutrons and the more basic particles that make them up. Its calculations and experiments are central to modern particle physics.
But there is a longstanding gap between physical success and mathematical rigor. A theory can work extremely well in calculations and experiments, yet still be difficult to construct as a mathematically rigorous quantum field theory satisfying accepted axioms.
In four-dimensional spacetime, interactions between particles, quantum fluctuations across scales, and the move from approximate calculations to strict definitions have made constructive quantum field theory a hard mathematical problem for decades.
OpenAI says its internal model followed the constructive quantum field theory program associated with mathematical physicist Tadeusz Balaban and established rigorous existence results for certain four-dimensional interacting quantum field theories containing gauge fields and fermions. The company said those results involve QCD and a unified extension that accommodates Standard Model gauge fields and fermions.
The work also includes an existence proof for pure Yang–Mills theory. In 2000, the Clay Mathematics Institute named “Yang–Mills existence and mass gap” one of its seven Millennium Prize Problems. That problem asks not only for existence of the quantum Yang–Mills theory, but also for proof of a positive mass gap. OpenAI said its announcement covers existence only, not the mass gap.
The names attached to the review effort stand out. Nobel Prize-winning physicist David Gross, who shared the 2004 Nobel Prize in Physics, will lead a working group that includes Michael Douglas, Arthur Jaffe, Igor Klebanov and Nikita Nekrasov. Gross was recognized for work related to asymptotic freedom, a key concept in understanding the strong interaction at very short distances. Jaffe, with Edward Witten, co-authored the formal statement of the Yang–Mills Millennium Prize Problem.
The release path here differs from the other two. OpenAI said the model generated the Yang–Mills existence proof on Sept. 9 and a QCD-related proof on Sept. 18. Because the material was complex and lengthy, the company waited until Oct. 16, when the Lean formalization of Yang–Mills existence was complete, before moving ahead with outside collaboration.
The physics working group chose to first organize, understand and write its own research text rather than release the AI-generated raw proof immediately. For now, the result closest to the foundations of modern physics still needs to be translated by specialists into a form the wider field can read, judge and build on.
OpenAI is separating proof generation from scientific understanding
Across the three announcements, the notable shift is not only the weight of the problems but also how OpenAI is handling the outputs.
When it released the 722 manuscripts on Oct. 6, OpenAI used a bulk-publication model: papers, proof files, a result catalog and some model reasoning summaries were posted to GitHub together. Mathematicians could read them directly, but the release also raised an immediate question. Who would check hundreds of manuscripts across multiple fields, who would explain them, and how would the community decide which ones contained genuinely new ideas?
This is not just a matter of workload. After the earlier release, three manuscripts were withdrawn because of notation errors in the proofs. Even with Lean formalization, researchers still need to determine whether the formalized theorems match the original conjectures and what new methods, if any, can transfer to other problems.
OpenAI is now using a striking phrase for these outputs: raw research data.
That framing separates generating a proof from producing scientific understanding. The first step may be done by AI. The second still requires researchers to verify arguments, rewrite explanations, trace prior work, assess contributions and place new tools inside the existing structure of a field.
That is why the Oct. 26 release came with three different arrangements:
- the mathematics result was published directly for the community to study;
- the computer science result was released publicly while the original ETH authors and other experts were invited to organize an independent review;
- the physics result was first handed to an expert group, which will decide how to publish after preparing its own research text.
OpenAI also laid out several collaboration principles. Outside participants are to join voluntarily and independently, without signing nondisclosure agreements, and not to seek academic authorship on OpenAI’s behalf because of model-generated discoveries. They should receive credit for the explanatory work, proof organization and follow-on research they themselves complete. The company also said it would support independent review and noted that it had previously committed $5 million to related Simons Foundation programs for workshops, conferences and research projects.
Behind the three release models sits the same question: if AI can generate large volumes of frontier research in a short period, how should the scientific community absorb that knowledge?
Publishing proofs directly lets more researchers engage at once, but it also pushes a large verification burden onto the community. Sending material first to expert teams can concentrate effort on difficult results, but it also means some researchers see the material earlier than others.
OpenAI appears to be adjusting to that tension. Compared with the one-time release of 722 mathematics manuscripts in early October, the company is now not only publishing results but also experimenting with review and collaboration structures tailored to different disciplines.
The ETH story offers a particularly symbolic scene. Nearly 30 years ago, Impagliazzo and Paturi helped formulate the hypothesis. Since then, hundreds of papers have used it to explore the limits of computational complexity. Now OpenAI’s model claims to have found a disproof, and those same scientists are joining peers to re-examine a theoretical foundation they helped build.
For OpenAI, the model has already generated and formalized the proofs. For the scientific community, the real work is still underway: checking the algorithms, understanding the new methods inside the proofs, and deciding how these results may affect future research.
From the 722 mathematics manuscripts to the latest cross-disciplinary claims, OpenAI is no longer only showing how many proofs AI can produce. It is also forcing a more practical question into view: when AI keeps proposing new scientific conclusions, how do humans turn them into knowledge that can be understood, verified and extended?
This article is based on a report from the WeChat public account “机器之心” (ID: almosthuman2014), credited to “关注AI的”.
