OpenAI has released another batch of scientific results, and this time it is aiming high. According to the company, its models have proved mathematical theorems in harmonic analysis, challenged a theory in computer science, and tackled quantum field theory. The company is also changing the way it publishes these discoveries. Instead of presenting finished scientific papers, it is handing the results to external experts, who are tasked with determining whether they hold up and explaining them to other scientists.
This is not the first announcement of its kind. In early October, OpenAI uploaded more than 370 mathematical results to GitHub. However, a group of advisers with whom the company had previously worked had already publicly criticized the fact that a private company was achieving breakthroughs using internal models that no one outside the company could test. The new trio of discoveries therefore comes with different rules governing who gets access to the data first and who receives credit for it.
Waves That Cannot Remain Concentrated at a Single Point
The first result concerns waves. The so-called local smoothing conjecture was proposed by mathematician Christopher Sogge in 1991 and describes how waves behave when concentrated in a single location. Put simply, it states that extreme concentration cannot last for long. Just as ripples on water spread outward after a while, the energy of a concentrated wave must eventually disperse again.
Mathematicians worked toward the proof step by step. In 2019, Larry Guth, Hong Wang, and Ruixiang Zhang proved the conjecture for two spatial dimensions. Wang and Zahl then proved the three-dimensional version of the Kakeya conjecture in 2025. It addresses the question of how small a set can be if it contains a line segment in every possible direction.
OpenAI claims that its models proved the conjecture for all spatial dimensions, thereby also solving the Kakeya conjecture and Stein’s restriction conjecture, according to the company. The model produced the proof on October 18 and converted it into the Lean language the very next day. Lean is a program that acts as a rigorous verifier. A mathematician enters a proof into it, and the computer checks whether every step follows logically from the previous one. The company published both the conventional written proof and its Lean version on GitHub.
Even computer verification, however, is not a cure-all. When OpenAI previously presented a Lean-verified solution to the Navier–Stokes problem, people reviewing it began to raise concerns. According to them, the program had actually confirmed a weaker claim than the one the company had originally described.
The Hypothesis Underpinning Hundreds of Studies
The second discovery concerns theoretical computer science. At first glance, it is an abstract question, but in reality a large part of the field depends on it. Scientists are studying the so-called satisfiability problem. Given a set of logical conditions, the task is to determine whether variables can be assigned values so that all the conditions hold simultaneously. Small problems can be solved quickly, but for large ones, the time required appears to grow at an explosive rate.
The Exponential Time Hypothesis, known among experts by the abbreviation ETH, states exactly this. It claims that no clever trick can significantly speed up such problems. Scientists have relied on it in hundreds of papers on graph theory, computational biology, databases, and computational geometry. If the new result holds up, the assumptions underlying these studies will have to be reconsidered.
OpenAI claims that its models have disproved the hypothesis. The refutation was produced on October 16, and the company formalized it in Lean one day later. The document bears the simple and confident title “The Exponential Time Hypothesis is false.”
Anyone expecting computers to immediately start solving difficult problems faster will be disappointed. Refuting the hypothesis does not automatically produce faster methods for existing computational problems, and scientists still need to determine what the result means in practice.
The result will now face rigorous scrutiny. The review will be led by Russell Impagliazzo and Ramamohan Paturi of the University of California, San Diego, together with Ryan Williams of MIT. OpenAI has promised to fund a conference devoted to examining the claim and has published the proof materials for independent review.
Physicists Want to Write Up the Results Themselves First
The third result is the boldest of all. It concerns the mathematical foundations of quantum field theory. According to OpenAI, the models produced proofs of the existence of theories with so-called gauge fields and fermions in four dimensions, including quantum chromodynamics, which describes the strong interaction between quarks and gluons. This is the force that holds atomic nuclei together. The company also speaks of a unified completion of the Standard Model of particle physics.
The work also touches on pure Yang–Mills theory, which lies at the heart of one of the Millennium Prize Problems posed by the Clay Mathematics Institute. It asks whether quantum Yang–Mills theory exists mathematically and whether it has a so-called mass gap.
For now, however, no one will see these results. The external group will be led by physicist David Gross and includes Michael Douglas, Arthur Jaffe, Igor Klebanov, and Nikita Nekrasov. The team first wants to prepare its own account and only then publish the materials produced by the model.
This approach reflects a new strategy across the company. OpenAI describes the manuscripts produced by its models as raw research data. To turn them into genuine scientific knowledge, it approached external experts and asked them to serve as so-called shepherds, assembling groups that would understand the new results. These are often the very scientists whose work the models built upon.
The company says it does not want academic credit for discoveries made by its internal models. That credit should go to the researchers who produce explanations, papers, and other materials around the results. It also acknowledges the weaknesses of this approach. According to the company, the selection of the first shepherds will inevitably influence who gets access to the work first and creates an information imbalance. It is therefore seeking feedback on how to broaden the group of participants.
According to OpenAI, all three groups are to work independently, and their members may publish the results freely. The company will also provide financial support through its previously announced five-million-dollar contribution to the Simons Foundation, which scientists are to use to organize workshops and conferences to evaluate the work as a whole.
Sources: openai.com, interestingengineering.com and gizmodo.com
