Author: OpenAI
Compiled by: Jiahuan, ChainCatcher
On October 26, OpenAI unveiled three research findings generated by its internal models, covering computer science, physics, and mathematics.
These results address fundamental questions that have been the subject of long-term research in related disciplines. OpenAI refers to the generated manuscripts as "raw research data" and invites the academic community to further validate, interpret, and conduct follow-up studies.
Computer Science: Is ETH Overthrown?
OpenAI stated that its system demonstrated that the Exponential Time Hypothesis (ETH) is false, a result subsequently verified in Lean.
Here, ETH refers to the Exponential Time Hypothesis in computer science, not Ethereum.
ETH (Exponential Time Hypothesis) is a "pessimistic conjecture" in computer science: for a certain class of logical problems, no matter how algorithms are improved, the worst-case runtime will grow exponentially with problem size, with no true shortcuts.
Thus, the overturning of ETH has reopened the question of how much faster these tasks can become, and may also point the way toward better algorithms. OpenAI also noted that substantial follow-up research is still needed to find such algorithms and translate theoretical advances into practical improvements.
For readers in the crypto space, a more direct question is: Will this affect existing cryptographic systems? The security of many cryptographic schemes relies on the difficulty of solving certain specific computational problems. Professor Ryan Williams from MIT asked OpenAI to assess the potential impact. According to his account, OpenAI has conducted the assessment and is reasonably confident that this result does not imply existing cryptographic systems have been broken.
Physics: Laying the Foundation for Quantum Field Theory
Physicists use quantum field theory to describe the world of particles with extremely accurate predictions, but its mathematical foundation is not rigorous.
OpenAI stated that the model provides rigorous "existence" proofs for pure Yang–Mills theory, quantum chromodynamics (QCD—the theory describing the strong force between quarks and gluons), and a grand unified theory incorporating the Standard Model gauge fields. In simple terms, this means providing a mathematical foundation for theories that have been in use for decades.
Currently, only the Yang–Mills portion has been verified in Lean; the QCD portion has not been mentioned in the announcement. The original AI-generated proof is not yet publicly available, as a group led by Nobel laureate David Gross is first studying it and will draft their own paper, so external parties cannot yet verify it.
Mathematics: Wave focusing and the "needle placement" problem
OpenAI stated that its models have proven the local smoothness conjecture in all dimensions, thereby solving the Kakeya conjecture and the Stein restriction conjecture in all dimensions.
The local smoothing conjecture, proposed by Sogge in 1991, studies the behavior of waves when they focus. A helpful analogy is water waves: several ripples moving toward the same point may momentarily combine to form a very high crest. However, how intense this instant is does not directly reflect the average behavior over the entire time period.
The local smoothing conjecture requires precise mathematical estimates to describe the relationship between these instantaneous extreme behaviors and time averages. Due to its fundamental significance, proving it would also impact other mathematical problems, including the Kakeya conjecture and the Stein restriction conjecture.
The Kakeya conjecture can be understood through the "needle placement" problem: imagine placing many needles of the same length and no thickness in space, such that there is at least one needle in every possible direction. These needles can be translated and overlap with each other. What is the smallest possible size of the set formed by these needles?
The key to measuring "size" here is the mathematical concept of dimension. In three-dimensional space, the Kakeya conjecture holds that such a set must have full three-dimensional measure; in higher-dimensional spaces, it must similarly have the corresponding full dimension. Dimension is a distinct concept from volume, so it should not be simply understood as how many cubic meters these needles occupy.
Previously, the two-dimensional local smoothness (2019) and the three-dimensional Kakeya conjecture (2025, by Wang and Zahl) had been proven separately. OpenAI’s newly announced result extends the proof of the local smoothness conjecture to all dimensions, thereby establishing the corresponding conjectures in all dimensions.
Regarding the above achievements, OpenAI stated that it does not claim academic credit for discoveries made by its internal models; however, contributions by researchers in understanding the results, writing papers, and creating materials should be recognized.
The company also noted that it previously announced a $5 million donation to the Simons Foundation. This funding is managed and distributed by the foundation to support workshops, academic conferences, and targeted programs that help researchers understand AI-generated outcomes and build upon them.