OpenAI announced three AI research achievements: "ETH" has been overturned, quantum field theory has been validated, and the Hanging Valley Conjecture has been solved?

Author: OpenAI

Compiled by: Jiahua, ChainCatcher

On October 26, OpenAI announced three research results generated by its internal models, involving computer science, physics, and mathematics.

These results touch on fundamental questions that have been long studied in their respective fields. OpenAI refers to the documents generated by the models as "raw research data" and invites the academic community to further verify, interpret, and conduct follow-up research.

Computer Science: Is ETH Overturned?

OpenAI stated that its system proved the Exponential Time Hypothesis (ETH) is false, a result that was 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 time consumption will grow explosively with the size of the problem, without any real shortcuts.

Therefore, the overturning of ETH reopens the question of "how much faster can these tasks actually be done," and may provide direction for finding better algorithms. OpenAI also pointed out that finding such algorithms and translating theoretical advances into practical improvements will require a lot of follow-up research.

For readers in the cryptocurrency space, a more direct question is: will this affect existing cryptographic systems? The security of many cryptographic schemes relies on certain specific computational problems being hard to solve. MIT professor Ryan Williams had asked OpenAI to check the relevant impacts. According to his account, OpenAI has conducted checks and is reasonably confident that this result does not imply that existing cryptographic systems are compromised.

Physics: Providing a "Foundation" for Quantum Field Theory

Physicists use quantum field theory to describe the particle world, predicting with extreme precision, but its mathematical foundation is not rigorous.

OpenAI stated that the model provided a rigorous "existence" proof for pure Yang-Mills theory, quantum chromodynamics (QCD, which describes the strong force between quarks and gluons), and a grand unified theory that includes the standard model gauge fields. In simple terms, it provides a mathematical foundation for a theory that has been used for decades.

Currently, only the Yang-Mills part has completed verification in Lean, and the QCD part has not been mentioned in the announcement. The original proof generated by AI is not publicly available yet, as it is being studied by a group led by Nobel laureate David Gross, who will write the document themselves, so external verification is temporarily not possible.

Mathematics: Wave Focusing and the "Needle Problem"

OpenAI stated that its model proved the local smoothness conjecture in all dimensions, thereby solving the hanging valley conjecture and the Stein restriction conjecture in all dimensions.

The local smoothness conjecture was proposed by Sogge in 1991, studying the behavior of waves when they focus. An intuitive analogy can be made with water waves: several ripples propagate towards the same position, and at a certain moment, they may superimpose to create a very high peak. However, how intense that moment is does not directly represent the average performance over the entire duration.

The local smoothness conjecture aims to use precise mathematical estimates to describe the relationship between this instantaneous extreme performance and the time average. Since this phenomenon has fundamental significance, proving it will also impact other mathematical problems, including the hanging valley conjecture and the Stein restriction conjecture.

The hanging valley conjecture can be understood through the "needle problem": imagine placing many needles of the same length and infinitesimal thickness in space, ensuring that there is at least one needle in every direction. These needles can be translated and can overlap with each other. How small can the set they form be compressed?

The key to measuring "size" here is the mathematical dimension. In three-dimensional space, the hanging valley conjecture posits that such a set must have full three-dimensionality; in higher-dimensional spaces, it must also have the corresponding full dimensionality. Dimension and volume are different concepts, so it cannot be simply understood as how many cubic meters these needles occupy.

Previously, the two-dimensional local smoothness (2019) and the three-dimensional hanging valley conjecture (2025, Wang Hong and Zahl) have been proven. The results announced by OpenAI this time advance the proof of the local smoothness conjecture to all dimensions, thereby obtaining conclusions for the related conjectures in all dimensions.

Regarding the above results, OpenAI stated that it does not claim academic credit for discoveries generated by its internal models; the contributions of researchers in understanding the results, writing documents, and producing materials should be recognized.

The company also mentioned that it had previously announced a donation of $5 million to the Simons Foundation. This funding is managed and distributed by the foundation for workshops, academic conferences, and special programs to help researchers understand the results generated by AI and conduct research based on them.