"Even Mathematicians Couldn't Solve Them for Decades": OpenAI Claims to Have Solved 3 of Humanity's Hardest Problems

New Results Announced in Computer Science and Physics
Nobel Laureates and Other Experts Begin Verification

There are claims that artificial intelligence (AI) has solved difficult problems that even world-renowned scientists had been unable to solve for decades. The claims follow an announcement by OpenAI, the developer of ChatGPT, that it had obtained new proofs for problems that had long remained unsolved in mathematics, computer science and physics. Observers say that, if the proofs are accepted, AI could develop beyond a tool that assists scientific research and become a research entity capable of producing new theories and discoveries.

OpenAI, the developer of ChatGPT, announced that it had found new proofs for problems in mathematics, computer science and physics that had remained unsolved for a long time. Reuters-Yonhap News Agency

On October 26 local time, OpenAI announced on its website that its own AI model had achieved new research results in three fields: the "Local Smoothing Conjecture" in mathematics, the "Exponential Time Hypothesis (ETH)" in computer science, and the "Mathematical Existence of Quantum Field Theory" in physics. OpenAI said that the model had generated new proofs for problems that had remained unresolved for a long time, going beyond simply summarizing existing research or reconstructing known theorems.

AI Claims Proof in All Dimensions for 35-Year-Old Mathematical Problem

In mathematics, OpenAI said it had solved the Local Smoothing Conjecture, proposed in 1991 by American mathematician Christopher Sogge. The conjecture holds that a phenomenon in which waves become extremely concentrated in a particular region of space cannot persist for a long time. Put simply, waves can become highly concentrated at a particular moment, but their average concentration over time is much lower than at that extreme moment.

On the 26th (local time), OpenAI announced on its website that its AI model had achieved new research results in three areas: the Local Smoothing Conjecture in mathematics, the Exponential Time Hypothesis (ETH) in computer science, and the mathematical proof of the existence of quantum field theory in physics. AP-Yonhap News

The problem is considered a major challenge in modern harmonic analysis and the study of partial differential equations. In 2019, mathematicians proved the conjecture in two-dimensional space, but whether it generally holds in higher dimensions remained an open question.

OpenAI said its model had found a proof that holds in all spatial dimensions by using a new auxiliary partial differential equation. According to OpenAI, if the proof is accepted, it could also resolve the "Kakeya Conjecture" and "Stein's Restriction Conjecture" in all dimensions, in addition to the Local Smoothing Conjecture.

The Kakeya Conjecture is a prominent problem that mathematicians have studied for many years. Research proving it in three dimensions was announced in 2025.

OpenAI said the AI generated the relevant proof on October 18 and formalized it the following day, October 19, using Lean, a mathematical proof verification system. The company posted the natural-language proof written by AI and the Lean proof materials online for researchers worldwide to review.

"A Theory Believed for Nearly 30 Years Was Wrong": Computer Scientists Also Take Notice

In computer science, OpenAI said it had obtained a proof that the Exponential Time Hypothesis, proposed in 1999, does not hold. The hypothesis states that the time required to solve certain computational problems must grow exponentially as the size of the problem increases. It has served as an important premise in computational complexity theory, which studies how efficiently computers can solve different problems. Hundreds of studies across fields such as graph theory, computational biology, databases and computational geometry have relied on the hypothesis to explain the computational difficulty of particular problems.

OpenAI said the AI generated a proof challenging the Exponential Time Hypothesis on October 16 and formalized it using Lean on October 17. If the result is ultimately accepted, it could lead to new approaches to some computational problems that have been considered difficult to solve quickly.

The review of the research will involve two of the researchers who first proposed the Exponential Time Hypothesis: Russell Impagliazzo and Ramamohan Paturi of the University of California, San Diego (UCSD), and Professor Ryan Williams of the Massachusetts Institute of Technology (MIT). The fact that researchers who proposed the hypothesis will directly review the AI's proof challenging it has drawn particular attention.

AI Proves, Humans Verify: Could the Approach to Scientific Research Change?

OpenAI also announced notable results in physics. It said it had obtained proofs establishing the mathematically rigorous existence of some four-dimensional quantum field theories, including quantum chromodynamics (QCD), which describes the interactions between quarks and gluons. QCD is a theory that explains the strong interaction binding particles inside atomic nuclei and is one of the core theories in the Standard Model of modern particle physics. Although the theory has successfully explained physical phenomena, formulating it as a rigorous mathematical system has long remained a challenge for both physics and mathematics.

OpenAI also proposed a new way to validate scientific research in announcing this achievement. Rather than treating AI-generated research findings as completed papers, the proposal is to regard them as raw data requiring verification and interpretation, which outside researchers can analyze and explain in terms of their academic significance. AP-Yonhap News.

OpenAI said the research also includes a proof of the existence of pure Yang-Mills theory. Yang-Mills theory is related to the "Yang-Mills Existence and Mass Gap" problem, one of the seven "Millennium Prize Problems" selected by the Clay Mathematics Institute. External review of the physics results will be led by David Gross, the 2004 Nobel Prize laureate in Physics. World-renowned theoretical physicists and mathematicians, including Michael Douglas, Arthur Jaffe, Igor Klebanov and Nikita Nekrasov, will also take part.

OpenAI said the AI model generated a proof of the existence of Yang-Mills theory on September 9 and obtained a proof related to quantum chromodynamics on September 18. It said the formalization of the Yang-Mills theory proof was completed on October 16. However, because the physics materials are extensive and complex, OpenAI will not immediately release the original AI-generated materials. Instead, it will publish them after the researchers have organized the content, in response to their request.

Alongside announcing these results, OpenAI proposed a new way to validate scientific research. Rather than treating AI-generated findings as completed papers, the approach views them as raw data requiring verification and interpretation, which external researchers then analyze and explain in terms of their academic significance. OpenAI also said it would not claim academic credit for the research results. External researchers will participate voluntarily and independently, without nondisclosure agreements, and will receive credit for their contributions to the verification and interpretation process. OpenAI also plans to support initiatives such as holding academic conferences to independently verify AI-generated research results and advance them through further research.

This content was produced with the assistance of AI translation services.