OpenAI Reports AI Research in Math, Science

OpenAI said on October 26, 2026, local time, that its internal AI models produced research results on difficult problems in three fields: harmonic analysis, computational complexity theory and quantum field theory. The company described AI-generated manuscripts as “raw research data” and outlined a process for releasing and checking them so researchers can understand, verify and build on the work.

In mathematics, OpenAI said a model generated a proof of the local smoothing conjecture in all dimensions, concerning how waves behave when they concentrate. The published formalization also describes results on Kakeya sets, collections of points containing a unit-length line segment in every direction. OpenAI said the work also leads to resolving the Kakeya conjecture and Stein restriction conjecture in all dimensions. The AI output was produced on October 18 and formalized in Lean the following day. Lean is a language and system for writing mathematical proofs in a form computers can verify. The natural-language manuscript and formalization materials are available on GitHub.

In computer science, OpenAI announced a refutation of the exponential time hypothesis (ETH), a conjecture that, in the worst case, solving 3-SAT—determining whether any assignment of true and false values satisfies a logical formula—cannot avoid exponential growth in computing time as the number of variables increases. The released materials describe a subexponential-time procedure that limits this growth; they do not show that the problem can be solved in polynomial time. A group led by Russell Impagliazzo and Ramamohan Paturi of the University of California, San Diego, and Ryan Williams of the Massachusetts Institute of Technology will verify and explain the work. Impagliazzo and Paturi proposed ETH. OpenAI released a natural-language manuscript and a Lean formalization. The refutation does not solve the P versus NP problem, and further research is needed to turn the theoretical result into practical speedups, the company said.

In physics, OpenAI said a model produced a mathematically rigorous existence proof for four-dimensional interacting quantum field theories, including quantum chromodynamics (QCD), which describes the strong interaction between quarks and gluons. The results also include an existence proof for pure Yang–Mills theory. However, the Clay Mathematics Institute’s Millennium Prize Problem also requires a proof of the “mass gap,” so the full problem was not solved by this announcement. A group led by Nobel laureate David Gross will interpret and explain the work. It chose to hold back the AI-generated materials while preparing its own account.

Release procedures differ by field. OpenAI published the mathematics materials for independent researcher review, while the computer science materials are being released alongside verification and explanation by an outside group. In physics, the outside group will decide when the materials are released. OpenAI said it did not require participants in the outside groups to sign nondisclosure agreements, and they may freely share or publish interim results and conclusions. The company said it is not seeking academic credit for discoveries made by its internal models.

OpenAI said on October 26, 2026, local time, that its internal AI models produced research results on difficult problems in three fields: harmonic analysis, computational complexity theory and quantum field theory. The company described AI-generated manuscripts as “raw research data” and outlined a process for releasing and checking them so researchers can understand, verify and build on the work.

In mathematics, OpenAI said a model generated a proof of the local smoothing conjecture in all dimensions, concerning how waves behave when they concentrate. The published formalization also describes results on Kakeya sets, collections of points containing a unit-length line segment in every direction. OpenAI said the work also leads to resolving the Kakeya conjecture and Stein restriction conjecture in all dimensions. The AI output was produced on October 18 and formalized in Lean the following day. Lean is a language and system for writing mathematical proofs in a form computers can verify. The natural-language manuscript and formalization materials are available on GitHub.

In computer science, OpenAI announced a refutation of the exponential time hypothesis (ETH), a conjecture that, in the worst case, solving 3-SAT—determining whether any assignment of true and false values satisfies a logical formula—cannot avoid exponential growth in computing time as the number of variables increases. The released materials describe a subexponential-time procedure that limits this growth; they do not show that the problem can be solved in polynomial time. A group led by Russell Impagliazzo and Ramamohan Paturi of the University of California, San Diego, and Ryan Williams of the Massachusetts Institute of Technology will verify and explain the work. Impagliazzo and Paturi proposed ETH. OpenAI released a natural-language manuscript and a Lean formalization. The refutation does not solve the P versus NP problem, and further research is needed to turn the theoretical result into practical speedups, the company said.

In physics, OpenAI said a model produced a mathematically rigorous existence proof for four-dimensional interacting quantum field theories, including quantum chromodynamics (QCD), which describes the strong interaction between quarks and gluons. The results also include an existence proof for pure Yang–Mills theory. However, the Clay Mathematics Institute’s Millennium Prize Problem also requires a proof of the “mass gap,” so the full problem was not solved by this announcement. A group led by Nobel laureate David Gross will interpret and explain the work. It chose to hold back the AI-generated materials while preparing its own account.

Release procedures differ by field. OpenAI published the mathematics materials for independent researcher review, while the computer science materials are being released alongside verification and explanation by an outside group. In physics, the outside group will decide when the materials are released. OpenAI said it did not require participants in the outside groups to sign nondisclosure agreements, and they may freely share or publish interim results and conclusions. The company said it is not seeking academic credit for discoveries made by its internal models.