OpenAI AI Discoveries: Kakeya Proved, ETH Refuted, Yang-Mills Existence

OpenAI says its internal models have proved the local smoothing and Kakeya conjectures in every dimension, refuted the Exponential Time Hypothesis, and constructed quantum chromodynamics rigorously in four dimensions, and it is handing all three results to outside academic groups to verify and explain. The announcement, Working with academic communities to build scientific understanding, was published on October 26, 2026. It is as much about how AI-generated mathematics should be released, credited and checked as it is about the results themselves.

TLDR

OpenAI announced three AI-produced results: a proof of Sogge’s local smoothing conjecture in all dimensions (which settles the Kakeya and Stein restriction conjectures), a disproof of the Exponential Time Hypothesis, and a rigorous existence proof for QCD, pure Yang-Mills theory and a grand unified completion of the Standard Model. The math and computer science results come with public Lean proofs, and the Yang-Mills part of the physics result has been formalized too. Instead of publishing papers, OpenAI asked outside experts to act as “shepherds,” and each field chose a different release strategy, from a full public dump to a physics group that is holding the AI artifacts back. OpenAI says it wants no academic credit, imposes no NDAs, and is funding a conference plus a $5 million Simons Foundation program to help researchers absorb the work.

Thoughts

The Exponential Time Hypothesis result is the one that rearranges a field overnight. ETH says satisfiability of Boolean formulas cannot be solved in subexponential time, and for 25 years it has been the load-bearing assumption behind “this problem has no fast algorithm” claims in graph theory, computational biology, databases and computational geometry. If the refutation holds, every one of those conditional lower bounds loses its footing at once. That does not mean the problems suddenly became easy. OpenAI is careful to say that finding faster algorithms and turning the theory into practice “will require significant further research.” What changes immediately is the map of what is known: hundreds of papers that said “assuming ETH” now describe a world that does not exist, and the search for better algorithms is open again in places researchers had stopped looking. It also matters who is checking it. Russell Impagliazzo and Ramamohan Paturi proposed the hypothesis with Francis Zane, and Ryan Williams has spent years refuting its stronger cousins. Having the people with the most reputation invested in ETH lead the examination is about as credible a verification setup as one could design.

The physics claim is the biggest and the least visible. A rigorous construction of four-dimensional Yang-Mills theory is half of a Clay Millennium Prize problem, and the announcement extends it to QCD with quarks and to the simplest grand unified theory containing the Standard Model’s fields. Note what is and is not claimed: the post describes existence, “part of” the Millennium problem, and says nothing about the mass gap, which is the other half. And unlike the other two results, the artifacts are not public. David Gross’s group, which includes Arthur Jaffe (who wrote the Millennium problem statement with Edward Witten), Michael Douglas, Igor Klebanov and Nikita Nekrasov, chose to write the result up themselves before releasing the AI output. That is a defensible choice for a proof long enough that OpenAI waited from September 9 to October 16 for the Lean formalization to finish. It also means that, for now, the most consequential claim in the announcement is the one outsiders can least check.

The release experiment is more interesting than any single result, because it admits that the bottleneck has moved. The proofs arrive in days and are formalized in a day or two; the Kakeya output came on October 18 and was in Lean on October 19. Understanding is now the slow part. OpenAI’s answer is money and structure: a conference for the ETH work at the group’s request, and a $5 million contribution to the Simons Foundation for workshops and programs about AI-produced results. The company is also candid about the cost of its own design. Picking shepherds “necessarily shapes who gets involved first and creates asymmetry of information.” A handful of senior people now sit closest to three landmark results, and early-career researchers in the same fields find out from a blog post. The three different strategies are, in effect, a live test of which arrangement spreads understanding fastest with the least gatekeeping.

The credit policy quietly redefines the job. OpenAI says it seeks no academic credit for discoveries made by its models, and that the human groups should be recognized for “the understanding they develop, the writeups they author, and the other materials they create.” It also calls the manuscripts “raw research data.” That framing treats the machine as an instrument, like a telescope or a particle detector, and the scientist as the person who turns its output into knowledge. It is a generous arrangement, and probably the only workable one, but it shifts prestige from discovering a theorem to explaining one. Whether hiring and tenure committees accept that shift will decide whether the best young mathematicians want to spend years digesting proofs they did not find.

Key Takeaways

OpenAI announced three AI-generated results on October 26, 2026, in harmonic analysis, computational complexity and quantum field theory.
Its models proved Sogge’s 1991 local smoothing conjecture in every dimension, which also resolves the Kakeya conjecture and Stein’s restriction conjecture in all dimensions.
Before this, local smoothing was proved only in two spatial dimensions (Guth, Wang and Zhang, 2019) and Kakeya only in three dimensions (Wang and Zahl, 2025). The new proof rests on a novel auxiliary partial differential equation.
OpenAI says its system proved the Exponential Time Hypothesis is false, a conjecture used in hundreds of papers to derive conditional hardness results.
The ETH disproof was found on October 16 and formalized in Lean on October 17. The natural language writeup and Lean proof are on OpenAI’s public GitHub.
Russell Impagliazzo, Ramamohan Paturi (both UC San Diego) and Ryan Williams (MIT) lead the ETH group. Impagliazzo and Paturi originally proposed ETH with Francis Zane.
The physics result is a rigorous existence proof for interacting four-dimensional gauge theories with fermions, including QCD and a unified completion of the Standard Model, built on Balaban’s approach to constructive field theory.
As a subproblem it proves existence of pure Yang-Mills theory, part of the Clay Millennium Prize problem. The announcement does not claim the mass gap.
Nobel laureate David Gross leads the physics group with Michael Douglas, Arthur Jaffe, Igor Klebanov and Nikita Nekrasov, and that group is holding back the AI artifacts to write the result up itself.
Each field picked a different release model: math released the raw AI output, computer science formed a group and published everything, physics formed a group and deferred release.
The collaborations run on four principles: transparency (no NDAs), no academic credit for OpenAI, recognition for the human groups, and independence (voluntary and unpaid).
OpenAI agreed to fund a conference on the implications of the ETH result and points to a $5 million contribution to the Simons Foundation for workshops and programs on AI-produced research.
OpenAI acknowledges that choosing shepherds creates an asymmetry of information and asks for feedback on how to broaden participation.

Detailed Summary

Three Fields, Three Release Strategies

OpenAI describes the AI manuscripts as raw research data that still has to be turned into scientific understanding. For this release it asked outside experts to act as shepherds and organize groups around each result. The math shepherd recommended releasing the AI output and letting researchers organize themselves. The computer science shepherds formed a group to explain the result while making everything public. The physics shepherds formed a group and asked OpenAI to defer to them on what gets released and when.

Four Principles for the Collaborations

The groups operate under four stated rules. Participants sign no NDAs and can discuss, share and publish freely. OpenAI claims no academic credit for its models’ discoveries, and being asked to steward a result does not imply the stewards made the discovery. The human groups should be recognized for the explanations and writeups they produce, and they will make sure prior researchers and techniques are properly credited. Participation is voluntary and unpaid, and members do not speak for OpenAI.

The Tradeoffs OpenAI Admits

The announcement is open about the downside of its own design. Shepherded groups help experts unpack results and place them in the wider literature. But choosing who shepherds first shapes who gets involved first, which gives a small group an information advantage. OpenAI says it does not yet know which of the three strategies will work best and asks for feedback.

Local Smoothing and the Kakeya Conjecture

The local smoothing conjecture, proposed by Christopher Sogge in 1991, says that extreme focusing of waves must be short-lived: at a typical time, focusing is much weaker than at the worst-case time. It implies both the Kakeya conjecture and Stein’s restriction conjecture. Guth, Wang and Zhang proved it in two spatial dimensions in 2019, and Wang and Zahl proved Kakeya in three dimensions in 2025. OpenAI’s models produced a proof for every dimension using a new auxiliary PDE. The output arrived on October 18, was formalized in Lean the next day, and was released publicly in the same way as OpenAI’s October 6 math release.

The Exponential Time Hypothesis Is False

ETH is a conjecture about how hard it is to decide whether a Boolean formula can be satisfied, closely tied to P versus NP. It was widely believed and is used throughout theoretical computer science to show that problems have no fast algorithms. OpenAI says its system disproved it, with a Lean-verified proof. The result reopens how much faster many tasks could become, though OpenAI stresses that real algorithms and practical speedups will take substantial further work.

Who Is Checking the ETH Result

The group is led by Russell Impagliazzo and Ramamohan Paturi of UC San Diego, who proposed ETH with Francis Zane and have written many of the key papers on it, and Ryan Williams of MIT, who has refuted strong versions of ETH and connected algorithms to lower bounds. At the group’s request, all raw data is public for independent verification and OpenAI will fund a conference on what the result means for the field. The group is inviting the community to help organize an open examination and to discuss verification, access and credit for AI breakthroughs.

Rigorous QCD and a Unified Standard Model

The physics result gives a mathematically rigorous existence proof for interacting four-dimensional quantum field theories with gauge fields and fermions. That covers quantum chromodynamics and the simplest grand unified theory containing the Standard Model’s fields, since the Standard Model alone cannot be extended to arbitrarily short distances. Along the way it proves that pure Yang-Mills theory exists, part of a Millennium Prize problem. The work builds on Tadeusz Balaban’s program in constructive quantum field theory, and asymptotic freedom is the key mechanism that makes the construction possible.

The Physics Group and Its Timeline

David Gross, who shared the 2004 Nobel Prize for asymptotic freedom, leads a group of Arthur Jaffe, Michael Douglas, Igor Klebanov and Nikita Nekrasov. Jaffe co-founded the Clay Mathematics Institute and wrote the Yang-Mills problem with Witten, and Douglas has reported on the problem’s status since 2004. The models produced a Yang-Mills proof on September 9 and a QCD proof on September 18, and OpenAI waited for the Lean formalization, finished October 16, before announcing. The group will write the result up before any AI artifacts are released.

Funding the Work of Understanding

OpenAI ties the collaborations to its recent $5 million contribution to the Simons Foundation, which the foundation manages and awards for workshops, conferences and special programs on AI-produced results. The company says it is committed to funding independent examination of all three results and continues to ask for feedback on how best to disclose discoveries.

Notable Quotes

“In our view, these manuscripts are best understood as a form of raw research data.”
OpenAI, on how it frames AI-generated proofs

“We do not seek academic credit for the discoveries produced by our internal models.”
OpenAI, stating the No Academic Credit principle

“On the other hand, choosing initial shepherds necessarily shapes who gets involved first and creates asymmetry of information.”
OpenAI, on the tradeoffs of its release process

“Our models produced a proof of the local smoothing conjecture in every dimension and therefore also resolve the Kakeya conjecture and Stein’s restriction conjecture in all dimensions.”
OpenAI, on the harmonic analysis result

“Surprisingly, our system has recently proved that the ETH is false, a result that was subsequently verified in Lean.”
OpenAI, on the Exponential Time Hypothesis

“Finding such algorithms, and turning theoretical advances into practical improvements, will require significant further research.”
OpenAI, tempering expectations about faster algorithms

“We hope that these results mark the beginning of a transition in which quantum field theory, the most important framework in modern theoretical physics, becomes a mathematically rigorous field.”
OpenAI, on the QCD and Yang-Mills result

Read the full announcement from OpenAI here: Working with academic communities to build scientific understanding.