# OpenAI's models say the Exponential Time Hypothesis is false > On 26 October 2026 OpenAI said its internal models proved the local smoothing conjecture in every dimension, refuted the Exponential Time Hypothesis and proved that quantum chromodynamics exists. Vlad Cealicu on what each claim means in plain words, what has been checked and what hasn't, who is checking it, why the Yang-Mills result isn't the Millennium Prize, and the questions about credit. - Canonical: https://www.anchorterminal.com/blog/openai-three-results - Markdown: https://www.anchorterminal.com/blog/openai-three-results.md (~3,100 tokens) - Slim: https://www.anchorterminal.com/blog/openai-three-results.min.md (~1,480 tokens, same facts, less prose, for token-sensitive contexts) - JSON: https://www.anchorterminal.com/blog/openai-three-results.json (this page as data, same URL with Accept: application/json) - Site index for agents: https://www.anchorterminal.com/llms.txt (full text: https://www.anchorterminal.com/llms-full.txt) - API: https://www.anchorterminal.com/api/v1/index.json - MCP: https://www.anchorterminal.com/mcp (the whole catalogue as tools, no key) - Licence: CC BY 4.0, credit Anchor Terminal and link this page (https://www.anchorterminal.com/terms/#data) - Updated: 2026-10-27 By Vlad Cealicu, 27 October 2026. 1,522 words, about 7 min to read. OpenAI published "Working with academic communities to build scientific understanding" on Monday 26 October 2026, and in it they say their internal models produced three results [[1]](#ref-1), a proof of the local smoothing conjecture in every dimension, which would also settle the Kakeya conjecture and Stein's restriction conjecture, a disproof of the Exponential Time Hypothesis, which hundreds of papers have used as an assumption, and a rigorous proof that quantum chromodynamics exists as a mathematical object, with the existence of pure Yang-Mills theory along the way. Any one of these would be the work of someone's career, and they came out of the same lab in about six weeks. This is crazy progress. None of it has been checked by the wider community yet. All three were formalised in Lean, a proof assistant that checks every step, and two of those Lean proofs are public. The physicists checking the third have kept the AI's materials back for now [[1]](#ref-1). - 3: results, in maths, computer science and physics - 3: formalised in Lean, two of them public - 35: years since Sogge proposed local smoothing, in 1991 - $5M: to the Simons Foundation, for people to study results produced by AI What OpenAI announced on 26 October 2026. Source [1]. ## What does "the Exponential Time Hypothesis is false" mean? The Exponential Time Hypothesis, ETH for short, is a guess about how hard one problem is. The problem is 3-SAT, where you get a long formula of true or false variables, grouped in clauses of three, and you have to say whether some assignment of the variables makes the whole thing true. Russell Impagliazzo and Ramamohan Paturi proposed in 1999 that 3-SAT can't be solved in subexponential time, so as the number of variables grows the time has to grow like a fixed number above one raised to the power of that count [[2]](#ref-2). Nobody proved it, and it was widely believed to be true [[1]](#ref-1). It mattered because so much was built on it. If ETH is true, then graph colouring, finding a clique, finding a Hamiltonian cycle and vertex cover can't be done in subexponential time either [[2]](#ref-2), and OpenAI counts hundreds of papers that derive hardness results from it, in graph theory, computational biology, databases and computational geometry [[1]](#ref-1). If ETH is false, those results lose the assumption they stood on, and the question of how fast these problems can be solved is open again. It doesn't mean P equals NP. A subexponential algorithm can still be far too slow to be polynomial, and P equals NP needs polynomial. It doesn't mean faster SAT solvers next week either, and OpenAI says turning it into practical improvements "will require significant further research" [[1]](#ref-1). The group checking it is led by Impagliazzo and Paturi, the people who proposed ETH, with Ryan Williams from MIT [[1]](#ref-1). In an open letter, as The Next Web reports, they wrote "Our involvement should not be taken as certification of the claim", and that if it's correct it "will change an important part of theoretical computer science" [[3]](#ref-3). They asked for all the raw data to be made public and for OpenAI to fund a conference on what it means for the field, and OpenAI agreed to both [[1]](#ref-1). The natural language proof and the Lean proof are on OpenAI's GitHub [[1]](#ref-1). ## What are local smoothing and Kakeya? Christopher Sogge proposed the local smoothing conjecture in 1991. It's about waves as they focus, and OpenAI's own description is that the most extreme focusing has to be short-lived, so at a typical moment the focusing is much smaller than at the worst moment [[1]](#ref-1). It sits under a lot of analysis, and it implies the Kakeya conjecture, the old question of how small a set can be that holds a needle pointing in every direction, and Stein's restriction conjecture [[1]](#ref-1). People have been getting closer. Guth, Wang and Zhang proved local smoothing in two spatial dimensions in 2019, and Wang and Zahl proved Kakeya in three dimensions in 2025 [[1]](#ref-1). The model's proof does every dimension, through what OpenAI calls "a novel auxiliary partial differential equation" [[1]](#ref-1). They got the output on 18 October, had it formalised in Lean the next day, and on the advice of the expert they spoke to released both and left mathematicians to organise around it themselves [[1]](#ref-1). ## Is the Yang-Mills result the Millennium Prize? No, and I think it's the claim most likely to get misreported. The Clay Mathematics Institute's problem asks for two things, a proof that a quantum Yang-Mills theory exists on four-dimensional space to the standard of constructive quantum field theory, and a proof that it has a mass gap, so the lightest particle it predicts has a mass above zero [[4]](#ref-4). The prize is $1 million [[4]](#ref-4). OpenAI claims existence. Their proof builds on Balaban's approach and covers quantum chromodynamics, the theory of quarks and gluons, and the simplest grand unified theory that holds the Standard Model's fields, with pure Yang-Mills as a subproblem [[1]](#ref-1). They call it "part of" the Millennium problem [[1]](#ref-1), and they don't claim the mass gap. The model wrote the Yang-Mills proof on 9 September and the QCD proof on 18 September, and OpenAI held the release until the Lean formalisation of Yang-Mills existence finished on 16 October [[1]](#ref-1). David Gross leads the group checking it, with Michael Douglas, Arthur Jaffe, Igor Klebanov and Nikita Nekrasov, and Jaffe wrote the Millennium problem with Edward Witten [[1]](#ref-1). The group decided to keep the AI's materials back and write the result up themselves [[1]](#ref-1) [[3]](#ref-3). A Lean proof is only as good as the statement it ends on and the definitions under it, and for a physics result that's where I'd look first. from the model to the announcement, as OpenAI tells it 1. **9 September** (proof). natural language proof that Yang-Mills theory exists [[1]](#ref-1) 2. **18 September** (proof). the proof for quantum chromodynamics [[1]](#ref-1) 3. **6 October** (release). OpenAI's previous maths release, the approach reused for local smoothing [[1]](#ref-1) 4. **16 October** (proof). the ETH disproof is found, and the Lean proof of Yang-Mills existence finishes [[1]](#ref-1) 5. **17 October** (lean). the ETH disproof is formalised in Lean [[1]](#ref-1) 6. **18 October** (proof). local smoothing in every dimension, in Lean the next day [[1]](#ref-1) 7. **26 October** (groups). the announcement, with the computer science and physics groups named [[1]](#ref-1) 8. **Next** (not yet). community verification, the write-ups and the conference ## Who gets the credit? OpenAI tried three ways of releasing them. The maths result went out as it is, the computer science result went to an outside group with everything public, and the physics result went to an outside group that decides what gets released [[1]](#ref-1). The groups work to four principles they call transparency, no academic credit, recognition and independence, which in practice means no NDAs, OpenAI doesn't ask for academic credit for what its models found, the groups get credit for the understanding and the write-ups, and taking part is voluntary and unpaid [[1]](#ref-1). Williams said on X, The Next Web reports, that the group took no confidentiality agreements and won't be paid for the review [[3]](#ref-3). The reviewers raised the uncomfortable part themselves in their letter, that credit may go to "those who publish first or have the most powerful tools" while checking and explaining the results gets less recognition [[3]](#ref-3), and OpenAI's own post says picking the first shepherds "creates asymmetry of information" [[1]](#ref-1). The models are internal, so nobody outside the company can rerun the search that found these proofs. OpenAI doesn't ask for academic credit, and I think it gets the headline anyway. ## What I'm watching I'm waiting for the ETH group's write-up and the conference, because an algorithm someone can run would tell us more than any headline, for the physicists' write-up whenever they decide it's ready, and for what people who know Lean find when they go through the definitions. Until then these are three claims, two with public Lean proofs, and I'll update this post when the people checking them say what they found. ## The terms, explained ### What is the Exponential Time Hypothesis? A conjecture from 1999 by Russell Impagliazzo and Ramamohan Paturi that 3-SAT, deciding whether a formula of true or false variables can be made true, can't be solved in subexponential time. Hundreds of hardness results assumed it. ### If ETH is false, does that mean P equals NP? No. ETH being false means 3-SAT can be solved faster than exponential time, which can still be much slower than polynomial time. P equals NP needs polynomial time. ### Did OpenAI solve a Millennium Prize problem? They claim part of one. The Yang-Mills problem asks for existence and a mass gap, and OpenAI claims existence, for pure Yang-Mills and for quantum chromodynamics. The mass gap isn't claimed, and nothing has been verified by the community yet. ### What is Lean? A proof assistant, software that checks every step of a proof written in its own language. A proof that compiles in Lean is correct for the statement and definitions it was written against, so people still have to check those match the problem. ### Who is checking the results? For ETH, Russell Impagliazzo, Ramamohan Paturi and Ryan Williams. For the physics, a group led by David Gross with Michael Douglas, Arthur Jaffe, Igor Klebanov and Nikita Nekrasov. The maths result was released for anyone to work on. --- More in [Opinion](https://www.anchorterminal.com/blog/opinion/index.md) ยท Older opinion [Slopcore will be a thing](https://www.anchorterminal.com/blog/slopcore.md) - Older: [Can you run GLM-5.3-Flash locally? Yes, on one big machine](https://www.anchorterminal.com/blog/run-glm-5-3-flash-locally.md) - All posts: https://www.anchorterminal.com/blog/index.md ## References Read on 27 October 2026. Quotes are as published. [1] OpenAI, "Working with academic communities to build scientific understanding", 26 October 2026. Source for the three results and their dates, the release strategies, the four principles, the people in each group, the Lean formalisations, the raw data and conference commitments, the hundreds of papers using ETH, the Sogge, Guth, Wang, Zhang and Zahl history, and the $5 million to the Simons Foundation. https://openai.com/index/building-scientific-understanding-with-academic-communities/ [2] Wikipedia, "Exponential time hypothesis". Source for who proposed it and when, what it states about 3-SAT, and the problems whose hardness follows from it. https://en.wikipedia.org/wiki/Exponential_time_hypothesis [3] Ana Maria Constantin, "OpenAI says its AI model has disproved the Exponential Time Hypothesis", The Next Web, 26 October 2026. Source for the reviewers' open letter, Ryan Williams on confidentiality agreements and pay, the physics group writing its own version, and the reviewers' concern about credit. https://thenextweb.com/news/openai-exponential-time-hypothesis-false-kakeya-yang-mills [4] Wikipedia, "Yang-Mills existence and mass gap". Source for what the Clay Mathematics Institute problem asks, existence and a mass gap, and the $1 million prize. https://en.wikipedia.org/wiki/Yang%E2%80%93Mills_existence_and_mass_gap --- Query this catalogue directly over MCP: https://www.anchorterminal.com/mcp (Streamable HTTP, no key). How: https://www.anchorterminal.com/agents/#mcp