# OpenAI's models say the Exponential Time Hypothesis is false (slim) > 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. - Full: https://www.anchorterminal.com/blog/openai-three-results.md (~3,100 tokens) · this version ~1,480 tokens · JSON https://www.anchorterminal.com/blog/openai-three-results.json · canonical https://www.anchorterminal.com/blog/openai-three-results - Index: https://www.anchorterminal.com/llms.txt · API: https://www.anchorterminal.com/api/v1/index.json · MCP: https://www.anchorterminal.com/mcp · Updated: 2026-10-27 - Licence: CC BY 4.0, credit Anchor Terminal and link this page (https://www.anchorterminal.com/terms/#data) 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. None of it has been checked by the wider community yet. - 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. It mattered because so much was built on it. It doesn't mean P equals NP. The group checking it is led by Impagliazzo and Paturi, the people who proposed ETH, with Ryan Williams from MIT [[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). People have been getting closer. ## Is the Yang-Mills result the Millennium Prize? No, and I think it's the claim most likely to get misreported. OpenAI claims existence. 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). 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 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). ## 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. ## 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. ### If ETH is false, does that mean P equals NP? No. ### Did OpenAI solve a Millennium Prize problem? They claim part of one. ### What is Lean? A proof assistant, software that checks every step of a proof written in its own language. ### Who is checking the results? For ETH, Russell Impagliazzo, Ramamohan Paturi and Ryan Williams. --- More in [Opinion](https://www.anchorterminal.com/blog/opinion/index.md) · 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. [2] Wikipedia, "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. [4] Wikipedia, "Yang-Mills existence and mass gap".