{"slug": "openai-s-claim-that-the-exponential-time-hypothesis-is-false-explained-what-eth", "title": "OpenAI's claim that the Exponential Time Hypothesis is false, explained: what ETH said, how fast the algorithm really is, and what the Lean proof checks", "summary": "OpenAI announced on October 26, 2026 that its internal models produced three research results, including a proof that the Exponential Time Hypothesis is false — a claim the company says was found by AI on October 16 and formalized in Lean on October 17, with the proof and Lean files public. The ETH-refuting result gives a deterministic algorithm that counts every 3-SAT solution faster than 2 to a fixed fraction of n in polynomial space, but the gain is so slow that, with hidden constants set to 1, it only beats today's best 3-SAT algorithm past about 540 million variables, and the Lean theorem proves a weaker bound that beats plain 2 to the n only when n has about 2,400 digits. A working group led by Russell Impagliazzo and Ramamohan Paturi of UC San Diego and Ryan Williams of MIT — two of whom proposed ETH — will verify and explain the proof, and OpenAI said it seeks no academic credit, will fund a conference on the ETH result, and contributed $5 million to the Simons Foundation for workshops on AI-produced results.", "body_md": "# OpenAI's claim that the Exponential Time Hypothesis is false, explained: what ETH said, how fast the algorithm really is, and what the Lean proof checks\n\nOn October 26, 2026, OpenAI said its models had produced three results, including an algorithm that refutes the Exponential Time Hypothesis. This page explains the hypothesis in plain words, puts the claimed running time into numbers, and reads the Lean theorem to show what it does and does not pin down.\n\n**This explains reporting by**\n\n[OpenAI, \"Working with academic communities to build scientific understanding\" (October 26, 2026)](https://openai.com/index/building-scientific-understanding-with-academic-communities/).\nRead the original first:\n\n[https://openai.com/index/building-scientific-understanding-with-academic-communities/](https://openai.com/index/building-scientific-understanding-with-academic-communities/)\n\n## In one minute\n\n- OpenAI says its unreleased models produced three results: ETH is false, the local smoothing conjecture holds in every dimension (which settles Kakeya), and QCD exists as a rigorous quantum field theory.\n- ETH said 3-SAT needs time that grows like 2 to the power of a fixed fraction of n. The paper gives a deterministic algorithm that counts every solution faster than that, in polynomial space.\n- The gain grows very slowly. With the hidden constants set to 1, the general bound only beats today's best 3-SAT algorithm past about 540 million variables. The real constants are not stated.\n- The Lean theorem proves a weaker bound. Even with constants of 1, it beats plain 2 to the n only when n has about 2,400 digits. It settles the math question, not any practical one.\n- A working group led by Russell Impagliazzo, Ramamohan Paturi and Ryan Williams will examine the proof. Two of them proposed ETH.\n\n## What OpenAI announced\n\nOn October 26, 2026, OpenAI announced three results in mathematics, computer science and physics, produced by internal models. It calls the manuscripts \"raw research data\" and asked outside experts to act as shepherds for each one.\n\n- Math: a proof of Sogge's 1991 local smoothing conjecture in every dimension, which OpenAI says also settles the Kakeya conjecture and Stein's restriction conjecture in all dimensions. The AI output came on October 18 and was formalized in Lean the next day. Both are public.\n- Computer science: a proof that the Exponential Time Hypothesis is false. The AI found it on October 16 and it was formalized in Lean on October 17. The proof and the Lean files are public, and a group led by Russell Impagliazzo and Ramamohan Paturi (UC San Diego) and Ryan Williams (MIT) will verify and explain it.\n- Physics: an existence proof for quantum field theories including QCD and a completion of the Standard Model, which includes pure Yang-Mills theory. A group led by David Gross, who shared the 2004 Nobel Prize in Physics, will write it up, and the AI artifacts are not released. OpenAI says the Yang-Mills Lean formalization finished on October 16.\n\nOpenAI says it seeks no academic credit, that the working groups are unpaid and free to publish, and that it will fund a conference on the ETH result. It also points to a $5 million contribution to the Simons Foundation for workshops on AI-produced results.\n\n## What the Exponential Time Hypothesis said\n\n3-SAT asks whether you can set n true-or-false variables so that every clause of three variables is satisfied. Trying every setting takes 2 to the n steps.\n\nSmarter methods do better, but only by a constant factor in the exponent. The paper lists Schoning's local search at (4/3) to the n, and a recent preprint by Jiang and Cai at about 1.307 to the n.\n\nImpagliazzo and Paturi, with Francis Zane, proposed ETH. The paper cites their 2001 work. It says 3-SAT needs 2 to the power of (some fixed fraction of n) in the worst case. No trick would ever make the fraction shrink toward zero. OpenAI's post says ETH has been used in hundreds of papers to show other problems are hard.\n\nThe Strong ETH is a stronger claim than ETH, so if ETH is false, the Strong ETH falls too. P versus NP is untouched: the paper itself notes that bounds like this do not carry over to all NP problems under general reductions.\n\n## What the paper claims\n\nThe paper, dated October 17 and revised October 25, gives a deterministic algorithm that counts every satisfying assignment of a k-CNF formula, for each fixed k of 2 or more, using polynomial space.\n\n- General formulas: time (n + m) to a fixed power, times 2 to the power of O(n times the square root of ln ln n over ln n).\n- Formulas with at most D times n clauses: 2 to the power of O(n over the square root of ln n).\n\nCounting is harder than deciding, so the same algorithm decides 3-SAT and can find a solution one variable at a time. The paper also derives faster exact counting of graph colorings, independent sets and perfect matchings, and a new circuit lower bound.\n\n## How fast is it, in numbers\n\nBig-O hides constants, and the paper does not state them. So the table below sets every hidden constant to 1. That is our assumption, and the real constants could be much larger:\n\n```\nExponent per variable (bits of work per variable), with every hidden constant set to 1\nn (variables)        general bound   sparse bound   Lean statement   PPSZ (best known)\n1,000                0.53            0.38           1.42             0.386\n1,000,000            0.44            0.27           1.86             0.386\n1,000,000,000        0.38            0.22           2.02             0.386\n\ngeneral bound  = sqrt(ln ln n / ln n)     paper, Theorem 1.1, any clause count\nsparse bound   = 1 / sqrt(ln n)           paper, Theorem 1.1, clauses <= D times n\nLean statement = (ln ln n)^2 / sqrt(ln n) the weaker bound the Lean theorem proves\nPPSZ           = log2(1.307031578)        Jiang and Cai's randomized 3-SAT bound, cited in the paper\n```\n\nRead the table this way. The best known 3-SAT algorithm costs about 0.386 bits of work per variable. The new general bound only gets under that at about 540 million variables, even with a constant of 1. The sparse bound, which covers formulas with a fixed number of clauses per variable, gets under it at about 800 variables with a constant of 1.\n\nThat is why OpenAI's own post says turning this into practical improvements \"will require significant further research\". The result is about how the cost grows forever, not about any solver you run today.\n\n## What the Lean proof checks\n\nThe repository ships a Comparator challenge, a short Lean file that states the theorem in plain Lean, and a separate solution library that proves it. The statement, exact_count_and_sat, says: for each k of 2 or more there is a multi-tape machine and positive constants C, A, S, d and e such that, on every k-CNF formula, the machine halts with the exact count and the right yes-or-no answer, within C times (input bits + 1) to the d, times 2 to the power of A times rho(n) steps, and polynomial space.\n\nHere rho(n) is n times (ln ln (n+16)) squared over the square root of ln (n+16). The paper calls this the weaker exponent, from its Section 6.4. It is still smaller than any fixed fraction of n in the long run, so it does refute ETH.\n\nTwo limits matter:\n\n- The constants are only said to exist. The statement never names them, so the proof gives no bound you can compute for a real formula.\n- rho(n) divided by n only drops below 1 when ln n passes about 5,500, which means n has about 2,400 digits. Below that, even with constants of 1, the proven bound is worse than trying every setting.\n\nThe challenge allows only Lean's three standard axioms. The solution folder holds 3,977 files, about 35.5 MB, by GitHub's file listing on October 27. We did not build it ourselves. The README gives the command: lake env comparator ComparatorChallenges/SubexponentialSAT.json.\n\n## Context from the repository\n\n- The openai/math repository holds 721 manuscripts in 373 families. Its README says about 42% of the top-line results are formalized in Lean.\n- Each result used three hours of ChatGPT Pro thinking compute on average, out of about 4,000 problems posed to the model.\n- On October 7, OpenAI withdrew three manuscripts after a sign error broke an argument, and revised 14 others. Results without a Lean check can be wrong, and the README says so.\n\n## Who is affected\n\n| Case | Status | \n|---|---|\n| Complexity theory papers that assume ETH | Hardness results proved by assuming ETH lose that support if the refutation holds. The working group is checking it. | \n| Strong ETH and results built on it | SETH implies ETH, so SETH falls with it if the proof holds. | \n| SAT solvers and the tools that use them | No change today. The gain only appears at sizes far past real inputs, with constants not stated. | \n| P versus NP | Untouched, per the paper itself. | \n\n## What to do\n\n- Read the Lean challenge statement listed under Sources. It is about 220 lines and is the exact claim the proof checks.\n- If you work in complexity, follow the Impagliazzo, Paturi and Williams working group, which invites the community into an open examination.\n- Do not change any solver, security estimate or product plan on this result yet.\n- Treat AI-produced proofs without a Lean check as drafts. The October 7 withdrawals show why.\n\n## What is still unknown\n\n- Whether the proof holds up. Verification by the working group has only started, and their involvement is not an endorsement.\n- Whether the Lean statement exactly matches the intended claim. A Lean check proves the statement as written, so experts still have to confirm it says what ETH is about. We read it and believe it does, but that is our reading.\n- The size of the hidden constants, which the paper and the Lean statement leave unnamed.\n- The physics materials are not public, so nobody outside the Gross group can check the QCD claim yet.\n- The table's numbers assume constants of 1. They show how the bounds grow, not real running times.\n\n## Sources\n\n[AI News Report](https://theainewsreport.com/)· every headline, every morning.", "url": "https://wpnews.pro/news/openai-s-claim-that-the-exponential-time-hypothesis-is-false-explained-what-eth", "canonical_source": "https://theainewsreport.com/2026-10-27-openai-exponential-time-hypothesis-false-runtime-lean-explained.html", "published_at": "2026-10-27 15:31:24+00:00", "updated_at": "2026-10-27 15:43:32.289572+00:00", "lang": "en", "topics": ["artificial-intelligence", "ai-research", "large-language-models", "machine-learning"], "entities": ["OpenAI", "Exponential Time Hypothesis", "Russell Impagliazzo", "Ramamohan Paturi", "Ryan Williams", "UC San Diego", "MIT", "Simons Foundation"], "also_reported_by": [], "alternates": {"html": "https://wpnews.pro/news/openai-s-claim-that-the-exponential-time-hypothesis-is-false-explained-what-eth", "markdown": "https://wpnews.pro/news/openai-s-claim-that-the-exponential-time-hypothesis-is-false-explained-what-eth.md", "text": "https://wpnews.pro/news/openai-s-claim-that-the-exponential-time-hypothesis-is-false-explained-what-eth.txt", "jsonld": "https://wpnews.pro/news/openai-s-claim-that-the-exponential-time-hypothesis-is-false-explained-what-eth.jsonld"}}