On October 26, 2026, OpenAI said one of its internal AI models had disproved the Exponential Time Hypothesis (ETH), a computer science belief that's stood since 1999. If the claim holds up, it's one of the biggest AI-meets-math stories of the year. It isn't confirmed yet, though. This guide covers the OpenAI ETH disproof claim, the rival result Anthropic reportedly posted a day later, and what the experts checking it are actually saying.
OpenAI says yes, but nobody outside the labs has confirmed it yet. On October 26, 2026, OpenAI claimed its model found an algorithm that solves 3-SAT in subexponential time, which would make the Exponential Time Hypothesis false. A Lean proof checker accepted a formal version of the argument. A review group led by the people who proposed ETH is now examining it, and it says its involvement isn't a certification. The result is a preprint, not a peer-reviewed paper.
- What is the Exponential Time Hypothesis?
- What OpenAI claimed, and when
- The rival result from Anthropic
- What "Lean-verified" does and doesn't mean
- The claim vs what reviewers say
- What this does NOT mean
- Why it matters if it's true
- The credit and process argument
- Reasons to trust it, reasons to wait
- How we researched this
- Frequently asked questions
- Our verdict
- Sources
What Is the Exponential Time Hypothesis?
The Exponential Time Hypothesis is a bet about how hard one famous puzzle really is. It says a problem called 3-SAT can't be solved much faster than brute force. Russell Impagliazzo and Ramamohan Paturi formulated it around 1999, and the 2001 journal papers with Francis Zane made it a standard tool in the field.
3-SAT works like this. You get a big list of small conditions, each linking three true/false switches. Something like "switch A is on, or switch B is off, or switch C is on." Your job is to find one setting of all the switches that satisfies every condition at once. Checking a proposed answer is quick. Finding one seems to get brutally slow as the number of switches grows.
๐ง In plain terms: ETH says that every time you add a few more switches, the work needed roughly multiplies. It never settles into something gentle. If OpenAI is right, there's a cleverer way around that wall than anyone believed for 27 years.
Why should anyone outside a theory lab care? Because researchers used ETH as a yardstick. According to Wikipedia's overview of ETH, it implies that many classic problems (graph coloring, vertex cover, maximum clique and others) have no subexponential algorithms. OpenAI says hundreds of papers lean on it to show that certain problems can't be sped up much.
What OpenAI Claimed, and When
Here's the timeline as OpenAI and the press tell it. The dates for the discovery and the Lean check come from OpenAI's own announcement, so treat them as the company's account.
What does the algorithm do? According to Lance Fortnow's summary, as relayed by StudyFinds, OpenAI claims 3-SAT can be solved, and its solutions counted, in 2o(n) time. That's subexponential, which is exactly what ETH says is impossible. The manuscript reportedly says the same algorithm also refutes the stronger Strong ETH.
OpenAI isn't claiming it settled P vs NP. It says the result "reopens questions" about how much faster these problems could get, and that practical gains "will require significant further research." The raw files sit in the company's public GitHub repository, and OpenAI's own write-up is on its announcement page.
๐ Worth knowing: OpenAI hasn't said which model produced the proof. You may see guesses online tying it to specific unreleased models. Those are guesses. The company only says "internal model."
The Rival Result From Anthropic
About a day after OpenAI's announcement, a second lab entered the story. Reports say Anthropic posted a repository called "A 2^o(n) Time Algorithm for 3-SAT," crediting an internal Claude model. Its README reportedly describes the result as purely asymptotic and says nothing about solving real formulas faster. Scientific American's headline calls the pair "dueling proofs."
Two things to keep straight. First, the coverage of Anthropic's result is thinner than OpenAI's, and early reports relied on social posts and a single news outlet before more coverage followed. We haven't independently verified the repository. Second, two labs landing on a similar headline result in the same week is striking, but it isn't proof that either one is right. Both are claims until the community says otherwise.
There's also a side dispute. As reported by Stanford Tech Review, citing TechCrunch, an Anthropic mathematician suggested OpenAI's work built on pre-existing results. That's one person's comment about a rival, and we're passing it along as reported, not as fact.
What "Lean-Verified" Does and Doesn't Mean
The phrase sounds like the end of the debate. It isn't. Lean is a proof assistant. You write a statement and a proof in its strict language, and Lean checks that every step follows from the rules. A Lean-checked proof is very hard to fake.
But Lean only checks the statement you wrote down. It can't tell you whether that formal statement says what the English paper says. Reviewers have to judge that match by hand. Reports also note a gap in the numbers: the bound Lean checked is weaker than the headline bound in OpenAI's paper. It's still subexponential, but it isn't the same claim.
๐ก The honest version: Lean passing means the logic inside the formal file is sound. It doesn't yet mean "ETH is false in the sense everyone cares about." Closing that last gap is exactly what the human reviewers are doing.
The Claim vs What Reviewers Say
The outside review is led by Impagliazzo, Paturi and Ryan Williams of MIT. They say they signed no NDAs and aren't paid. In an open letter they wrote that being involved isn't the same as certifying the claim. OpenAI agreed to release all the raw data and fund a conference on the result. Here's how the main questions line up.
| Question | What OpenAI says | What reviewers and reports say |
|---|---|---|
| Is ETH false? | Yes, 3-SAT is solvable in subexponential time | Not confirmed. Review is ongoing |
| Is it verified? | A Lean proof checker accepted it | Lean checks logic, not that the formal statement matches the paper |
| Peer reviewed? | Manuscripts are "raw research data" | No. These are posted manuscripts |
| Faster in practice? | Needs "significant further research" | Asymptotic only, no practical solver change |
| Solves P vs NP? | Doesn't claim that | Still open |
| Breaks encryption? | Reasonably confident no cryptosystem is broken | RSA and elliptic-curve security don't depend on ETH |
What have the named experts said so far? Paturi told the Washington Times his first reaction was surprise, and he called the result a potential major advance if it holds. Williams, as reported through TechCrunch, found parts of the proof routine and a couple of parts unfamiliar, and doesn't think OpenAI's algorithm is likely the best possible. Fortnow wrote that the claim doesn't reduce running times of NP problems in general. Those are paraphrases of secondhand reports, so check the originals before quoting anyone.
What This Does NOT Mean
Headlines like "AI breaks 27-year-old barrier" invite big leaps. Here's the grounded version.
- It doesn't prove P = NP. A subexponential algorithm is still far slower than a polynomial one. If ETH fails, that says nothing about P vs NP.
- It doesn't break online security. Mainstream encryption like RSA and elliptic curves doesn't rest on ETH. Williams reportedly said OpenAI checked this and was reasonably confident no cryptosystem is broken.
- It won't speed up your software. Subexponential is an asymptotic idea about huge inputs. Stanford Tech Review ran a rough estimate with hidden constants set to 1 and found the Lean-checked bound only beats brute force beyond an astronomically large number of variables. Treat that as one analyst's back-of-envelope figure, not a measured benchmark.
- It doesn't mean every NP problem just got easy. The paper doesn't show that general NP problems have subexponential algorithms.
- It doesn't mean the proof is correct. It means two labs say they have one, and experts are checking.
Why It Matters If It's True
For theorists, a lot of "this problem can't be sped up" results were proven assuming ETH. If ETH falls, those conditional results lose their footing, even if the underlying problems stay hard for other reasons. The review group's open letter makes the point that reductions built on ETH keep their value even if some readings of them change.
For everyone else, the bigger story is the process. A frontier AI lab says a model produced a serious research-level result in days, and a formal checker accepted the logic. That's a different kind of claim from "AI scored well on a benchmark." It's also why we're watching the review so closely. If it survives, it changes what people expect AI research assistants to do. If it doesn't, it's a useful reminder of why checking matters. For more on how fast frontier models are moving, see our explainers on GPT-5.6 Sol and OpenAI's GPT-6 Astra, and our ChatGPT vs Claude vs Gemini vs DeepSeek comparison.
The Credit and Process Argument
The math isn't the only thing people are arguing about. OpenAI chose who sees its results first, and some mathematicians didn't like it. Reporting summarized by explainx says critics objected to a lab picking early reviewers, and OpenAI itself acknowledged that its choice creates an "asymmetry of information." The review group also warned that credit could end up with whoever publishes first or owns the strongest tools.
The same OpenAI announcement covered two other claims, in harmonic analysis and quantum field theory. We're not covering those here. They have their own review processes and their own status.
For a wider look at how governments and labs are reframing AI progress, read our piece on why the White House is calling it "Super Intelligence", and our report on Claude Fable 5's US government ban.
Reasons to Trust It, Reasons to Wait
How We Researched This
We want to be upfront about what we did and didn't do. We did not check the proof. We're not a math lab, and we haven't run OpenAI's or Anthropic's code. What we did was read the primary announcement and the public repository references, cross-check dates and claims across several independent news and analysis outlets, and note where a detail came from only one source.
Our rules for this article: attribute every claim to who made it, say "claims" and "reports" instead of "proves" and "confirms," flag single-source details, and skip anything we couldn't tie to a source. We'll update this page when the review group publishes a verdict. If you spot an error, please tell us.
Frequently Asked Questions
The Exponential Time Hypothesis (ETH) is an unproven assumption that 3-SAT can't be solved in subexponential time. In other words, the work needed grows at least exponentially with the number of variables. Russell Impagliazzo and Ramamohan Paturi formulated it around 1999, and many "can't be sped up" results in computer science assume it.
OpenAI claims it did, but that hasn't been independently confirmed. The company announced on October 26, 2026 that an internal model found a subexponential algorithm for 3-SAT. A Lean checker accepted a formal version, and outside experts are reviewing it. The work is a posted manuscript, not a peer-reviewed paper.
Reports say Anthropic posted a repository called "A 2^o(n) Time Algorithm for 3-SAT" around October 27, crediting an internal Claude model. It reportedly claims a similar headline result and says it's purely asymptotic. Coverage is thinner than for OpenAI's claim, and we haven't independently verified it.
No. A subexponential algorithm is still much slower than a polynomial one. Failure of ETH doesn't say anything about P vs NP, and OpenAI doesn't claim to have solved it. Lance Fortnow wrote that there's still no known approach to settling P vs NP.
No. Common systems like RSA and elliptic-curve cryptography don't depend on ETH. Ryan Williams reportedly said OpenAI checked this and was reasonably confident no cryptosystem is broken. Your bank and your passwords aren't affected by this result.
Lean is a proof assistant that checks every logical step of a formal proof. If Lean accepts it, the logic inside that file is sound. It doesn't check whether the formal statement matches the claim in the paper, so human experts still have to confirm that match.
A group led by ETH's proposers Russell Impagliazzo and Ramamohan Paturi, along with Ryan Williams of MIT. They say they signed no NDAs and aren't paid. Their open letter says their involvement isn't a certification of the claim.
OpenAI hasn't said. It describes the source only as an internal model. Any specific model name you see online is a guess. Anthropic credits an internal Claude model for its own result and also hasn't given a public product name.
Not in practice. The result is asymptotic, meaning it concerns extremely large inputs. One analysis estimates the Lean-checked bound beats brute force only at an astronomically large number of variables. OpenAI itself says practical gains would need significant further research.
Researchers use ETH as a baseline to show that many problems can't be solved much faster than known methods. OpenAI says hundreds of papers rely on it. If it fails, some of those conditional results lose their motivation, though the problems may still be hard for other reasons.
Nobody has given a date. Checking a long, novel proof by hand can take weeks or months, and formal peer review takes longer. Watch for a statement from the review group and for the planned conference OpenAI agreed to fund.
Our Verdict
This is a serious claim, not a stunt. A Lean checker accepted a formal version, the people who invented ETH are reviewing it, and a second lab reports something similar. But "serious" isn't "settled." The Lean bound is weaker than the headline, the match between formal and informal statements still needs human sign-off, and the reviewers say they aren't certifying anything. Our advice: stay interested, stay skeptical, and don't repeat "ETH is dead" as fact until the review group says so. Whatever the outcome, none of it touches your encryption, and none of it changes P vs NP.
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Sources
- OpenAI: building scientific understanding with academic communities (primary announcement)
- OpenAI math repository on GitHub (manuscripts and Lean files)
- Wikipedia: Exponential time hypothesis (background)
- Lean theorem prover
- Scientific American: Dueling proofs from OpenAI and Anthropic astound scientists
- Washington Times: Did AI just solve a 30-year-old computer science puzzle? Experts are checking
- StudyFinds: Did OpenAI bots disprove the Exponential Time Hypothesis?
- Stanford Tech Review: OpenAI, Anthropic say Exponential Time Hypothesis is false
- explainx: OpenAI's three claims and the Lean collaborations
