OpenAI Exponential Time Hypothesis disproof: what it means

OpenAI says one of its internal models has proved the Exponential Time Hypothesis false. The model found a way to count the solutions of a k-SAT formula in subexponential time, faster than complexity theorists have assumed for decades. Nobody has confirmed it yet. Two of the people who proposed the hypothesis are now organizing an open review with a third theorist, and the Lean proof is public, so anyone can look at it.

The posts

The news broke Monday afternoon. OpenAI scientist @danintheory (Dan Roberts) wrote "We are announcing three new discoveries in mathematics, computer science, and physics," and linked the company's post on working with academic communities. His post passed 650,000 views within hours.

A minute earlier, MIT professor @rrwilliams (Ryan Williams) had posted that "an internal model at OpenAI has produced a claimed refutation of ETH!" He says he, Russell Impagliazzo and Mohan Paturi were asked to organize "an open and independent examination of this claim." That lineup matters: Impagliazzo and Paturi proposed the hypothesis in the first place, with Francis Zane, and Williams has spent years probing its stronger variants.

@AndrewCurran_ gave the plain-language version. His list of the three announcements includes the line "The Exponential Time Hypothesis is false."

What ETH is, and what OpenAI claims

ETH is a bet about 3-SAT, the classic puzzle of finding true/false values that satisfy a set of three-variable clauses. It says that some fixed exponential, roughly 2 to the power cn for n variables, is a floor no algorithm can get under. Hundreds of papers build on it.

According to the scope document in OpenAI's repository, the Lean-checked theorem says that for any fixed k of at least 2, a deterministic machine can exactly count the satisfying assignments of a k-CNF formula using polynomial space and subexponential time. The formal bound is about 2 to the power of n(log log n)²/√log n, times a polynomial. A manuscript dated October 25 claims tighter bounds. Since counting solutions is at least as hard as finding one, this would sink ETH.

A few limits are worth keeping in mind:

• P vs NP is untouched. The running time is still superpolynomial, just smaller than any 2^(cn).
• You can't download a faster SAT solver out of this. OpenAI says practical algorithms "will require significant further research."
• Peer review hasn't happened. Williams describes the group's job as examining the claim, which is different from endorsing it.

OpenAI also agreed to two requests from the group: release the raw data behind the result, and fund a conference on what it means.

The other two results

The same announcement claims a proof of the local smoothing conjecture in every dimension. If it holds, the Kakeya and Stein restriction conjectures fall with it. The physics claim is a rigorous existence proof for four-dimensional quantum field theories including QCD, with pure Yang-Mills existence along the way, which is part of a Clay Millennium Prize problem. A group led by David Gross is checking that one and has not released the AI artifacts yet.

How to look at the proof yourself

Everything sits in the public openai/math repo (Apache-2.0). It now holds 721 manuscripts in 373 families, and the README says about 42% of top-line results are formalized in Lean. For the ETH claim:

To run the check, install comparator, landrun and lean4export and put them on your PATH. Then, from the lean/ folder, run lake update, lake exe cache get, and lake env comparator ComparatorChallenges/<challenge>.json. The repo's own example uses QuasiRiemannHypothesis.json. The README suggests compiling small pieces rather than the whole library, which can run into memory-map limits on Linux.

Nobody needs to run Lean to get something out of this, though. Read 378.md next to the manuscript's abstract and you'll see how far the machine-checked statement goes and where the paper's sharper claims begin.

My read

OpenAI's math work has been in the news all year. In August it was the $2,000 batch of solved open problems, and in July the model that kept escaping its sandbox. This lands two days after the company said it now trains frontier models offline. What I find most credible here is the process. A Lean proof, with the hypothesis's own authors running the review, beats a press release by a wide margin. I'd still wait for that review before repeating "ETH is false" as settled.

What a business can take from this: the most useful AI output comes with a checker attached. Lean does that for proofs. In a company, the checker might be a test suite, a reconciliation report or a person signing off. If you want help working out where AI can do real work in your operations and how you'd confirm it got things right, New Face Design's free process audit starts there.