OpenAI releases model-written preprint claiming the Exponential Time Hypothesis is false
OpenAI published on 26 October 2026 a preprint by an internal model giving a subexponential algorithm for counting k-SAT solutions, with a Lean formalization.
By Emre A.Published
Tech AI LatestOpenAI added a manuscript titled The Exponential Time Hypothesis is False to its public math repository on 26 October 2026. Produced by an unreleased internal model, it gives a deterministic algorithm that counts satisfying assignments of k-CNF formulas in subexponential time and polynomial space, which would refute a central assumption of complexity theory. A Lean formalization covers the main bound. Russell Impagliazzo, Ramamohan Paturi and Ryan Williams will lead an independent review.
Key facts
- Released: 26 October 2026 in the openai/math repository; manuscript dated 17 October, revised 25 October
- Claim: deterministic exact counting of k-CNF solutions in time (n+m)^O(1) times 2^(n/(log n)^(1/2+o(1))), polynomial space
- Consequence: refutes the Exponential Time Hypothesis, which says 3-SAT needs 2^Ω(n) time
- Verification: Lean formalization of a slightly weaker subexponential bound; independent review led by Impagliazzo, Paturi and Williams
- Author line: OpenAI; produced by an unnamed internal model
- Not stated: peer review status, practical speed on real SAT instances
What OpenAI released
OpenAI added two manuscripts with Lean formalizations to its public math repository on 26 October 2026, according to the repository's history file. One is titled The Exponential Time Hypothesis is False. Its author line reads OpenAI, and the repository says its manuscripts were produced by an unnamed internal model.
The paper gives a deterministic algorithm that exactly counts the satisfying assignments of any k-CNF formula. For fixed k, on n variables and m clauses, it runs in time (n+m) to a constant power times 2 to the n divided by (log n) to the power one half plus o(1), and uses polynomial space. Because 3-SAT is a special case, the authors say this refutes the Exponential Time Hypothesis (ETH), which holds that deciding 3-SAT needs time exponential in n in the worst case.
The second manuscript proves Sogge's local smoothing conjecture for the wave equation in every dimension from two up, from which the repository says the Kakeya maximal conjecture follows.
How it is being checked
The repository's Lean documentation says the formal proof shows a deterministic machine counts solutions in subexponential time and polynomial space, with a slightly weaker exponent than the paper's sharpest bound. Lean checks the logic of a formal statement, so the open question is whether that statement captures the claim fully.
OpenAI says an independent working group of computer scientists led by Russell Impagliazzo, Ramamohan Paturi and Ryan Williams will examine and explain the proof, and that the group asked for the materials to be made public immediately. Impagliazzo and Paturi introduced ETH in 2001; the paper cites their work.
The repository is candid about risk. It calls the manuscripts "raw research data, rather than polished academic papers" and notes that unformalized results "could have issues". On 7 October it withdrew three manuscripts after a sign error was found.
What changed and why it matters
ETH underpins a large body of complexity theory: many lower bounds for algorithms are stated as "unless ETH fails". If the result holds, those conditional bounds lose their footing, and the paper itself derives new consequences, including subexponential approximation algorithms for Max-k-SAT. The improvement is subexponential rather than polynomial, so it does not settle P versus NP and is unlikely to change everyday SAT solving.
For AI research the significance is the source. Earlier machine-assisted maths results, such as Microsoft Research's KLS work and Anthropic's Lean proof of Fermat's Last Theorem, formalised or advanced known problems. This is a model-produced claim against a widely believed conjecture, released for public checking before peer review.
What to watch next
The key event is the working group's assessment. Watch also for independent re-checks of the Lean statement, any corrections in the repository's version history, and journal submission. More research news is under AI Research.
What changed
| Before | The Exponential Time Hypothesis, that 3-SAT needs exponential time, was a standard unproven assumption in complexity theory |
|---|---|
| After | An OpenAI model-written preprint with a partial Lean formalization claims a subexponential counting algorithm that refutes it, pending independent review |
| Who it affects | Complexity theorists, researchers whose results assume ETH, and anyone assessing AI as a source of new mathematics |
Questions people ask
- What is the Exponential Time Hypothesis?
- A conjecture that deciding 3-SAT on n variables requires time 2 to the power of a constant times n in the worst case. Many algorithmic lower bounds assume it.
- Has the OpenAI ETH proof been verified?
- Partly. A Lean formalization covers a subexponential counting bound. An independent group led by Russell Impagliazzo, Ramamohan Paturi and Ryan Williams is reviewing the proof. It has not been peer reviewed.
- Does this mean P equals NP?
- No. The claimed algorithm is subexponential, not polynomial, so it does not resolve P versus NP.
Sources
- openai/math: History, October 26, 2026 (OpenAI, 26 Oct 2026)
- The Exponential Time Hypothesis is False (preprint) (OpenAI, 25 Oct 2026)
- Lean formalization scope: The Exponential Time Hypothesis is False (OpenAI, 26 Oct 2026)
Reference pages
- OpenAI company
Spotted an error? See the corrections policy and tell the editor. Facts in this story come from the sources listed above; vendor claims are attributed.
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