Has OpenAI's Exponential Time Hypothesis proof been verified?

OpenAI's Exponential Time Hypothesis result has a Lean formalization in the openai/math repository, which checks a subexponential-time, polynomial-space counting algorithm for fixed-width CNF formulas. Human review is still under way: an independent group led by Russell Impagliazzo, Ramamohan Paturi and Ryan Williams is examining the proof, and OpenAI published the materials so others can examine them too.

What exactly does OpenAI's ETH manuscript claim?

The OpenAI manuscript dated October 25, 2026 claims a deterministic algorithm that exactly counts satisfying assignments of k-CNF formulas in time 2 to the power O(n·log log n/√log n) times a polynomial, using polynomial space. Because that is below 2^(cn) for 3-SAT, OpenAI says the case of 3-SAT refutes the Exponential Time Hypothesis.

Does the OpenAI result prove that P equals NP?

No. The OpenAI algorithm is still superpolynomial, so the Exponential Time Hypothesis result does not show that P equals NP. ETH is a stronger assumption: it says 3-SAT needs exponential time 2^(cn). A subexponential algorithm breaks ETH while leaving the P versus NP question open.

What did OpenAI prove about the Kakeya conjecture?

OpenAI's second October 2026 manuscript establishes Sogge's local smoothing conjecture for the Euclidean wave equation by using an auxiliary partial differential equation. According to the openai/math release history, that result also yields the Kakeya maximal and dimension conjectures and Stein's spherical restriction estimates, and it ships with a Lean formalization.

OpenAI · 2026-10-26 · major

OpenAI says its model refuted the Exponential Time Hypothesis — Lean-checked

OpenAI released AI-written proofs that claim the Exponential Time Hypothesis is false and that settle Sogge's local smoothing conjecture, with Kakeya results. Both ship with Lean formalizations; outside experts will check them.

GitHub card for the openai/math repository of AI-written manuscripts

OpenAI's internal model claims a subexponential algorithm for counting SAT solutions, which would break a core assumption of complexity theory.

Quick facts

Maker	OpenAI
CS result	Exponential Time Hypothesis is false (claimed)
Math result	Sogge's local smoothing; Kakeya maximal and dimension conjectures
Verification	Lean formalizations in openai/math
ETH reviewers	Group led by Impagliazzo, Paturi and Williams
Manuscript date	October 25, 2026

What is it?

OpenAI added two new AI-written manuscripts to its public openai/math repository on October 26, 2026, both with Lean formalizations. The headline one is titled 'The Exponential Time Hypothesis is False'. The other proves Sogge's local smoothing for the Euclidean wave equation, which also yields the Kakeya maximal and dimension conjectures and Stein's spherical restriction estimates.

How does it work?

The complexity result is a deterministic algorithm that exactly counts the satisfying assignments of k-CNF formulas with a fixed clause width. The Lean statement proves a running time of 2 to the power O(n·(log log n)²/√log n) using polynomial space — slower than polynomial, but faster than any 2^(cn), which ETH says is impossible for 3-SAT. The analysis proof instead builds an auxiliary partial differential equation to reach local smoothing.

Why does it matter?

The Exponential Time Hypothesis underpins many hardness results, so a valid refutation would force researchers to revisit lower bounds that assume it. OpenAI made the materials public at the request of an independent group led by Russell Impagliazzo, Ramamohan Paturi and Ryan Williams, who are examining the proof. Early Hacker News discussion is mostly skeptical, so treat the claim as unconfirmed until that review reports.

Who is it for?

complexity theorists, analysts, formal-verification researchers

Frequently asked questions

Has OpenAI's Exponential Time Hypothesis proof been verified?: OpenAI's Exponential Time Hypothesis result has a Lean formalization in the openai/math repository, which checks a subexponential-time, polynomial-space counting algorithm for fixed-width CNF formulas. Human review is still under way: an independent group led by Russell Impagliazzo, Ramamohan Paturi and Ryan Williams is examining the proof, and OpenAI published the materials so others can examine them too.
What exactly does OpenAI's ETH manuscript claim?: The OpenAI manuscript dated October 25, 2026 claims a deterministic algorithm that exactly counts satisfying assignments of k-CNF formulas in time 2 to the power O(n·log log n/√log n) times a polynomial, using polynomial space. Because that is below 2^(cn) for 3-SAT, OpenAI says the case of 3-SAT refutes the Exponential Time Hypothesis.
Does the OpenAI result prove that P equals NP?: No. The OpenAI algorithm is still superpolynomial, so the Exponential Time Hypothesis result does not show that P equals NP. ETH is a stronger assumption: it says 3-SAT needs exponential time 2^(cn). A subexponential algorithm breaks ETH while leaving the P versus NP question open.
What did OpenAI prove about the Kakeya conjecture?: OpenAI's second October 2026 manuscript establishes Sogge's local smoothing conjecture for the Euclidean wave equation by using an auxiliary partial differential equation. According to the openai/math release history, that result also yields the Kakeya maximal and dimension conjectures and Stein's spherical restriction estimates, and it ships with a Lean formalization.

Try it

git clone https://github.com/openai/math