OpenAI said Monday that an unnamed internal model produced a proof that the Exponential Time Hypothesis is false, with a natural-language writeup and a Lean formalization now posted to its public GitHub math repository, according to a company post, The Verge and Al Jazeera. The claim is unverified and under independent review.
The Exponential Time Hypothesis is a longstanding conjecture about how hard Boolean satisfiability problems are. OpenAI said it has been widely believed and used in hundreds of papers for conditional hardness results. A working group led by Russell Impagliazzo and Ramamohan Paturi of UC San Diego, who helped propose the hypothesis, and Ryan Williams of MIT will examine the proof. OpenAI said it made the raw materials public immediately at the group’s request.
Williams, an MIT computational complexity expert, told Al Jazeera that a confirmed ETH refutation would be a major development. “The ETH has been widely believed true since the late 1990s and a great deal of theoretical computer science was built on it being true,” Williams said. “We are working to verify and examine their proof and determine if it holds up.” He said the claimed proof “proposes a mysterious new method to more efficiently solve a variety of fundamental problems that have a wide range of potential applications,” with any real-world benefits likely longer-term.
In the same drop, OpenAI said the model also produced a proof of Sogge’s local smoothing conjecture in every dimension, with materials on GitHub, and a mathematically rigorous existence proof for certain interacting quantum field theories in four dimensions, including quantum chromodynamics. OpenAI said the physics artifacts are not being released yet while a shepherd group writes them up. Alex Iosevich, a University of Rochester mathematics professor who specializes in harmonic analysis, told Al Jazeera that OpenAI appeared to have “cracked” the local smoothing conjecture in a “historic development.” He said the news was “a shock at first” and that he is now “filled with joy and optimism” about AI in theoretical math. Those broader claims are likewise pending verification.
The ETH claim sits alongside OpenAI’s earlier math-result release and its September Navier-Stokes Millennium claim. Independent verification is still pending.
Sources
- OpenAI: Working with academic communities to build scientific understanding (October 26, 2026)
- Al Jazeera (John Power): AI makes discoveries in mathematics, computer science, physics, OpenAI says (October 27, 2026)
- The Verge (Richard Lawler): OpenAI announces three big science discoveries and a new way to work with academics (October 26, 2026)
- GitHub: openai/math (ETH writeup and Lean formalization)