OpenAI and Anthropic Put the Exponential Time Hypothesis Under Review
OpenAI has released proposed solutions to three long-standing mathematical problems, including a claimed refutation of the Exponential Time Hypothesis (ETH), a foundational assumption in computational complexity. Anthropic subsequently released its own claimed refutation. The computer-science community is now organizing an open review of the proofs and their implications.
MIT professor Ryan Williams is coordinating part of that response with UC San Diego researchers Russell Impagliazzo and Ramamohan Paturi, who first published ETH in 1999. They issued an open letter seeking feedback from researchers interested in the questions and in the norms that should govern AI contributions to scientific work.
What ETH says about hard problems
ETH belongs to the broader family of questions around P versus NP. In simplified terms, the question asks whether every problem whose proposed solution can be verified quickly can also be solved quickly. A completed Sudoku grid is easy to check, for example, while finding a valid grid can be time-consuming.
Most researchers assume that P does not equal NP. ETH supplies a more specific basis for that intuition: for a class of difficult logical puzzles, even the fastest algorithm requires exponentially more time as the number of variables increases. The OpenAI and Anthropic claims concern algorithms that solve such puzzles in sub-exponential time, faster than ETH permits.
If the work withstands scrutiny, it would not itself settle P versus NP. But Williams said it could make computer scientists assign more weight to the possibility that P equals NP and reconsider established intuitions about the tractability of hard NP problems. He also stressed that OpenAI's algorithm is unlikely to be the final or best possible algorithm.
A review process with wider consequences
Williams described parts of OpenAI's proof as routine, while identifying two or three unusual sections that warrant collective examination. OpenAI handled releases differently across the three disciplines: mathematicians received material through GitHub, computer scientists requested public proofs and support for a conference, and physicists formed a working group to decide how their result should be released.
The review matters because ETH has been used in many scientific papers over the past decade. If it is false, researchers will need to revisit work that relies on the hypothesis and determine which theoretical foundations remain appropriate. The task extends beyond checking an individual algorithm to understanding how assumptions should change.
Cryptography remains an immediate question
Complexity assumptions also underpin cryptographic systems. Better algorithms can matter where security depends on certain problems remaining impractical at particular key sizes. Williams said he asked OpenAI science lead Dan Roberts whether the claimed result affected crypto systems; OpenAI had checked and was reasonably confident it did not imply that any systems were now broken.
For businesses, the immediate implication is not to assume a cryptographic failure, but to monitor independent verification closely. Security, research and engineering teams should identify where their work depends on complexity assumptions, distinguish a claimed proof from an accepted result, and be ready to reassess technical exposure if the review validates the algorithms.