OpenAI and Anthropic Claim Refutation of the Exponential Time Hypothesis: What It Means for Computer Science

Refutation of ETH and Anthropic's Response

OpenAI published solutions to three major mathematical problems, one of which is the unexpected refutation of the Exponential Time Hypothesis (ETH), a long-standing assumption in computer science regarding which problems are fundamentally difficult to compute. As with the laboratory's previous publications, the scientific community reacted with a mix of excitement and concern: Anthropic mathematician Levent Alpöge suggested that OpenAI relied on existing work, and a few hours later Anthropic released its own refutation of ETH.

What is ETH and Why It Matters

ETH relates to the class of problems known as P versus NP: for decades, researchers have tried to understand whether any problem whose solution can be easily verified (NP) can also be easily solved (P). A simple example is Sudoku: a completed solution can be checked quickly, but finding it takes considerable time. Cryptographic systems are built on this principle: with a key, a message is decrypted easily, while breaking an effective cipher takes too long to be practical. Most scientists believe that P ≠ NP, and ETH served as evidence for this: it asserts that even the fastest algorithm requires exponentially more time as the number of variables increases in a certain class of complex logical problems. The claims by OpenAI and Anthropic are that their models have discovered algorithms that solve such problems in 'sub-exponential' time—faster than previously thought possible. If true, ETH is refuted, potentially opening the path to algorithms that generally achieve P = NP.

How the Community Organizes Verification

This time OpenAI structured the publication of results differently for three disciplines: mathematicians received the work via GitHub, computer scientists formed a working group and requested public publication of the proofs and funding for a conference, while physicists created a group to determine the order of releasing their results. The organizer of the computer science response is MIT professor Ryan Williams, who previously criticized OpenAI's approach, calling the October release of proofs 'a DDOS attack on mathematicians'. Nevertheless, he agreed to collaborate: the laboratory asked him, along with Russell Impagliazzo and Ramamohan Paturi from UC San Diego—the authors of the 1999 ETH paper—to help interpret the solution. The first step was an open letter calling on scientists to provide feedback and discuss norms for AI participation in their work. 'Some parts of the proof can be considered routine, but there are two or three foreign parts—these are exactly what made me think: this is something the entire community needs to work on together,' Williams told TechCrunch.

Consequences: From Cryptography to Revisiting Scientific Papers

Over the past decade, ETH has been incorporated into numerous scientific papers that will need to be re-evaluated if the laboratories' claims are confirmed—one reason why Williams pushed for OpenAI to support a conference. Beyond scientific questions, cybersecurity concerns arise: an improved algorithm could mean that cryptographic systems with certain key sizes should be considered compromised. Williams reported that his first question to OpenAI's scientific director, Dan Roberts, was whether they had assessed the impact on cryptographic systems, and according to him, they had checked and were reasonably confident that no vulnerabilities were introduced. Williams himself remains cautious: 'Every computer scientist should probably slightly increase the plausibility of P = NP, but the weight each assigns is their own. This challenges many of our intuitions about how solvable difficult NP problems are. However, this is likely not the final answer—the OpenAI algorithm is probably not the best possible one.'