# OpenAI and Anthropic Both Say Their AI Broke a 27-Year-Old Computer Science Hypothesis > Source: > Published: 2026-10-27 20:23:04+00:00 On Monday OpenAI announced that one of its internal models had refuted the Exponential Time Hypothesis, a belief that has been part of how computer scientists think about hard problems since 1999. A few hours later Anthropic put out its own refutation of the same hypothesis. Two AI labs, one result, same day. That is either a remarkable coincidence or a sign that this is now an industry where proofs ship on a release schedule. Probably both. I’ll start with the math, because it is simpler than it sounds. Take a logic puzzle called 3-SAT. You get a long list of true-or-false variables and a set of rules, each rule involving three of them, and you have to find a way to set the variables so that every rule is satisfied. The dumb approach tries every combination. With n variables that’s 2 to the n combinations, so each extra variable doubles the work. Clever algorithms have been doing better than that for a long time, but only by shrinking the exponent. The hypothesis, [as Wikipedia summarizes it](https://en.wikipedia.org/wiki/Exponential_time_hypothesis), says you can’t do fundamentally better: any algorithm for 3-SAT still needs time that grows exponentially in n in the worst case. What the labs claim is a “sub-exponential” algorithm, meaning one whose running time grows more slowly than any exponential of that kind. Nobody needs it to be fast in practice. It just has to exist. Why would anyone care? Because the hypothesis does a lot of work. Researchers use it as an input: assume ETH, and you can show that hundreds of other problems in graph theory, databases and geometry can’t be solved quickly either. Ryan Williams, the MIT professor organizing the field’s response, [told TechCrunch](https://techcrunch.com/2026/10/27/frontier-ai-labs-say-their-models-refuted-the-exponential-time-hypothesis-what-does-that-mean/) that many papers from the last decade relied on it and “will need to be revisited” if the refutation stands. Think of it as a load-bearing assumption. Someone has just said it isn’t load-bearing, and now everyone gets to go back and check which walls are still standing. The obvious next question is whether this means P = NP, the famous one with a $1 million Millennium Prize attached. No, not directly. In a blog post, the applied mathematician John D. Cook pointed out that “the status of P = NP is still unknown,” adding: “While 3-SAT does not take exponential time, it still takes more than polynomial time as far as anyone knows.” Williams goes a bit further than that. “Every computer scientist should probably put a little more weight on the plausibility of P = NP, but one’s weight may vary,” he said, while also saying he doesn’t think OpenAI’s algorithm “could possibly be the best algorithm you can get.” The Fields medallist Timothy Gowers called the result “very surprising and consequential” on X, and Gary Marcus, who is usually skeptical of AI claims, said it could be “a big deal if correct,” [according to Business Insider](https://www.businessinsider.com/smart-people-openai-exponential-time-hypothesis-2026-10). ## Who checks the homework The part I find most interesting is how the proof gets checked, because that’s where the incentives sit. OpenAI asked Williams to help interpret the result, together with Russell Impagliazzo and Ramamohan Paturi of UC San Diego. Impagliazzo and Paturi are two of the people who first proposed the hypothesis, so they have been asked to referee the claim that it’s false. Their first step was an open letter asking for feedback, and it says: “Our involvement should not be taken as certification of the claim. A result of this significance requires careful, independent scrutiny.” According to [The Next Web](https://thenextweb.com/news/openai-exponential-time-hypothesis-false-kakeya-yang-mills), the reviewers are unpaid and signed no non-disclosure agreements. OpenAI says the model found the disproof on October 16 and that it was machine-checked in the Lean proof assistant on October 17. OpenAI agreed to the group’s requests to release all the raw data and to fund a conference, and it gave $5 million to the Simons Foundation for workshops on AI-produced results. It also says it wants no academic credit, which should go to whoever writes up the explanations. The letter adds a warning that credit may end up with whoever publishes first or has the strongest tools, while the work of verifying and explaining results gets undervalued. Put that in plain terms. A trillion-dollar-ish industry produces a proof at near-zero marginal cost. Three senior academics then check it for free. The payment for checking is the honour of checking, plus a conference. Lean can confirm that a formal statement was proved, but humans still have to confirm that the formal statement is the thing everyone meant by ETH. That second job is the expensive one, and nobody is pricing it. OpenAI has refined its process somewhat. Earlier this month it published hundreds of mathematics manuscripts at once, which Williams called a “DDOS attack on mathematicians.” Monday’s release came with three headline results, and each field got different handling. The mathematicians got their proof dropped on GitHub. The computer scientists formed a working group and asked for public proofs and a conference. The physicists formed a working group that will decide how their result comes out. Williams said some parts of the ETH proof are “maybe you could say routine, but then yes, there are two or three alien-looking parts,” and that those parts made him think “the whole community needs to work on together.” ## Meanwhile, at the other lab Then there is Anthropic. Its mathematician Levent Alpöge suggested that OpenAI’s solutions were based on pre-existing work, and hours later Anthropic released its own refutation. Arguing about who got there first is an old academic sport, and Anthropic has [been on the other side of it](https://rews.cc/a/anthropic-s-ai-found-a-crispr-like-system-a-copenhagen-biolo-a46b8a) before. What’s new is that the competitors are two companies [with every reason to want to look smarter than each other](https://rews.cc/a/sam-altman-thinks-the-other-ai-lab-is-the-one-with-the-god-p-460250), and that the result in dispute could have been unthinkable a month ago. Joshua Fonseca Rivera, an AI safety research fellow at Anthropic, wrote on X that “the bitter lesson should make this unsurprising,” and then added: “I know this logically, but... wow, we really don’t know anything.” There is also a practical worry. Faster algorithms for this class of problem could, in principle, mean that some cryptographic systems using certain key sizes count as broken. Williams said one of the first things he asked OpenAI science lead Dan Roberts was whether anything is now insecure. “Apparently, they checked, and they were reasonably confident that this didn’t … imply any breaks.” That is reassuring, as long as you’re comfortable with the company that produced the result also being the one that checked whether it breaks the internet. Williams sounds mostly optimistic, though he has real concerns about how the next generation of scientists will be trained. AI, he said, is forcing researchers to ask: “if all the things I used to work on are now low-hanging fruit for AI, how can I raise the bar for the kind of scientific questions I want to ask?” Dan Shipper, the chief executive of the startup Every, put it differently: “I regret not getting a Ph.D. in frontier math,” he wrote, adding “(Not that I ever considered getting one but still).” Given how the checking currently pays, the regret seems misplaced.