The hypothesis that fell

On October 16, an internal OpenAI model found what no mathematician or computer scientist had been able to find for decades: a counterexample to the Exponential Time Hypothesis (ETH), a conjecture formulated in 1984 that had become one of the invisible pillars of computational complexity theory. By October 25, the result had been verified in Lean—the formal proof language that has proven unbeatable at automatic validation—and published to public repositories. Less than 24 hours later, Anthropic announced its own independent result, also refuting the ETH.

The news rippled through the academic world like a bomb. ETH is not just another conjecture lurking in some corner of graph theory. It is a statement about the minimum time cost to solve problems in fundamental complexity theory, including the famous 3-SAT problem (Boolean satisfiability with three-variable clauses). If ETH is true—and for 42 years, the entire community operated as if it were—then no algorithm can solve 3-SAT in subexponential time. If it is false, as both companies claim, then the barrier separating problems "decidedly hard" from problems "maybe solvable" dissolves.

What ETH claimed

The Exponential Time Hypothesis, proposed in its standard form by Russell Impagliazzo and Ramamohan Paturi in 1998, states the following: for the 3-SAT problem with n variables, any deterministic algorithm needs, in the worst case, time proportional to 2^(cn) for some constant c > 0. In more tangible terms, this means that as the number of variables grows, computational time grows so explosively that for sufficiently large instances, even the world's best computers would be stuck forever.

The practical implication is enormous. ETH serves as the foundation for hundreds of "fine-grained complexity" results in fields ranging from computational biology to databases, from computational geometry to graph theory. Many of these results state: "if problem X cannot be solved faster than time t, then problem Y also cannot"—and the entire chain of deductions rests on ETH. If the hypothesis falls, it doesn't automatically mean those results are wrong. It means the proofs that depended on them need to be re-examined.

The two companies' responses

What made the story even more extraordinary was the speed with which Anthropic responded. The Anthropic team, led by mathematician Levent Alpöge, published on October 27 its own refutation of ETH, based on the work of an internal model at the company. The differences between the two results—and between the styles of the two teams—became a case study in how scientific discovery can be accelerated by autonomous agents.

OpenAI took a different approach. Instead of simply publishing the result, the company formally requested that the ETH creators themselves—Impagliazzo, Paturi, and Ryan Williams of MIT—organize an independent investigation. The group received full access to the proofs, both in natural-language form and in Lean formalization, and announced on October 26 that it would conduct an open, independent review.

Anthropic, meanwhile, published the paper in a public repository (hexagonmath.org) and put it through traditional peer review. The Anthropic model's work differed in technical detail but converged on the same conclusion: ETH, in its standard form, is false.

Why scientists are cautious

The scientific community's reaction was one of contained astonishment. Gary Marcus, an independent AI researcher, called the result "a potentially huge development"—but also reminded that refuting ETH does not automatically prove that P = NP. These are different claims, and a substantial gap remains between them. ETH is a statement about the exact growth rate of execution time; the P vs NP conjecture is about whether every verifiable-in-polynomial-time solution can also be found in polynomial time.

Joshua Fonseca Rivera, an AI safety researcher at Anthropic, captured the sentiment: "we may have underestimated computational systems." But he also emphasized the need for independent verification. The Lean verification confirms that the formal proof is valid—but that doesn't close the case. A paper published on October 26, available on arXiv (arxiv.org/pdf/2610.08144), argues that AI-written proofs, when translated into Lean, may contain semantic ambiguities that the automatic verifier does not detect.

Ryan Williams, the MIT mathematician involved in the review, was more direct: "If OpenAI's conclusions prove correct, computer scientists will have to take the possibility of P = NP a little more seriously." But he also stated publicly that he had not believed ETH was true even before the announcement—suggesting the result may be more confirmation than radical surprise.

The broader context: AI solving Millennium problems

This was not the only scientific claim from OpenAI in recent weeks. In September, the company announced that it had used 10,000 AI agents over 88 hours to attack the Navier-Stokes problem—one of the Clay Mathematics Institute's Six Millennium Prize Problems, with a US$ 1 million prize. The result, while controversial regarding authorship (mathematician Tristan Buckmaster alleged OpenAI had scooped him before he could publish his own work), marked the first time an AI system solved a genuinely open problem in mathematics.

In May 2026, the same model had cracked an 80-year-old conjecture by Paul Erdős about planar distance between points—a result human mathematicians called "a seismic moment for AI's mathematical capability." And in October of the same year, OpenAI announced that its models had produced a rigorous proof of the existence of Yang-Mills theory in four dimensions, another part of a Millennium Problem.

The sequence of results raises a question that the scientific community cannot ignore: if a system of AI can refute conjectures, solve Millennium problems, and produce verifiable proofs across multiple fields—mathematics, theoretical physics, computer science—what is the next field that AI will dismantle?

What comes next

OpenAI has committed to funding a conference to discuss the implications of the ETH refutation and donated US$ 5 million to the Simons Foundation to support workshops and conferences on AI-generated results. The group led by Impagliazzo, Paturi, and Williams promised to publish a report with its verdict.

Meanwhile, the impact is already being felt. Papers that depended on ETH as a basis for computational hardness results are being revisited. Research groups in computational geometry and graph theory have already announced that they are re-examining their assumptions. And in the field of cryptography—where the difficulty of certain problems is the basis for all digital security—the news has generated a discomfort that no specialist can yet quantify.

The question is: if AI can refute fundamental hypotheses with the same ease with which it previously solved differential equations, what remains for human mathematicians to discover—and what does this mean for the future of scientific investigation as a whole?

Sources: WIRED, Scientific American, The Decoder

✓ Independent sources cross-checked and verified before publishing