Two of the largest artificial intelligence companies on the planet — OpenAI and Anthropic — announced this week that their internal AI models produced results that refute the Exponential Time Hypothesis (ETH), a conjecture proposed in 1999 that has served as the bedrock for decades of research in computational complexity. If the claim holds, it would represent a result requiring a complete revision of hundreds of scientific papers and textbooks on algorithms.
The story began on October 16, when an OpenAI model found what the company describes as a refutation of the ETH. The following day, the company's team formalized the argument using Lean, a theorem prover widely used by mathematicians. On October 26, OpenAI published a blog post announcing the results, and hours later Anthropic released an independent, parallel result — credited to "Claude (Anthropic)" — that had found the core algorithm without any human input on the same date.
What is the Exponential Time Hypothesis
The Exponential Time Hypothesis is a conjecture about the SAT problem (Boolean Satisfiability Problem), widely considered the most fundamental problem in computational complexity theory. The SAT problem asks whether there exists an assignment of truth values to the variables of a logical formula that makes the entire formula true. For formulas in conjunctive normal form (CNF), the corresponding problem — 3-SAT — is NP-complete, meaning that if any algorithm can solve it in polynomial time, then P = NP.
The ETH goes beyond the binary "P versus NP" question and makes a more specific claim: it proposes that 3-SAT requires exponential time — that is, no algorithm can solve the problem in subexponential time, which grows as 2^o(n), where o(n) is any function that grows faster than c^k·n for some constant k. In practical terms, the ETH establishes a "floor" under exponential time, a boundary of computational difficulty that the scientific community has assumed to be true for three decades.
Fedor Fomin of the University of Bergen in Norway told New Scientist the discovery left him "surprised." "It would require a complete revision of several areas of algorithms," he said, noting that a textbook he wrote on the subject would become obsolete if the finding holds.
How the models arrived at the result
According to OpenAI, its model produced a deterministic procedure that exactly counts satisfying assignments of fixed-width CNF formulas using subexponential time and polynomial space. The company published a Lean formalization on GitHub and described the work as part of three scientific discoveries announced simultaneously — the other two addressed harmonic analysis and quantum field theory.
Anthropic, for its part, stated that its Claude model identified the "core algorithm" without any human input. Anthropic researchers subsequently helped improve the presentation of the result, which also includes a Lean formalization available in a public repository.
Both companies date the core discovery to October 16. The coincidental timing raises questions about whether the two models followed similar approaches — or whether the nature of the problem makes a certain path of inquiry inevitable when one has a mathematical reasoning system powerful enough.
The scientific community's reaction
The academic community responded with caution. Computational complexity researcher Richard Williams of MIT, who had publicly criticized OpenAI's "proof dumps" on previous occasions as a "DDoS attack on mathematicians," agreed to collaborate with the frontier labs to interpret the new results. He asked OpenAI to fund a conference on the topic, arguing that "you need to understand, beyond the algorithms, what should the foundations have been before."
Levent Alpöge, a mathematician at Anthropic, initially suggested that OpenAI might be basing its solutions on pre-existing work. Hours after publishing, Anthropic announced its own independent result — which, instead of confirming the plagiarism speculation, reinforced the independence of the discovery.
The ETH's own creators — including Russell Impagliazzo and Ramamohan Paturi of the University of California, San Diego, who published the original hypothesis in 1999 — are leading a working group to examine the proofs. OpenAI published an open letter asking for feedback from interested scientists and proposing a discussion of norms around AI contributions to academic work.
Why this matters (if it's true)
The ETH is not merely an abstract conjecture. It has been used in hundreds of papers as the basis for "conditional lower bounds" — statements about how hard certain problems are that depend on other assumptions. If ETH is refuted, many of those conclusions become unusable and must be rebuilt from scratch.
The impact would extend beyond pure theory. Cryptography algorithms, optimization, formal verification, and machine learning often assume that certain problems cannot be solved faster than exponential time. If there are subexponential algorithms for 3-SAT, this calls into question not only theoretical complexity but the practical security of protocols that depend on computational difficulty.
What makes this case different from other purported AI "breakthroughs" in mathematics is the existence of a machine-checkable proof. Rather than merely describing an algorithm in natural language — which has previously led to contested results — both companies have made Lean formalizations available, allowing any researcher to verify each step logically. This accelerates the review process compared to traditional handwritten proofs that could take months or years to be examined.
That said, the process is far from concluded. As noted by John D. Cook on his blog, "the status of P = NP is still unknown: while 3-SAT does not take exponential time, it still takes more than polynomial time as far as anyone knows." Refuting ETH does not automatically prove P = NP — it only means that 3-SAT is easier than previously thought. The frontier between subexponential and polynomial time remains intact.
What to expect in the coming days
The community is now in a period of intensive verification. The group led by the original ETH creators should produce a verdict in the coming weeks. OpenAI has already asked the community to fund a dedicated conference on the topic. Meanwhile, hundreds of complexity researchers are reviewing the Lean formalizations.
Regardless of the final result, this episode has already changed how the community thinks about the role of artificial intelligence in scientific discovery. For the first time, two of the largest labs in the sector independently announced results on the same fundamental problem — and both attributed the core discovery to their models, not to humans. If ETH is refuted, it will be a feat that is human in the traditional sense (theorems were still formulated and verified by people), but an unprecedented achievement in the role of AI as a partner in theoretical research.
Sources: TechCrunch, Progressive Robot, New Scientist, OpenAI Blog, John D. Cook
✓ Independent sources cross-checked and verified before publishing
