On 26 October 2026, OpenAI announced three results produced by its models in mathematics, computer science and physics. Among them is a striking theoretical claim: an algorithm that would refute the Exponential Time Hypothesis (ETH), an assumption proposed in 1999 that underpins conditional hardness results in hundreds of papers. Hours later, Anthropic published its own refutation of the same problem, attributed to an internal Claude model.
What was announced must be classified precisely: it is a claim by the labs, not a result already confirmed by the scientific community. Neither proof has yet gone through complete peer review or published independent verification. OpenAI describes its manuscripts as raw research data that still need to be turned into scientific understanding.
What is the Exponential Time Hypothesis
In simplified terms, ETH states that the logic problem known as 3-SAT cannot be solved in subexponential time in the worst case. Both labs claim deterministic algorithms that count the satisfying assignments of formulas in subexponential time, which, if correct, would overturn the hypothesis.
According to reports, both labs date the central discovery to the same day, 16 October 2026. OpenAI says its result was formalized in the Lean proof assistant the next day, and Anthropic says its internal model found the core algorithm with no human input, using about 14 million output tokens, also on 16 October. Both Lean formalizations were published on GitHub alongside the manuscripts.
Who is checking the proofs
Verification is in the hands of outside experts, called “shepherds” by OpenAI. For ETH, the group is led by Russell Impagliazzo and Ramamohan Paturi of the University of California, San Diego — two of the hypothesis's original authors — and Ryan Williams of MIT. In an open letter, the three made clear that their involvement should not be taken as certification of the claim. Williams described parts of OpenAI's proof as unfamiliar-looking and argued that the whole community needs to examine the material together.
Practical effect: none, for now
Even if the proofs hold, the practical effect is, for now, nil. OpenAI's own paper reportedly states that the result does not directly bear on the P versus NP question and yields no practical algorithms for NP-complete problems. Anthropic's paper reportedly notes that its bound only beats brute force for astronomically large formulas, without changing solvers used in practice. OpenAI also told reviewers, according to TechCrunch, that its initial checks indicate no break of current cryptographic systems.
Alongside ETH, OpenAI announced two other results: a proof of Sogge's local smoothing conjecture in every dimension, with implications for the Kakeya and Stein restriction conjectures, and mathematical existence results for four-dimensional quantum field theories, including quantum chromodynamics (QCD) and pure Yang-Mills theory. The physics material, however, was not fully released: the group led by Nobel laureate David Gross chose to write its own interpretation before publication, a decision publicly questioned by an advisory group at the Institute for Advanced Study.
The episode marks a new chapter in AI-assisted science: models generate proof candidates at unprecedented speed, but validation remains the bottleneck — and this time it is happening in public, with the hypothesis's original authors leading the check.
Sources: WebProNews; Tech Times; CellCog; Dealroom; OpenAI's mathematics repository on GitHub; ETH discussion group's open letter.
Image credit: Wikimedia Commons, author Behnam Esfahbod. Diagram of the P, NP, NP-complete and NP-hard complexity classes, used here to illustrate the theoretical context — it is not material from OpenAI or Anthropic.
Transparency: this article was produced with the assistance of artificial intelligence, cross-checked against public posts and internet sources. Claims about the proofs are presented as statements by the labs, not as already validated results. Consult the original sources before citing the results.
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