On October 16, 2026, two artificial intelligence models from rival laboratories arrived independently at the same conclusion: a mathematical hypothesis that had served as the foundation of theoretical computer science for nearly three decades was wrong. Twelve days later, on October 26, OpenAI and Anthropic publicly announced what could be one of the most significant results in computational complexity theory since the original formulation of the P versus NP problem.
The Exponential Time Hypothesis (ETH) was proposed in 1999 by Russell Impagliazzo and Ramamohan Paturi of the University of California, San Diego, along with Francis Zane. In simple terms, the hypothesis states that no algorithm can solve the classic problem of 3-SAT in subexponential time — that is, faster than 2 raised to an exponent proportional to the number of variables, as that number grows. The 3-SAT problem, which consists of determining whether there exists a combination of truth values for the variables of a logical formula that makes the entire expression true, is NP-complete. This means that if someone found a polynomial-time algorithm for it, they would automatically solve all problems in NP — the famous P = NP.
ETH was far more modest than P = NP. Instead of claiming the problem is intractably hard, it suggested only that the improvement over exhaustive search would be limited. Even the best known algorithms before 2026, such as the Monien-Speckenmeyer branching algorithm (1985), solved 3-SAT in time proportional to the golden ratio φ raised to the number of variables — better than brute force, but still exponential. The hypothesis became the basis for hundreds of papers that established conditional computational hardness results in areas as diverse as graph theory, computational biology, databases, and computational geometry.
When OpenAI announced its discovery, the community reaction was immediate and intense. Ryan Williams, a professor at MIT specializing in computational complexity, organized an independent examination group that includes ETH's own authors — Impagliazzo and Paturi — as well as himself. The same day, the team published an open letter calling for public scrutiny of the result. Williams, who had previously described OpenAI's "proof dump" method as a "DDOS attack on mathematicians," agreed to collaborate this time, but made clear: our involvement should not be taken as certification of the claim. A result of this significance requires careful, independent scrutiny.
What OpenAI claimed
According to OpenAI's announcement, one of its internal models produced a natural-language refutation of ETH on October 16. The following morning, the proof was verified in Lean — a theorem-proving assistant that verifies mathematical proofs independently, line by line. The company published both the model-generated text and the Lean formalization on GitHub.
The approach combined sparsification (decomposing arbitrary formulas into smaller subproblems), spectral replacement (operator approximation via eigenvalues), and rational interpolation (reconstructing Boolean functions from sampled points). The final result, according to OpenAI, is a deterministic algorithm that, for every ε > 0, solves 3-SAT in time C_ε · 2^(εn) — technically subexponential.
Hours later, Anthropic published its own work, titled "A 2^o(n) Time Algorithm for 3-SAT," with "Claude (Anthropic)" listed as the author. The Claude research model used 14 million output tokens to discover the core algorithm, without direct human assistance in the discovery phase. Anthropic's research team helped only to refine the presentation. What drew attention was that Anthropic's proof also addressed the counting problem (#3-SAT), not just decision.
Both teams arrived at the same conclusion independently, on the same day — which, according to analysts, suggests that frontier models from different laboratories now possess comparable skills for original theoretical research. The details of the arguments, however, differ: each team used distinct methods and produced separate Lean formalizations, checked by independent verifiers.
What this means — and what it doesn't
The crucial question is that, at the time of writing, neither proof has undergone peer review. Both have been verified by Lean, which provides substantial mechanistic confidence, but does not eliminate the possibility that subtle gaps exist in the logic. Williams and Impagliazzo's group is organizing a dedicated conference to examine the work.
Even if confirmed, the practical consequences are nearly nil. Anthropic explicitly states that its algorithm only surpasses exhaustive search for formulas with more than approximately 10^(10,000) variables — an astronomical number that will never appear in any practical problem. For solving satisfiability in banks, chips, or logistics optimization, nothing changes tomorrow.
From a theoretical standpoint, however, the implications are enormous. ETH underpins hundreds of conditional results in algorithmic complexity. If it falls, many of those results will need to be reevaluated. Bart Jansen of Eindhoven University of Technology described the work as "ingenious," "spectacular" and "awe-inspiring — but also humbling." "It really makes you think about the future of mathematics, algorithms, and academia."
Leslie Goldberg of the University of Oxford summed it up: "Assuming that it is confirmed, this is an enormous breakthrough." At the same time, she warns that the result does not resolve P = NP: disproving ETH does not prove P = NP, it only means that the conditional proof of P ≠ NP based on ETH doesn't work. The barrier to P = NP remains.
The crisis of AI-assisted mathematics
The ETH discovery came alongside two other claims published by OpenAI on October 26: the demonstration of the "local smoothing" conjecture (which, if confirmed, would settle the Kakeya conjecture in all dimensions) and work on quantum field theory that touches on the million-dollar Millennium problem regarding the existence of Yang-Mills theory. Nobel physicist David Gross leads an external group that examined the physics work and decided not to release it publicly while preparing its own account.
OpenAI's approach of publishing discoveries in rapid succession — over 700 mathematical papers on October 6 — had already drawn criticism. Williams called the earlier dump a "DDOS attack on mathematicians." This time, the company changed its format: it created discipline-specific working groups, opened raw data, and agreed to fund conferences. It was an improvement, but it does not resolve the fundamental question: when an AI produces an original result, who is the author? How does academia validate discoveries generated by models trained on existing literature?
Anthropic's mathematician, Levent Alpöge, publicly suggested that OpenAI was basing its solutions on pre-existing work. If confirmed, this would raise even more delicate questions about the authenticity of AI contribution. Anthropic's own paper cites OpenAI's work, which suggests that the team was aware of the rival discovery before publishing theirs — which explains the temporal proximity.
What to watch
The path forward is now academic scrutiny. Williams's group will examine the Lean proofs line by line, looking for gaps and testing whether the logic holds in edge cases. If the demonstrations are confirmed, computational complexity theory will need a complete review — hundreds of papers reevaluated, textbooks rendered obsolete. If they are refuted, the result will serve as a case study on the current limits of language model formal reasoning.
Ultimately, what is striking is not necessarily the refutation of ETH itself, but the fact that ETH was challenged by an AI. For the first time, an artificial system has produced a verifiable theoretical discovery in one of the most abstract fields of mathematics. If the proof holds, it will not be the end of computational complexity — it will be the beginning of a new era in which human knowledge advances not through a single genius, but through machines that think in patterns humans have yet to see.
Sources: TechCrunch, New Scientist, Vanda Team
✓ Independent sources cross-checked and verified before publishing
