On October 26, 2026, OpenAI and Anthropic released preprints in public repositories containing machine-checkable proofs in Lean of subexponential algorithms for SAT problems, refuting ETH — the 1999 hypothesis by Impagliazzo and Paturi that 3-SAT cannot be solved faster than c^n. The internal correctness of the proofs is verified mechanically, but the question remains open as to whether the formal theorem specifically negates ETH, a matter currently being investigated by a working group led by MIT professor Ryan Williams. For production, cryptography, and SAT solvers, nothing changes for now, but for the first time, two competing frontier labs have simultaneously published an equally loud fundamental claim with verifiable artifacts.

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What happened

On October 26, 2026, OpenAI released a 67-page preprint titled "The Exponential Time Hypothesis Is False," dated October 25, in the public repository openai/math. It contains a complete machine-checkable proof in Lean: according to the company, its internal model found a deterministic algorithm that counts the number of satisfying assignments of a k-CNF formula for k >= 2 in time (n+m)^{O_k(1)} * 2^{n/(log n)^{1/2+o(1)}} using polynomial memory. On the same day, Anthropic published a 49-page article titled "A 2^o(n) Time Algorithm for 3-SAT" in the anthropics/formal-math repository: an internal research Claude model provided a deterministic algorithm for #3-SAT in 2^{O(n log log n / sqrt(log n))}, formalized in Lean 4 against Mathlib. The contrast with the previous record is striking: the best known algorithms for 3-SAT had constant bases of the exponent, approximately 1.328^n for deterministic and 1.308^n for randomized, whereas the new bounds are subexponential for the first time. However, OpenAI acknowledges that the bound proven in Lean is weaker than the one claimed, and key procedural details from Anthropic — approximately 14 million output tokens and algorithm search without human involvement — remain self-reported by the team. The Hacker News post had 1 point and 0 comments at the time of data collection, and the press cycle ran through TechCrunch on October 27 and The New York Times on November 6.

Context

ETH — the Exponential Time Hypothesis — was formulated in 1999 by Russell Impagliazzo and Ramamurty Paturi and asserts that 3-SAT cannot be solved faster than c^n. Hundreds of works on problem hardness, including Clique, Vertex Cover, and Independent Set, are built on this hypothesis, which is why it is called the "workhorse" of complexity theory: if the refutation holds, an entire layer of fine-grained complexity will need to be reconsidered, not just one theorem. The second part of the context is methodological. The main difference from typical "AI made a breakthrough" news is that both articles contain formalization in Lean, with Anthropic using Lean 4 against Mathlib, so the internal correctness of the derivations is checked by a mechanical program rather than by trust in the lab. The bottleneck shifts to the fidelity of the formulation: the question is no longer whether the derivation is correct, but whether the proven theorem specifically negates ETH. The third part is protection against typical interpretation errors: refuting ETH is not the same as P = NP. Both claimed bounds, 2^{n/(log n)^{1/2+o(1)}} for OpenAI and 2^{O(n log log n / sqrt(log n))} for Anthropic, remain superpolynomial, the bounds of the two teams differ by lower-order factors like log log n and o(1) terms, and, according to sources, neither proof claims P = NP.

Why this matters for the industry

Directly in production, nothing changes right now: there is no API, no implementation, no real-time execution measurements, and no changes in inference, latency, or cost. There is no practical gain for SAT solvers; the advantage over brute force only manifests on inputs with tens of thousands of variables, and cryptographic systems, according to the participants themselves, are not threatened: OpenAI expressed "reasonable confidence" that the new algorithms do not break current systems. The real impact occurs within science: the complexity theory community is mobilized for verification, and MIT professor Ryan Williams has organized a working group and workshops with the original hypothesis authors Impagliazzo and Paturi to check whether the formal statements in Lean correspond to ETH itself. If the claim is confirmed, hundreds of hardness results built on ETH, including Clique, Vertex Cover, and Independent Set, will require review. The signal for engineering and product teams is twofold. First: the combination of Lean and Mathlib unexpectedly became a critical resource for both frontier labs, which is a direct indicator of demand for such competencies. Second: a working pattern has been confirmed where an autonomous model works on a problem for hours, and the final artifact is an object checkable by a deterministic verifier — Lean, Mathlib, a compiler, or a test runner; "model proposes — verifier checks" pipelines make sense to implement now, preserving artifacts of all attempts, not just showcase ones. Third: there is a new research economy where two competing labs independently arrived at the same fundamental claim on the same day, and the only reasonable action for a team today is to read the preprints in openai/math and anthropics/formal-math and set up an internal tracker for the conclusions of Williams' working group.

Why this matters for users

For the reader, the form of publication is significant: for the first time, two frontier labs independently arrived at the same fundamental refutation and published it with machine-checkable proofs, rather than just a story about how the "model reasoned." The preprints are open in the openai/math and anthropics/formal-math repositories, and anyone can re-verify the internal correctness of the proofs themselves by running Lean, without relying on press releases. No immediate benefit should be expected: no tools are being released, services are not getting faster, passwords do not need to be changed, and cryptography, according to the authors, has not been broken. Going forward, it is worth watching three things: the conclusions of Williams' working group on whether the formal theorem specifically negates ETH; the analysis of "alien-looking parts," i.e., unusual fragments of the proof; and whether the bound 2^{n/(log n)^{1/2+o(1)}} holds up under transfer to other NP problems via polynomial reductions. And a backup reference in case of loud retellings: versions claiming that "AI solved the P vs NP problem" are factually incorrect; both claimed bounds remain superpolynomial.

What is currently unknown / limitations

These are lab claims that have not undergone peer review, so it is premature to consider the refutation of ETH a fact. OpenAI explicitly acknowledges that the bound proven in Lean is weaker than the one claimed in the title, so the exact formulation of the result may shift during review. Anthropic's procedural details — approximately 14 million output tokens and algorithm search without human involvement — are self-reports without external verification. The open history contains only successful showcase artifacts: there is no statistics on attempts, failed runs, or the degree of implicit control over the search, which means the "model proposes — Lean checks" combination cannot be considered a reliably validated template. The key unknown is the fidelity of the formalization, i.e., whether the theorem proven in Lean actually negates ETH specifically; this is what Williams' working group is checking, and its conclusions are not yet available. Finally, there is no implementation and no independent performance measurements: it is unknown on which inputs the new algorithms actually outperform the deterministic record of 1.328^n and whether the open bound transfers to other NP problems via polynomial reductions.

Sources

Author

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