On October 16, 2026, an internal OpenAI model generated a 67-page mathematical proof claiming that the Exponential Time Hypothesis (ETH) is false. On the same day, Anthropic's Claude independently reached the same conclusion without any human intervention, consuming 14 million output tokens. Both documents were then translated into the Lean formal language for machine checking, and both passed.
According to Lance Fortnow, author of the complexity theory blog Computational Complexity and a professor at Georgia Tech, the algorithms from the two organizations can solve the k-SAT problem in $(n+m)^{O(1)} \cdot 2^{O(n \log\log n / \sqrt{\log n})}$ time—a regime known as subexponential time (2^o(n)), which in theory directly refutes ETH's core assertion that "no algorithm faster than exponential time exists for 3-SAT." Fortnow wrote: "I have not personally checked this proof, but it is now safe to consider the Exponential Time Hypothesis finished."
What Is ETH, and Why Does It Matter
The Exponential Time Hypothesis was jointly proposed by Russell Impagliazzo, Ramamohan Paturi, and Francis Zane in 2001. Its core claim is that the 3-SAT problem requires exponential time in the worst case and that no essentially faster algorithm exists. Although this hypothesis is not equivalent to P≠NP, over the following 25 years it was used as a premise in hundreds of papers on algorithmic complexity, becoming one of the de facto cornerstones of theoretical computer science.
For this reason, when OpenAI publicly released its GitHub repository on October 26, publishing both the natural-language proof and the Lean formalized version, the reaction from the algorithm theory community was not joy but caution. The original ETH authors Impagliazzo and Paturi, along with Ryan Williams of MIT, immediately issued a public statement: "A result of this magnitude requires rigorous independent review." The three organized an independent review panel, and Williams explicitly stated that he "did not sign any confidentiality agreement and did not receive any compensation."
Lean Passes: What Was Actually Checked
Lean is a mathematical theorem-proving assistant that can step through and check the formal validity of every logical inference. OpenAI completed the Lean formalization the day after generating the natural-language proof.
What Lean checks is that "this Lean code is logically consistent starting from its premises," not that "these premises are mathematically equivalent to the full statement of ETH." According to the notes in the public Lean repository, the algorithmic bound checked by this formalized version is "slightly weaker than the main mathematical claim"—that is, there is a technical gap between the proposition proved in Lean and the proposition in the paper's main text, and the equivalence between the two has not itself been formally proved.
Independent researchers Ryan O'Donnell and Shiva Kintali subsequently each released simplified follow-up proofs, deliberately sacrificing optimal bounds for clarity—the specialist community is working to retranslate this highly compressed, machine-assisted proof into a form that humans can follow.
Both Companies Reached the Same Conclusion Independently on the Same Day
One detail disclosed by Anthropic: during the solving process, Claude worked "without any human input" and used 14 million output tokens. That scale means Claude's chain of reasoning far exceeded the single-session workload of any human mathematician—it amounted to a draft of roughly 30,000 pages.
The two companies independently reached the same subexponential bound $2^{n/\sqrt{\log n}}$ on the same day. This more likely indicates that, at the current level of AI capability, this mathematical path has entered the model's "reachable region," and that no one had previously attempted it in a targeted way. Rather than a breakthrough, this is more like a measurement of the capability boundary—how large a search space can AI navigate to find a valid mathematical construction without a human roadmap?
Is the Cryptography Panic Justified?
The security of current mainstream public-key cryptography systems (RSA, elliptic-curve cryptography) rests on the hardness of integer factorization and discrete logarithm problems, not directly on ETH. Refuting ETH means that a particular class of algorithmic-bound assumptions fails, but it does not directly imply that integer factorization has become easy.
A more direct quantification comes from Anthropic's own paper: for 3-SAT instances with five clauses per variable, for its algorithm to outperform brute-force search in practical terms, the problem size would need to reach $10^{10000}$ variables. What this number means is that even if the proof holds, the algorithm would produce no practical improvement in SAT-solving efficiency on any foreseeable computing device on Earth. There is currently no direct threat to existing cryptographic infrastructure.
AI-Assisted Mathematics: The Gap in Trust Mechanisms
This incident reveals a systemic problem in the field of AI scientific discovery, rather than a flaw in a single event.
The traditional mathematical community's mechanism for building trust is: experts read the proof, find gaps, peer review, and time settles the matter. This mechanism is under strain when faced with high-density proofs generated by AI. When a proof is 67 pages long, constructed step by step by a machine, has passed formal checking, but exceeds most experts' real-time reading speed, how does a consensus that something "has been proved" form?
The Lean system offers a new path: shifting trust from "whether experts can understand it" to "whether the formal system is self-consistent." But this path has two premises that must both hold: first, the formal statement in Lean must faithfully reproduce the original mathematical proposition; second, the reliability of the Lean system itself must command sufficient trust. The former remains an open question; the latter has a strong consensus in theoretical computer science, but it is not absolute.
What the original ETH authors Impagliazzo and Paturi, together with Williams, are doing is essentially repairing this gap: adding a human review layer staffed by the people best able to judge "whether the formal statement is equivalent to the original proposition." This layer should not be an after-the-fact patch; it should be a standard procedure for AI-assisted mathematical discovery.
Independent Judgment
Based on the evidence currently available, the following judgments can be made:
What is highly likely to hold: The AI models have indeed made substantive mathematical progress along a particular technical path in theoretical complexity. The fact that the Lean check passed and that top complexity researchers are willing to invest time in independent review (without compensation) are both positive signals. Lance Fortnow's cautious optimism represents the current mainstream mood in the theory community.
The gap that remains open: The equivalence between the Lean formal statement and the full mathematical statement of ETH has not yet been formally proved. This is not a minor gap; it is precisely the central work of the entire independent review. Until then, "ETH has been refuted" is a high-confidence expectation, not a confirmed conclusion.
The bigger picture: This incident demonstrates the real capability boundary of AI in mathematical discovery—it can complete highly technical mathematical constructions without a human roadmap, and it can do so extremely quickly. But it also exposes the fragility of existing systems for checking knowledge: when AI's output speed exceeds human review speed, how the scientific community can absorb these results in time without lowering its standards will be the central institutional challenge of the coming years, as important as the difficulty of the mathematics itself.
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