AI Disproves 27-Year-Old Exponential Time Hypothesis, OpenAI and Anthropic Say

Artificial intelligence has done something no human mathematician managed in 27 years: disprove a foundational assumption of computer science. OpenAI and Anthropic said on October 26, 2026, that their AI models had disproved the Exponential Time Hypothesis, a conjecture from 1998 that underpins hundreds of papers in theoretical computer science, and that both companies reached the result independently, on the same day.

The announcement, reported by The New York Times on November 6, stunned even the mathematicians who proposed the hypothesis. Russell Impagliazzo, who formulated it with Ramamohan Paturi in 1998, called it "very shocking news" and "a shock that the work was not done by a human being."

Key Points

• OpenAI said on October 26, 2026, that its internal models produced three new results: a disproof of the Exponential Time Hypothesis, a proof of the local smoothing conjecture settling the Kakeya conjecture in every dimension, and a rigorous existence proof for four-dimensional Yang-Mills theory and quantum chromodynamics.
• Anthropic independently claimed the same disproof on October 16, from an internal Claude research model that used about 14 million output tokens.
• OpenAI says its model found the disproof on October 16 and formalised it in Lean on October 17, publishing a 67-page manuscript, "The Exponential Time Hypothesis Is False", with public Lean files on GitHub. Anthropic published a 49-page paper, "A 2^o(n) Time Algorithm for 3-SAT".
• The hypothesis, proposed in 1998 and 1999 by Russell Impagliazzo and Ramamohan Paturi with Francis Zane, holds that 3-SAT cannot be solved in subexponential time. Anthropic says more than 800 papers rely on it or stronger assumptions.
• Neither proof is peer-reviewed or independently certified yet. Independent examination is being led by Impagliazzo, Paturi and MIT's Ryan Williams, and the result does not prove P=NP, which remains open.

What Happened

The story broke in two stages. On October 16, OpenAI's internal model produced a natural-language disproof of the hypothesis, formalised in Lean the next day. Anthropic's internal Claude research model reached the same conclusion independently on October 16, generating about 14 million output tokens in the process.

Both companies went public on October 26. OpenAI published a 67-page manuscript, "The Exponential Time Hypothesis Is False", with the Lean files on GitHub, while Anthropic published a 49-page paper, "A 2^o(n) Time Algorithm for 3-SAT". The New York Times reported the developments on November 6, and the two companies' simultaneous, independent results are what turned a research announcement into a landmark moment for artificial intelligence.

Background and Context

The Exponential Time Hypothesis, or ETH, is one of the load-bearing assumptions of modern computer science. Proposed by Impagliazzo and Paturi in 1998, with Francis Zane, it states that 3-SAT, deciding whether a logical formula with three variables per clause can be satisfied, cannot be solved in subexponential time. As these problems grow, the time needed grows exponentially, and no clever algorithm can escape that.

That assumption matters because hundreds of papers build on it; Anthropic puts the figure at more than 800. Researchers have used it to prove other problems hard and to map the landscape of computational complexity. If the hypothesis is false, a large body of work built on it must be re-examined. The claim is carefully bounded: it does not prove P=NP, the most famous open problem in computer science, which remains unsolved.

The Exponential Time Hypothesis: Why the Disproof Matters

The disproof matters first for computer science itself. If 3-SAT admits an algorithm running in 2^o(n) time, faster than the exponential lower bound the hypothesis asserted, then the map of which problems are hard and which are tractable shifts. Algorithms researchers assumed could never exist may be possible, and hardness results that depend on the hypothesis would need rebuilding on other foundations.

Impagliazzo pointed to the longer horizon. He told The New York Times the result could one day point to faster chip design, faster drug discovery, or new ways of breaking encryption, though there are no immediate practical applications. The value for now is theoretical: a wall researchers believed was solid may have a door in it.

Reactions and Quotes

Impagliazzo's reaction carried the most weight, because he co-authored the hypothesis. "Very shocking news," he told The New York Times on November 6, adding it was "a shock that the work was not done by a human being." His surprise was not only at the result but at its author: a machine, not a mathematician.

Both companies have framed their claims as results awaiting confirmation, not settled fact. Neither proof has been peer-reviewed or independently certified, and the mathematics community will need time to work through the manuscripts and the Lean formalisations. Verification is being led by Impagliazzo, Paturi and Ryan Williams of the Massachusetts Institute of Technology, the researchers best placed to spot an error.

Why It Matters

The episode matters because it changes who can do mathematics. For centuries, proof has been an exclusively human activity. If AI models can produce original proofs of major conjectures, the pace and authorship of mathematical discovery change at once, and human mathematicians shift toward checking, guiding and interpreting what machines produce.

It also matters as a test of how science handles machine-made claims. The Lean formalisation is central: because the proof is written in a language a computer can check, errors can in principle be found mechanically. That combination, machine-generated proof plus machine-checkable verification, may become the template for judging AI mathematics.

What Happens Next

The immediate next step is verification. Impagliazzo, Paturi and Williams are leading the independent examination of both proofs, and the wider community will pore over the manuscripts and public Lean files in the coming weeks. If the proofs hold, the result moves from claim to theorem, and the re-examination of the papers built on the hypothesis begins. If a flaw is found, the attempt still shows how close AI has come to frontier mathematics.

Frequently Asked Questions

What is the Exponential Time Hypothesis?
A 1998 conjecture by Russell Impagliazzo and Ramamohan Paturi, with Francis Zane, holding that 3-SAT cannot be solved in subexponential time. Hundreds of computer science papers build on it.

Did AI really prove it false?
OpenAI and Anthropic both say their models disproved it independently on October 16, 2026, and both have published papers. Neither proof is peer-reviewed or independently certified yet.

Does this mean P=NP?
No. The result does not prove P=NP, the most famous open problem in computer science, which remains unsolved.

What are the other two results?
A proof of the local smoothing conjecture, settling the Kakeya conjecture in every dimension, and a rigorous existence proof for four-dimensional Yang-Mills theory and quantum chromodynamics.

When will we know if the proofs are correct?
There is no fixed timetable. Mathematicians are examining both papers and the public Lean formalisations, and their verdict will come over the coming weeks and months.

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