Explainer · Complexity theory · October 27, 2026

OpenAI and Anthropic say AI refuted the Exponential Time Hypothesis. What that means

Computer scientists have long built on an assumption called the Exponential Time Hypothesis: that a classic puzzle named 3-SAT can never be solved much faster than by trying a fixed fraction of all possibilities. On October 26, 2026, OpenAI posted a paper titled "The Exponential Time Hypothesis is False," produced by an internal AI model. Soon after, Anthropic posted its own paper, saying a Claude research model found a similar algorithm on October 16 with no human input. Both come with proofs checked by Lean, software that verifies math step by step. Experts are now checking them.

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The short version

  1. 3-SAT asks whether you can set a row of on/off switches so that a list of small rules all hold. Trying every setting doubles the work with each switch.
  2. The hypothesis said the best possible algorithm still needs 2 to the power of (some fixed fraction of the switches). The fraction could be small, but never zero.
  3. Both labs claim an algorithm where that fraction keeps shrinking as puzzles grow, about 1 divided by the square root of the log of the number of switches. That refutes the hypothesis, if the proofs hold.
  4. It does not show P equals NP, and Anthropic says it does not make real-world solvers faster.
The same story in 75 seconds.

1 What happened

On October 26, 2026, OpenAI announced three results from an unnamed internal model: one in math (the local smoothing conjecture, which would also settle the Kakeya conjecture in every dimension), one in computer science (the Exponential Time Hypothesis is false), and one in physics (the existence of quantum chromodynamics). OpenAI researcher Dan Roberts wrote that "our system has recently proved that the ETH is false, a result that was subsequently verified in Lean." The 67-page paper on OpenAI's GitHub is dated October 17 and revised October 25.

Soon after, Anthropic posted a 49-page paper, A 2o(n) Time Algorithm for 3-SAT, credited to "Claude (Anthropic)." It says: "The core algorithm was found on October 16 by an internal Claude research model with 14M output tokens and no human input." Anthropic researchers then worked with Claude on the write-up. The paper notes that OpenAI's model, "independently and concurrently," claims a similar result.

OpenAI asked MIT complexity theorist Ryan Williams to lead a group to verify and explain the result, with Russell Impagliazzo and Ramamohan Paturi, who proposed the hypothesis (with Francis Zane) in the first place.

2 How it works: the shrinking exponent

Write the time an algorithm takes on a 3-SAT puzzle with n switches as 2e. Brute force has e = n: every new switch doubles the work. Decades of clever algorithms cut that down. The best previous bound OpenAI's paper cites, by Jiang and Cai, is about 1.307n, which is e = 0.386 × n.

The hypothesis said that's the kind of improvement you can get, and no more: there is some fixed share c above zero, and no algorithm can do better than 2c × n. The new papers claim an exponent of roughly n / √(log n). Its share of n is 1 / √(log n), which keeps falling forever. So whatever floor c you pick, the new algorithm eventually goes under it.

The exponent as a share of the number of switches, for brute force, the best previous bound, and the claimed new algorithm
Brute force
21,000
Best previous bound
2386
Claimed new bound
2380

The new curve drops below a 25% floor at about 9 million switches.

Constants and lower-order terms are left out, as the papers leave them unspecified, so the numbers show the shape, not real running times. Our arithmetic, checking the bound the papers state.

Two things make this both huge and modest. Huge: many results in complexity theory say "this problem can't be solved faster, unless the hypothesis is false." Those results now lose their footing. Modest: 2n / √log n is still enormous. It is far slower than any polynomial, so it says nothing about P versus NP. Anthropic's paper puts it plainly: "our result says nothing about solving formulas in practice."

3 What we checked ourselves

We didn't check the proofs; that is the experts' job, and it will take time. We did the arithmetic on the bound both papers state, ignoring the constants they leave open. The new exponent is about 47% of n at 100 switches, 38% at a thousand, 27% at a million, and 19% at a trillion. It falls slowly but never stops, which is exactly what the hypothesis said could not happen. With constants ignored, it passes the best previous bound (38.6%) at around 800 switches.

4 Reactions

  • "Very surprising and consequential that it is false."

    Timothy Gowers, Fields Medal mathematician, @wtgowers, October 26

  • "Maybe the biggest surprise I have seen during my 20 years following theoretical computer science."

    Sebastien Bubeck, OpenAI researcher, @SebastienBubeck, October 26

  • "We are working to verify and examine their proof and determine if it holds up."

    Ryan Williams, MIT, to Al Jazeera, October 27. He posted the paper and its bound on X.

  • "p still probably doesn't equal np"

    Greg Brockman, OpenAI president, @gdb, October 26

5 Timeline

  1. Anthropic says a Claude research model found the core algorithm.
  2. Date on OpenAI's paper (revised October 25).
  3. OpenAI announces three results, including that the ETH is false, and posts the paper and Lean proof on GitHub.
  4. Anthropic posts its paper and a Lean proof of 1,050 modules, checked by Lean and a second, independent checker.
  5. Ryan Williams says his group is working to verify the proof.

6 What it means

If the proofs hold, a widely used assumption in complexity theory is gone, and researchers will have to rework which "no faster algorithm exists" results still stand. Anthropic's paper notes that it also refutes a stronger version, the Strong Exponential Time Hypothesis. It does not settle P versus NP: as OpenAI president Greg Brockman put it, "p still probably doesn't equal np."

It is also a milestone for AI in research: two labs say their models found the same kind of breakthrough on their own. Lean proofs make that easier to trust, but not automatic: Anthropic writes that its Lean proof "was written by AI models and reviewed by other instances of the same models; at the time of writing no person is known to have checked it." Lean confirms the proof is valid; people still have to confirm the statement says what the paper claims.

7 Questions people are asking

Does this mean P equals NP?

No. P equals NP would need a polynomial-time algorithm, like n3. The new bound is still far slower than any polynomial. Disproving the hypothesis removes one route to proving P is not NP, but doesn't settle it.

Will my SAT solver or my encryption get faster or weaker?

Not from this. The result is about how the running time grows for enormous puzzles, with constants left open. Anthropic says it "says nothing about solving formulas in practice."

Is it confirmed?

Not yet by independent people. Both proofs were checked by Lean, but experts led by Ryan Williams are still verifying that the proofs show what the papers claim.

Did both labs really find it at the same time?

Anthropic says its model found its algorithm on October 16. OpenAI's paper is dated October 17. Neither date has been independently confirmed.

What is Lean?

A proof assistant: software where every step of a proof is written in a formal language and checked by a small, trusted program. If it accepts the proof, the theorem as stated follows from the axioms.

8 Updates

  • This page went up. It covers everything known as of October 27, 2026.

9 Sources

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