OpenAI and Anthropic Say Their Models Refuted the Exponential Time Hypothesis

OpenAI said on October 26, 2026 that one of its internal models had proved the Exponential Time Hypothesis (ETH) false, a claim it backed with a machine-checked Lean proof. Anthropic released its own refutation, credited to an internal Claude research model, hours later, TechCrunch reported.

The ETH, first published by Russell Impagliazzo and Ramamohan Paturi in 1999, says that 3-SAT, a basic logic puzzle, cannot be solved much faster than by trying an exponential number of possibilities. OpenAI said in its announcement that hundreds of papers in graph theory, computational biology, databases and computational geometry use the ETH to derive hardness results. Both labs had their proofs checked by the Lean proof assistant, but neither had been verified by outside complexity theory researchers as of October 27, 2026.

What exactly did OpenAI and Anthropic claim?

Both labs claim a deterministic algorithm that solves 3-SAT on n variables in 2^o(n) time, meaning its running time grows more slowly than any fixed exponential, which is exactly what the ETH rules out. Both algorithms go further and count the satisfying assignments of a formula, a harder problem than deciding whether one exists.

OpenAI's paper, dated October 17 and revised October 25, covers formulas with clauses of any fixed width k and uses polynomial memory. Anthropic's paper, dated October 26, states a running time of 2 to the power of n log log n divided by the square root of log n, up to a constant, for counting 3-SAT solutions. Before these papers, the fastest deterministic 3-SAT algorithm ran in about 1.328^n time, according to Anthropic's paper.

Anthropic's paper also says the result refutes the stronger Strong Exponential Time Hypothesis (SETH) and implies faster algorithms for graph problems such as Independent Set, Clique and Vertex Cover. It adds that the result "does not bear on the P vs NP question directly," and its GitHub page says the result says nothing about solving formulas in practice.

How do the two refutations compare?

The two papers report the same shape of running time in n and say their core result was found on the same day, October 16, 2026. Anthropic's paper describes OpenAI's work as independent and concurrent, and says its own core algorithm came from an internal research model using 14 million output tokens with no human input, after which Anthropic researchers worked with the model on the presentation.

Anthropic mathematician Levent Alpöge suggested OpenAI was basing its solutions on existing work before Anthropic released its own paper, according to TechCrunch.

Who is checking the proof?

A working group led by Russell Impagliazzo and Ramamohan Paturi of UC San Diego and Ryan Williams of MIT will examine and explain OpenAI's result, OpenAI said. In an open letter, the three write that their involvement "should not be taken as certification of the claim" and invite other researchers to examine the argument independently.

At the group's request, OpenAI published the AI-written proof and its Lean formalization immediately and committed to fund a conference on what the result means for the field. OpenAI says group members are unpaid, signed no nondisclosure agreements and do not speak for the company, and that it does not seek academic credit for the discovery. The letter also proposes a virtual meeting before the end of 2026 and lists Anthropic's refutation among its resources.

Williams told TechCrunch he asked OpenAI science lead Dan Roberts whether the algorithm breaks any cryptographic systems, and that OpenAI said it had checked and was reasonably confident it did not.

Why is OpenAI releasing results this way?

OpenAI says it is testing three release strategies after feedback from academics: direct release for a math result, an outside working group plus public files for the ETH result, and a group that controls release for a physics result. The change follows criticism of its October 6, 2026 release of hundreds of AI-written math manuscripts (The Verge), which Williams called a "DDOS attack on mathematicians," according to TechCrunch.

The same October 26 announcement claims a proof of the local smoothing conjecture in every dimension and an existence proof for quantum chromodynamics, with a group led by Nobel laureate David Gross shepherding the physics work. OpenAI also points to a $5 million contribution to the Simons Foundation for researchers who examine AI-produced results. OpenAI's own advisory panel on mathematics and AI had recommended a day earlier that labs not give selected people first access to such results, The Verge reported.

What we don't know yet

• Whether either proof survives independent review, and how long the working group expects that review to take.
• Whether OpenAI's and Anthropic's algorithms rest on the same core technique.
• When the OpenAI-funded conference will take place and who will organize it.
• Whether faster algorithms with practical value will follow from the result.