Openai AI Model Challenges Longstanding Complexity Theory Conjecture
WHO:
Al Jazeera - Alex Iosevich, a University of Rochester mathematics professor specializing in harmonic analysis, said OpenAI appeared to have “cracked” the local smoothing conjecture in a “historic development.”
Al Jazeera - Iosevich said the announcement initially shocked him, but he was optimistic AI could help science make “giant leaps”; he also said he felt “a twinge of sadness” and expected theoretical mathematics to “change drastically.”
Al Jazeera - Ryan Williams said the ETH proof proposes “a mysterious new method” for solving fundamental problems more efficiently.
The Next Web - OpenAI said an internal AI model produced a claimed refutation of the Exponential Time Hypothesis (ETH).
The Next Web - Russell Impagliazzo and Ramamohan Paturi of UC San Diego, and Ryan Williams of MIT, will organize an open, independent examination of the ETH claim. Impagliazzo and Paturi first proposed ETH with Francis Zane.
The Next Web - Impagliazzo, Paturi and Williams said, “Our involvement should not be taken as certification of the claim. A result of this significance requires careful, independent scrutiny.”
The Next Web - Williams said the reviewers signed no non-disclosure agreements and will not be paid.
The Next Web - OpenAI said it does not seek academic credit for the discoveries; it said researchers who explain the results should receive recognition.
The Next Web - Nobel laureate David Gross will lead the physics group, which includes Michael Douglas, Arthur Jaffe, Igor Klebanov and Nikita Nekrasov.
WHAT:
The Verge - OpenAI’s local-smoothing result also yields the Kakeya maximal conjecture and Stein’s spherical restriction estimates throughout the interior of the conjectured range.
The Verge - OpenAI’s ETH paper describes an algorithm that exactly counts satisfying assignments of fixed-width CNF formulas in subexponential time and polynomial space.
Al Jazeera - OpenAI said harmonic analysis researchers had previously proved the local smoothing conjecture only in three dimensions.
Al Jazeera - OpenAI described the work as “mathematically rigorous” proof concerning quantum chromodynamics.
Al Jazeera - OpenAI said three expert groups served as “shepherds” and that different release approaches involved tradeoffs, including potential information asymmetry and the initial groups’ influence over participation.
Al Jazeera - Al Jazeera reported Anthropic announced a discovery of a new enzyme said to resemble CRISPR the previous month; scientists reacted with both enthusiasm and skepticism about its significance.
The Next Web - OpenAI said ETH had been “widely believed to be true.”
The Verge - The ETH paper’s deterministic algorithm exactly counts satisfying assignments of fixed-width CNF formulas in subexponential time and polynomial space.
The Next Web - Christopher Sogge proposed the local smoothing conjecture in 1991.
The Verge - OpenAI’s local-smoothing proof uses an auxiliary partial differential equation to establish Sogge’s result for waves in every spatial dimension of at least two.
The Verge - The local-smoothing proof also yields the Kakeya maximal and dimension conjectures, and Stein’s spherical restriction estimates throughout the interior of the conjectured range.
The Verge - OpenAI’s ETH paper presents a deterministic algorithm that exactly counts satisfying assignments of fixed-width CNF formulas in subexponential time and polynomial space; its 3-SAT case refutes ETH.
The Next Web - OpenAI announced three claimed AI-produced results: a refutation of ETH, a proof of the local smoothing conjecture, and a proof that certain four-dimensional quantum field theories exist.
The Next Web - ETH concerns the difficulty of solving Boolean satisfiability, a problem closely related to P versus NP. Researchers have used it to establish hardness results in graph theory, computational biology, databases and other fields.
The Next Web - OpenAI’s claimed local smoothing proof applies in every dimension and, if correct, would also settle the Kakeya conjecture and Stein’s restriction conjecture in all dimensions.
The Next Web - The physics result includes a proof that pure Yang-Mills theory exists; the reported theories include quantum chromodynamics, which describes the strong force between quarks and gluons.
The Next Web - OpenAI agreed to release all raw data on the ETH result and fund a conference for the research community. It also gave $5 million to the Simons Foundation for workshops and conferences on AI-produced results.
Techmeme - OpenAI described the announcements as discoveries in harmonic analysis, complexity theory and quantum field theory.
WHERE:
Al Jazeera - OpenAI is based in San Francisco.
The Next Web - The ETH reviewers are affiliated with UC San Diego and MIT; the reported quantum field theories are in four dimensions.
WHEN:
The Next Web - OpenAI’s internal model found the claimed ETH disproof on 16 October; the company checked it in Lean the next day.
No dates included because the articles’ dates are not future dates.
WHY:
Al Jazeera - Williams said a confirmed ETH refutation could potentially lead to future societal benefits, although he did not expect short-term real-world applications.
The Next Web - Mathematicians raised similar concerns about credit for AI-discovered results in the Leiden declaration in June.
The Next Web - The three computer scientists said independent scrutiny is necessary for a result of this significance.
The Next Web - The researchers raised concerns that AI-discovered results may be credited to those who publish first or possess the most powerful tools, while verification and explanation receive less recognition.
HOW:
Al Jazeera - OpenAI said it consulted three expert groups acting as “shepherds” on how to release its findings, and acknowledged that choosing initial shepherds shapes who participates first and creates information asymmetry.
The Next Web - OpenAI said its model found the ETH disproof on 16 October and that the company checked it the next day using Lean, a language for machine-verified proofs. OpenAI published the model’s write-up and Lean proof on GitHub.
The Next Web - OpenAI also released a write-up and Lean proof for the local smoothing result. The physics group did not release its AI-produced material and will write up the work itself.
The Next Web - OpenAI said mathematicians had proved local smoothing in two dimensions in 2019 and the Kakeya conjecture in three dimensions in 2025; it also said it had published 722 mathematics papers on 6 October.
The Next Web - OpenAI said in September it would not claim the Millennium Prize for its Navier–Stokes proof.
(UPDATED October 27, 2026 18:00 UTC)
OpenAI says its AI system has produced major results in mathematics and physics, including a claimed disproof of the Exponential Time Hypothesis, a proof of the local smoothing conjecture, and a result showing certain four-dimensional quantum field theories exist. The math findings could also resolve long-standing problems in harmonic analysis, while the ETH claim describes a faster way to count solutions to certain Boolean formulas; OpenAI says the work was checked with Lean, a proof-verification system. Independent computer scientists will scrutinize the ETH result, and mathematicians say the claims need careful review as researchers debate how credit should be shared for discoveries made with AI.
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(UPDATED October 27, 2026 13:21 UTC)
OpenAI says an internal AI model produced three major mathematical and physics results, including a refutation of the widely believed Exponential Time Hypothesis, a proof of the local smoothing conjecture, and a proof that certain four-dimensional quantum field theories exist. The company says it machine-checked the ETH proof and released it for review, while an independent group of computer scientists stressed that their examination is not a certification; a separate team of physicists will assess the physics result. Mathematicians have called the claims potentially historic but say they require careful scrutiny, as questions grow over who receives credit when AI produces discoveries.
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(UPDATED October 27, 2026 09:08 UTC)
OpenAI says its AI model produced a result that would refute the Exponential Time Hypothesis, a widely used assumption about the difficulty of solving certain Boolean problems, and says it checked the work with machine-verified proof software. Computer scientists Russell Impagliazzo, Ramamohan Paturi and Ryan Williams will lead an independent review, stressing that their involvement is not an endorsement; OpenAI says it will release raw data and fund research gatherings. The company also announced claimed proofs of the local smoothing conjecture and the existence of certain four-dimensional quantum field theories, though the physics team has not released its material.
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(UPDATED October 27, 2026 04:09 UTC)
OpenAI says an internal AI model produced three major results: a claimed refutation of the Exponential Time Hypothesis, a proof of the local-smoothing conjecture, and a proof that certain four-dimensional quantum field theories exist. The company says it checked the ETH proof using Lean and released the proof materials, while mathematicians Russell Impagliazzo, Ramamohan Paturi and Ryan Williams are organizing an independent review and caution that their involvement does not certify the claim. OpenAI says it will release raw data on the ETH result and support community scrutiny, as researchers raise concerns about ensuring recognition goes to those who verify and explain AI-produced discoveries.
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(UPDATED October 27, 2026 02:15 UTC)
OpenAI announced three claimed AI-generated breakthroughs: a refutation of the Exponential Time Hypothesis, a proof of the local smoothing conjecture, and a proof that certain four-dimensional quantum field theories exist. Computer scientists Russell Impagliazzo, Ramamohan Paturi and Ryan Williams will lead an independent review of the ETH claim, stressing that their involvement is not an endorsement; OpenAI says it will release the raw data and fund research gatherings. OpenAI says it checked the ETH proof with Lean, a tool for machine-verified proofs, while the physics group, led by Nobel laureate David Gross, will prepare its own account of that work.