OpenAI claims AI proofs in math, computing and physics, now under review

OpenAI announced on October 26, 2026 that its internal models produced three new research results in mathematics, computer science and physics, covering harmonic analysis, complexity theory and quantum field theory. The most consequential claim is that its system proved the Exponential Time Hypothesis (ETH), a widely assumed limit on how fast certain logic problems can be solved, is false, a result OpenAI says was verified in Lean, a software tool that checks mathematical proofs step by step.

None of the three results has yet been independently confirmed. Outside working groups are examining two of them, and OpenAI describes the AI-written manuscripts as “a form of raw research data” rather than finished science.

The three results OpenAI is claiming

The announcement arrives less than a month after OpenAI’s mass release of manuscripts it said solved long-standing mathematics questions. Engadget describes that earlier release as more than 700 pages of solutions and progress on 372 major math problems.

Field	Claimed result	Release approach	Who is reviewing
Mathematics (harmonic analysis)	Proof of Sogge’s local smoothing conjecture in every dimension, implying the Kakeya and Stein restriction conjectures	AI output and Lean proof released on GitHub	Open to any researcher
Computer science (complexity theory)	The Exponential Time Hypothesis is false	Natural-language proof and Lean formalisation released on GitHub	Russell Impagliazzo, Ramamohan Paturi and Ryan Williams
Physics (quantum field theory)	Existence proof for certain four-dimensional quantum field theories, including quantum chromodynamics	AI materials withheld; group writing its own account	Group led by David Gross

Mathematics: local smoothing and the Kakeya conjecture

In 1991, Christopher Sogge proposed the local smoothing conjecture, which concerns how waves behave as they become focused; it states that the most extreme focusing must be short-lived. OpenAI says the conjecture implies both the Kakeya conjecture and Stein’s restriction conjecture. Guth, Wang and Zhang proved local smoothing in two spatial dimensions in 2019, and Wang and Zahl proved the Kakeya conjecture in three dimensions in 2025. The Verge notes that NYU professor Hong Wang received a Fields Medal a few months ago for resolving the three-dimensional Kakeya case.

OpenAI says its models proved local smoothing in every dimension, and therefore resolve Kakeya and Stein’s restriction conjecture in all dimensions, using a new auxiliary partial differential equation. It obtained the AI output on October 18 and formalised it in Lean the following day.

The wording of the scope varies. OpenAI’s GitHub summary, as quoted by The Verge, says the proof covers “every spatial dimension at least two” and yields Stein’s spherical restriction estimates “throughout the interior of the conjectured range”, a more qualified description than the blog’s “all dimensions” phrasing.

Physics: quantum chromodynamics

OpenAI describes the physics result as a “mathematically rigorous existence proof of certain interacting quantum field theories containing gauge fields and fermions in four dimensions, including Quantum Chromodynamics (QCD) and a unified completion of the Standard Model of particle physics”. QCD describes how quarks and gluons interact. Secondary accounts frame it differently: one explainer calls it a proof of existence for Yang-Mills theory that does not address the “mass gap” portion of the related Millennium Prize problem, while daily.dev describes it as a proof that pure Yang-Mills theory exists. Because the materials are not public, the claim cannot yet be compared against the text.

Why the ETH claim draws the most attention

ETH is a conjecture about the computational complexity of Boolean formula satisfiability, the problem of deciding whether a logic formula can be made true, and it is closely related to P versus NP. OpenAI notes that it has been used in hundreds of papers to derive hardness results in graph theory, computational biology, databases and computational geometry. Impagliazzo and Paturi, both of UC San Diego, first published it in 1999.

According to OpenAI’s GitHub summary, the paper gives a deterministic algorithm that exactly counts the satisfying assignments of fixed-width CNF formulas in subexponential time and polynomial space; applying it to 3-SAT refutes ETH. In plain terms, ETH said the fastest algorithm for a class of logic puzzles needs exponentially more time as variables are added, and the claim is that a model found a faster-than-exponential method.

This does not settle P versus NP. Applied mathematician John D. Cook wrote that “the status of P = NP is still unknown,” adding that 3-SAT “still takes more than polynomial time as far as anyone knows”. A long technical thread summarised by daily.dev makes the same point: reductions to 3-SAT can inflate problem size enough to erase the gain for other NP problems.

OpenAI is not alone. Anthropic mathematician Levent Alpöge suggested OpenAI was building on pre-existing work, and hours later Anthropic released its own refutation of ETH, TechCrunch reports. Business Insider also reports Anthropic’s same-day release.

A new release process, and its critics

OpenAI says it asked outside experts to act as “shepherds,” and they recommended three different strategies: release the math output openly, form an external group to explain the computer science result while publishing the AI output, and let a physics group decide on release. OpenAI says it does not yet know which approach will prove most useful.

The collaborations follow four principles, according to OpenAI:

• Transparency: no nondisclosure agreements; participants can discuss and publish freely.
• No academic credit: OpenAI does not seek credit for discoveries by its internal models.
• Recognition: the groups get credit for the understanding and write-ups they produce.
• Independence: participation is voluntary and unpaid, and members do not speak for OpenAI.

OpenAI concedes the trade-off: “choosing initial shepherds necessarily shapes who gets involved first and creates asymmetry of information”. The AGMAI panel of math advisers that OpenAI assembled recommended a day earlier that AI companies not designate people to receive output first, citing unequal access, borrowed reputation and biased selection; it urged releases on scholarly repositories with Lean proofs, details of how problems were chosen and funding for people working to understand the results.

The computer science group released an open letter asking for input on three questions, including “How should credit recognize both new discoveries and the work of checking, fixing, simplifying, explaining, and extending them?”. Williams, who called OpenAI’s earlier October release a “DDOS attack on mathematicians,” says he pushed OpenAI to support a conference because the result “is going to affect many researchers’ works”. The physics group has published nothing yet but has provided an email address for questions.

What independent experts say so far

Reviewers are careful to separate involvement from endorsement. Williams “has not certified the company’s results,” Business Insider reports, and he has previously said he does not believe in ETH. He told Al Jazeera: “We are working to verify and examine their proof and determine if it holds up”.

“Some parts [of OpenAI’s proof] are, maybe you could say routine, but then yes, there are two or three alien-looking parts, and it’s exactly these parts that I saw that made me think, wow, this is something that the whole community needs to work on together.” — Ryan Williams, to TechCrunch

Williams also said he does not think “OpenAI’s algorithm could possibly be the best algorithm you can get”. Fields Medallist Timothy Gowers wrote that it is “Very surprising and consequential that it is false”, and Gary Marcus called it potentially “a big deal if correct”.

On the math result, University of Rochester professor Alex Iosevich, whose PhD adviser was Sogge, told Al Jazeera OpenAI appeared to have “cracked” the conjecture, adding: “I feel a twinge of sadness as a result”.

Market-news service Newsquawk offers a broader caution: vendor claims of foundational scientific results from AI labs have a mixed record, with a substantial share later needing correction, restating known results or containing gaps in formal proofs.

What it means for builders

Nothing changes in production systems this week. Williams told Al Jazeera the development would not have real-world applications in the short term, though it could lead to future benefits. OpenAI itself says the refutation may help guide the search for better algorithms, but that finding them and turning theory into practical improvements is still ahead.

Three practical points stand out:

• Hardness assumptions: Many papers that relied on ETH will need revisiting if the claim holds, Williams said. Teams that cite ETH-based lower bounds when arguing a problem cannot be solved faster should treat those arguments as under review.
• Security: Williams asked OpenAI science lead Dan Roberts whether any cryptosystems were affected; OpenAI checked and was “reasonably confident” there were no breaks, Williams said.
• Model capability signal: Two labs claiming the same result on the same day suggests frontier models are being pointed at open research problems, with formal verification in Lean as part of the output.

The models that produced these results are described only as internal models; the sources give no indication they are available through an API. For OpenAI’s publicly released models, see Unicorn API’s coverage of GPT-6.1 Terra.

What to watch next

• The ETH working group’s verdict: whether Impagliazzo, Paturi and Williams confirm, correct or reject the proof, and whether the conference Williams requested goes ahead.
• Comparison with Anthropic’s proof: whether the two refutations use the same method, and how Alpöge’s point about pre-existing work is resolved.
• The physics write-up: what David Gross’s group, which includes Michael Douglas, Arthur Jaffe, Igor Klebanov and Nikita Nekrasov, eventually publishes.
• Math community scrutiny: whether harmonic analysts confirm the full scope of the local smoothing claim, given the differing descriptions.
• Release norms: whether OpenAI shifts towards the AGMAI panel’s recommendations for future releases.