OpenAI's Kakeya, ETH and Yang–Mills claims: what they say and what stays open

On 26 October 2026, OpenAI announced three results produced by its internal models:

• Mathematics. A proof of Sogge's local smoothing conjecture in every dimension, which would also settle the Kakeya and Stein restriction conjectures in all dimensions. A Lean formalization has been released.
• Computer science. A refutation of the Exponential Time Hypothesis (ETH), the widely used assumption that 3-SAT needs exponential time. A Lean proof is public, and a group led by Russell Impagliazzo, Ramamohan Paturi and Ryan Williams is organizing an open examination of it.
• Physics. A rigorous existence proof for 4D pure Yang–Mills theory, QCD and a unified completion of the Standard Model, building on Balaban's approach. A group led by David Gross is shepherding the work, and the AI-produced materials have not been released yet.

A few hours later, at 03:07 UTC on 27 October, Anthropic posted its own refutation of ETH to GitHub: a paper credited to "Claude, Anthropic" and a Lean 4 formalization. Both companies say their models found the result on the same day, 16 October.

If these hold up, each would be a landmark. But the most interesting part of the announcement, for us, is not the results. It is how they were released, and how hard it already is to say who claimed what, when, and who checked it.

Below: what each problem asks and what exactly was claimed, the two ETH claims side by side, why P versus NP and the Yang–Mills mass gap stay open, the three release strategies and the criticism of them, and what we changed on Papex. Every source was checked on 27 October 2026. None of the three results has been independently confirmed yet.

What is the local smoothing conjecture?#

It is a statement about how waves focus. A wave can concentrate sharply at the instant its rays meet, the way light gathers at the focus of a lens. OpenAI's announcement summarizes the conjecture like this: "the most extreme focusing must be short-lived. The amount of focusing at a typical random time is always much smaller than what can happen at the worst-case time."

Christopher Sogge proposed it in 1991. The opening of OpenAI's paper states the question more precisely: "A wave arriving from a thin spherical shell may have a much larger L^p norm at the instant when its rays come to a focus than it had initially. Sogge asked how many derivatives of the initial data are needed to control the norm averaged over a time interval." The conjecture names that number for every dimension and every exponent p. Averaged over a short stretch of time, a wave is smoother than its worst instant suggests, and the conjecture says by how much.

Before October, the conjecture was proved in two space dimensions. OpenAI dates that proof, by Larry Guth, Hong Wang and Ruixiang Zhang, to 2019; it appeared in the Annals of Mathematics in 2020. OpenAI's paper claims every dimension from two upwards, and says its proof rests on "a novel auxiliary partial differential equation".

Why would local smoothing settle the Kakeya conjecture?#

Because local smoothing is strong enough to imply it. A Kakeya set (also called a Besicovitch set) is a set that contains a line segment of length one pointing in every direction. Sōichi Kakeya asked in 1917 how small such a set can be, and Abram Besicovitch showed in 1920 that in the plane it can have zero area, according to Wikipedia's summary. The Kakeya conjecture says that even so, such a set cannot be thin in another sense: in n dimensions its Hausdorff dimension must be n. Hong Wang and Joshua Zahl proved the three-dimensional case in 2025, a result OpenAI's post cites. Every higher dimension was open.

OpenAI's paper derives a list of consequences from local smoothing. Its abstract: "We also prove the Kakeya and Nikodym maximal conjectures; in particular, every Euclidean Kakeya set has full Hausdorff dimension." It also claims the Bochner–Riesz conjecture and sharp bounds for Stein's square function, two other long-standing questions in Fourier analysis.

The fine print on restriction. The announcement says the result resolves "Stein's restriction conjecture in all dimensions". That conjecture is about Fourier transforms of functions that live on curved surfaces such as the sphere. The paper words its own claim more narrowly: "We obtain Fourier restriction estimates for the sphere, the compact paraboloid, and the truncated cone throughout the interior of the conjectured range". The repository's change log uses the same phrase. Whether "the interior of the conjectured range" covers every form of the conjecture is a question for specialists. It is the kind of detail an announcement compresses and a record should keep.

What the Lean proof covers. OpenAI's scope note says the formalization bounds the wave in every dimension n ≥ 2 and every 2 ≤ p < ∞, gives an L^n bound for the Kakeya maximal operator, and proves that "every subset of ℝ^n containing a unit segment in each direction has Hausdorff dimension n". A Lean proof checks that the formal statements follow from the axioms. It does not check that they are the statements mathematicians mean; that comparison is done by people.

How it was released. OpenAI says it obtained the AI output on 18 October and formalized it in Lean the next day. On an expert's advice it released the output directly, as it did with its manuscripts of 6 October, and left mathematicians to organize their own efforts to explain it. The Kakeya and restriction conjectures are not in Papex's catalogue of open problems, so no Papex page changed for this result.

What is the Exponential Time Hypothesis?#

It is a bet about the hardest version of a simple puzzle. In 3-SAT you get a formula made of clauses, each an OR of up to three variables or their negations, joined by AND. The question is whether some assignment of true and false to the variables makes every clause true. Checking a proposed answer is easy. Finding one, as far as anyone knew, could take time that grows exponentially with the number of variables n: trying every assignment takes 2^n steps.

Since the 1980s, cleverer algorithms have lowered the base of that exponential. Anthropic's paper gives the previous records: 1.328^n for deterministic algorithms and 1.308^n for randomized ones, up to constant factors. The Exponential Time Hypothesis, stated by Russell Impagliazzo and Ramamohan Paturi in "On the Complexity of k-SAT" (2001), says this kind of progress has a floor: there is some fixed δ > 0 such that no algorithm solves 3-SAT in 2^(δn) time. You can shrink the base, but you cannot get below exponential.

ETH mattered because so much was built on it. OpenAI's announcement says it "has been used in hundreds of papers to derive conditional computational hardness results across a wide variety of fields including graph theory, computational biology, databases, computational geometry, and more". A typical result reads: if ETH is true, then this graph problem cannot be solved faster than a certain bound. Its stronger cousin, the Strong Exponential Time Hypothesis (SETH), was used the same way for problems such as edit distance between strings. Papex's ETH page covers the history and the evidence that was offered for it.

What exactly did OpenAI and Anthropic claim about ETH?#

Both claim the same thing in substance: a deterministic algorithm that counts the satisfying assignments of a formula, not just decides whether one exists, in time that grows more slowly than 2^(δn) for any fixed δ > 0. Both give the same leading term in the exponent, n·log log n/√log n.

Anthropic's paper says the gain is unlikely to appear "unless n > 10^7", far beyond formulas anyone could check by brute force. It also says the two results were found "independently and concurrently", citing OpenAI's paper. OpenAI's announcement does not mention Anthropic.

OpenAI's paper lists what else falls with ETH: "Theorem 1.1 refutes ETH, SETH, Gap-ETH, and NSETH", along with the counting version, #ETH. Some of the hardness results built on these hypotheses lose their footing. The paper adds that "some of the same lower bounds follow from weaker hypotheses that may still hold", such as the Orthogonal Vectors Conjecture.

Same day, no shared clock. Both companies date the discovery to 16 October. Both dates are the companies' own statements; the first public files are the GitHub commits of 26 and 27 October. An independent timestamp on the day would not settle who had the idea first, but it would show that a particular file existed by then. That is the kind of question that is easy to answer at the time and hard to answer a year later.

The examiners say they are not certifying it. Impagliazzo, Paturi and Williams, two of whom proposed ETH, wrote an open letter to colleagues: "Our involvement should not be taken as certification of the claim. A result of this significance requires careful, independent scrutiny". They want the examination open "without conferring ownership of the effort on a company or a selected group of researchers, including ourselves", and they list Anthropic's refutation among its resources, next to OpenAI's. Their view of the stakes: "If the claim is correct, it will change an important part of theoretical computer science."

Does refuting ETH settle P versus NP?#

No. Refuting ETH means 3-SAT can be solved faster than exponential time. Faster than exponential is not the same as polynomial, so P ≠ NP remains entirely possible.

Both papers say so. OpenAI's: "runtime bounds of the form 2^o(n) do not transfer under general polynomial-time reductions …, so our result does not imply that general NP problems admit subexponential time algorithms." Anthropic's: "our result does not bear on the P vs NP question directly, nor does it yield any practical algorithms for NP-complete problems."

The P versus NP problem asks whether every problem whose answers can be checked quickly can also be solved quickly, meaning in polynomial time. A running time such as 2^(n/√log n) is far smaller than 2^n, but still far larger than any polynomial. Our Millennium Prize explainer has the background on P versus NP and the other six problems.

What did OpenAI claim about Yang–Mills and QCD?#

An existence proof, not a mass gap. The Clay Mathematics Institute's official problem description, by Arthur Jaffe and Edward Witten, asks for two things at once: "Prove that for any compact simple gauge group G, a non-trivial quantum Yang–Mills theory exists on ℝ^4 and has a mass gap Δ > 0." The mass gap means "there must be some constant Δ > 0 such that every excitation of the vacuum has energy at least Δ". Jaffe and Witten explain why it matters: the classical Yang–Mills equations describe "massless waves that travel at the speed of light", yet a mass gap is "necessary to explain why the nuclear force is strong but short-ranged".

OpenAI's announcement claims "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". It adds: "As a subproblem, this work also includes a proof of the existence of pure Yang–Mills theory, which is part of the Quantum Yang–Mills Theory Clay Mathematics Institute Millennium Prize problem." The word "part" is accurate. The announcement does not mention the mass gap.

OpenAI says its models produced the Yang–Mills existence proof on 9 September and the QCD proof on 18 September, and that it finished formalizing Yang–Mills existence in Lean on 16 October. None of it is public. David Gross, who shared the 2004 Nobel Prize in Physics for asymptotic freedom, leads a group with Michael Douglas, Arthur Jaffe, Igor Klebanov and Nikita Nekrasov. Jaffe co-wrote the Clay problem with Witten. According to OpenAI, the group "has decided to hold off on releasing the AI-produced artifacts and instead work on writing it up themselves."

Not everyone agrees with that choice. On 26 October the Advisory Group on Mathematics and Artificial Intelligence (AGMAI), a group of mathematicians hosted at the Institute for Advanced Study, called on OpenAI "to make public without delay the natural-language and Lean outputs behind its announced existence proofs". Its reasoning: "OpenAI have said they delayed release until the Lean formalization was complete. By their own account this is now complete, so there is no scientific reason for delaying the release any further."

Clay considers a proposed solution only after publication in a qualifying outlet, a wait of at least two years and general acceptance, so nothing changes at Clay soon. The Yang–Mills existence and mass gap problem stays open on Papex, as it is on Clay's list.

Three results, three release strategies#

OpenAI did not release all three the same way. External experts recommended a different approach for each field:

• For the mathematics result, publish the AI output and let researchers organize their own efforts to understand it.
• For the computer science result, publish the output and form an outside group to verify and explain it.
• For the physics result, form an outside group and let it decide what to release and when.

OpenAI says openly that it does not yet know which approach works best, and asks for feedback. It also sets out principles for the collaborations: it seeks no academic credit for the discoveries, and the human working groups should be recognized for the understanding, writeups and materials they produce. Participants are not asked to sign nondisclosure agreements, take part unpaid, and speak only for themselves.

Anthropic took a fourth route for its ETH result: the paper and Lean proof went straight onto its own GitHub, with no working group. The folder calls itself a "Research artifact. Not maintained and not accepting contributions." About two hours later, Anthropic deposited the paper on Hexagon, a repository for research in the mathematical sciences and theoretical computer science, as hexagon:2610.00577v1. The listed author is Anthropic, as an organization, and the deposit carries two declarations: "Primarily AI-generated", and "Some parts of this contribution, but not all, have been checked and understood by a human."

The day before OpenAI's announcement, AGMAI published a statement on exactly this choice, "Disadvantages of Designating Mathematicians for AI Output Releases". It grants the appeal: early expert eyes, a better first writeup, a point of contact. But it recommends "that AI companies do not adopt this practice for future releases", for five reasons it names: delayed or unequal access, denied agency, biased selection, borrowed reputation, and that it is unnecessary. On borrowed reputation: "designated mathematicians' names end up vouching for the release and shielding the AI company from criticism." It stresses that this is no criticism of the mathematicians involved, who "have acted in good faith and done valuable work". The ETH examiners raise the same tension in their letter: "Early access can help researchers check a result before release, but it also creates opportunities unavailable to others."

That is a real attempt to separate kinds of contribution. It also shows how little infrastructure science has for doing so.

Discovery, verification and explanation are different contributions#

In a traditional paper, these three things usually arrive together, from the same people, in one document. With AI-produced results they come apart:

• The claim comes from a model, on a specific date, in a specific output.
• Verification comes from a Lean formalization, and from experts checking that the formal statement matches the conjecture people actually care about.
• Explanation comes later, from people who turn a long proof into understanding the field can build on.

Each step has different authors, different dates and different evidence. ETH already shows it twice over: two models, one discovery date, two papers posted about six hours apart, and a group of examiners who say their involvement is not a certification. Today most of this lives in blog posts, GitHub repositories, shared documents and email threads. A year from now, it will be hard to answer simple questions: when was this first claimed, which version was verified, and who did the work of making it understandable?

What we changed on Papex today#

Papex tracks 190+ open problems, from the Riemann hypothesis to the function of sleep. The Exponential Time Hypothesis was one of them. Each problem shows the status its source gives it, never a verdict of ours. Until today, all but one were open. The exception was Navier–Stokes, which the Clay Mathematics Institute has listed as active since OpenAI's claim in September.

That is no longer enough. A claimed resolution now has its own status, shown on every problem card and page:

• Open: the source lists the problem as unsolved.
• Claimed: a resolution has been announced. The page names who claimed it, when and where, links the writeup and any formal proof that has been released, and names whoever is checking it.
• Under review: the body behind the problem, such as a prize committee, is weighing a claim.
• Resolved: the source treats the question as settled.

The ETH page is now marked Claimed. It lists OpenAI's and Anthropic's claims, each with its date, writeup and Lean proof, and names the group organizing the examination. Papex does not decide whether the refutation holds. When the examination reports, its findings go on the page.

Two related problems did not change status, and the reasons matter:

• P versus NP stays open. A subexponential algorithm for 3-SAT is not a polynomial one, as both papers say.
• Yang–Mills existence and mass gap stays open. The announcement claims existence, but does not address the mass gap. We have added the claim to the page as a partial result, with a note.

Open questions#

Some questions this release raises, and that we don't think have settled answers yet:

• When is a result verified? Is a Lean proof enough, or does it also need experts to confirm the formal statement says what we think it says?
• What should be on the record for an AI-produced result: the model, the date, the exact output, the prompts, the failed attempts?
• If a model makes the discovery and people make it understandable, how should each be credited, and where should that credit live?
• When two labs claim the same result on the same day, what evidence of the date should count?

For mathematics, AGMAI has offered one answer to the second question. Its summary of its September recommendations asks labs to post AI output "on scholarly repositories that are not controlled by the posting AI lab, have citable identifiers, and record modifications", together with "the name of the model, prompts, chains of thought, number of attempts, compute used, and costs". The ETH examiners ask the third question in their own words: how credit should "recognize both new discoveries and the work of checking, fixing, simplifying, explaining, and extending them".

OpenAI's release is a useful experiment in answering these in public. The answers will need a durable record, not just a press release.

That record is what we are building at Papex: hypotheses, findings and publications linked to the people, tools and versions behind them, signed and dated independently. You can follow the open problems, and the status of each claim, at papex.org/explore/problems.

Sources#

• Working with academic communities to build scientific understanding (OpenAI, 26 October 2026), checked 27 October 2026 (Internet Archive copy of 26 October 2026; the live page blocked automated access)
• Local smoothing for the Euclidean wave equation (OpenAI, 25 October 2026), checked 27 October 2026
• Lean scope note 079, local smoothing (openai/math, GitHub), checked 27 October 2026
• The Exponential Time Hypothesis is False (OpenAI, 17 October 2026, revised 25 October 2026), checked 27 October 2026
• Lean scope note 378, ETH (openai/math, GitHub), checked 27 October 2026
• openai/math README and history (GitHub), checked 27 October 2026
• Sharing AI progress in mathematics (OpenAI, 6 October 2026), linked from the 26 October announcement
• A 2^o(n) Time Algorithm for 3-SAT (Claude, Anthropic, 26 October 2026), checked 27 October 2026
• anthropics/formal-math, eth folder README and commit 16a5ce41 (GitHub, 27 October 2026), checked 27 October 2026
• A 2^o(n) Time Algorithm for 3-SAT, hexagon:2610.00577v1 (Hexagon, 27 October 2026), checked 27 October 2026
• Impagliazzo, Paturi and Williams, letter to colleagues on the ETH claim (October 2026), checked 27 October 2026
• Recommendation for Immediate Release of Yang–Mills AI Output (AGMAI, 26 October 2026), checked 27 October 2026
• Disadvantages of Designating Mathematicians for AI Output Releases (AGMAI, 25 October 2026), checked 27 October 2026
• General Guidelines (AGMAI, 29 September 2026), checked 27 October 2026
• Jaffe and Witten, Quantum Yang–Mills Theory (Clay Mathematics Institute), checked 27 October 2026
• Yang-Mills & the Mass Gap (Clay Mathematics Institute), checked 27 October 2026
• Rules for the Millennium Prize Problems (Clay Mathematics Institute), checked 27 October 2026
• Impagliazzo and Paturi, On the Complexity of k-SAT (Journal of Computer and System Sciences, 2001), checked 27 October 2026
• Guth, Wang and Zhang, A sharp square function estimate for the cone in ℝ^3 (Annals of Mathematics, 2020), checked 27 October 2026
• Kakeya set (Wikipedia), checked 27 October 2026