OpenAI solved open math problems, it says: local smoothing, the Exponential Time Hypothesis and QCD, and what has actually been checked

OpenAI solved open math problems, it says: in an announcement on Monday, October 26, 2026, the company said its AI solved open problems in mathematics, computer science and physics. According to OpenAI's public repository and reports of the announcement, an unreleased internal model proved Sogge's local smoothing conjecture in every dimension, refuted the Exponential Time Hypothesis (ETH), a widely believed assumption in complexity theory, and produced existence proofs for interacting quantum field theories, including quantum chromodynamics (QCD). These are much bigger claims than the KLS conjecture proof that credited ChatGPT in early October 2026, and they arrive with a different kind of evidence for each.

This story is based on OpenAI's October 26 announcement, "Working with academic communities to build scientific understanding," its openai/math GitHub repository (its history log, manuscript map, the ETH preprint and the Lean scope notes for both new results), the ETH working group's open letter, reports by Al Jazeera, Interesting Engineering, Quartz and The Next Web, Peter Woit's October 2026 blog post, OpenAI's May 20, 2026 post, and coverage of the earlier claims by TechCrunch and Quanta Magazine, checked October 28 and 29, 2026.

The short version

• Three claims: local smoothing for the wave equation in every dimension (harmonic analysis), a disproof of ETH (computer science), and existence proofs for four-dimensional gauge theories including QCD (mathematical physics).
• Machine-checked: the math and computer-science results come with Lean formalizations, according to OpenAI's repository. OpenAI says it also finished a Lean formalization of the Yang–Mills part of the physics work on October 16, but it hasn't been released.
• Human review: OpenAI asked outside experts to act as "shepherds." Named groups are examining the ETH and physics results; for the math result, OpenAI released the output and left researchers to organize their own efforts. None has published a verdict. The physics group is holding the materials back to write its own explanation first.
• Early reactions: a harmonic analyst called the local smoothing result "historic"; MIT's Ryan Williams said his group is "working to verify" the ETH proof; Woit called the physics press release hard to interpret.
• Track record: an October 2025 Erdős claim was deleted after mathematicians objected; a May 2026 Erdős disproof was endorsed by outside mathematicians; OpenAI withdrew three of its own manuscripts on October 7, 2026 over a sign error.

Which problems does OpenAI say its AI solved?

Local smoothing (harmonic analysis). OpenAI's manuscript, dated October 25, 2026, says it proves "Sogge's local smoothing conjecture for the Euclidean wave equation in every spatial dimension at least two," and derives a chain of famous consequences: the Bochner–Riesz conjecture, the Kakeya and Nikodym maximal conjectures, and Fourier restriction estimates for the sphere "throughout the interior of the conjectured range." In particular, it says, "every Euclidean Kakeya set has full Hausdorff dimension." Before this, OpenAI's announcement notes, local smoothing had been proved in two dimensions and the Kakeya conjecture in three. OpenAI's repository already held September manuscripts claiming Kakeya results in three and four dimensions and critical local smoothing in three.

The Exponential Time Hypothesis (computer science). ETH says that solving 3-SAT, the classic hard logic puzzle, on n variables needs time exponential in n in the worst case. OpenAI's paper, "The Exponential Time Hypothesis is False," dated October 17 and revised October 25, gives "a deterministic algorithm for exactly counting the satisfying assignments of an arbitrary k-CNF formula" that runs in subexponential time, about 2 to the power of n divided by the square root of log n, using polynomial space. Counting solutions also decides whether one exists, which is why the paper says the result "refutes" ETH. The paper notes that it does not show that general NP problems have subexponential algorithms.

Quantum field theory (physics). OpenAI's announcement describes "a mathematically rigorous existence proof of certain interacting quantum field theories containing gauge fields and fermions in four dimensions," including QCD, the theory of quarks and gluons, and "a unified completion of the Standard Model," which it explains as the simplest grand unified theory that includes the Standard Model's gauge fields and fermions. As a subproblem, OpenAI says, the work includes a proof that pure Yang–Mills theory exists, which is part of one of the Clay Millennium Prize problems. OpenAI says its models produced a natural-language Yang–Mills proof on September 9 and a QCD proof on September 18, 2026, and that it finished a Lean formalization of Yang–Mills existence on October 16.

Last verified: October 29, 2026

Has anyone independently verified the proofs?

Not yet. OpenAI says it asked outside experts to act as "shepherds" and that they advised a different release approach for each result. OpenAI says participation is voluntary and unpaid, with no nondisclosure agreements. None of the groups has reported back.

• ETH: OpenAI's repository history says "an independent working group of computer scientists led by Russell Impagliazzo, Ramamohan Paturi, and Ryan Williams will examine and explain the proof," and that "at the group's request" the materials were made public immediately. OpenAI's announcement says it also agreed to the group's request to fund a conference on the claim. In an open letter to their field, the three wrote: "Our involvement should not be taken as certification of the claim." Williams, of MIT, told Al Jazeera: "The ETH has been widely believed true since the late 1990s." He added: "We are working to verify and examine their proof and determine if it holds up."
• Local smoothing: Alex Iosevich, a harmonic analyst at the University of Rochester, told Al Jazeera that OpenAI appeared to have "cracked" the conjecture in a "historic development." He said the conjecture was posed by his PhD adviser, Chris Sogge. That is a first reaction, not a completed check. OpenAI says the expert it consulted advised releasing the output and leaving researchers free to organize their own efforts, so no review group is named for this result.
• Physics: a group led by physicist David Gross with Michael Douglas, Arthur Jaffe, Igor Klebanov and Nikita Nekrasov will shepherd the work, OpenAI says. The group decided to hold off on releasing the AI-produced material and to write it up themselves.

Mathematicians are not all waiting quietly. Peter Woit, whose Not Even Wrong blog is hosted by Columbia University's mathematics department, wrote that "from this press release, it's hard to know exactly what they're claiming to have proved." He pointed out that the press release doesn't mention the mass gap, the second half of the Millennium Prize problem, and doesn't say what "existence" means. He noted that only Jaffe in the physics group is a specialist in rigorous quantum field theory, and reported that a group of mathematicians has called for the materials to be released immediately.

A second ETH claim. OpenAI's isn't the only one. In a commit dated October 27, 2026 (UTC), Anthropic added a paper titled "A 2^o(n) Time Algorithm for 3-SAT," credited to Claude, with a Lean 4 formalization, to its anthropics/formal-math GitHub repository. Anthropic's README says the result "was discovered by an internal research Claude model," that the Lean formalization "was written by AI models," and that the result "is purely asymptotic and says nothing about solving formulas in practice." The ETH working group's open letter lists Anthropic's refutation among its resources. No outside verdict has been published on it either.

Did OpenAI use Lean or another formal proof system?

Yes for the two public results, with an important caveat about scope. OpenAI's repository history says both new manuscripts were added "with accompanying Lean formalizations," and its manuscript map links a Lean scope note for each. A Lean proof is checked by a small, trusted program, so if the formal statement is right, the proof is right. The open question is always whether the formal statement says what the paper claims, a point our explainer on OpenAI's Navier–Stokes claim makes in detail.

The scope notes show where they differ:

• ETH: the Lean formalization proves that a deterministic machine counts k-CNF solutions in time with exponent of order n·(log log n)² / √(log n), using polynomial tape. That is still subexponential, so it would still refute ETH, but it's weaker than the paper's sharper bound, which the note says comes from the October 25 manuscript.
• Local smoothing: the Lean formalization covers the half-wave local smoothing estimate in every dimension n ≥ 2 for Schwartz initial data, plus Bochner–Riesz, spherical restriction, Stein square-function and Kakeya maximal bounds, and the full Hausdorff dimension of Kakeya sets.

OpenAI says its model produced the ETH result on October 16 and local smoothing on October 18, with each formalized in Lean the next day. Across the whole repository, OpenAI says about 42% of its top-line results are formalized; the rest rely on human checking.

How does this compare with OpenAI's earlier math claims?

OpenAI's record on math claims is mixed, which is why outside checking matters here.

October 2025, the Erdős episode. Then-OpenAI VP Kevin Weil posted that GPT-5 "found solutions to 10 (!) previously unsolved Erdős problems," TechCrunch reported on October 19, 2025. Thomas Bloom, who runs the Erdős Problems website, called it "a dramatic misrepresentation": the problems were listed as open only because he didn't know of a solution, and "GPT-5 found references, which solved these problems." Google DeepMind CEO Demis Hassabis wrote: "This is embarrassing." Weil deleted the post.

May 20, 2026, the unit-distance disproof. OpenAI said an internal reasoning model disproved Erdős's 1946 conjecture that n points in the plane have at most n^(1+o(1)) unit-distance pairs. This time it brought in outside mathematicians first: Tim Gowers wrote a companion paper and Bloom a companion note, according to OpenAI. Gowers later told Quanta Magazine that for a human-written paper he "would have recommended acceptance without any hesitation." OpenAI's post doesn't mention a Lean proof.

September and October 2026, the catalogue. After the September 8 Navier–Stokes claim, covered in our builders' explainer, OpenAI's repository grew to 721 manuscripts in 373 families by October 26. It describes them as "raw research data, rather than polished academic papers," says "some of the unformalized results could have issues," and records corrections: on October 7, 2026, it withdrew three manuscripts after "a sign error invalidates a stabilization-trace cancellation argument," and revised 14 others. The repository even includes a September 23 manuscript claiming to prove Khot's Unique Games Conjecture, which the ETH paper cites.

The pattern is that OpenAI now publishes more of its evidence, including Lean proofs and corrections, than it did in 2025. It also announces results before the people it asked to check them have finished.

What it means

For mathematicians and computer scientists, two of the three claims are open for inspection now, and the Lean proofs shift the question from "is the proof correct?" to "does the formal statement match the conjecture?" For the physics claim, there is nothing to inspect yet. If the ETH disproof holds, it would overturn an assumption behind many conditional lower bounds in algorithms research; Williams's group is the one to watch. Fields medalist Richard Borcherds' decision to stop grading homework for correctness shows how quickly AI's math ability is changing practice, and Anthropic's formal verification work shows the same Lean-style checking applied to software.

What's confirmed, reported and uncertain

• Confirmed (OpenAI's announcement and repository): the two new manuscripts, their dates and abstracts; the Lean scope notes; the named ETH and physics groups and the three release approaches; the physics group's decision to write up the work first; the production and formalization dates OpenAI gives; the conference funding; the October 7 withdrawals; the 721-manuscript catalogue; the ~42% formalization share. These are OpenAI's statements about its own work, not outside confirmation that the proofs are right.
• Confirmed (others' own documents): the ETH working group's open letter; Anthropic's separate ETH paper and Lean files in its repository.
• Reported: Iosevich's and Williams's comments (Al Jazeera); Woit's critique of the physics release.
• Uncertain: whether any of the proofs survives expert review; what exactly the physics claim covers, including the mass gap; when the physics materials will be released; which model produced the results (OpenAI's repository calls it "an unnamed internal OpenAI model").