On October 26, OpenAI announced that its internal models produced three research results in mathematics, computer science and physics, including a claimed disproof of the Exponential Time Hypothesis, a longstanding complexity conjecture. According to OpenAI, the models also proved local smoothing in every dimension, settling the Kakeya and Stein restriction conjectures, and proved quantum chromodynamics and a unified Standard Model completion exist. OpenAI calls the manuscripts raw research data, says their correctness awaits independent scrutiny, and has handed each result to an external expert group. The mathematics and computer science materials are public on GitHub; the physics group writes its result up itself first.
What The Three Results Claim
The mathematics result concerns harmonic analysis. Christopher Sogge proposed the local smoothing conjecture in 1991, about wave focusing: OpenAI says the most extreme focusing must be short-lived, so focusing at a typical time is far weaker than at the worst case. The conjecture, OpenAI says, also implies the Kakeya conjecture and Stein’s restriction conjecture. OpenAI says its models proved local smoothing in every dimension, using a novel auxiliary partial differential equation, resolving both in all dimensions. The output came on October 18, 2026, was formalised in Lean the next day, and is public on GitHub.
The computer science result concerns the Exponential Time Hypothesis, a conjecture about Boolean formula satisfiability, closely related to P versus NP. OpenAI says the widely believed hypothesis has underpinned conditional hardness results in hundreds of papers. According to OpenAI, its system proved the hypothesis false, and the result was subsequently verified in Lean. The model produced the disproof on October 16, 2026, and OpenAI formalised it in Lean on October 17, 2026. OpenAI says the refutation reopens questions about faster algorithms, while warning that practical improvements need significant further research.
The physics result gives a rigorous existence proof of interacting quantum field theories with gauge fields and fermions in four dimensions, including quantum chromodynamics and a unified completion of the Standard Model, according to OpenAI. It includes, as a subproblem, a proof of pure Yang-Mills existence, part of the Clay Mathematics Institute’s Millennium Prize problem on quantum Yang-Mills theory. OpenAI says the proof concerns the simplest grand unified theory including the Standard Model’s gauge fields and fermions, since the Standard Model cannot be extrapolated to arbitrarily short distances. The Yang-Mills proof came on September 9, 2026, and the quantum chromodynamics proof on September 18; OpenAI finished the Yang-Mills Lean formalisation on October 16, 2026.
How Independent Experts Will Check Them
OpenAI asked external experts to act as shepherds, organising groups to understand the results, and the shepherds advised three different release strategies. For mathematics, the model’s output is released directly. For computer science, an external group will explain the result while the output stays public. For physics, an external group will explain the result, and OpenAI will defer to that group on releasing materials.
The computer science group is led by Russell Impagliazzo and Ramamohan Paturi of UC San Diego, who proposed the hypothesis with Francis Zane, and Ryan Williams of MIT, according to OpenAI. OpenAI agreed to the group’s requests for public raw data and conference funding. The three wrote in an open letter that their involvement should not be taken as certification of the claim, adding that it requires careful, independent scrutiny. Williams said the group signed no nondisclosure agreements and will not be paid.
The physics group is led by David Gross, whose 1973 asymptotic-freedom work helped establish quantum chromodynamics and earned him a share of the 2004 Nobel Prize in Physics, according to OpenAI. Michael Douglas, Arthur Jaffe, Igor Klebanov and Nikita Nekrasov also join the group. It will hold off releasing the model’s artifacts and write up the result itself. OpenAI says all three collaborations follow four principles: transparency without nondisclosure agreements; no academic credit sought by OpenAI; recognition for the groups’ own writeups; and voluntary, unpaid, independent participation. The collaborations complement OpenAI’s recently announced $5 million Simons Foundation contribution for workshops on model-produced results.
Why It Matters
OpenAI’s announcement matters because each claim, if it survives scrutiny, would close or reopen questions in its field. A false Exponential Time Hypothesis would unsettle hundreds of conditional hardness results. A proof of local smoothing in every dimension would settle two further conjectures long resisted by mathematicians. A rigorous quantum chromodynamics existence proof would firm up particle physics’ mathematical foundations. But it settles none of these questions: the manuscripts are raw research data, the reviewers’ role is not certification, and the physics artifacts are not yet public. What is new is the release experiment itself. The announcement arrived on the same day OpenAI launched GPT-6.1 Terra for computer use. The test now is whether independent experts can reproduce the work, or find its flaw.
Background
This announcement follows OpenAI’s October 6, 2026 mathematics release, which used the same strategy of publishing the model’s writeup alongside a Lean formalisation, according to OpenAI. It also comes after OpenAI resumed frontier training with no internet access following a September security scare. OpenAI says it does not yet know which approach will prove most useful and has invited feedback. OpenAI treats model-produced results as material the academic community must convert into understanding, rather than finished science.
Source: OpenAI: “OpenAI science announcement” ; Unite.AI: “OpenAI three proofs”
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