OpenAI discoveries in three separate fields were announced on Monday 26 October 2026, and each one targets a problem that experts have worked on for decades. In mathematics, OpenAI says an internal model proved Sogge’s local smoothing conjecture in every dimension, which would also settle the Kakeya conjecture. In computer science, it says the model proved that the Exponential Time Hypothesis is false. In physics, it says the model produced a rigorous existence proof for quantum chromodynamics and pure Yang–Mills theory in four dimensions.
None of the three OpenAI discoveries has been confirmed by independent experts yet. What is new this time is the way OpenAI is releasing them. Instead of simply posting the papers, it asked senior researchers to act as “shepherds” for each result, and they chose three different release strategies. The physics group, led by Nobel laureate David Gross, has not released the AI-produced material at all.
There is a twist, too. A few hours after OpenAI’s post, Anthropic published its own refutation of the Exponential Time Hypothesis, found by an internal Claude model on the same day, 16 October. This guide explains what each of the OpenAI discoveries claims, how strong the evidence is, what the Anthropic paper changes, and what it all means for researchers and businesses.
Table of contents
- The Three OpenAI Discoveries at a Glance
- Three Release Strategies for Three OpenAI Discoveries
- OpenAI Discoveries in Mathematics: Local Smoothing and Kakeya
- The Exponential Time Hypothesis Is False, Says OpenAI
- Anthropic’s Claude Found the Same Result on the Same Day
- OpenAI Discoveries in Physics: Yang–Mills and QCD
- How Strong Is the Evidence Behind the OpenAI Discoveries?
- Who Gets Credit for the OpenAI Discoveries?
- What the OpenAI Discoveries Mean for Businesses and Developers
- What Happens Next With the OpenAI Discoveries
- OpenAI Discoveries FAQs
- References
The Three OpenAI Discoveries at a Glance
OpenAI’s announcement, titled Working with academic communities to build scientific understanding, opens plainly: “We are announcing three new discoveries in mathematics, computer science, and physics about harmonic analysis, complexity theory, and quantum field theory.”
The short version
Each of the OpenAI discoveries comes from an unnamed internal model. OpenAI calls the outputs “a form of raw research data” rather than finished papers. For two of the three results, the model’s write-up and a machine-checked Lean proof are already public in the company’s GitHub repository. For the physics result, they are not. OpenAI has not said how the model was built or trained, including whether reinforcement learning on proofs played a part.
The table below sets out the three claims side by side.
| Field | Claimed result | AI output date | Public material |
|---|---|---|---|
| Mathematics | Local smoothing in every dimension, implying Kakeya and restriction results | 18 October 2026 | Paper and Lean proof |
| Computer science | The Exponential Time Hypothesis is false | 16 October 2026 | Paper and Lean proof |
| Physics | Existence of four-dimensional QCD and pure Yang–Mills theory | 9 and 18 September 2026 | Withheld by the expert group |
Where the OpenAI discoveries sit in the wider release
These results follow OpenAI’s 6 October release of 722 mathematics manuscripts, which we covered in OpenAI Just Posted Hundreds More Results on Major Math Problems. The repository’s history file shows that three of those papers were withdrawn on 7 October after a sign error, leaving 719.
The 26 October update adds two manuscripts, taking the catalogue to 721 manuscripts in 373 families. The same update also adds a Lean formalisation of an earlier result, the rational Hodge conjecture for CM abelian varieties, which the blog post itself does not mention.
Why this batch is different
Earlier releases were document drops, which drew sharp criticism from mathematicians, as we reported in OpenAI Keeps Bulldozing Mathematicians. This time OpenAI says its “release process continues to evolve in response to feedback from the associated academic communities.” The OpenAI discoveries of 26 October are the first to come with named outside experts attached before release.
Three Release Strategies for Three OpenAI Discoveries
OpenAI says it “reached out to external experts and asked them to act as shepherds, organizing groups to understand the new results.” Each shepherd picked a different way to handle the material.
Mathematics: release the raw output
For the local smoothing result, the expert advised OpenAI to “release the AI output, leaving researchers free to organize their own efforts to explain the result.” This copies the approach of the 6 October release, and it makes this the most open of the OpenAI discoveries. Anyone can download the paper and the Lean files today.
Computer science: an expert group, with everything public
For the Exponential Time Hypothesis, OpenAI formed an external group of collaborators “while making the AI output publicly available for anyone to examine and build on.” The group asked for the raw data to go public immediately, and OpenAI agreed.
Physics: the expert group controls the release
For the quantum field theory result, the group “decided to hold off on releasing the AI-produced artifacts and instead work on writing it up themselves.” So of the three OpenAI discoveries, this is the only one that outsiders cannot yet read.
| Result | Shepherds | Release strategy |
|---|---|---|
| Local smoothing | One unnamed expert, consulted | AI output released as it stands |
| Exponential Time Hypothesis | Impagliazzo, Paturi, Williams | Group examines; all materials public at once |
| QCD and Yang–Mills | Gross, Douglas, Jaffe, Klebanov, Nekrasov | Group writes it up; AI material held back |
The four principles behind the collaborations
OpenAI lists four rules for the outside groups. Transparency: participants sign no nondisclosure agreements and can publish freely. No academic credit: OpenAI does not seek credit for discoveries made by its models. Recognition: the groups should be credited for the explanations they write. Independence: taking part is voluntary and unpaid, and members do not speak for OpenAI.
The tradeoff OpenAI admits
OpenAI is candid about the downside. Choosing the first shepherds “necessarily shapes who gets involved first and creates asymmetry of information.” In other words, the people asked to steward the OpenAI discoveries see the material first, which is the very problem early access creates. The company says it does not yet know which of the three approaches will work best.
OpenAI Discoveries in Mathematics: Local Smoothing and Kakeya
The mathematics result sits in harmonic analysis, the study of how waves and signals break down into simple parts. It is the most “classical” of the OpenAI discoveries, and the one with the longest chain of human results behind it.
What local smoothing says
In 1991, Christopher Sogge proposed the local smoothing conjecture. OpenAI explains it simply: when waves become focused, “the most extreme focusing must be short-lived.” The focusing seen at a typical moment is much smaller than the worst case. Because this pattern runs so deep, the conjecture implies other famous problems, including the Kakeya conjecture and Stein’s restriction conjecture.
From Sogge to Wang and Zahl
Progress came slowly. In 2019, Larry Guth, Hong Wang and Ruixiang Zhang proved local smoothing in two spatial dimensions. In 2025, Hong Wang and Joshua Zahl proved the Kakeya conjecture in three dimensions, one of the biggest results in the field in years. An earlier OpenAI manuscript, dated 24 September, had already claimed local smoothing in three dimensions.
What the new proof claims
Among the OpenAI discoveries, this is the broadest single claim. OpenAI says its models “produced a proof of the local smoothing conjecture in every dimension.” The paper, dated 25 October and 59 pages long, also claims the Bochner–Riesz conjecture, the Kakeya and Nikodym maximal conjectures, and that “every Euclidean Kakeya set has full Hausdorff dimension.” OpenAI says the proof rests on “a novel auxiliary partial differential equation.”
A careful reading of “restriction conjecture”
The blog post says the result resolves Stein’s restriction conjecture “in all dimensions.” The repository is more careful. Its history file says the proof gives Stein’s spherical restriction estimates “throughout the interior of the conjectured range”, which leaves the endpoint cases open. Readers comparing coverage of the OpenAI discoveries should note that difference.
How fast it was checked
OpenAI says it obtained the AI output on 18 October and formalised it in Lean the next day. Lean is a proof assistant: software that checks every logical step of a proof. The Lean scope page for the result lists formal statements for local smoothing, Bochner–Riesz bounds, the Kakeya maximal estimate and full Hausdorff dimension of Kakeya sets.
The Exponential Time Hypothesis Is False, Says OpenAI
The computer science claim is the one that has drawn the most attention. Of all the OpenAI discoveries, it touches the most published work, because the Exponential Time Hypothesis (ETH) underpins hundreds of papers.
What the Exponential Time Hypothesis says
The ETH concerns 3-SAT, the problem of deciding whether a logic formula with n true-or-false variables can be satisfied. Trying every option takes 2n steps. The ETH, proposed by Russell Impagliazzo, Ramamohan Paturi and Francis Zane around 1999, says no algorithm can do fundamentally better: 3-SAT needs time exponential in n. It is a stronger assumption than P ≠ NP.
Why so many results depend on it
Researchers use the ETH to show that other problems cannot be solved quickly, so this part of the OpenAI discoveries reaches well beyond one paper. OpenAI 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.” Anthropic’s paper puts a number on it: a literature search found “more than 800 papers” with at least one result proved only under ETH or a hypothesis that implies it. That reach is why this is the most closely watched of the OpenAI discoveries.
The claimed algorithm
OpenAI’s paper, 67 pages long and dated 17 October (revised 25 October), gives “a deterministic algorithm for exactly counting the satisfying assignments of an arbitrary k-CNF formula.” Its running time grows like 2 raised to n divided by roughly the square root of log n. That is slower than any polynomial, but faster than 2δn for every fixed δ, which is exactly what refuting the ETH requires.
Further consequences the paper claims
The paper also claims a new circuit lower bound for the class ENP, subexponential algorithms for counting graph colourings and perfect matchings, and arbitrarily accurate subexponential approximations for Max-k-SAT. It stresses one limit: the result “does not imply that general NP problems admit subexponential time algorithms.”
Who is checking it
The computer science group is led by Russell Impagliazzo and Ramamohan Paturi of UC San Diego, two of the original proposers of the ETH, and Ryan Williams of MIT. OpenAI notes that Williams has refuted strong versions of the ETH and is known for linking algorithms to lower bounds. On Bluesky, Williams wrote that the three “signed no NDAs and will receive no compensation.”
The open letter: “not certification”
The group published an open letter to the theoretical computer science community. Its key sentence is blunt: “Our involvement should not be taken as certification of the claim. A result of this significance requires careful, independent scrutiny.” They also wrote: “If the claim is correct, it will change an important part of theoretical computer science.”
The letter asks the community for help organising the examination and links a shared folder, a feedback form and a discussion site, ethdiscussion.blog. It also proposes a virtual meeting before the end of 2026. At the group’s request, OpenAI has agreed to fund a conference on the result.
Anthropic's Claude Found the Same Result on the Same Day
The most surprising part of this story was not in OpenAI’s post at all. The open letter’s resource list includes a link titled “Anthropic: Refutation of ETH”, which points to a new folder in Anthropic’s formal-math repository.
What Anthropic published
The folder holds a 49-page paper, “A 2o(n) Time Algorithm for 3-SAT”, credited to “Claude, Anthropic” and dated 26 October 2026. GitHub records show the pull request was opened at 02:46 UTC on 27 October and merged at 03:07 UTC, a little over five hours after Williams’s first post. The README says: “The result was discovered by an internal research Claude model.”
The provenance note
The paper’s provenance section is striking: “The core algorithm was found on October 16 by an internal Claude research model with 14M output tokens and no human input.” It adds: “Independently and concurrently, an internal model at OpenAI claims to have proven a similar result.” OpenAI’s date for its own ETH disproof is also 16 October. Both labs say their systems found it on the same day.
Two ETH papers compared
The two papers reach the same headline bound, and both use tensor contraction. OpenAI’s main text uses a different route, then reproves its bounds by tensor contraction in an appendix. The comparison below uses only what each document states.
| Detail | OpenAI | Anthropic |
|---|---|---|
| Title | The Exponential Time Hypothesis is False | A 2^o(n) Time Algorithm for 3-SAT |
| Discovery date given | 16 October 2026 | 16 October 2026 |
| Length | 67 pages | 49 pages |
| Headline time bound | 2^O(n log log n / √log n) | 2^O(n log log n / √log n) |
| Bound proved in Lean | Weaker: 2^O(n (log log n)² / √log n) | The main theorem’s bound |
| Counts solutions | Yes, in polynomial space | Yes (#3-SAT) |
| Expert examination | Impagliazzo, Paturi, Williams group | Not stated |
Why a second proof matters
For the OpenAI discoveries, a rival lab reaching the same result with a separate system is not proof that it is correct. But it is meaningful evidence. Each paper comes with its own Lean formalisation, written separately, and both claim a machine-checked proof that ETH is false. For the reviewing group, two write-ups also give two routes into the argument.
A short recipe for other AI systems
Anthropic’s README includes something unusual: a short paragraph that, it says, lets “Claude Opus 5.5 at high effort” rederive the refutation. It is careful about scope: what the model rederives “is only that 3-SAT can be decided in time O(2δn) for every δ > 0, not the sharper bounds.” That alone is enough to refute ETH. It is a notable claim about how a compressed idea can be expanded back into a full proof.
OpenAI Discoveries in Physics: Yang–Mills and QCD
The physics claim is the most ambitious of the OpenAI discoveries, and the least visible. It concerns the mathematical foundations of quantum field theory, the framework behind modern particle physics.
What OpenAI claims
OpenAI says the result “gives a mathematically rigorous existence proof of certain interacting quantum field theories containing gauge fields and fermions in four dimensions.” These include quantum chromodynamics (QCD), the theory of quarks and gluons and the strong force, and “a unified completion of the Standard Model.” The work builds on Tadeusz Balaban’s approach in constructive quantum field theory.
The Yang–Mills subproblem
As part of the work, OpenAI says it proved the existence of pure Yang–Mills theory, which describes gluons interacting only with each other. That is “part of the Quantum Yang–Mills Theory Clay Mathematics Institute Millennium Prize problem.” The Clay problem carries a $1 million prize.
What the physics claim does not say
The word “part” matters. The Clay problem asks for two things: a rigorous construction of quantum Yang–Mills theory and a proof of the “mass gap”, the property that its particles have positive mass. Clay notes that experiments and simulations suggest a mass gap, “but no proof of this property is known.” OpenAI’s announcement does not claim a mass gap proof. In September, OpenAI also said it would not claim the prize for its earlier Navier–Stokes result.
The Gross group
David Gross leads the group shepherding the physics side of the OpenAI discoveries. His 1973 work on asymptotic freedom helped establish QCD and earned him a share of the 2004 Nobel Prize in Physics. The other members are Michael Douglas, who has reported on the problem’s status since 2004; Arthur Jaffe, who formulated the Clay problem with Edward Witten and co-founded the Clay Institute; Igor Klebanov, who directs the Simons Collaboration on Confinement and QCD Strings; and Nikita Nekrasov, winner of the 2023 Dannie Heineman Prize for Mathematical Physics.
A slower timeline
This result took the longest to check. OpenAI says the natural-language proof for Yang–Mills existence was produced on 9 September and the QCD proof on 18 September. Because of “the length and complexity of the work,” it waited for the Lean formalisation, which finished for Yang–Mills on 16 October. OpenAI does not give a Lean date for QCD or the Standard Model completion.
The chart compares how long each of the OpenAI discoveries took to go from AI output to a finished Lean formalisation, using the dates OpenAI gives.
Bars are scaled to the longest gap: 1 day is about 3% of 37 days.
How Strong Is the Evidence Behind the OpenAI Discoveries?
A Lean proof is strong evidence, but it is not the end of the story. Here is what the checks do and do not establish for the OpenAI discoveries.
What a Lean check proves
Lean confirms that a proof follows logically from its stated assumptions, step by step. Anthropic’s README explains that its build takes about 8 minutes on 24 virtual CPUs, the kind of check anyone can repeat on rented cloud computing capacity. In its fuller check, a second, independently written kernel then re-checks the proof. That removes most worries about a hidden slip in a long argument.
The statement problem
The weak point is the statement itself. Anthropic’s README puts it clearly: “Lean checks the proofs, but no machine checks that this file says what the paper says.” If a formal statement quietly differs from the famous conjecture, a perfect proof proves the wrong thing. That is why human experts still need to read the definitions closely, for OpenAI and Anthropic alike.
The scope can be narrower than the headline
OpenAI’s own Lean page for the ETH paper shows this in practice. Lean proves a slightly weaker running time than the paper’s sharpest bound. The weaker bound still refutes the ETH, so the headline claim is covered, but not every number in the paper is machine-checked.
OpenAI’s track record
OpenAI’s earlier releases show why caution is sensible. On 7 October, it withdrew three manuscripts after “a sign error invalidates a stabilization-trace cancellation argument,” and revised 14 others. At that point it said about 42% of its top-line results were formalised (300 of 719). The 26 October OpenAI discoveries both come with Lean proofs, which puts them at the stronger end of that range.
| Result | Lean proof | Public | Independent confirmation |
|---|---|---|---|
| Local smoothing | Yes (19 October) | Yes | None yet |
| Exponential Time Hypothesis | Yes, plus Anthropic’s separate proof | Yes | Under examination |
| Yang–Mills existence | Yes (16 October) | No | Group writing it up |
| QCD and Standard Model completion | No date given | No | Group writing it up |
Paper length as a rough guide
The papers are long, which affects how fast experts can check them. The chart below compares the public PDFs by page count.
Bars are scaled to the longest paper: 59 is 88% of 67, and 49 is 73% of 67.
Who Gets Credit for the OpenAI Discoveries?
The OpenAI discoveries arrive in the middle of a heated debate about how AI-generated results should be handled. Several of the people closest to this release have spoken about it directly.
OpenAI gives up academic credit
OpenAI says plainly that it wants no academic credit for the OpenAI discoveries: “We do not seek academic credit for the discoveries produced by our internal models.” The repository README now adds: “We defer to the math community completely on matters of credit, precedence, and concurrent work.” With Anthropic claiming the same ETH result, that last phrase may soon be tested.
The open letter’s harder questions
The Impagliazzo, Paturi and Williams letter warns that “recognition may go mainly to those who publish first or have the most powerful tools, while the work of checking, explaining, and building on results receives less credit.” It asks how early access, conflicts of interest and the use of unpublished material should be governed.
Access to the models
Williams also raised access. “We do not have access to their internal models. We only got the PDF,” he wrote on Bluesky, adding that he believes “all mathematicians should be given access to (at least) Astra-level models.” That refers to GPT-6 Astra, one of OpenAI’s public frontier models.
Terence Tao’s “Math 2.0” warning
On 6 October, Terence Tao argued on Mathstodon that solutions to open problems “are now being harvested at large scale in an unsustainable fashion.” He said a “Math 2.0” era must value exposition and community building, not just raw problem solving. The shepherd model in the 26 October OpenAI discoveries looks like a direct response to that criticism.
Money for the checkers
OpenAI points to its “recently announced $5 million contribution” to the Simons Foundation, which will fund workshops and conferences on AI-produced results. Reports describe it as part of a $15 million initiative co-funded by OpenAI, Anthropic and the foundation. We covered the earlier debate over OpenAI’s training data in Mathematicians Want Proof OpenAI Didn’t Use Their Work.
What the OpenAI Discoveries Mean for Businesses and Developers
Most organisations will never touch the Kakeya conjecture. But the OpenAI discoveries still carry practical lessons, especially for anyone who relies on SAT solvers, formal methods or AI research claims.
Your SAT solver will not get faster
The ETH result is purely theoretical. Anthropic’s paper says its bound “does not improve on the 2n of exhaustive search unless n > 1010,000” for formulas with five clauses per variable, and that brute force is “already impractical for n = 100.” Its conclusion: “our result says nothing about solving formulas in practice.” Scheduling, chip verification and logistics tools will keep using the same solvers.
Hardness assumptions in security and planning
Some complexity arguments in research rest on the ETH. If the ETH part of the OpenAI discoveries holds up, those conditional results lose their footing, though the techniques behind them keep their value, as the open letter notes. Everyday cryptography relies on the hardness of specific problems, and neither paper claims any practical attack on it.
Formal verification is becoming a core skill
The bigger signal is the speed of Lean checking: one day for two of the three OpenAI discoveries. Machine-checked proofs are moving from niche to normal. For teams building safety-critical software, the same tools and habits, writing precise specifications and proving code meets them, are worth investing in.
How to read an AI research claim
These releases also offer a simple checklist. Ask whether the full output is public, whether a formal proof exists, what exactly the formal statement says, and whether named independent experts are examining it. A claim that passes all four, like the ETH result, deserves more weight than a press release alone.
For AI strategy teams
The OpenAI discoveries, and the fact that two labs found the same ETH result on the same day, say something about where frontier AI systems are. Research-level reasoning is no longer a one-lab story. Our recent coverage of The Anthropic S-1 Filing Is Also an AI Doomsday Primer shows how seriously the labs themselves take that pace.
What Happens Next With the OpenAI Discoveries
All three OpenAI discoveries now enter a long checking phase. Here is what to watch.
The ETH examination
The computer science group plans to share findings on the ETH part of the OpenAI discoveries as it goes. Its letter proposes a virtual meeting before the end of 2026, and OpenAI has agreed to fund a conference. With two papers to compare, early commentary may come quickly.
The physics write-up
The Gross group has not said when it will publish. OpenAI says it hopes the result marks “the beginning of a transition in which quantum field theory, the most important framework in modern theoretical physics, becomes a mathematically rigorous field.” Until the group releases its account, outsiders cannot check the claim.
The repository history
OpenAI keeps a public changelog of withdrawals and fixes. Anyone following the OpenAI discoveries should watch the history file, since corrections appear there first. Earlier context on how the company organised its math work is in OpenAI Forms Math Advisory Group as Its AI Resolves More Than 100 Open Problems.
OpenAI Discoveries FAQs
What did OpenAI announce on 26 October 2026?
Three results from an internal model: a proof of local smoothing in every dimension (implying the Kakeya conjecture), a disproof of the Exponential Time Hypothesis, and an existence proof for four-dimensional QCD and pure Yang–Mills theory.
Have the OpenAI discoveries been verified?
Local smoothing, the ETH disproof and Yang–Mills existence all have machine-checked Lean proofs, but OpenAI gives no Lean date for QCD. None has been confirmed by independent experts yet, and the ETH group says its involvement is not certification.
Did OpenAI solve a Millennium Prize problem?
No. The OpenAI discoveries include the existence part of the Yang–Mills problem, but the Clay problem also requires a proof of the mass gap, which OpenAI does not claim.
Did Anthropic find the same ETH result?
Anthropic says an internal Claude model found a 2o(n) algorithm for 3-SAT on 16 October, the same date OpenAI gives. Its paper calls the two efforts independent and concurrent.
Will the ETH result make computers faster?
Not in practice. None of the OpenAI discoveries changes everyday computing, and Anthropic’s paper says the improvement only appears for astronomically large inputs, far beyond any real problem.
Who is reviewing the results?
Russell Impagliazzo, Ramamohan Paturi and Ryan Williams lead the ETH group. David Gross leads the physics group with Michael Douglas, Arthur Jaffe, Igor Klebanov and Nikita Nekrasov.
References
Working with academic communities to build scientific understanding (OpenAI)
openai/math release history (GitHub)
The Exponential Time Hypothesis is False: Lean scope (OpenAI)
A 2^o(n) Time Algorithm for 3-SAT (Anthropic formal-math)
Open letter on the ETH claim (Impagliazzo, Paturi and Williams)
ETH Refutation: Examination and Discussion
OpenAI says its AI model has disproved the Exponential Time Hypothesis (The Next Web)
OpenAI claims major proofs in math, computing and quantum physics (Interesting Engineering)
Yang–Mills and the Mass Gap (Clay Mathematics Institute)
The Kakeya set conjecture in three dimensions (Wang and Zahl, arXiv)
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