OpenAI Overturns a 30-Year Pillar of Computer Science and Solves the Hanging谷 Conjecture

OpenAI topples a 30‑year foundation of computer science and also solves the Hang‑Gu conjecture

The discussion sparked by 722 mathematical manuscripts is still ongoing, and OpenAI has presented scientists with an even more surprising result.

This time, it claims to overturn a hypothesis in theoretical computer science that has been widely believed for nearly 30 years. The two researchers who originally proposed the hypothesis are now preparing to gather peers to examine the AI‑generated counterexample.

On October 26, OpenAI announced new achievements in mathematics and computer science and physics: proving the local smoothing conjecture in all dimensions, refuting the Exponential Time Hypothesis (ETH), and establishing rigorous mathematical existence proofs for four‑dimensional quantum field theories, including quantum chromodynamics. OpenAI President Greg Brockman called them three major scientific discoveries on X, marveling that AI is beginning to help humanity deepen its understanding of the universe.

Less than three weeks ago, OpenAI publicly released 722 math manuscripts generated by internal models, spanning 372 result families and 17 research areas. We reviewed that list at the time and selected the 30 most talked‑about classic problems. The four‑dimensional hanging‑valley conjecture was one of them.

Now the hanging‑valley problem has resurfaced, but it’s no longer the most surprising news among those results. What has pulled the entire theoretical computer science community into the discussion is another abbreviation: ETH.

A hypothesis that has existed for nearly 30 years,

is being challenged by AI

To understand ETH, you can start with the example below.

Suppose there is a combination lock, and each digit of the code can only be 0 or 1. We don’t know the correct code, but we have several clues: the first and second digits cannot both be 0; the second digit is 0, or the third digit is 0; the first, third, and fourth digits cannot all be 1.

The question is, can we find a set of digits that satisfies all the clues? For example, 1010 is a code that meets the conditions.

In computer science, such a choice that can only take the values 0 or 1 can be represented by Boolean variables, corresponding to “false” and “true.” If each clue is written as a logical condition consisting of at most three variables or their negations connected by “or,” and we ask whether there exists an assignment that makes all conditions true simultaneously, we obtain the classic  3-SAT problem.

A four‑digit password is easy to solve, requiring at most 16 combinations to check. But if the password has 100 digits, the possible combinations increase to 2^100. Although clever algorithms need not try each one individually, how fast a computer can solve a general 3‑SAT problem has always been an important question in theoretical computer science.

In the late 1990s, Russell Impagliazzo, Ramamohan Paturi, and Francis Zane proposed the Exponential Time Hypothesis (ETH).

Its core claim is:There is no algorithm that can solve all 3‑SAT instances in sub‑exponential time. In other words, no matter how algorithms improve, the worst‑case running time cannot be reduced to 2^{o(n)}, where n denotes the number of Boolean variables.

For nearly 30 years since then, ETH has become a fundamental assumption in theoretical computer science. Researchers often take it as a premise to derive lower bounds on computation time in areas such as graph algorithms, database queries, and computational biology. Hundreds of research papers have relied on this assumption to establish conditional hardness results.

And this time, an internal OpenAI model has produced an algorithm that might break this barrier.

According to the published proof, for fixed-width CNF Boolean formulas, the model constructs a deterministic algorithm that not only can decide whether a satisfying assignment exists, but also can precisely count the number of satisfying assignments. Taking the previous password puzzle as an example, determining whether there is a password that fits the clues only requires finding one solution; precise counting, however, can compute exactly how many passwords satisfy all the conditions.

Even more crucial is the speed. For related problems where the number of clauses is proportional to the number of variables, OpenAI gave a runtime bound of the form 2^{O(n/√log n)}  (omitting polynomial factors), which is sub‑exponential time.

If the proof holds, it would mean that ETH is falsified. This does not imply that all hard problems can be solved quickly, but a theoretical barrier once thought insurmountable may have been breached.

Even more dramatically, the next people responsible for reviewing this result.

OpenAI invited Impagliazzo and Paturi from the University of California, San Diego, and Ryan Williams from MIT to organize an external research team. The first two are the original proposers of ETH, and Williams has long studied lower bounds in algorithms and computational complexity.

Scientists who proposed this hypothesis nearly 30 years ago are now personally involved in checking the AI‑generated counterexample.

According to OpenAI’s timeline, the model generated a natural‑language counterexample to ETH on October 16, completed a Lean formalization the next day, and publicly announced the results on October 26. Williams and others then announced an open, independent review.

Explanation of the Lean formalization of OpenAI’s publicly disclosed refutation result for the Exponential Time Hypothesis (ETH).

A hypothesis that has been widely used in theoretical computer science for nearly 30 years may now be rewritten by AI. Its originator is now examining this proof together with peers.

Last time it was the 4D Hang Valley,

This time it extends directly to all dimensions

Turning the focus back from computer science to mathematics, longtime readers will encounter a familiar name: Kakeya.

In the previous 722 manuscript reports, we introduced this needle‑originated mathematical problem: how large a region is needed for a needle to sweep through all directions? This seemingly simple geometric question later gave rise to the conjecture on the dimension of high‑dimensional sets and forged deep connections with harmonic analysis and partial differential equations.

In 2025, Wang Hong and Joshua Zahl resolved the three‑dimensional Kakeya set conjecture. Earlier, OpenAI’s result series No. 074 claimed to further prove the three‑dimensional Kakeya maximal conjecture and the four‑dimensional Kakeya dimension conjecture, but at that time those manuscripts had no corresponding Lean formalizations.

This time, OpenAI’s entry point has changed.

The new results first target the Local Smoothing Conjecture proposed by mathematician Christopher Sogge in 1991. It studies how waves focus during propagation: at a given instant, a wave may become unusually concentrated at a point; but when the behavior over a period of time is considered together, such extreme concentration cannot be sustained indefinitely.

It can be thought of as ripples meeting on a water surface. At a particular instant, wave peaks may overlap to become unusually high at one location, but as the waves continue to propagate, that extreme peak quickly changes. Local smoothing aims to use precise inequalities to quantify how much additional regularity time averaging can provide.

This is not a purely mathematical game of studying waves. The behavior of the wave equation, restriction estimates in Fourier analysis, and the geometric structure of Kakeya sets are all underpinned by an interrelated theory.

In 2019, Larry Guth, Wang Hong, and Zhang Ruixiang proved the local smoothing conjecture in two dimensions. OpenAI now claims that the model discovered a new auxiliary partial differential equation, using it to extend the proof to all spatial dimensions.According to OpenAI’s description, this result further solves the Kakeya conjecture and Stein restriction conjecture in every dimension.

From the previous specific results in three and four dimensions to this claim of a unified conclusion covering all dimensions via local smoothing, the change is quite striking.

The timeline is also tight: OpenAI said the model was proven on October 18, and the Lean formalization was completed on October 19. The natural‑language manuscript and the formal proof have been made public. Unlike ETH, OpenAI did not assign a prior review panel for the mathematics; instead, researchers were given direct access to the public materials to organize their own explanations and verification.

OpenAI’s publicly released Lean formalization of the local smoothing conjecture explains the estimate for local smoothing in all spaces of dimension two and higher, and proves that the associated hanging‑valley set has full Hausdorff dimension in each relevant dimension.

Challenges in Physics:

Can the theories we use be rigorously constructed?

If ETH is concerned with how fast computation can be, and Galois and local smoothing are concerned with the nature of space and waves, then the third result directly touches the foundations of modern physics describing the microscopic world.

Quantum Chromodynamics (QCD) describes the strong interaction between quarks and gluons. Physicists use it to study protons, neutrons, and the more fundamental particles that compose them; related calculations and experiments form an important pillar of modern particle physics.

But there is an interesting discrepancy here: a theory can be extremely successful in physical calculations and experiments, yet constructing it rigorously as a mathematically axiomatic quantum field theory is a different matter.

In four-dimensional spacetime, particle interactions, the ever‑present quantum fluctuations at various scales, and the passage from approximate calculations to rigorous definitions have long posed enormous mathematical challenges for constructive quantum field theory.

OpenAI now claims that, following the constructive quantum field theory program of mathematical physicist Tadeusz Balaban, its internal model has established rigorous existence proofs for certain four‑dimensional interacting quantum field theories that include gauge fields and fermions, involving QCD and a unified extension that accommodates the Standard Model gauge fields and fermions.

This work also includes a proof of existence for the pure Yang–Mills theory. In 2000, the Clay Mathematics Institute listed “Yang–Mills existence and mass gap” as one of the seven Millennium Prize Problems. The problem requires not only proving the existence of quantum Yang–Mills theory but also establishing a positive mass gap. OpenAI’s announcement this time concerns the existence part and does not claim to have solved the mass gap.

The most eye‑catching aspect of the physics side is the list of scientists who will take over the research.

2004 Nobel laureate in Physics David Gross will lead a working group that includes Michael Douglas, Arthur Jaffe, Igor Klebanov, and Nikita Nekrasov. Gross won the prize for his work on asymptotic freedom, which is the key to understanding the behavior of the strong interaction at extremely short distances; Jaffe co‑authored the formal statement of the Yang–Mills Millennium Problem with Edward Witten.

This release method also differs from the previous two. OpenAI disclosed that the model generated a proof of the existence of Yang–Mills as early as September 9, and a QCD‑related proof on September 18. Because the material is complex and voluminous, the company waited until October 16 to complete a Lean formalization of the Yang–Mills existence proof before moving forward with external collaboration.

The physics working group chose to first organize, understand, and write their own research manuscript, temporarily keeping the AI‑generated original proof private. In other words, this result, which is closest to the foundations of modern physics, still needs to be translated by the expert team into a long‑form proof that peers can read, evaluate, and build upon.

After AI writes the proof,

Scientists begin to take on the next piece of work

Looking back at these three achievements, the truly noteworthy change, besides the weight of the problems themselves, is the way OpenAI handles these results.

When OpenAI announced 722 manuscripts on October 6, it used a centralized public release model: the papers, proof files, result indexes, and partial model inference summaries were all placed on GitHub. Mathematicians could read them directly, but they also had to face a practical problem:Hundreds of manuscripts spanning different disciplines—who will check them, who will interpret them, and how can we determine which contain truly new ideas?

This is not merely a question of workload. After the previous batch of manuscripts was made public, three were withdrawn due to symbol errors in the proofs. Moreover, even with Lean formalization, the academic community still needs to determine whether the formalized theorems accurately correspond to the original conjectures and what new methods introduced by the proofs can be transferred to other problems.

OpenAI now uses a very interesting term for these AI outputs:original research data.

This framing separates “producing proofs” from “forming scientific understanding.” The former can be done by AI first; the latter requires researchers to review the arguments, rewrite explanations, trace prior work, assess contributions, and integrate the new tools into the existing knowledge framework of the discipline.

Therefore, the three outcomes on October 26 correspond to three different arrangements:The mathematics part is released directly, allowing the community to organize research on its own;The computer science part is released with a proof, while inviting the original ETH proposer and domain experts to conduct an independent review;The physics part is first handed over to an expert team, and after they finish organizing the material, a decision will be made on how to publish it.

OpenAI also proposes several collaboration principles: external members participate voluntarily and independently, without signing nondisclosure agreements, and they do not seek academic credit for OpenAI based on model discoveries; they should receive recognition for the explanations, proofs, compilations, and subsequent research they complete. The company also commits to supporting independent review and has previously announced a $5 million investment in related Simons Foundation initiatives for workshops, academic conferences, and research projects.

Behind the three publishing approaches lies essentially the same question:When AI can generate a large volume of cutting‑edge research results in a short time, how should the scientific community digest this new knowledge?

Directly publishing proofs allows more researchers to engage immediately, but it also means the massive review workload must be shouldered by the community itself. Handing it over to an expert team can concentrate effort on organizing complex research outcomes, yet it also gives some people earlier access to the material than other researchers.

OpenAI clearly recognized this issue as well. Compared with the one‑time release of 722 math manuscripts in early October, this time it not only published the research results but also began to try to establish appropriate review and collaboration mechanisms for different disciplines.

The ETH story happens to provide a rather symbolic scenario.

Nearly 30 years ago, Impagliazzo and Paturi helped propose this conjecture, and since then hundreds of research papers have taken it as a premise to explore the complexity boundaries of computational problems. Today, OpenAI’s model claims to have found a counterexample, and the two scientists, together with their peers, are re‑examining the theoretical foundation they once helped establish.

For OpenAI, the model has completed the generation and formalization of the proof. For the scientific community, the real research work continues: reviewing the algorithm, understanding the new methods in the proof, and assessing how these results will affect subsequent research.

From 722 mathematical manuscripts to research results now spanning three disciplines, OpenAI is showing not only how many proofs AI can produce, but also addressing a more practical question:As AI continually proposes new scientific conclusions, how can humans transform these conclusions into knowledge that can truly be understood, verified, and further developed.

Reference links:

https://x.com/gdb/status/2114837059462627533

https://openai.com/index/building-scientific-understanding-with-academic-communities/

© THE END 

For reprints, please contact this public account for permission

Submit articles or request coverage: liyazhou@jiqizhixin.com