Just a few weeks after dropping hundreds of AI-generated math breakthroughs on a field already in crisis, OpenAI is piling on more.
Yesterday, the company dropped three massive proofs touching math, physics and computer science, sending shockwaves rippling through those fields. Experts are still staggering.
The releases continue to contradict the recommendations of the Advisory Group on Mathematics and Artificial Intelligence (AGMAI), the expert panel OpenAI organized in response to widespread criticism from mathematicians. The large language model (LLM) that produced the new proofs remains proprietary, and the exact prompts used, time spent computing, and “chain-of-thought outputs” have not been made public. Two of these results coincide with closely related work involving OpenAI’s competitor, Anthropic.
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OpenAI’s approach to two of the three results also directly violates a clarifying statement AGMAI published just a day prior, by designating specific experts to shepherd the results into the public sphere. One of those papers, relating to the “million-dollar” Yang-Mills problem (one of seven “Millennium Problems” set by the Clay Mathematics Institute in 2000), has still not been made public. AGMAI has already called upon OpenAI to publish it, writing that “there is no scientific reason for delaying the release any further.”
Here’s a quick breakdown of the three new results and why they matter.
The Kakeya conjecture is about swirling a stylus around in the air cleverly—so that it points in every direction once but sweeps out as little volume as possible. Specifically, the conjecture says that the volume cannot be less than a certain number. Surprisingly, this simple geometry problem is one of the most important conjectures in harmonic analysis, the study of how all kinds of mathematical objects can be built from waves.
Two mathematicians proved the conjecture in three dimensions last year, in one of the biggest math breakthroughs of this century. The feat won Hong Wang a Fields Medal—math’s highest honor—at this summer’s International Congress of Mathematicians in Philadelphia. But the problem remained open in higher-dimensional spaces.
In last month’s “slop drop,” OpenAI’s internal model solved the four-dimensional Kakeya problem. Now, it’s solved the problem in every dimension, proving that there’s a limit to how small a space a high-dimensional stylus can take up.
The Yang-Mills problem, one of the five remaining open million-dollar Clay Millennium Problems, is about particle physics. Specifically, it asks whether the mathematics that underpins the “Standard Model” of particle physics is actually sound.
This question comes down to two parts, referred to as “existence” and “mass gap.” The Clay Institute’s definition of the problem asks for answers to both. Today, OpenAI claimed to prove the first of the two—that the theory, called a “Yang-Mills theory,” is mathematically consistent in all relevant cases: it “exists.” The part of the Clay problem that remains open is more quantitative, asking whether a certain number in the theory, called the “mass gap,” is always bigger than zero.
In the late 1980s, long before the Clay Institute existed, the mathematician Tadeusz Balaban published a series of massive, dense papers sketching out a path to prove the "existence" part of the problem. As voluminous as “Balaban’s program” is, it only codifies the problem’s broad contours—filling in the gaps to construct a mathematically rigorous proof would demand far more laborious and exhaustive calculations. The project is so immense—and Balaban’s papers are so difficult to parse—that no one has carried it out more than three decades later. “He basically did it,” says Roland Bauerschmidt, a mathematician at New York University. “But I think there’s no one who understands these papers and can tell whether they’re correct or not.”
Recently, some experts began using LLMs to churn out the necessary tomes of tedious mathematical steps, sensing that a race was on to finish Balaban’s program. Working with Anthropic employee Levent Alpöge, Ethan Sussman, a mathematician at Northwestern University, created and publicized a public GitHub repository to do just that. They were making considerable progress, and believed they were near finishing the program. OpenAI’s claim—with no publicly available proof—appears to have beaten them to the finish line.
The exponential time hypothesis (ETH) is about computational complexity—the science of ranking computer science problems by how hard they are.
Specifically, ETH suggests that a famous task called the 3-SAT problem cannot be solved in less than exponential time. This means that as the number of variables in the problem increases, any algorithm that solves it will run exponentially slower. OpenAI’s proof provides an algorithm to solve 3-SAT that is just a tiny bit faster than exponential time, refuting ETH.
ETH is important because of its relationship to “P versus NP,” another of the Clay Institute’s Millennium Problems, and widely considered one of the most difficult, with major implications for modern encryption.
First suggested in 1999, ETH can be thought of as an even stronger version of P≠NP, and has been treated as a bedrock tenet of computer science by most experts. Because 3-SAT is a problem computer scientists know to be central to the debate, ETH would show that there are problems that are not only fundamentally hard to solve, but also scale up so brutally that they are effectively unassailable.
Although the new result doesn’t necessarily translate to a transformative breakthrough about P versus NP, the potential effects are so sweeping that OpenAI has tapped outside experts to vet its claim. Russell Impagliazzo and Ramamohan Paturi, two computer scientists at the University of California–San Diego who first suggested ETH, as well as Ryan Williams, a computer scientist at the Massachusetts Institute of Technology, are now working with OpenAI to perform an independent investigation of the result. The company will also sponsor a computer-science conference about the proof as part of this investigation, according to Williams.
The result hasn’t been without controversy, either. Mere hours after OpenAI’s announcement, Anthropic posted its own independent result disproving ETH, based on work by an internal Anthropic model.
This is a breaking news story and will be updated.
