Text by Tian Yu

Edited by Shen Xiao

On October 6 at 6:00 PM Eastern Time, OpenAI unveiled 722 mathematical research manuscripts crafted by an unpublished internal model. These manuscripts encompassed 372 findings across 17 disciplines, with 162 including Lean-formalized proofs for their main conclusions.

Company officials stated that each successful result required computational resources equivalent to three hours of ChatGPT Pro's processing.

The purported achievements include breakthroughs on the Four-Dimensional Kakeya Conjecture, issues related to the Generalized Riemann Hypothesis, the Unique Games Conjecture, the Isomorphism Problem for Free Group Factors, Hilbert's Tenth Problem over the Rationals, the Hodge Conjecture for CM Abelian Varieties, and the Mahler Conjecture, among others.

However, OpenAI has a reputation within the mathematical community for making bold claims with limited transparency, and this announcement has been met with considerable skepticism.

An OpenAI spokesperson informed Scientific American that the company has not yet publicly released a new model, and nearly all results were generated after issuing a single instruction to a single agent. If true, this suggests that unprecedented mathematical capabilities may soon be within reach of the general public.

Skepticism quickly surfaced. Doubts were first raised on September 8, when OpenAI claimed to have resolved the Navier-Stokes Millennium Problem.

The Navier-Stokes equations, which describe the motion of fluids such as water and air, are crucial in engineering and physics.

The crux of the problem lies in determining whether a three-dimensional fluid, starting smoothly and evolving according to the Navier-Stokes equations, will remain normal and smooth or if it could suddenly develop 'singularities'—points where velocity or vorticity becomes infinitely large—within a finite time.

OpenAI stated, "Our system provided both an analytical proof and a Lean-formalized proof, demonstrating that an initially stationary and smooth fluid can develop singularities in finite time. The fluid is subjected to smooth external forces, and throughout the entire process from rest to singularity formation, its energy remains finite. This result proves that propositions 'C' (and 'D') in the official statement of the Millennium Prize Problem hold, thereby solving the Navier-Stokes Millennium Prize Problem."

The Clay Mathematics Institute established this problem in 2000 as one of seven global Millennium Prize Problems, offering a $1 million reward. OpenAI has no intention of claiming the prize.

The agent system was launched on September 1. Approximately 88 hours after the first agents were activated on September 5, the solution emerged. The subsequent Lean-formalized verification was completed by GPT-6 Astra in about 17 hours. Throughout this process, the agents exchanged roughly 4.9 million messages and consumed about 300 billion output tokens.

Just 12 hours before OpenAI's official announcement, mathematician Tristan Buckmaster of New York University and his collaborator Levent Alpöge, who works at Anthropic, publicly disclosed similar findings, sparking a priority dispute.

The duo had previously employed AI models like Claude and Codex in their research, achieving key results on August 15 and completing Lean verification by August 22. Buckmaster claimed that upon hearing rumors, OpenAI rapidly deployed numerous AI agents and released their findings.

Buckmaster argued that OpenAI mishandled the announcement, attribution, and priority of the results, alleging that the company had requested Alpöge's exclusion. OpenAI denied using their private data and asserted that the research was conducted independently. The priority dispute remains unresolved.

Mathematicians have stated that leading AI companies "give the impression of gangster-like behavior."

On September 21, the company announced the formation of an independent advisory panel of mathematicians to recommend responsible practices for releasing AI-generated computational results.

The panel advised OpenAI to disclose the model, specific prompts, and computational time behind the results. It particularly criticized AI companies for using "proprietary internal models" that are not widely accessible to mathematicians.

However, OpenAI chose to release only the average computational time per problem and some additional statistics, without providing any prompts.

A more significant issue is that Lean verification does not equate to mathematical confirmation of these results.

Valerio Capraro, an associate professor at the University of Milano-Bicocca, posted that recent research found at least two discrepancies between OpenAI's Navier-Stokes paper and its Lean code:

The paper's estimate required only four additional input derivatives, while the Lean version needed five, weakening the formalized result. Additionally, the pressure-flux estimate in the paper used different bounds and proof methods.

Capraro noted that it is unclear whether these discrepancies invalidate the entire proof, but they raise a critical question: OpenAI continually announces revolutionary breakthroughs, yet no one knows if the proofs are correct or if they prove what they claim.

As Capraro pointed out, this highlights the limitations of Lean verification. Lean can only check for formal logical errors in the mathematical derivation, not whether the content proves the original problem or whether the result is truly novel.

This is why mathematicians demand full disclosure of the model, prompts, and computational process. The public needs to know if the code runs, to compare the paper, formalized statements, and final proof item by item, and to confirm that all three address the same mathematical question.

OpenAI's claim that "each result consumes about 3 hours of ChatGPT Pro's computational power" does not mean AI solved a mathematical problem in 3 hours.

Three hours represents the average computational effort for successfully retained results. The model also underwent problem screening, directional exploration, repeated attempts, and result organization. An OpenAI spokesperson admitted to Scientific American that some findings may have required multiple attempts.

MIT mathematician Andrew Sutherland stated, "Unless they release the model and people can reproduce their results, I think any claim of solving a problem in one shot with a single agent should be regarded as unverified. We should demand evidence."

Mathematics is never just about obtaining answers. Terence Tao once wrote that in an era of heavy AI use, "remembering what it feels like to be a child" is crucial; in his view, pure mathematics relies especially on curiosity, exploration, and a sense of wonder.

AI is shifting the focus of mathematical activities: As generating answers becomes easier, understanding results, asking good questions, judging their significance, and integrating new findings into human knowledge become more important.

This focus is being tested by "proof overload." After releasing 722 mathematical manuscripts, OpenAI withdrew three due to a symbol error and revised 14 others; the repository currently lists 719.

Mathematicians lack the time to read all 700+ manuscripts, many of which are hundreds of pages long. This reflects the mathematical community's deepest concern about the AI race: AI is generating "proof overload," yet humans lack the time and capacity to understand these proofs.

Verification has always been one of the most mentally taxing parts of mathematical research.

After the Kepler Conjecture proof was accepted, it took over a decade for formal verification; the Four Color Theorem's answer was widely accepted, but trusting, checking, and reconstructing the computational process took decades; the Finite Simple Group Classification Theorem relies on a knowledge network requiring generations to maintain, patch, and organize...

OpenAI's release of over 700 manuscripts at once compresses this slow digestion process into a single night.

Some netizens called this release method "dumping unverified work on mathematicians," while others optimistically argued that even if only half the results hold, this batch could still transform mathematical research.

"If we want answers to these mathematical questions, I see no reason for companies to keep them secret," said Daniel Litt, a mathematician at the University of Toronto. "In my view, this is good for mathematical progress."

In "A Grave Misalignment Between AI and Mathematics," signed by Terence Tao and several Fields Medalists, it is noted that solving problems is merely a tool in mathematical research; more important goals include conceptual understanding, theoretical development, talent cultivation, and knowledge preservation.

OpenAI shows no signs of slowing down, as it aims to provide mathematicians with the most powerful tools possible. However, for those who pursue mathematics as an end in itself, the consequences of the AI race remain unpredictable.