Level 1 — Absolute Beginner

OpenAI is a company that makes AI. OpenAI says one of its AI models solved a very hard math problem. The problem is about how water and air move. It is called the Navier-Stokes problem. People tried to solve it for more than 100 years.

There is a prize of one million dollars for this problem. A group called the Clay Mathematics Institute offers the money. OpenAI says its AI will not take the prize money.

OpenAI used about 10,000 AI helpers together. They worked for 88 hours. They wrote the proof in a special computer language called Lean.

But there is a problem too. A math professor named Tristan Buckmaster says he worked on this same problem for almost a year. He worked with a scientist named Levent Alpöge, who works at a company called Anthropic. Buckmaster says OpenAI did not give Alpöge enough credit.

AI model

A computer program trained to think, write, or solve problems like a person.

math problem

A question in mathematics that needs to be solved or proven.

prize

Money or a reward given to someone who does something great.

proof

A clear explanation that shows a math idea is true.

scientist

A person who studies and researches a subject, like math or science.

credit

Recognition given to a person for the work they did.

solve

To find the correct answer to a problem.

company

A business that makes or sells products, like AI tools.

Level 2 — Elementary

OpenAI announced that a powerful, unreleased AI model created a proof for one of the hardest problems in mathematics. The problem is about the Navier-Stokes equations, which describe how liquids like water and gases like air move and flow. Mathematicians have tried to fully understand this problem for over 100 years.

This is one of the seven famous Millennium Prize Problems. The Clay Mathematics Institute offers one million dollars for solving each of these problems. OpenAI's model showed that, under certain conditions, the math can develop a point where it stops working normally in a finite amount of time. OpenAI said it would not personally claim the prize money.

To build the proof, OpenAI used around 10,000 AI agents working together for 88 hours. The final proof was written and checked using a computer language called Lean, which mathematicians use to make sure a proof has no mistakes.

Soon after the announcement, a disagreement began. Tristan Buckmaster, a mathematician at New York University, said he and Levent Alpöge, a researcher at Anthropic, had spent almost a year working on this exact kind of problem using AI. Buckmaster said they had reached their own solution by August 22, 2026, and he accused OpenAI of rushing to finish first while leaving Alpöge out of the credit.

equation

A mathematical statement showing that two things are equal.

finite

Having a limit or an end, not going on forever.

mathematician

A person who studies numbers, shapes, and math problems for a living.

agent

In AI, a program that can act and make decisions to complete a task.

claim

To say that something belongs to you or that you deserve it.

accuse

To say that someone did something wrong.

condition

A situation or requirement that must be true for something to happen.

researcher

A person whose job is to study a topic carefully to learn new things.

Level 3 — Intermediate

OpenAI has announced that an internal, unreleased AI model, said to be significantly more capable than the company's public GPT-6 Astra model, produced a proof addressing one of mathematics' oldest open questions: whether the three-dimensional Navier-Stokes equations can develop a singularity, a point at which the math effectively blows up, within a finite amount of time. The Navier-Stokes existence and smoothness problem, which describes how fluids such as water and air behave, has resisted a complete solution for more than a century.

The problem is one of seven Millennium Prize Problems, each carrying a one million dollar reward from the Clay Mathematics Institute. According to OpenAI, its internal system coordinated roughly 10,000 AI agents over 88 hours to construct the proof, which was then formalized in Lean, a programming language mathematicians use to verify that a proof is logically airtight. OpenAI stated that it would not personally seek the prize money.

The announcement quickly became entangled in a dispute over credit. Tristan Buckmaster, a mathematician at New York University, said that he and Levent Alpöge, a researcher at Anthropic, had spent close to a year using AI models to pursue proofs related to fluid blow-up, and that they had arrived at a solution of their own by August 22, 2026. Buckmaster alleged that OpenAI accelerated its own effort after learning of his progress, and that the company tried to minimize Alpöge's role in the outcome.

OpenAI researcher Sebastien Bubeck offered a different account, saying the company began investigating the problem after hearing rumors that Anthropic had made headway on it, and he denied that OpenAI's researchers or its AI agents ever had access to Buckmaster's prompts or draft proof. In its published write-up, OpenAI credited Buckmaster and Alpöge for concurrent, independent work on a closely related problem involving the forced Euler equations, and it offered to recognize their priority in a joint announcement.

singularity

In math, a point where a formula or system breaks down or produces an undefined result.

formalize

To put an idea or argument into a strict, official, and checkable form.

verify

To confirm that something is true or correct through careful checking.

dispute

A disagreement or argument between two or more sides.

allege

To claim that something is true, often without full proof yet.

concurrent

Happening at the same time as something else.

independent

Done separately, without being influenced by or copying another party's work.

priority

The right to be recognized as first, especially in discovery or invention.

Level 4 — Advanced

OpenAI has claimed a milestone that, if it withstands scrutiny, ranks among the most consequential results ever attributed to an artificial intelligence system: an internal, unreleased model, described as substantially more capable than the company's public GPT-6 Astra release, has produced a proof concerning the three-dimensional Navier-Stokes equations, the partial differential equations that govern the motion of viscous fluids. The proof establishes that these equations can, under certain conditions, develop a singularity, a point at which their solutions cease to behave smoothly, within a finite span of time. The Navier-Stokes existence and smoothness problem has stood among the discipline's most stubborn open questions for well over a century.

It is one of the seven Millennium Prize Problems, each carrying a one million dollar award from the Clay Mathematics Institute, and OpenAI has said it will not personally pursue the prize money. To construct the proof, the company reports orchestrating a swarm of roughly 10,000 AI agents across 88 hours of continuous work, with the final argument formalized in Lean, a proof assistant language that allows mathematicians to mechanically verify that every logical step holds, eliminating the possibility of a subtle human or machine error slipping through unnoticed.

The triumph was almost immediately complicated by an unusually public dispute over priority. Tristan Buckmaster, a mathematician at New York University, said that he and Levent Alpöge, a researcher affiliated with Anthropic, had spent nearly a year deploying AI models toward proofs of fluid blow-up phenomena, and that the pair had independently reached a solution of their own by August 22, 2026. Buckmaster went further, alleging that OpenAI, upon learning of his progress, moved swiftly to complete and publish its own result while attempting to sideline Alpöge's contribution from the record.

OpenAI's account, offered by researcher Sebastien Bubeck, diverges sharply on both timeline and conduct: Bubeck said the company turned its attention to the problem only after hearing rumors that Anthropic researchers had made meaningful progress, and he explicitly denied that any OpenAI researcher or AI agent had ever accessed Buckmaster's prompts or draft proof. In its formal write-up, OpenAI credited Buckmaster and Alpöge with concurrent, independent work on a closely related question involving the forced Euler equations, and proposed recognizing their priority through a joint announcement, a gesture that has so far done little to settle a controversy that touches on authorship, competitive pressure between AI labs, and the still unresolved question of how credit should be assigned when machines, not just people, do the mathematical work.

partial differential equation

A mathematical equation involving rates of change across more than one variable, often used to describe physical systems like fluid flow.

viscous

Having a thick, sticky quality that resists flowing easily, a property of many real fluids.