OpenAI claims AI solved Navier-Stokes millennium prize problem amid credit controversy

Sep 8, 2026

OpenAI announced that an unreleased next-generation model solved the Navier-Stokes existence and smoothness problem, a historic Millennium Prize math puzzle. However, the breakthrough is overshadowed by allegations from NYU mathematician Tristan Buckmaster. Buckmaster claims OpenAI rushed to solve the problem after learning of his private, AI-assisted research with Anthropic's Levent Alpöge, and pressured him to strip Alpöge of credit. OpenAI denies using their data, stating its proof was generated independently.

OpenAI Navier-Stokes proof announcement

▪OpenAI announced on September 8, 2026, that an unreleased next-generation internal model solved the Navier-Stokes existence and smoothness problem, one of the seven Millennium Prize problems
▪OpenAI stated that its Navier-Stokes proof shows that the three-dimensional Navier-Stokes equations can develop a singularity in finite time, meaning the equations can theoretically become unreliable
▪OpenAI stated that its Navier-Stokes proof was produced in 88 hours by approximately 10,000 coordinated AI agents and was formalized in the Lean programming language
▪OpenAI stated that it does not intend to claim the $1 million Millennium Prize from the Clay Mathematics Institute for its Navier-Stokes solution

Buckmaster credit dispute allegations

▪OpenAI's Sébastien Bubeck and other executives denied inspecting Tristan Buckmaster and Levent Alpöge's Codex prompts, stating they did not see any of the pair's work before it was released publicly
▪Tristan Buckmaster alleged that OpenAI's Sébastien Bubeck twice proposed removing Anthropic researcher Levent Alpöge from authorship because Alpöge's affiliation with a rival AI lab was annoying
▪OpenAI acknowledged in a blog post that while unlikely, the company cannot rule out that de-identified data derived from Tristan Buckmaster and Levent Alpöge's usage of OpenAI products helped improve its models
▪Tristan Buckmaster raised concerns that OpenAI's model may have been trained on or accessed his and Levent Alpöge's private Codex sessions, which contained all their drafts throughout the project
▪NYU mathematician Tristan Buckmaster published a statement alleging that OpenAI pressured him, tried to remove his co-author Levent Alpöge from their joint paper, and threatened his career after learning of their AI-assisted Navier-Stokes research

AI mathematical research methods

▪OpenAI mathematician Ven Chandrasekaran stated that OpenAI's mathematical solution differs fundamentally in its approach and precise results from the work produced by Tristan Buckmaster and Levent Alpöge
▪Tristan Buckmaster and Levent Alpöge conducted their fluid-dynamics research using a mix of AI models, relying primarily on OpenAI's Codex and Anthropic's Claude

Millennium Prize problem significance

▪The Clay Mathematics Institute established the seven Millennium Prize Problems in the year 2000, offering a $1 million reward for the solution to each fundamental mathematical question
▪The Navier-Stokes equations describe the physical movement of fluids and gases, and are widely used to model weather, ocean currents, and climate behavior
▪Prior to OpenAI's September 2026 announcement, only one of the seven Millennium Prize Problems had ever been successfully solved

Compute cost trends

▪OpenAI head of research Mark Chen stated that the compute power required to solve the Navier-Stokes problem cost in the millions of dollars
▪OpenAI's week-long mathematical research effort consumed 300 billion output tokens, which would equal $22.5 million worth of compute if charged at current GPT-6 Astra rates

AI capability scaling patterns

▪OpenAI developer Noam Brown stated that while hitting 87.5 percent on the ARC-AGI benchmark cost $500,000 using OpenAI o3, the GPT-6 Astra model can achieve the same result for approximately $20
▪OpenAI stated that its unreleased internal model solves nearly three times as many open mathematics problems as GPT-6 Astra, with the performance gap widening as compute scales up

Debatable claims

▪AI companies should not train models on user data from developer tools
▪The Clay Mathematics Institute should award Millennium Prizes to AI-generated proofs
▪OpenAI's rush to solve Navier-Stokes after hearing of academic progress was unethical