OpenAI Publishes Navier-Stokes Proof but Declines Millennium Prize
OpenAI has made a significant breakthrough by publishing its proof of the Navier-Stokes equation, one of mathematics’ most challenging open problems. However, they have stated that they will not be claiming the associated $1 million Millennium Prize.
The Details:
OpenAI has released a write-up, a PDF paper, and a formalization in Lean with a public GitHub repository. This formalization is crucial as it allows for machine verification of the mathematics. They utilized GPT-6 Astra to complete 17 hours of additional work for this proof.
The key achievement is demonstrating that an initially smooth fluid at rest can develop a singularity in finite time, resolving a statement within the Millennium Prize formulation. Yet, OpenAI explicitly states:
"We do not intend to claim the Millennium Prize for this result."
This decision provides a clear signal regarding the strength of their claim compared to the prize’s criteria.
The Method and Scale:
The proof involved approximately 10,000 concurrent agents working for about 88 hours from September 1st to 5th. This massive effort consumed 2.7 million messages and around 130 billion output tokens, utilizing an internal system described as significantly more capable than GPT-6 Astra.
The proof was not produced by a single model but by a swarm of models running in parallel, marking a shift from focusing on individual model intelligence to the computational resources available. This suggests that future breakthroughs of this nature will require similar levels of resource allocation.
Credit and Recognition:
Initially, OpenAI denied any connection to concurrent work by Levent Alpöge and Tristan Buckmaster. However, the published page now credits both men for their contributions, indicating a shift in the credit dispute.