OpenAI Purportedly Solves Navier-Stokes Problem via 10,000 Collaborating Agents

IT Home, September 9 news: OpenAI announced on local time September 8 that its unreleased internal AI model (with performance far exceeding the newly released GPT-6 Astra) has successfully solved the "Navier-Stokes existence and smoothness problem," one of the seven "Millennium Prize Problems" established by the Clay Mathematics Institute in 2000. This problem, which concerns whether the Navier-Stokes equations describing fluid motion can break down, has puzzled mathematicians for about 90 years.

Prior to this, Anthropic announced last month that an unreleased internal Claude research model had made significant progress toward solving the Riemann hypothesis, raising the long-standing lower bound for the proportion of zeros on the critical line of the Riemann zeta function from 41.6% to 67.2%. The news shocked the industry.

▲ Schematic of locally incompressible motion. Orange indicates faster angular rotation speed; cyan indicates slower angular rotation speed. Circulation velocity also depends on radius. The trajectory shows inward spiraling motion and axial stretching.

OpenAI stated that the team generated the proof in about 88 hours through the collaboration of roughly 10,000 AI agents. The proof shows that an initially smooth, stationary fluid can develop a "singularity"—meaning the flow velocity grows infinitely—within a finite time.

OpenAI provided an analytical proof alongside a formal verification in the Lean programming language. The verification was completed by GPT-6 Astra in about 17 hours.

▲ The internal model used by OpenAI far outperforms GPT-6 on open mathematical problems

Core of the Proof and Verification

OpenAI researcher Sébastien Bubeck called this "an astonishing culmination of the trajectory seen over the past twelve months."

The proof achieves a singularity by designing an inward-swirling and stretching "vortex" whose energy remains finite as required by the laws of physics. The proof is established under the premise of smooth external forces, corresponding to cases "C" and "D" in the official problem statement.

Controversies and Inquiries

However, the announcement quickly sparked controversy. Tristan Buckmaster, a mathematics professor at New York University, and Levent Alpöge, a researcher at Anthropic, had published research results on related fluid equations just the day before.

Buckmaster questioned whether OpenAI had redirected resources to the "force-driven" approach he had been exploring only after hearing of their progress. He expressed concern that research data stored by him and Alpöge within OpenAI's Codex models might have been used for training or reference, but received no definitive answer.

OpenAI researcher Bubeck denied the allegations, emphasizing that neither researchers nor AI agents accessed specific user data. However, OpenAI also acknowledged in a statement that it "cannot rule out" that anonymized data from their use of OpenAI products helped improve the model.

Broader Implications

This event highlights the complex issues arising as AI tackles the pinnacles of human intellect. Renowned mathematician Terence Tao, while comparing such problems to "beacons" that attract scientific endeavor, also warned that if AI can solve the hardest problems without deep human involvement, it might weaken human understanding of mathematics.

It is estimated that the compute cost could be as high as $15 million to $22.5 million (currently about 101 million to 151 million RMB). OpenAI stated that it does not intend to claim the $1 million prize from the Clay Mathematics Institute (IT Home note: currently about 6.728 million RMB), but the incident has sparked widespread discussions regarding credit attribution, data privacy, and the safety of AI development.

Related reading: "Unreleased New Model 'Strikes Gold': Anthropic Announces Major Breakthrough by Claude in Tackling Riemann Hypothesis" "Google's AI Framework AlphaProof Nexus Solves 2 Mathematical Problems Unresolved for 56 Years" "AI Disproves Famous Geometric Conjecture: OpenAI Announces Solution to 80-Year-Old Math Problem" "OpenAI GPT-4b Solves Nobel-Level Problem: Human Cells 'Rejuvenated', Reversal Efficiency Surges 50-Fold"