A Millennium Problem, Cracked by AI?!
The mathematics and AI communities may be on the verge of an unprecedented seismic event.
Just now, Tristan Buckmaster, a mathematician specializing in fluid dynamics and partial differential equations at New York University, made a major announcement—
A mysterious model internal to OpenAI has solved a key challenge regarding the "Navier–Stokes equations."
It is claimed that this is very likely the "next-generation Bel."
It directly produced a rigorous, 100-page mathematical proof in one go! However, Tristan revealed he has not yet obtained this AI-generated manuscript.
Currently, Tristan's 4-page manuscript is exploding within the community, with everyone's screens filled with two words—"shock."
What has the entire internet even more excited is the 100-page proof that OpenAI is expected to release in the coming days.
OpenAI Poised to Claim a Victory for a Millennium-Century Problem
Discussing fluid dynamics inevitably leads to the "Navier–Stokes equations" (N-S equations). They are humanity's "Physics Bible" for describing the real fluid world.
In the mathematics community, it is also a towering, seemingly insurmountable peak.
In 2000, the Clay Mathematics Institute listed it as one of the "Millennium Prize Problems."
Whoever can completely solve this problem will receive a $1 million prize and the highest honor in the academic world.
To put it in the context of our daily lives, the N-S equations study how water and air move.
Whether it's the swirling milk foam in a cup of coffee, the airflow over an airplane wing, or the roaring ocean currents deep in the Pacific, it's all related to them.
The underlying governing principles are established by this set of partial differential equations formulated by 19th-century mathematicians Navier and Stokes:
However, many years have passed, and mathematicians have still not fully resolved a core question:
Will the solutions to these equations remain stable at all times, or could they "blow up"—exhibit singular behavior—under certain conditions?
Who would have thought that OpenAI's mysterious model has cracked it.
According to Tristan, this model directly generated a proof manuscript up to a hundred pages long, completing the proof of blow-up for the N-S equations under "external forcing."
Does Including External Forcing Count as a True Solution?
It's worth noting there's a qualifier this time: "external forcing."
Think of it as an external influence still acting on the fluid while it moves. But the mathematical definition of an external force is a precisely defined function; one cannot simply substitute the intuition of stirring with a rod.
One might ask: Allowing external forcing, isn't that relaxing the problem?
The key point is that the official Clay problem statement already permits this route.
In the formal formulation given by Fefferman, items C and D allow for the construction of smooth initial data and smooth external forces that satisfy the conditions, to prove the non-existence of a global smooth solution meeting the specified criteria.
Therefore, if OpenAI's proof indeed satisfies all conditions, then the result with external forcing could also solve the official prize problem.
Three Mathematical Breakthroughs, All Announced on the Same Day
Simultaneously, another batch of AI mathematics results was made public.
Tristan and Levent Alpöge announced that, with the help of LLMs, they obtained finite-time blow-up results for three classes of fluid equations under smooth external forcing.
Now, the paper and Lean formalization code are all online.
The three results they published this time correspond to the incompressible porous medium equation, the Boussinesq equations, and the three-dimensional incompressible Euler equations, respectively.
In simple terms, they involve fluid flowing through porous materials, the interplay between buoyancy and flow, and the motion of ideal fluids ignoring viscosity, respectively.
The common breakthrough is advancing the existing external forcing blow-up route to the case of "smooth external forcing."
This is crucial because if the external force itself is very rough, it's hard to tell whether the flow's instability stems from its internal mechanism or from problems already embedded in the external input.
Elevating the forcing to smooth eliminates this source of roughness.
The research approach for this work draws from a long-term research program by Diego Córdoba and Luis Martínez-Zoroa.
Tristan took up this route and continued pushing it forward with the help of Claude, Codex, and GPT-5.6 Sol.
Even Astra was deployed for the final stages of organizing the manuscript and reviewing the arguments.
Almost immediately, Fields Medalist Terence Tao gave high praise, calling it an "outstanding achievement."
According to Tao's interpretation, the authors repeatedly add very small, highly localized, high-frequency perturbations to an existing flow.
Here, "high-frequency" refers to tiny structures that change rapidly in space. Think of it as superimposing an increasingly fine texture onto a large-scale flow.
The clever part is: the large-scale flow amplifies these fine structures, while the back-reaction of these fine structures on the large-scale flow is strictly controlled.
Once the new structure grows strong enough, it takes over the local dynamics at the right moment, creating the conditions for the next round of even finer perturbations.
Round after round, the changes progress to smaller scales, ultimately constructing a solution that blows up in finite time.
The most difficult part is calculating clearly for each transition: where to place the perturbation, how fast it grows, the magnitude of errors, and whether the entire process can be seamlessly connected.
Tao mentioned that in the model case for Boussinesq, the core evolution of amplitude and frequency can be reduced to a rather simple ordinary differential equation.
However, technical details like spatial truncation still result in a paper that, even simplified, is 76 pages long.
This also explains why the breakthrough for Euler cannot directly announce the conquest of Navier–Stokes.
The extra viscous term νΔu in Navier–Stokes smooths out fine structures. The higher the perturbation frequency, the stronger the viscous dissipation typically becomes.
The aforementioned method relies precisely on the continuous strengthening of high-frequency structures. To extend it to Navier–Stokes, one must prove that the blow-up mechanism can still hold even as viscosity continuously weakens these structures.
Tao's assessment is that, in principle, nothing seems to prevent this route from being extended further, and one cannot even rule out the eventual possibility of removing the external forcing.
However, he also clearly emphasized that implementing this plan would face an enormous number of technical difficulties.
This is a judgment on the research's prospects; there remains a critical step before announcing the proof's completion.
OpenAI's Preemptive Move Sparks Online Debate
Adding to the drama, an experience Tristan disclosed in an accompanying statement pushed the debate in another direction—
How exactly did OpenAI's result come about?
According to his account, the route of "smooth external forcing" is a very specific research line pioneered by Córdoba and Martínez-Zoroa.
He and Alpöge had been progressing along this line privately, with almost no other teams working on it.
Therefore, when OpenAI told him that their internal model had also tackled the "externally forced" scenario, he was particularly shocked.
The story begins on September 3rd.
At that time, the entire internet was buzzing with rumors that Anthropic's Claude had solved a major mathematical problem, reportedly the N-S equations.
Coincidentally, that same day, the progress of Tristan and Alpöge's research reached someone inside OpenAI.
Tristan subsequently contacted a prominent mathematician at OpenAI, explained the situation, and specifically emphasized that this was his personal collaboration with Alpöge, unrelated to any institution.
The other party replied the same day, hoping to learn more details and also offering computational support.
Tristan originally proposed a call the following week, but OpenAI hoped to talk sooner. By September 6th, Sébastien Bubeck joined the communication, and the three had two phone calls that afternoon.
NYU Mathematician Tristan Buckmaster
It was during these calls that they heard the news of the "hundred-page proof."
At that moment, Tristan immediately asked, "Had they used their prior project drafts, session data, or other user data placed in Codex for training?"
The response was that the OpenAI model did not access user data; when he pressed further on the "training question," he received no reply.
Unusual Request: Remove Anthropic Employee's Authorship
Next, when discussing how to publish, Tristan and OpenAI reached another impasse.
At that time, OpenAI presented two options—
One, Tristan's team publishes their Euler result first, and OpenAI publishes the "Navier–Stokes result" the next day;
The other, they write the paper independently, publish the Navier–Stokes result, and note that the result came from an internal OpenAI model.
Among these, the most unacceptable to him was the issue of authorship.
Bubeck explicitly requested twice to remove Alpöge's name from authorship, precisely because Alpöge is an employee of Anthropic.
Anthropic Employee Levent Alpöge
Everything now awaits OpenAI's response tomorrow; the truth will then come to light.
Mathematics may be facing its own "AlphaGo moment."
If this hundred-page proof ultimately withstands scrutiny, and the research process can also confirm the AI's independent contribution, then there will be a heavyweight instance of AI autonomously pushing the boundaries of human knowledge.
Back then, AlphaGo made a move unexpected by human Go players. Now, AI might be writing the next line that human mathematicians have not yet written.
Of course, this moment still requires the complete proof and a clear account of the result's origins.
Mathematics' history can be rewritten with AI's participation, but every step must add up.
References:
https://x.com/imjustnewatai/status/2097199843445739565?s=20
https://cims.nyu.edu/~tristanb/statement.pdf
https://mastodon.social/@tristanbuckmaster/117233413705701198
https://mathstodon.xyz/@tao/117233527638291447
This article is from the WeChat public account "新智元" (New Zhi Yuan); author: ASI启示录; editor: Taozi
