Tristan Buckmaster and the New Partnership Between Human and Artificial Intelligence
By Ralph Losey.
Imagine receiving an AI-generated legal brief so poorly written and organized that you are tempted to stop reading. The argument wanders, important points are buried under unnecessary complications, and some passages seem to make no sense at all. Yet buried inside is a genuinely original legal argument that even the best lawyers had overlooked. Suppose the argument might actually be correct. Would you use it? Of course. But first you would have to understand it, verify the reasoning, authorities, and evidence, clean up the language, and present it clearly enough for a judge to understand its significance.
In September 2026, something remarkably similar happened in mathematics, except that the hidden idea concerned one of the most difficult problems ever posed. Frontier AI did not merely assist the mathematicians. It produced an original mathematical connection that leading experts had missed and carried that insight into a proposed, formally checked solution to the Navier–Stokes Millennium Prize Problem. We should call this what it is: artificial intelligence has attained superintelligence in one field, Mathematics. It can now do some mathematical work beyond the reach of the best unaided human mathematicians. Buckmaster’s delight, not fear, is part of the story. Lawyers should pay attention. Someday it may happen in our field.
This three-part series follows the evidence and then asks what it means. Part One tells the human story of New York University mathematics professor Tristan Buckmaster, who had been working on the Navier–Stokes problem for years, and his friend Levent Alpöge, a mathematician at Anthropic. It then turns to thousands of coordinated AI agents, the Navier–Stokes breakthrough itself, and the evidence of mathematical superintelligence. Part Two turns to the flood of 722 AI-generated manuscripts, famed UCLA mathematics professor Terence Tao’s Math 2.0, the difference between proof and understanding, education, attribution, corporate power, and the future of science. Part Three returns to lawyers: the jagged frontier of AI capabilities, the Hybrid human-AI method, agent teams, verification, confidentiality, accountability, and a concrete Rule 37(e) investigation. The point is not merely to learn how to delegate to super-smart AI, much less worship the machine. It is to learn how extraordinary machine intelligence can be placed under real human direction.
The Worst Mathematics the Expert Had Ever Seen, and Yet
Tristan Buckmaster has spent much of his professional life studying some of the most difficult problems in mathematics. Tristan is a professor at New York University’s Courant Institute of Mathematical Sciences. His specialty is the mathematics of fluid motion, including difficult questions concerning how fluids behave under extreme conditions. This is at the core of the famous Navier–Stokes problem. Mathematicians have been trying to solve this for almost a century, and in 2000 the Clay Mathematics Institute promised a $1 million prize to anyone who did.
On September 7, 2026, Buckmaster and Alpöge formally announced the forced-Euler results they had been developing. See NYU’s later report, Tristan Buckmaster Constructs Finite-Time Blowup for 3D Euler with Smooth Forcing. Their work did not solve the Navier–Stokes problem, but it was an important step toward a possible solution.
The next day, September 8, 2026, OpenAI made its initial announcement on the social media site X that its internal system had found a solution to the Navier–Stokes problem. The company said a decentralized swarm of roughly 10,000 autonomous AI agents, using a not-yet-released frontier model, worked for about 88 hours and produced a 166-page proof and Lean formalization. OpenAI’s initial announcement did not name Buckmaster or Alpöge. The omission ignited a priority dispute because rumors of their forced-Euler work had helped trigger the laboratory race, although OpenAI says its agents did not see their unpublished work and that its proofs differ significantly. Many people, including Buckmaster, do not believe that denial. The timing was too coincidental. The Navier–Stokes priority controversy drew a flurry of public responses, many of them negative.
At the time, Professor Buckmaster was working informally with mathematicians connected to both OpenAI and Anthropic on related problems. He was among the first outside experts invited by OpenAI to publicly examine the AI-generated proof. Buckmaster’s reaction was unexpected:
It was the worst mathematics I’ve ever seen in my life, but it was actually correct!
In an October 2, 2026, World Science Festival video interview conducted by physicist Brian Greene, Professor Buckmaster explained that the AI-generated proof was almost incomprehensible, containing irrelevant calculations, unnecessary complications, and passages that seemed to make little sense. Yet buried in this extraordinary confusion, Buckmaster saw what he considered a mathematically valid argument resolving the Navier–Stokes problem. The Clay Mathematics Institute has said the problem “apparently” has been settled, but its established process for evaluating the proof and assigning credit is deliberately unhurried. Buckmaster expects the proof to survive that review. OpenAI says it will not claim the million-dollar prize; its stated goal was to demonstrate the progress of its AI models.
In the World Science Festival interview, physicist Brian Greene asked whether Buckmaster had personally checked the entire proof.
Buckmaster answered, “No”. He shook his head and smiled as if to say, ‘are you kidding?’ “How, then, could you be so confident?” Greene asked. The answer involved other AI agents and Lean, a computer program that checks mathematical proofs according to precisely defined rules of logic. The researchers used Lean to check a formalized version of the mathematical argument even though its original presentation was extraordinarily difficult for humans to follow. A successful Lean check establishes that the encoded statement follows from its stated assumptions. It does not, by itself, prove that the formalization perfectly captures every claim in a long prose manuscript. Humans must still examine that correspondence.
Here is the core of OpenAI’s September 8 announcement on X.
We’re sharing a solution to the Navier-Stokes Millennium Prize Problem, one of the deepest problems at the frontier of mathematics. The proof was produced by a group of agents, using an OpenAI next-generation model significantly more capable than GPT-6 Astra. The problem concerns whether the description of smooth three-dimensional fluid motion modeled by the Navier-Stokes equations can break down. It has remained unresolved for roughly 90 years. …
The group (OpenAI) produced an analytical proof and Lean formalization that via Navier-Stokes dynamics a fluid can develop a singularity in finite time. The solution is a vortex, a spinning swirl of fluid, that spirals inward and gets increasingly elongated, like spaghetti.
The announced Navier–Stokes solution used a smooth external force and a vortex that spirals inward while stretching vertically. What startled Buckmaster was not merely the scale of the computation, but a mathematical connection the agents had found that he had never seen. OpenAI still denies that Buckmaster’ work influenced its result. He in turn denies that the supervising mathematicians at OpenAI correctly understand the meaning of their agents’ work. The distinction between producing a proof and understanding what it means will become important later in our story, especially for lawyers.
For centuries, mathematicians have relied on exceptional human intellectual ability to explore the boundaries of mathematical knowledge. They develop theories, exchange ideas, challenge assumptions, and sometimes spend entire careers pursuing problems such as Navier–Stokes that have defeated previous generations.
Now AI is making original contributions to that process. Even more remarkably, coordinated groups of AI agents can investigate mathematical questions simultaneously, developing and testing ideas at speeds that astonish experienced researchers.
Buckmaster believes mathematics has reached a moment comparable to IBM’s Deep Blue defeating world chess champion Garry Kasparov in 1997 and that the implications go well beyond mathematics.
A machine had demonstrated superior performance at an intellectual activity long regarded as a supreme test of human ability.
But this story is not simply about machines surpassing people. It is about what becomes possible when exceptional human mathematicians learn to collaborate with extraordinarily capable artificial intelligence.
Buckmaster is an especially interesting and well-qualified person to understand that transformation.
Meet Tristan Buckmaster: The Laughing Mathematician
A mathematical star like Buckmaster, immersed in an intellectual world most of us seldom encounter, might seem difficult to relate to. Yet his personality is anything but inaccessible.
When Brady Haran began his September 29, 2026, Numberphile interview with Buckmaster, the professor began laughing when asked if he was ready to start. Haran first asked how he was feeling after the extraordinary events of the preceding weeks. Buckmaster’s answer revealed something quite human about a mathematician who had suddenly become internationally famous. Media outlets all over the world were seeking interviews, trying to stir up controversy between him and OpenAI, whereas Buckmaster’s real interests were in mathematics and computer science.
In this whirlwind of events, he was also helping to care for a two-month-old baby at home. During the week of the mathematical breakthroughs and corporate controversies, he estimated that he had averaged approximately two hours of sleep per night. Colleagues would come to his university office and find him asleep at his desk.
Haran jokingly asked whether Buckmaster felt as though he had been mugged by 10,000 AI agents. The exchange is worth watching. Buckmaster’s laughter, animated expressions, and sleep-deprived manner help you understand the man.
There is something appealing about this Max Planck Institute-trained math wizard whose research has attracted the attention of the world’s largest AI laboratories during the past few years. Then, after he apparently helped one of them reach this breakthrough, the attention of the whole media world turned to this sleep-deprived academic.
Tristan Buckmaster has been interested in computers since childhood. In the Numberphile interview, he recalls that at thirteen he was experimenting with graphics processing units, or GPUs, and writing computer graphics software. In that case, he was probably using Nvidia chips. He initially wanted to become a computer game programmer. Like many people, I had a similar dream inspired by video-game graphics, but, alas, mine was diverted by law and the responsibilities of fatherhood. Unlike me, Tristan was a super-genius type who grew up in Melbourne, Australia. He skipped twelfth grade to go straight to Monash University in Melbourne to study computer science. Then a professional mathematician in his family encouraged him to pursue mathematics instead. I wonder what would have happened if his family inspiration had been a lawyer.
Young Tristan then switched to a mathematics program at the Australian National University in Canberra, Australia. From there he went to the University of Bonn in Germany for his master’s degree in mathematics. He was then accepted into the prestigious University of Leipzig/Max Planck Institute for Mathematics in the Sciences in Leipzig, Germany, where he obtained his Ph.D. in 2014. From there he went into teaching, ultimately becoming a full professor at New York University’s Courant Institute of Mathematical Sciences. A few years later, he returned to advanced computing through his work with artificial intelligence. He started with DeepMind in London, where he was under contract as a consultant, and then worked with Anthropic and OpenAI as an independent adviser without contracts or nondisclosure agreements. More on that later.
One particular mathematical question has been an intellectual guide throughout his career: the famous Navier–Stokes problem. Buckmaster called it his North Star.
He explained that he had not spent his career directly attempting to solve that enormous problem. It was too big a problem. Instead, he pursued smaller, more accessible questions that might eventually lead toward it. This is often how important research progresses. Scientists investigate questions they can reasonably approach while keeping larger objectives in view.
Buckmaster was not primarily motivated by the Millennium Prize money. In fact, he did not expect the problem to be solved for many years. In his interview with Greene, he explained that his ambition had always been to participate in the history of solving the problem, not necessarily to be the person who delivered the final answer. He just hoped to live long enough to see the solution. As it turns out, he accomplished his ambition, although not in the way he expected.
To appreciate what happened, we need to understand the mathematical question. Fortunately, that does not require learning differential equations, although the basic ideas behind such calculations are straightforward.
The Million-Dollar Mathematical Question
The equations developed by Navier and Stokes are among the most useful in science. They describe the motion of fluids, including water and air, and help engineers design airplanes, predict weather, and understand blood circulation.
To oversimplify, think of the Navier–Stokes equations as the mathematical engine inside a flight simulator. Feed the equations information about air pressure, velocity, and other conditions, and they calculate what the air should do next. The Millennium Prize question asks whether that mathematical engine can ever crash. Could perfectly smooth starting conditions cause the fluid’s calculated velocity to grow to infinity in a finite time?
If that happened, the airplane would not crash. The calculation would. Real air cannot move infinitely fast. An infinite answer would mean that the equations had reached the limit of their smooth mathematical description and could no longer tell us what happened next.
That matters because Navier–Stokes is one of the foundations of fluid science. Discovering where the equations can fail tells scientists what the model can and cannot guarantee. It may eventually lead to a better understanding of turbulence, extreme vortices, and other difficult fluid behavior. For now, however, this is primarily a historic result in pure mathematics, not a discovery that makes existing airplanes unsafe or weather forecasts useless.
For roughly ninety years, nobody could prove whether such a mathematical breakdown was possible. On September 8, 2026, OpenAI announced that its AI agents had found one. The agents designed a smooth external force that drives the calculated fluid into a singularity.
The Clay rules allowed the Millennium Prize problem to be resolved with one such example. OpenAI did not have to show that ordinary air or water behaves this way. It needed only to prove that the equations permit such a breakdown under some mathematically valid conditions. The illustration below shows the external force and vortex used in its construction.
OpenAI’s result leaves a harder and more natural question unanswered: could the equations break down on their own, without a specially designed outside force? The proposed proof does not answer that question. Clay’s rules did not require it, so the proof could still resolve the Millennium Prize problem. But the distinction helps explain why this is historic mathematics rather than an immediate discovery about airplanes, weather, or other real fluids.
Physicist Amin Doostmohammadi of the Niels Bohr Institute at the University of Copenhagen put the point well in an October 14, 2026, editorial in Communications Physics, The Navier–Stokes Equations Blow Up, and Planes Still Fly:
For pure mathematics, this could be a major event. For physics, it changes much less. …
Knowing that an idealized equation can reach a singularity is useful. What matters physically is what happens near that region and whether real systems ever get there.
Real fluids consist of molecules. Mathematical descriptions treating them as perfectly continuous substances eventually cease to apply at very small scales. The proposed singularity does not demonstrate that actual fluids reach infinite velocity, nor does it make existing engineering applications unreliable. But the mathematical structure may someday help scientists understand extreme vortices, turbulence, or other difficult physical conditions. That is how mathematical and scientific research often works: practical value can emerge later and in unexpected places.
What interests me most is how Buckmaster and AI worked together to help solve this extraordinarily difficult mathematical puzzle.
The Human Ideas Behind the Breakthrough
Great mathematical discoveries seldom appear from nowhere. Researchers build upon earlier ideas, recognize unexpected connections, and combine established knowledge in new ways.
Two mathematicians, Diego Córdoba and Luis Martínez-Zoroa, developed an approach for constructing mathematical singularities through increasingly concentrated disturbances. Their work supplied essential building blocks for the subsequent breakthroughs. Buckmaster recognized the potential of their approach and began investigating how it could be extended. He is unusually emphatic about giving these researchers credit.
In his September 8, 2026, public statement, Buckmaster expressly identifies Córdoba and Martínez-Zoroa as the originators of the research program. He even argues that Martínez-Zoroa deserves consideration for a Fields Medal, one of mathematics’ highest honors.
The AI systems therefore entered a research program already shaped by human discoveries. Their contribution was to extend that work with new mathematical ideas.
Greene asked Buckmaster directly whether AI had produced unexpected ideas that he considered creative. Buckmaster answered emphatically: “Yes, absolutely.” Buckmaster sees AI as participating in a comparable creative process.
Buckmaster and Alpöge: A New Way of Doing Mathematics
Buckmaster had been experimenting with AI for years.
An early research project involving an undergraduate student and Antarctic ice sheets introduced him to using neural networks to explore complicated mathematical models. He later collaborated with Google DeepMind on related research involving extreme fluid behavior.
Those investigations did not solve the Millennium Prize problem. But they demonstrated that machine learning could help researchers explore mathematical possibilities that would otherwise demand enormous human effort.
More capable language models then began generating mathematical proofs, creating an entirely different opportunity for Buckmaster’s research.
Around fall 2025, Buckmaster began working closely with Levent Alpöge, a mathematician employed by Anthropic. Their research was a personal collaboration, not an official Anthropic corporate project. Alpöge brought mathematical expertise and something else of enormous importance: a sophisticated system for coordinating AI agents to investigate mathematical problems. Buckmaster strongly emphasizes this contribution. He and Alpöge became quite proficient at supervising and directing agents in mathematical research.
In Numberphile interview, Buckmaster described his first meeting with OpenAI after the 10,000-agent discovery. Marketing personnel initially leading the meeting claimed that someone had simply pasted the famous problem into a prompt and pressed Enter. Buckmaster rejected that account as ridiculous. He knew better. The OpenAI mathematicians and technical experts then took over and admitted he was right: the agents had first investigated related problems and received additional prompts. OpenAI’s published account now describes a far more complex hybrid procedure involving multiple groups of agents, enormous computational resources, and continuing direction by human experts.
Yes, the OpenAI mathematicians were always there supervising, guiding the hordes of agents they had “unleashed” or, as Buckmaster likes to say, “spawned.” They selected different versions of the problem, assigned a related Euler problem as an intermediate step, supplied the resulting Euler solution to the Navier–Stokes agents, encouraged different groups to explore different approaches, and used Codex to consolidate the strongest intermediate results. The agents then explored possibilities and contributed ideas the humans could not investigate nearly as quickly, if at all, on their own.
This was an advanced version of what I have long called the Hybrid method: knowledgeable humans directing, learning from, and working with artificial intelligence.
Instead of asking one AI system to answer a question, researchers can ask many agents to investigate the same question or assign different questions to different agents. Some approaches may fail while others succeed. Promising results can guide additional investigations and assignments. The central innovation is not merely running more AI systems. It is organizing them to conduct a productive investigation under knowledgeable human direction.
I will not dwell here on the controversy over OpenAI’s alleged copying and secret use of the work of Buckmaster and Alpöge. The press has focused on the dispute, a real tempest in a teapot (excuse the turbulent analogy). If you want to learn more, see Buckmaster’s September 8 public statement and the video interview below with Professor Buckmaster, conducted by physicist Brian Greene.
After describing the controversy, Buckmaster concludes at 1:11:18 to 1:11:33 of the World Science Festival video that the important thing is:
We had gone beyond human intellect and reached this Deep Blue Kasparov moment. And it was time as a community to think what that means and the implications of that. I think the implications go well beyond mathematics!
The Deep Blue Moment: Human and Machine Creativity
After wading through the agents’ messy work, Buckmaster recognized a genuinely clever mathematical connection. It was a very human kind of creative insight, an aha moment he had never before seen from generative AI. His analysis of the proof remained preliminary, but he regarded the connection as a genuine conceptual advance.
Buckmaster compared the historical significance of OpenAI’s accomplishment to IBM’s Deep Blue defeating world chess champion Garry Kasparov in 1997. Kasparov’s defeat did not end human chess. Grandmasters continued competing; in fact, chess is now more popular than ever. The specialized chess engines developed since then have become valuable tools for studying and improving the game. I go to Chess.com every day for fun, especially for the chess puzzles and AI tutoring.
The Deep Blue Moment for agentic AI does not establish that an AI system experiences mathematical understanding exactly as a human does, nor does it establish consciousness or wisdom. It does establish that AI can contribute genuinely new mathematical ideas and, on some of the hardest tasks in the field, outperform the best unaided human mathematicians. That is what I mean by superintelligence in one field, Mathematics. The claim is domain-specific, not a claim of universal competence or general superintelligence, as Part Two of this series will explain.
Deep Blue did not become generally intelligent by defeating Kasparov. Likewise, the new mathematical systems remain jagged and fallible, but their achievement is broader than a chess victory. Coordinated frontier models are producing original work across difficult areas of mathematics, formalizing some results, and accelerating research at a speed and scale no human team can match. The rest of this series examines the limits and consequences without denying what has happened.
Had the Millennium Problem Finally Been Solved?
On September 11, the Clay Mathematics Institute issued an announcement acknowledging that the Navier–Stokes problem had apparently been settled. The Institute emphasized that evaluating the proof and determining appropriate credit would proceed through its established process. That process is deliberately unhurried. OpenAI has published a proposed proof and its Lean formalization, but final mathematical acceptance and any official prize determination require further review. OpenAI has also stated that it does not intend to claim the million-dollar prize.
Whatever the final review determines, the larger development is already significant. AI systems have contributed to constructing and formally verifying mathematical arguments that previously lay beyond the reach of human researchers working alone. But what that achievement ultimately means depends on more than the validity of a difficult proof.
Near the end of his World Science Festival interview, Greene asked Buckmaster how long the recent sequence of mathematical advances might have taken through more traditional research methods. Buckmaster estimated that the work could have required five to ten years.
That was his informed judgment, not a measurable scientific finding. Nobody can know precisely how an alternative history of mathematical research would have unfolded. Nevertheless, it illustrates the acceleration Buckmaster believes he has witnessed.
Buckmaster finds these events inspiring. So should we. Buckmaster has neither rejected artificial intelligence nor surrendered his intellectual responsibilities to it. Neither should we. Buckmaster has developed exceptional expertise in his domain of mathematics. He has recognized promising ideas from earlier researchers, collaborated with Alpöge, a mathematician with deep AI expertise, and learned to direct increasingly capable AI systems. We should do the same in our legal domain. Buckmaster’s experience shows that human intellectual abilities can be enlarged through sophisticated collaboration with artificial intelligence. We should do the same in law.
From One Breakthrough to a Flood
Buckmaster’s story began with one extraordinary proof. It did not remain a single story for long. On October 6, OpenAI released 722 mathematical manuscripts at once. Days later, additional claims arrived in mathematics, computer science, and theoretical physics. The bottleneck was no longer only finding proofs. It was deciding which results were sound, what they meant, who deserved credit, which questions mattered, and whether human institutions could absorb the flood.
That is where Part Two begins. The evidence of mathematical superintelligence becomes harder to dismiss as the volume and range of AI-generated work expand. The human task also becomes clearer. The next installment examines proof versus understanding, education, attribution, corporate power, and the struggle to keep human independence and judgment at the center.
Ralph Losey Copyright 2026. All Rights Reserved
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