Navier–Stokes, AI and maths integrity

WHEN OpenAI announced on September 8, 2026, that it had solved the Navier–Stokes smoothness problem, the news rent the social media space. Within days of the announcement, the claim generated extensive discussion across X (formerly Twitter) and LinkedIn. Scientists were not surprised by the hullabaloo that greeted the announcement. The Navier–Stokes smoothness problem is one of the seven famous Millennium Prize Problems set by the Clay Mathematics Institute. It has stumped mathematicians for nearly a century and carries a $1 million reward. The equations are classified under fluid mechanics, widely considered one of the most notoriously difficult branches of physics because they compel scientists to track an infinite number of moving, changing points that constantly interfere with each other. Added to the difficulty are the equations’ non-linearity, which means that a fluid’s current motion directly alters its future motion, and turbulence (a characteristic that fluids exhibit). Speaking of turbulence, a famous quip attributed to Werner Heisenberg, the Nobel Prize-winning pioneer of quantum mechanics, captures the difficulty of understanding the phenomenon: “When I meet God, I am going to ask him two questions: Why relativity? And why turbulence? I really believe he will have an answer for the first.” The joke underscores how intractable turbulence has proved to be, suggesting (humorously) that even God might lack a clear answer as to why turbulence exists.

The equations are named after French engineer and physicist Claude-Louis Navier and British physicist George Gabriel Stokes, who developed them in the 19th century. The equations apply Newton’s second law of motion as a momentum balance to a moving, continuous fluid, accounting for pressure, viscosity and, where relevant, forces such as gravity. Despite the lack of a proof of Navier–Stokes smoothness, science and engineering continue to use the equations. Climate modellers find the equations useful in forecasting climate and weather. Astrophysicists rely heavily on them to understand the formation of stars, galaxies and planets, and the interaction of fluids with gravity. In aerospace and automotive engineering, engineers employ them to design aircraft and race cars.

Until OpenAI’s announcement, the equations (useful as they have been, and despite having stood the test of time) had yet to be proven incapable of ‘blowing up.’ In fluid mechanics, blowing up means that, at some point in finite time, the equations predict a fluid speed that becomes infinite. At such a singularity, the equations cease to provide a meaningful description of the fluid’s motion. Put another way, the Navier–Stokes problem concerns whether the equations always yield smooth, finite solutions, or whether they can develop singularities: points at which the predicted fluid motion becomes unbounded, and the equations cease to hold. OpenAI apparently debunked the smoothness hypothesis by demonstrating that the equations could blow up, although the result is still pending peer review. The demonstration was carried out over 88 hours by coordinating around 10,000 AI agents in a ‘swarm compute’ effort. Although OpenAI described the computing cost as amounting to millions of dollars, Karthik Duraisamy of the University of Michigan estimated its retail-price equivalent at approximately $6 million for the output alone and about $10 million in total.

Following OpenAI’s announcement, leading mathematicians, in an open letter signed by 26 Fields Medalists, read the riot act about OpenAI’s modus operandi in solving mathematical problems. They argued in the letter, “A Severe Misalignment of AI in Mathematics,” that the practice of benchmarking AI models by deploying them on celebrated mathematical problems is detrimental to the science of mathematics because “solving problems is only a tool and proxy for achieving the primary goal of conceptual understanding and insight.” The mathematicians further warned that intellectual work would be imperilled if the construction of proofs without human-like comprehension (a widely recognised characteristic of AI in mathematics) were to become the new normal.

Their concerns are valid. However, before the mathematical community calls on AI companies to alter their practices, it must first put its own house in order. Prior to AI, the mathematical community had been inundated with accusations relating to research integrity, exclusion and elitism, publication fraud and metrics gaming, among others. Perhaps Grigori Perelman, the reclusive Russian mathematician who proved the Poincaré conjecture (also a Millennium Prize Problem), is an apposite illustration. Between 2002 and 2003, Perelman posted three preprints of his solution on arXiv, circulated summaries of his work to mathematicians working in the same field and presented his work in lectures, but he never submitted the papers to a journal. He subsequently declined the Fields Medal in 2006, despite entreaties from Sir John M. Ball, the then President of the International Mathematical Union, and in 2010 he also refused the Clay Mathematics Institute’s $1 million prize. Although Perelman did not attribute his rejection of the prize to a 2006 paper by Huai-Dong Cao and Xi-Ping Zhu (a paper whose claim to present the first complete written proof of the Poincaré conjecture was widely seen as diminishing Perelman’s contribution), the episode may have reinforced his disillusionment with the mathematical community.

When Sylvia Nasar and David Gruber interviewed Perelman in 2006, he expressed profound disquiet about the mathematical community, stating, “It is not people who break ethical standards who are regarded as aliens. It is people like me who are isolated.” He bluntly said the Fields Medal “was completely irrelevant for me,” adding that “everybody understood that if the proof is correct then no other recognition is needed.” Perelman’s further damning remarks that “there are many mathematicians who are more or less honest. But almost all of them are conformists. They are more or less honest, but they tolerate those who are not honest” reflect the distrust he might harbour towards the mathematical community. It is glaring that some cunning mathematicians continue to exploit ways of earning credit for original contributions.

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•Olaniyi holds a Master’s of Science in Energy Policy.

While producing an original proof is straightforward to understand, the second route—identifying a significant gap in a researcher’s proof—has sometimes been misconstrued as merely “filling gaps in exposition.” In mathematics, making another person’s proof more explicit or readable is exposition. It does not qualify as an identification of a genuine mathematical gap (such as a missing or mistaken argument), and therefore cannot be a claim to originality. The boundary between clarifying a proof (exposition) and finding or fixing a significant flaw (identifying a mathematical gap) can be fluid, but an expert merely filling in standard, routine calculations or algebraic manipulations that the original author omitted for the sake of brevity is doing mathematical exposition, as long as the underlying logic, strategy and conceptual breakthroughs of the original author are fully intact.

A reform the scientific community desperately needs is to give professional credit to colleagues who audit, verify or write expositions of existing research. Those who verify complex proofs that may run to hundreds of pages and require months or years of scrutiny can be rewarded through career advancement, tenure or monetary compensation. While rewards remain heavily skewed towards original, groundbreaking results alone, peer review of highly complex papers will continue to weaken, high-profile errors will occasionally slip into published journals, and the temptation to dress up exposition as discovery will persist.

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Rewarding exposition and verification would hugely reduce the unscrupulous, subterranean attempts at achieving recognition. It would also answer the Fields Medalists’ own concern. AI will produce more proofs, longer proofs and proofs that no single person fully understands. OpenAI’s proof of the Navier–Stokes smoothness problem alone runs to 166 pages, assembled by 10,000 agents, and it is only the beginning. Checking such work, and explaining it, is precisely where conceptual understanding and insight will be built. A profession that rewards verification and exposition will be ready for that future. One that rewards only priority will keep quarrelling over credit while the machines do construct the proofs. Therefore, before demanding that AI companies change a practice that may threaten intellectual work, the mathematical community must first cleanse itself of the Augean stables besetting it and safeguard the integrity of its own foundations.