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A group of French scientists sees OpenAI unjustly claiming it has solved the mathematical Navier-Stokes problem and sounds the alarm. A handful of tech moguls are establishing a private monopoly over cognition and knowledge.
« Reality considered partially unfolds in its own general unity as a pseudo-world apart, an object of mere contemplation. The specialization of images of the world is fulfilled in the world of the autonomous image, where deceit has deceived itself. The spectacle in general, as the concrete inversion of life, is the autonomous movement of non-life. »
Guy Debord, The Society of the Spectacle
It is now time to offer a critical synthesis on generative and agentic artificial intelligence, on the social project underlying its development, and on what it does to the university, to the sciences, and to democracy. The companies developing these sociotechnical systems aim to establish a private monopoly over cognition and knowledge, built on the plunder of the entirety of humanity’s cultural, textual, and visual output.
Unlike every technology that has previously claimed it would upend the whole of social structures, the narratives accompanying the promotion of AI rest not on a promise of human progress and prosperity, but on a dystopia already in motion. It is obvious that the development of AI answers no real, concrete need of humanity – unlike the need to insulate buildings, expand the use of decarbonized energy, and curb the collapse of living systems.
Given this absence of any use value, the apparatus of propaganda, addiction, and attention-capture – in which AI excels, as X and Grok demonstrate – rests on the demand that we turn a nuisance into a need, adapt to it, and help relay its supposedly inevitable rise. The progressive social contract gives way here to a contract of allegiance, which promises its collaborators survival amid the coming collapse. These narratives aim to hypnotize by trapping us in a single alternative: board the moving train, or be crushed. Consequently, any critical discourse built on an apocalyptic nightmare narrative ends up, despite itself, reinforcing the techno-deterministic story that Big Tech intends to impose.
The publication of an AI-assisted preprint concerning a certain problem posed by the Navier-Stokes equations has led to media coverage that may horrify anyone who has spent time on these questions.
We affirm the necessity of opposing Big Tech’s propaganda with a scientific counter-narrative. In the first part (below), we offer a scientific perspective on the problem posed around the Navier-Stokes equations in question. This problem is difficult to understand, and our efforts at explanation will likely not be enough to show the gulf between OpenAI’s output, lacking any scientific integrity, and the true nature of scientific work; at the very least, we hope this exercise illustrates one of the great dangers of generative AI: a particular form of alienation consisting in the illusion of understanding through the mere act of deferring to it.
This properly scientific counter-narrative is not meant to cast doubt on the problem-solving capabilities of generative and agentic AI. That kind of denial – one painfully recalls the sneering over ChatGPT’s arithmetic errors – has wasted the time the academic community needed for thought and coordination, and more broadly that of everyone whose work and profession rest on knowledge, cognition, and truth-telling. Rather, it is meant to show that what is at stake here is not science, and that the nature of this important moment for humanity is in no way a « Deep Blue-Kasparov » moment for the sciences – if one remembers the moment chess players were definitively outcompeted by computers. There was indeed a moment when it became clear that machines would definitively surpass even the best-trained humans at chess. Even though there is a game aspect in mathematics and a mathematical dimension of chess, mathematics is of a fundamentally different nature – and that is precisely the point we need to unpack. What we are dealing with in Open-AI’s mathematics-based branding is a moment of propaganda, exemplifying a whole economy of attention, and, underlying it, a total eclipse of reason. We have here evidence of the predatory, destructive intent of Big Tech’s leadership, and our current endeavor consists in turning this moment into a wake-up call.
It has become clear that Big Tech has the capacity to drain mathematics dry by brute-forcing formal proofs tailored to a large number of problems formalized by the scientific community. This cognitive extractivism proceeds through the plunder of the scientific literature and of dissertation topics posted online, but also through the predation of ideas that researchers – reduced to the role of mere « prompters » – hand over out of addiction or naivety. But solving a mathematical problem is not an end in itself: it is a beacon, a marker of new conceptual understanding. The problems given to young mathematicians in training serve above all to teach them to travel within the discipline and to formulate new questions, rather than adding new theorems to the stock of already proven ones. By « solving » these problems on an assembly line, AI companies aim to destroy this geography of the discipline – the very thing that enables that travel, and therefore our capacity to identify research’s new frontiers. What goes on with the « GPT-solves-Navier-Stokes » event – and will be analyzed in details in the next section – is rather the obfuscation of mathematical understanding by the delivery of innumerable proven theorems.
Mathematics cannot be reduced to the technical force that turns an idea into a proof, nor to the mass production of « True/False » statements. Proof is indeed an important component of mathematics, but it requires time for analysis and contextualization – time that algorithmic haste destroys, raising serious questions of attribution and plagiarism along the way. Attempts, detours, verification, integration, transmission of ideas, collective work – these are all precious components. It is this long and difficult human process that turns a raw idea into knowledge the community can absorb and that can ultimately be taught to a student. Today, the mere rumor of an emerging line of research can trigger a brute-force attack by predatory AI companies, threatening to destroy centuries of « open science » tradition by pushing researchers to hide their work. However, a purely extractivist solution, without understanding of its own process or integration into the chain of human transmission, has no scientific value; it amounts to a final answer that has forgotten why the question was worth asking in the first place. The case of Navier-Stokes « solved » by GPT illustrates it perfectly well, as we will show.
For many reasons pertaining to the general political situation as well as to years of neoliberal research policies in various OECD countries, scientific research and the university are now confronted with their own contradictions. The GPT-Navier-Stokes episodes therein functions as a wake-up call to engage in this reflection and open a conversation in academia overall. Time has therefore come for collectively discussing the meaning and role of university education and scientific research. The sciences are not abstractions existing outside society; they rest on a craft-like dimension, on the social and human transmission of knowledge and methods. In turn, technologies do not evolve autonomously, inevitably or neutrally. Technologies such as generative and agentic AI likewise do not exist outside society. AI is, above all, a sociotechnical apparatus, the development of which directly serves the economic, political, and social interests of the industry’s corporate leadership. It is therefore essential that each of us answer, for ourselves, not the question « Should we board on the train? » – the answer to which is obviously no – but rather: what kind of world do these companies intend to produce? And what kind of world do we want to see come into being, one in which the sciences as a common good and the university as a pillar of democracy find their rightful place?
The Navier-Stokes moment : A stand-alone result is not genuine science.
Drawing on the conceptual edifice built by seventeenth-century scholars, from Kepler to Newton by way of Hooke, to formulate universal gravitation and mathematize the motion of the planets, generations of physicists joined forces to model the complexity of our environment in the same way, and in particular the evolution of fluid bodies (air, water, magma, honey…). The first equations proposed by Euler in 1755 had an important limit: they neglected the viscosity of fluids, i.e., the property that explains why honey flows more slowly than water down a tilted spoon. As d’Alembert pointed out, this omission prevented the model from accounting for essential physical phenomena such as drag (which slows a car or a cyclist) and lift (which allows birds and airplanes alike to fly). Over the course of the eighteenth and nineteenth centuries, several scientists – among them Bernoulli, Navier, Stokes, and Saint-Venant – gradually refined Euler’s model. The names of Navier and Stokes remain attached today to the equations of mathematical physics used to capture these viscous fluids. These equations describe the fluid’s velocity field, a quantity depending upon both space (position) and time. They govern the dynamics of an incompressible viscous fluid, setting aside for instance the propagation of acoustic waves, and are used in numerical simulations as well as to interpret experiments and observations of natural phenomena. Then, it is possible to add a term to the Navier-Stokes equations whose only reality is that of a mathematical abstraction – this is called a forcing term.
Even so, like any mathematical object, these equations allow for a purely mathematical study, entirely independent of the equations’ usefulness to the physical sciences and to engineers. Thus, at the start of the twentieth century, mathematicians examined three fundamental properties of such a set of equations and its solutions – existence, uniqueness, and stability, which are the hallmarks of what Hadamard (1902) called a well-posed problem.
(Existence) Can we be sure that these solutions actually exist? One difficulty is that the nature of the Navier-Stokes equations does not allow for a general formula providing an explicit, analytical solution.
(Uniqueness) Can we be sure that these solutions are unique –that is, that for a given initial configuration of the fluid, there is only one solution to these equations?
(Stability) Does an initial configuration close to a given configuration lead to a nearby solution?
In 1934, Jean Leray showed that if the fluid’s initial velocity has finite energy, then a finite-energy solution exists for all later times. Although Leray left the question of global uniqueness open, he nonetheless managed to prove that, under additional regularity assumptions on the initial data (and on any external forcing), a unique, stable solution exists – but it exists only for a short time. Yet, his theorem says nothing about the solution’s later behavior: does it lose its uniqueness? Does it diverge locally to infinity? Since then, many mathematicians have tackled these questions of uniqueness and finite-time singularities, which are two related but distinct issues.
Let’s define these difficult concepts.
Stability. Stand a broom upright on its bristle end. The vertical equilibrium position exists but is unstable. The tiniest deviation from that equilibrium grows until the broom falls. The same holds in fluid mechanics: the fact that a solution to the equations exists does not mean it occurs in reality; in many situations, infinitesimal (arbitrarily small) perturbations grow and lead to an entirely different flow.
Finite-time singularity. Consider a ping-pong ball bouncing on a table, making its very particular noise: the ball suddenly stops (in finite time) after the frequency of the bounces has grown greater and greater (it is said to diverge). Now consider a drop forming at the spout of a badly closed tap. A thread of water stretches, forming a neck (or a « waist », if you prefer) between the tap and the small accumulation of water that will become the drop, at the point where the thread’s diameter is smallest. This minimal diameter shrinks over time until the precise instant it reaches zero. At that point there is no longer any continuity between the drop, which has detached, and the small bead of water remaining at the tap’s spout. This is called a finite-time singularity. In physics, near the singularities of equations, something other than what the equation predicts always happens. For the ping-pong ball, one must account for its deformation, its roughness, and so on. For drop formation, the thinning only continues until a neck one molecule wide remains. One last example: the collapse of a massive star at the end of its life under gravity, as predicted by Einstein’s equations of general relativity, leads to a regime where quantum effects become significant, and the result is not a singularity but the formation of a black hole.
Now, notice that providing the answer to these questions of regularity, uniqueness, and stability for the Navier-Stokes equations can in no way be presented as « solving » these equations. Twentieth- and twenty-first-century fluid mechanics is full of such « solutions », measured against experiments aimed at understanding every aspect of the flow of liquids and gases. To name just a few examples of natural phenomena: the turbulent flow of rivers, breaking waves, falling raindrops, the flight of birds, the swimming of fish, tornadoes, the formation of heat domes, the propagation of sound… Physicists and fluid mechanics researchers often rely on elegant simplification methods that overlap in part with those used in mathematics to exhibit finite-time singularities: these are known as self-similar solutions and asymptotic methods.
Let’s define a self-similar solution using an example. Place a drop of honey on a plate; it spreads out until it forms a thin puddle. Its height decreases over time while its radius increases, but its shape remains the same once you abstract away questions of size. By photographing the honey dome from the side, one can produce the image of the dome at a later moment by stretching it horizontally and compressing it vertically. The thinning of the water thread before a drop detaches satisfies this same property of invariant shape (though of varying spatial dimensions). This is called a self-similar asymptotic solution: it is identical once rescaled.
From this point of view, the regularity of the Navier-Stokes equations appears as a non-problem, with the whole art of the physicist’s approach lying in adapting the equations to the regimes in which they are applied. To take just one example, one far more fundamental to humanity than AI: the equations used to describe atmospheric or oceanic dynamics – dynamics that are fundamental to climate – require adding to Navier-Stokes an equation for internal energy (or, equivalently, for temperature). It must be stressed that reality, experimentation, measurement, and clever experimental settings remain paramount in most discoveries in this field of knowledge. Decades from now, we will still need fluid mechanics experiments to gain access to reality.
Let us return to the properly mathematical questions. In 1964, following on from Leray, Fujita and Kato established conditions guaranteeing global-in-time existence and uniqueness (requiring in particular sufficiently small initial data and forcing), as well as blow-up criteria. « Blowing up » here means that the peripheral velocity of vortices diverges to infinity while the size of the core tends to zero. As anyone can observe in a sink or bathtub, a flow converging toward the drain amplifies the rotation of the draining vortex.
This work was gradually refined, notably by Kato in 1984, then by Cannone, Meyer, and Planchon in 1994. Finally, in 2001, Koch and Tataru identified the minimal regularity to impose on the initial condition or forcing to guarantee existence and uniqueness of solutions over a short time, and for all time if the data and forcing are sufficiently small. Work over the past few decades has rested on the conjecture that it might be possible to exhibit blow-up self-similar solutions, drawing on vortex dynamics. While the possibility of a purely self-similar explosion was ruled out in a 1994 paper by Nečas, Růžička, and Šverák, mathematical solutions have since been sought close to these self-similar structures, so as to produce a reinforcement of this amplifying stretching, running counter to generic behavior. In the compressible case, such solutions – which are the exception rather than the rule (they are called pathological) – have been found in various settings, the most remarkable being the « implosion » constructed in 2023 by Merle, Raphaël, Rodnianski, and Szeftel. In parallel, work on non-uniqueness, such as that of Jia and Šverák in 2014, Guillod and Šverák in 2017, and Albritton, Brué, and Colombo in 2021, has pursued the same line of inquiry. Further advances have been made by studying slightly different, « simpler » equations, notably by Córdoba and Martínez-Zoroa in 2024.
Now, drawing inspiration from this research program, Buckmaster and Alpöge have in recent months turned to generative and agentic artificial intelligence to formalize solutions in models increasingly close to incompressible Navier-Stokes, starting with the Euler equations (the inviscid version of Navier-Stokes). OpenAI’s rank-and-file staff were then tasked by the company with pre-chewing the formalization of the problem so it could be fed to the AI, in order to produce the construction of an explosive Navier-Stokes solution with an ad hoc forcing term. The result turns out to be devoid of elegance, originality, and new ideas, but does amount to a technical feat. The proof was « verified » by having the same team produce ad hoc modules for the Lean proof assistant — modules that have yet to be shown to actually accomplish the required task.
Beyond the sense of weirdness, several remarks should be made, regarding the nature of what has been produced. The Open-AI preprint was not submitted for peer review: its sole purpose is to produce an effect of propaganda and stupefaction, relying on scientific illiteracy and on its uncritical amplification by the media echo chamber. To produce its propaganda effect, it erases the hundreds of researchers who have spent thousands of hours discussing ideas at a blackboard, publishing papers that contributed meaning to this line of questions – including ideas around axisymmetry and similarity solutions, building a forcing term that produces non-uniqueness, and studying neighboring models. This proof is a sabotaged, ethics-free technical step within a long-term collective undertaking. It does not replace past results but sits within a continuity. This preprint would have been rejected on account of its sheer appropriation of ideas and knowledge produced by a scientific community, which needs the long time of maturation and understanding. This preprint would also have been rejected simply because of its stated author (OpenAI), which erases all the company’s employees who contributed to formalizing the technical problem, as well as the plundering of human knowledge involved.
And last but not least, the preprint offers no perspective whatsoever, even though the interesting questions raised along this mathematical thread remain wide open. What of explosive solutions? Do they require an ad hoc, pathological forcing? Is the solution exhibited stable? Are there finite-time singularities in Navier-Stokes without forcing? Is the constructed forcing an isolated point in the space of possibilities? This preprint shows no interest whatsoever in the mathematical problem itself – rather, it almost disdains it, in favor of showing off GPT’s technical strength.
Now, let’s put the whole timing in its context, and it becomes more meaningful: the preprint Navier-Stokes paper by Open-AI amounts to exploiting the fetishization of prizes (the Clay Foundation’s money, here, but the same could be said of Fields Medals, Nobel Prizes, and a fortiori the Swedish central bank’s prize) for propaganda purposes. Indeed, far more than the Navier-Stokes equations, what worries Big Tech’s leadership is stock market valuation, the threat of a crash, and the public’s rejection of data centers as a midterm election issue.
Coming back to the science: if the problem could be well posed, it is also because it is only one among countless problems, most of which are not formalized. Thus turbulence – the name given to the flow regime exhibiting spatial and temporal fluctuations, a regime that is ubiquitous in the real world – is a far more important problem in hydrodynamics, the field in which Navier-Stokes equations do apply. But the questions raised by turbulence cannot be cast in a form eligible for a prize. In research, questions rarely pre-exist their resolution and formalization. Research is not an endeavor that can be reduced to mere technique.
Thus, one should emphasize that the proof constructed by exhibiting a finite-time Navier-Stokes singularity with ad hoc forcing has strictly no bearing whatsoever on either fluid mechanics or the real world. The technical proof provided by AI doesn’t have the slightest importance for physics.
By contrast, the paths taken by mathematicians during their research – the process, the errors, the intermediate results, and the tools built along the way – were, and are, necessary to the development of fluid mechanics, which relies on experiment and on the mathematical meta-language. In this predatory attack on mathematics by Big Tech, physics – which seems beyond AI’s reach insofar as it is grounded in the material world – stands entirely in solidarity with the mathematical community.
Ultimately, we stand far from any supposed « Deep Blue–Kasparov moment » for the sciences (as some wanted to describe the situation). In the end, all that remains of these automated productions is the bitter aftertaste of shoddy work, a reflection of the determination of AI’s pure players to « disrupt » mathematics through its inexorable enshittification. It is essential to expose the propaganda that presents Big Tech’s predation of research as inevitable. This means refusing to decouple scientific questions from Big Tech’s interests, even as the world of finance grows alarmed at the likely absence of any return on investment.
RogueESR (Enseignement Supérieur Recherche — higher education research) is a collective of academics that was founded in 2017 in a response to what they saw as a neoliberal assault in French universities.
