Fields Medalist Terence Tao has laid out a scenario in which one of the most famous open problems in mathematics gets “solved,” and mathematics is worse off for it.
In a six-part thread, Tao used the Navier-Stokes global regularity problem — a Millennium Prize-level question about whether the equations governing fluid flow can spiral into a singularity in finite time — to argue that an AI system solving a landmark problem is not automatically a win for the field. If the solution arrives as a sealed black box, with the AI company behind it keeping the path to the answer hidden, Tao says the “solve” could actually poison the problem as a source of future progress, rather than spark it.
It’s a notable turn in tone. Tao has for years been one of the more optimistic and hands-on mathematicians experimenting with AI tools, and he opens the thread by noting that Navier-Stokes regularity was, until recently, shaping up to be one of the best case studies for AI-assisted mathematics done right.
The problem nobody actually needs solved
Tao is careful to point out that almost nobody is losing sleep over the Navier-Stokes regularity problem for practical reasons. Computational fluid dynamics is already a mature engineering discipline, used daily in weather modeling and climate science, and its strengths and blind spots are well understood regardless of what the theoretical proof eventually says. A clean regularity result, or a concrete example of blow-up, would be intellectually satisfying but wouldn’t change how meteorologists forecast next week’s storm.
What actually justifies decades of effort on the problem, in Tao’s telling, is everything the attempts have produced along the way. He lists the Leray-Hopf weak solutions, the Gagliardo-Nirenberg-Ladyzhenskaya inequalities, the Prodi-Serrin partial regularity theorems, the Beale-Kato-Majda blow-up criterion, and the Escauriaza-Seregin-Šverák conditional regularity result as examples of foundational tools that emerged from people trying and failing to close out the problem. He also points to his own work connecting the equations to Turing-complete “fluid computation,” which unexpectedly opened a link to symplectic topology. The broader field of turbulence theory, he adds, has been shaped philosophically by these efforts even where there’s no direct technical link.
A plausible playbook for finding a blow-up
Tao says the mathematical community’s consensus has shifted: the expectation now is that Navier-Stokes regularity is false, and that specific initial conditions exist which blow up in finite time. He sketches out a four-step strategy that’s increasingly looking viable:
- Design a nearly self-similar template (“ansatz”) for what a finite-time blow-up solution should look like.
- Numerically locate an approximate solution that satisfies the ansatz up to a tiny, computable error term.
- Show that, in the right renormalized coordinates, the ansatz is stable around that numerical solution — meaning it could in principle be nudged into an exact solution if the error is small enough.
- Confirm the actual error falls inside the threshold the stability argument requires.
Every one of those steps, Tao says, is brutally complicated on its own, and they all depend on each other. Many candidate templates fail outright — some violate conservation of energy — while others only get ruled out after enormous amounts of computation.
That’s exactly the kind of workflow, Tao argues, where a combination of machine-learning-driven simulation, rigorous interval arithmetic, formal verification, and LLM-generated candidate ansätze — all steered by human mathematicians learning from each failed attempt — could plausibly crack the problem. He expects the resulting proof to be so complex that no human could check it by hand, with the Lean formalization potentially becoming one of the largest formal proof files ever produced.
Where it goes wrong
Tao’s concern isn’t the size or ugliness of the eventual proof — it’s who gets to see the road that led there. The value of the exercise, he argues, was never really going to be the final theorem. It’s the process: starting from one ansatz, discovering precisely why it fails, adjusting it, and iterating — all without knowing the destination in advance, since knowing the answer up front would kill the incentive to explore “dead ends” that often turn out to be instructive failures.
The scenario Tao is now worried about is one where an autonomous AI system, backed by massive compute, runs that entire iterative loop internally and privately, and simply hands over a finished ansatz and a completed proof — while the company running it keeps the actual search process out of public view. In that world, Tao says, one of the most famous open problems in mathematics gets checked off, but the field gains almost nothing from it. A third party could try to reverse-engineer some insight out of the finished proof after the fact, but Tao calls that a far less efficient path to the same understanding.
His broader point is about what pure mathematics problems are actually for. Unlike curing a disease or improving an engine’s efficiency, a problem like Navier-Stokes regularity isn’t usually pursued because mathematicians desperately want that specific answer — it’s pursued because the attempt tends to generate new tools and ideas that outlast the specific result. Solve it prematurely, opaquely, and without transparency into the method, Tao warns, and the exercise can flip from a net positive for the field into a net negative.
The backdrop: AI has been racking up real wins in math
Tao’s warning lands at a moment when AI is racking up a string of real, headline-grabbing wins in math. Google DeepMind’s AlphaProof Nexus autonomously cracked nine previously unsolved problems from a famous list of open questions posed by the mathematician Paul Erdős — two of which had stumped mathematicians for 56 years — for just a few hundred dollars of compute each, with every step double-checked by Lean, software that verifies a proof’s logic line by line. Harmonic’s Aristotle claimed a roughly 30-year-old problem from the same list, though mathematicians cooled on the achievement once it came out that it was the easier of two related problems Erdős had actually posed. OpenAI’s GPT-5.2 was credited with independently solving one such problem, and separately with cracking another that had sat open for 44 years — a result Tao himself called “perhaps the most unambiguous instance” of AI solving an open math problem to date, even after it turned out someone had found a similar answer decades earlier using different methods.
Not every headline has held up, though, which is really Tao’s whole point. OpenAI’s earlier model, GPT-5, was publicly called out by Google DeepMind’s CEO Demis Hassabis after it turned out the model hadn’t actually solved several of those open problems at all — it had just found existing answers buried online that even the person maintaining the official problem list hadn’t come across. OpenAI’s next model, Astra, later claimed 10 solved problems and this time backed every one with a machine-verified proof, around the same stretch of time that Anthropic’s Claude Fable 5 was credited with helping disprove an 87-year-old conjecture about a certain type of mathematical function, by finding a single, fairly short counterexample. Add in gold-medal-level scores at the 2025 International Math Olympiad from both OpenAI and Google — a competition long thought to be out of AI’s reach — and it’s easy to see why Tao keeps circling back to transparency. Several of these “breakthroughs” got walked back or corrected within days of being announced, and that kind of public back-and-forth is exactly the scrutiny a closed, black-box proof would never have to survive.
Tao’s evolving read on AI in math
The Navier-Stokes thread doesn’t come out of nowhere. Tao has been tracking, in public, how fast his own estimates of AI’s mathematical ability have had to be revised. As we have covered, Tao described AI’s mathematical ability as comparable to a “mediocre, but not completely incompetent” graduate student as recently as September 2024. By early 2026, at a conference held at UCLA’s Institute for Pure and Applied Mathematics, his assessment had shifted enough that he declared AI ready for primetime in math and theoretical physics, on the grounds that it now saves more time than it wastes. The same piece notes his view that as AI drives down the cost of routine problem-solving, the scarce skill for a mathematician becomes picking the right problem, designing the workflow, and checking the output — a different job description than most current PhD programs are built to train for, which is part of why Tao has separately argued that graduate math education needs a rethink.
Taken together with the Navier-Stokes thread, the throughline in Tao’s recent commentary is consistent: he’s not arguing AI can’t or shouldn’t do serious mathematics — by his own account it increasingly can. His concern is narrower and more structural — that the way a result is produced and disclosed determines whether it advances the field or just closes a scoreboard entry. A landmark proof produced transparently, with humans able to trace the failed attempts and dead ends along the way, extends mathematics. The same proof produced inside a closed, proprietary AI harness, Tao argues, risks being mathematically “solved” while leaving the field no further ahead than before.