Fields Medalist Terence Tao — widely regarded as one of the best living mathematicians — has weighed in on the fluid dynamics results published today by NYU’s Tristan Buckmaster and Anthropic’s Levent Alpöge, calling the work a “remarkable achievement” and saying he sees no obvious obstacle to the same methods eventually being pushed all the way to Navier-Stokes, one of math’s seven Millennium Prize Problems.
Tao’s reaction matters because he isn’t just a bystander commenting on a viral math story — he’s one of the small number of people in the world qualified to actually evaluate whether this kind of claim holds up. His endorsement, even a measured one, is the closest thing this result has gotten to independent expert validation so far.
Catching up: what was claimed
As we covered earlier today, Buckmaster and Alpöge say they proved that several fluid equations — 3D incompressible Euler, Boussinesq, and incompressible porous media (IPM) — can develop a “blow-up” (a point where some physical quantity spirals to infinity in finite time) when a smooth external force is applied. This builds on a multi-year research program by mathematicians Diego Córdoba and Luis Martínez-Zoroa, who had previously shown blow-up was possible only with rougher, less physically realistic forcing. Getting the forcing term all the way down to smooth is the harder, more meaningful version of the result — and it’s what Buckmaster and Alpöge say they achieved, with heavy assistance from AI models, formally checked using the Lean proof-verification system.
Buckmaster’s public statement also included serious allegations that OpenAI attempted to pressure him during the release process — claims we detailed in our earlier piece, and which remain unconfirmed by OpenAI.
What Tao actually said
Tao confirmed the technical headline: Alpöge and Buckmaster took the Córdoba–Martínez-Zoroa approach and pushed it from a partial, “subcritical regularity” version of the forcing term all the way to a fully smooth one — something the earlier work by the same authors hadn’t managed. He noted the arguments have been formalized in Lean, which functions as a machine-checked guarantee that the logic holds together.
Notably, Tao said he spoke to Buckmaster directly by phone to have the ideas explained to him — a detail he pointedly contrasted with AI-mediated communication, calling the human conversation a refreshing change of pace. He also acknowledged that the proofs themselves rely heavily on AI-generated content, though he said the authors have spent recent weeks reworking that material into a more presentable form, and noted they were forced to publish earlier than they’d planned because of the external pressure described in Buckmaster’s statement.
How the proof actually works, in plain terms
Tao gave a genuinely useful sketch of the underlying strategy, which is worth unpacking for anyone without the relevant background.
Imagine you’re trying to build a fluid flow that spirals out of control. The Córdoba–Martínez-Zoroa method does this by repeatedly layering small, localized “high-frequency” ripples on top of a larger, smoother, “low-frequency” background flow — a bit like adding finer and finer detail to a sketch. This is philosophically similar to a well-known mathematical technique called convex integration, which has been used for years to build strange, non-intuitive solutions to fluid equations.
The twist, per Tao, is the direction the energy flows. Normally in these constructions there’s feedback in both directions between the coarse and fine layers. Here, there’s essentially none flowing back from the fine ripples into the coarse background. Instead, the coarse background does one job: it exponentially amplifies the fine ripples, boosting them until, at exactly the right moment, they become strong enough to take over the dynamics and set up the next round of amplification. Repeat that process enough times, faster and faster, and the result is a genuine finite-time blow-up.
Tao noted that for the Boussinesq equation (a simpler cousin of Euler often used as a testing ground), the amplitude and frequency of these ripples end up following a remarkably simple mathematical rule — but that turning that clean idea into an airtight proof still requires an enormous amount of technical bookkeeping, which is why the Boussinesq paper alone reportedly runs 76 pages even after the authors tried to trim it down.
Could this crack Navier-Stokes?
This is the question everyone actually wants answered, and Tao’s take is cautiously optimistic but not a prediction of imminent success. He said he sees nothing in the current method that would fundamentally block extending it to Navier-Stokes itself, and floated that there’s even a real possibility the forcing term could eventually be removed entirely — which would move the result meaningfully closer to the actual Millennium Prize problem, since the official version asks about unforced fluids.
But he was equally clear that a large number of technical difficulties still stand between here and there. His framing was that he wouldn’t be shocked if someone eventually powered through those difficulties with enough compute and AI assistance — but he was candid that grinding out that extension doesn’t particularly interest him personally. What he says he’s actually excited about is something more traditionally mathematical: digesting the new proof technique and figuring out what genuinely new ideas it contains, independent of whether it eventually bags the full prize.
The bigger picture
Tao’s reaction lands as a meaningful, if measured, validation of a story that’s been swirling with unverified claims and rumors of AI labs racing each other toward a Millennium Prize problem. He isn’t declaring Navier-Stokes solved, and he isn’t backing away from acknowledging how much AI shaped the work — but he is saying the mathematics itself looks real, sits on solid formal-verification footing, and represents a genuine advance on a hard, well-known program. He added that he’s looking forward to more talks and write-ups from the authors as the ideas get properly digested by the field — a fairly ordinary thing for a mathematician to say, and, in the middle of this week’s chaos, maybe the most normal part of the whole story.