Clay printed seven problems in 2000 and put a million dollars under each of them, which is a very human way of saying: we have been stuck here long enough that money has become a joke we tell ourselves. One of the seven was Navier-Stokes. The equations are from the nineteenth century. They describe water, air, the curl behind a boat. Mathematicians could write them on a blackboard in a minute and then spend a career not knowing whether a smooth splash in three dimensions stays smooth forever.
I have been thinking about that blackboard this week, because the rumor is that a language model finally moved the chalk. Maybe it did. The file is not on my desk. What I do have, and what this piece is actually about, is a shorter, stranger history: the last few years, in which models that were trained to continue text started returning objects that had not existed before. Not summaries of Wikipedia. New constructions. New bounds. Proofs a machine would accept. A kernel that now runs in a data centre.
Humanity's hardest problems are a messy category. Some of them are on Clay's poster. Some of them are a combinatorist's private obsession. Some of them are a multiplication you have been doing the same way since 1969. I want to walk through the ones that actually came back, in the order they came back, because the week only makes sense if you can see the trail that led to the lake.
A grid with no three in line
Terence Tao once called the cap set problem his favourite open question, which from Tao is not small talk. You are looking for the largest set of points in a high-dimensional grid such that no three lie on a line. It is the sort of question that sounds like a puzzle for a rainy afternoon until you learn that brute force, past a certain size, exceeds the number of atoms you have to count with.
In December 2023, DeepMind published FunSearch. The name is almost unserious. The method is not. A language model, PaLM 2, was asked to write little programs that might construct a cap set. A second, unromantic piece of software ran those programs and threw away the ones that failed. The survivors went back into the prompt. Evolution, with a chatbot as the mutation operator.
What came out, in some dimensions, were the largest cap sets anyone had found. The authors called it the first time a large language model had produced a new piece of verifiable knowledge on a notorious scientific problem. Jordan Ellenberg, who had already given the area one of its important bounds, looked at the programs and said he learned something. Not from a list of points. From the code that built them.
There is a second, quieter trophy in the same paper: better rules for packing bins, the problem behind shipping containers and the way a data centre decides where to put a job. FunSearch did not "solve packing." It found heuristics that used fewer bins than the ones people had been using on well-studied distributions. I like this pairing. One object is a jewel in combinatorics. The other is a slightly tighter warehouse. Both are new. Both can be checked by running them.
If you only remember one mechanical fact from the last three years of model-and-math, remember this one. The model proposes. Something that cannot be sweet-talked decides whether the proposal is real. Every later success is a variation on that refusal.
The exam they built to find us
The International Mathematical Olympiad is not a Clay problem. It is worse, in a different direction. Six questions, two days, four and a half hours a session, written for the strongest pre-university mathematicians on earth. Countries send their best. Most contestants do not take gold. A handful, some years, take all forty-two points.
For a long time this was a way of hunting for humans. Then it became a way of hunting for whether a machine could do the kind of mathematics that does not have a multiple-choice back of the book.
In 2024, in Bath, DeepMind's AlphaProof and AlphaGeometry 2 solved four of the six problems and scored 28, silver. They did not do it in English. Specialists translated the questions into Lean and into a geometry language; the systems searched; the answers came back as objects a proof assistant would sign. It took days. Problem six, the one almost nobody touched, fell to AlphaProof. I still find that detail more moving than the medal colour. The hardest question in the room, solved in a formal language, by a system that had trained itself the way AlphaZero trained itself on games: play, check, play again.
A year later the translation step dropped out. An advanced Gemini, running a Deep Think mode that explores several lines at once, wrote natural-language proofs inside the contest clock and scored 35 out of 42. Official IMO graders marked it gold. OpenAI, the same summer, reported the same score from a general-purpose reasoning model, marked by three former medalists rather than the official desk. Five problems each. Problem six still standing, just, like a last human hill.
By 2026 the hill was gone. Several systems, some through official grading, some through independent runs, were reported at a perfect 42. The exam that was supposed to sort the species had become a timing and cost story. That is not the same as having understood fluids. It is the sound of a lock opening on a door we had used to define talent.
I do not know what we will use next to find mathematicians. I know we will need something. The olympiad was never the work. It was the entrance.
Forty-nine
In 1969, Volker Strassen showed you could multiply two 2 by 2 matrices with seven multiplications instead of eight, and the world of algebraic complexity has been arguing about the bill ever since. For 4 by 4 complex matrices, 49 scalar multiplications (Strassen, used twice) sat there as the thing to beat, in that particular setting, for a very long time.
In May 2025, DeepMind's AlphaEvolve, an evolutionary coding agent sitting on Gemini models, found 48.
If you are not an algebraist this will not make your hands shake. If you are, someone just moved a number that had been furniture. AlphaTensor, an earlier DeepMind system, had already hunted matrix multiplication and found gains in binary arithmetic. AlphaEvolve found this one in the complex case, and it did it as a side effect of a more general machine: propose whole programs, score them automatically, keep the ones that win.
The same system, pointed at more than fifty open problems in analysis, geometry, combinatorics, number theory, rediscovered the known best in about three quarters of them and improved the known best in about a fifth. One of the improvements was a kissing-number construction in eleven dimensions: 593 non-overlapping spheres touching a central one. Newton and Gregory quarrelled about the three-dimensional version in 1694. The quarrel about higher dimensions has been running, with gaps, for three centuries. AlphaEvolve put a new lower bound on the table you can look at.
Then, because Google is Google, the same loop went to work on the company's own plumbing. A scheduling heuristic that recovered, on average, 0.7 percent of worldwide Borg compute. A Verilog rewrite in a matrix-multiplication circuit, accepted into a future TPU. A 23 percent speedup on a Gemini kernel that shaved a percent off training time. Up to 32.5 percent on a FlashAttention implementation at the level of GPU instructions humans usually will not touch. In 2026 DeepMind said the agent had become ordinary infrastructure: cache policy in two days that used to take months, circuits for a quantum processor with far lower error than the handmade baseline, work with Tao on Erdős problems.
I am aware that 0.7 percent does not belong on Clay's poster. I am also aware that Clay's poster never paid the electricity bill. Some of humanity's toughest problems are "does this proof exist." Some of them are "can we keep doing this at all." The models have been at both.
The lake
Navier-Stokes is the one on the poster that looks like weather. Clay asked for existence and smoothness, or for a breakdown: do the equations that govern a viscous incompressible fluid, in three dimensions, with smooth data, stay smooth, or can they explode in finite time. Charles Fefferman wrote four official versions. Two of them set the external force to zero and ask the solution to last forever. Two of them allow a smooth force and ask you to exhibit a crash.
The Euler equations are the same family with the viscosity turned off. They are not on the prize list. They are, if anything, ruder.
This week Tristan Buckmaster and Levent Alpöge put three Lean-verified results on the table: finite-time blow-up, under a smooth force, for incompressible porous media, for Boussinesq, and for three-dimensional incompressible Euler. They had been walking a path opened by Diego Córdoba and Luis Martínez-Zoroa, who had already made blow-ups with rougher forcing. The language models (Claude, Codex, GPT-5.6 Sol, and Astra for the writeups) were how two people pushed that path onto smooth forcing and onto Euler in a year that would, a decade ago, have been a career.
Buckmaster said the first generated proof he was sent was the most horrendous he had ever read. They checked it in Lean on 22 August anyway. I have sat with that sentence longer than with any launch post this year. The object was ugly. The object was true. The species that used to require beauty as a proxy for truth now has a second proxy, and it compiles.
Around the same days, OpenAI told Buckmaster that an internal model had produced a manuscript, about a hundred pages, of forced Navier-Stokes blow-up, the Clay breakdown options with a smooth force. I have not seen it. Clay's page still says unsolved. If that manuscript is what they say it is, it belongs at the end of this history, not in a footnote. If it is not, the Lean files for Euler are already the most important fluid-dynamics objects a language model has helped into the world. I can live in that uncertainty. The lake has moved either way.
What is still on the wall
Riemann's hypothesis is still on the wall. Yang-Mills. The remaining Clay problems. Unforced Navier-Stokes, the version where nobody is allowed to push the fluid from outside. P versus NP, which is the poster behind the poster. Most of the Erdős list. Almost all of the questions a working mathematician would write down if you asked, without warning, what they were actually stuck on.
The models have not become Hilbert. They have become a new kind of junior colleague: tireless, uneven, sometimes horrendous on the page, and occasionally in possession of a construction nobody in the room had.
I used to think the story would be that a chatbot, one afternoon, typed a proof of something famous and we all went home. That was a science-fiction shape I had borrowed. The real shape, looking backward from tonight, is slower and more physical. Someone writes a skeleton. A model fills it. A checker that cannot be flattered throws most of it away. Once in a while a cap set gets larger, a multiplication count drops by one, a sphere packing in eleven dimensions grows, an olympiad script hits forty-two, a fluid blows up in Lean, a data centre gets a fraction of itself back.
The problems that were supposed to outlast us have not all fallen. A few of them have started returning objects we can hold. That was the thing I did not know how to imagine in 2020, when the same family of models was still losing to grade-school arithmetic in public. I do not know which object comes back next. I know the method that has been working, and I know it does not care whether the problem is on a poster or in a warehouse.
If you have a file from this week I should have opened, send it. The history is still being written, which is a ridiculous sentence, and also, for the first time in my life, a literal one.