Ten open maths problems fell overnight, for two thousand dollars
Andrew Wiles wept after seven years of isolation solving Fermat. Keep that image, because it now belongs to the past.
Look at Andrew Wiles's face as he breaks down in tears.
That is what a person looks like when they finally defeat a problem that resisted the greatest minds for three and a half centuries. Seven years of struggle and isolation, nearly losing his mind, compressed into those tears.
Keep that image.
What happened
On 1 August 2026, OpenAI announced that its forthcoming model Astra had produced solutions to ten open problems in mathematics and theoretical computer science, each unsolved for ten years or more.
They span group theory, von Neumann algebras, high-dimensional geometry, quantum complexity, lattice cryptography and extremal combinatorics.
The headline result: an explicit construction of a non-sofic group, a question open since Mikhail Gromov introduced the notion of soficity in 1999.
The last number is the important one
This gets skipped in most coverage, and it is everything.
The solutions were not presented as claims. They came with proof certificates in Lean 4, published on GitHub under Apache 2.0.
The repository's sorry count is zero. Meaning not one inferential step is unproven. Anyone wanting to check does not need to trust anybody; they run the files through the compiler, which either accepts them or does not.
That changes the nature of the argument entirely. We are no longer discussing a model that asserts. We are discussing a proof that is machine-checked.
And another figure deserves a pause: two thousand dollars. Not a million, not a hundred thousand. Less compute than a month's salary.
What this means for you
If you research: machine-checkable proof tooling has stopped being a luxury. Learning Lean or something like it is now an investment in your career.
If you study: the part that used to be measured in years of isolation can be accelerated. What remains is choosing the question, which was always the harder half.
If you are outside the field: take the general lesson. When output becomes automatically verifiable, the last argument against relying on these systems collapses. The question becomes: which other fields tolerate verification that strict?
In closing
Wiles's tears may have been the last breath of romance in science. But that two thousand dollar bill did not only solve ten problems. It ended the human monopoly on the thrill of mathematical discovery.
And honestly: I do not know whether that is something to celebrate or to mourn. I suspect it is both.
Common questions
- What is the Astra model?
- A forthcoming OpenAI model not yet publicly released. On 1 August 2026 the company announced it had produced solutions to ten mathematics problems open for a decade or more.
- How do we know the solutions are correct?
- They ship with proof certificates in Lean 4, and the Lean compiler either accepts a proof or rejects it, with no room for opinion. The repository has zero unproven steps.
- What did it cost?
- Roughly two thousand dollars in compute for all ten solutions, according to OpenAI.
- Does this mean mathematicians are replaced?
- No. What accelerated is producing the proof. What remains is choosing which question is worth asking, which was always the harder part.
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