Finn Brooks: Two thousand dollars.
Finn Brooks: That's — wait, I need to just say that again — two thousand dollars. Total. To resolve ten open problems in mathematics.
Finn Brooks: Not one. Ten.
Finn Brooks: August 1st, 2026, OpenAI drops a paper called "Ten advances in mathematics and theoretical computer science," and that number just won't leave me alone.
Finn Brooks: Two grand at API rates.
Finn Brooks: And that's OpenAI's own figure, by the way. Not some back-of-the-napkin estimate somebody ran on Twitter.
Finn Brooks: The model is Astra — an internal build, hasn't been publicly released — and what it produced includes… okay, the first-ever explicit construction of a non-sofic group. A problem that Mikhail Gromov basically opened in 1999 when he introduced soficity in the first place, and that just… sat there for twenty-seven years. Completely unsolved.
Finn Brooks: Plus a disproof of Alain Connes's rigidity conjecture on von Neumann algebras.
Finn Brooks: Plus three problems from the Paul Erdős catalogue — specifically problems 146, 180, and 183, where 183 is on multicoloured Ramsey numbers.
Finn Brooks: Noam Brown — OpenAI research scientist, works on reasoning and reinforcement learning — called this "a major step for scientific reasoning."
Finn Brooks: And every single one of these results comes with a Lean 4 machine-checkable certificate and a 249-page manuscript, which means anyone can run the verification themselves. No trust required, no waiting on peer review — the proof checker either accepts it or it doesn't.
Finn Brooks: Here's the actual tension though: the Lean 4 certificates are real and independently runnable — that part's not in dispute — but the $2,000 figure, and exactly where Astra ends and human scaffolding begins… those two things nobody outside OpenAI can check.
Finn Brooks: Lean 4 is important — and I want to be precise here, because this is the part that actually changes the game.
Finn Brooks: A Lean 4 certificate means the proof is encoded in machine-readable code and a proof checker verifies it line by line — not a human, not a peer reviewer who maybe skimmed section three, an automated system that either accepts the logic or it doesn't.
Finn Brooks: You can run it yourself.
Finn Brooks: That's genuinely new — like, the trust model just shifted.
Finn Brooks: It's not 'believe OpenAI,' it's 'run the checker.' Which is a real thing.
Finn Brooks: But — hang on — there's a ceiling to that, and I don't think it gets talked about enough.
Finn Brooks: Lean 4 verifies that the formal proof is internally consistent. It does NOT verify that the formal statement you fed it actually captures what mathematicians meant by the problem.
Finn Brooks: That gap is everything.
Finn Brooks: And here's where it gets uncomfortable — OpenAI says it 'helped prepare the manuscripts and formalize the proofs in Lean.' Meaning the same organization that generated the math also decided what the math means when it gets encoded. That editorial boundary is completely unaudited.
Finn Brooks: Think about that. The non-sofic group problem — 27 years open since Gromov introduced soficity in 1999 — Astra constructs one, and the certificate says the logic is airtight. But did the formalization faithfully capture what the group theory community was actually asking? Nobody outside OpenAI can verify that.
Finn Brooks: Timothy Gowers — not some AI booster, not a doomer — publicly declined to sign the Leiden Declaration, which had over 3,000 mathematicians sign it. He didn't dismiss it. He engaged with it, wrestled with it. That tension he's sitting in is exactly this.
Finn Brooks: What does it mean when proofs are cheap and abundant but the question of whether you proved the RIGHT thing gets harder, not easier?
Finn Brooks: That's the proof-abundance paradox in real time — Astra knocks out Connes's rigidity conjecture, three Erdős problems, all of it, and the individual result doesn't deflate exactly, but the weight shifts to a question the Lean certificate cannot answer.
Finn Brooks: More solved. Less certain we solved the right version. That's — yeah, that's the thing I keep landing on.
Finn Brooks: And all of that — the proof abundance, the formalization gap, the trust model shifting — that's the backdrop. But the forward pressure, the actual thing to watch right now? It's Astra. Specifically the fact that Astra is still an internal build. Not released. Not reproducible.
Finn Brooks: Outside researchers cannot independently run the same model. Can't retrace the steps. Can't even begin to ask whether the formalization of, say, Ehrhart's volume conjecture or that sphere-packing upper bound — first improvement since 1978, by the way — actually captured what the community intended. The Lean 4 certificate tells you the logic is airtight. It does not tell you the question was encoded faithfully. And right now? Nobody outside OpenAI can check.
Finn Brooks: Which is where Sam Altman's Washington trip gets really interesting to me.
Finn Brooks: The same week a federal frontier-model review framework came due, Altman is in the room demoing Astra to policymakers. An unreleased model. Presenting discovery infrastructure to regulators before the mathematics community has independently reproduced a single one of these ten results. Think about that order of operations.
Finn Brooks: The policy conversation is moving faster than the reproducibility check. That's the gap.
Finn Brooks: And look — there's a prior here that makes this more complicated, not less. In May 2026, an unreleased OpenAI model generated a disproof of the Erdős unit-distance conjecture during internal testing. A week later, human researchers took that core proof technique and disproved another major conjecture with it. AI-to-human transfer, working. Pattern established. But — and this is the thing — it all happened on OpenAI's timeline. The community responded to what OpenAI chose to surface, when OpenAI chose to surface it.
Finn Brooks: The reproducibility test is the named event here. Not a vague 'we'll see.' A specific pass/fail: can an independent mathematician take one of these ten results — the non-sofic group construction, Connes, any of the three Erdős problems — and reproduce it by a different method, then confirm the Lean formalization was actually faithful to the intended problem? That's the check. If it passes, the trust model holds. If the formal statement silently shifted the goalposts? The $2,000 figure and the certificates start looking like marketing.
Finn Brooks: Now here's the move that blurs all of this — ChatGPT for Academic Researchers. OpenAI's program giving 100,000 scientists and mathematicians free access to their models, announced right alongside the ten proofs.
Finn Brooks: One read: that accelerates the reproducibility test. More mathematicians with access, more chances someone pokes a hole or confirms fidelity. Maybe. But the other read — the uncomfortable one — is that you're enrolling the mathematics community into OpenAI's infrastructure before the independence question is even settled. Before anyone outside has reproduced result one.
Finn Brooks: And if that enrollment happens first… the reproducibility test doesn't disappear, but who's running it, and on whose tools, gets a lot harder to answer cleanly.
Finn Brooks: And that enrollment question — I'm always coming back to that, actually.
Finn Brooks: A hundred thousand researchers handed access to ChatGPT models, right as the community is still trying to figure out whether Astra's formalizations were even faithful to the problems they claimed to solve.
Finn Brooks: That's not a small thing.
Finn Brooks: Because the independence of the reproducibility check depends on who's running it and on what tools — and if the tools are OpenAI's, that independence gets genuinely complicated.
Finn Brooks: Not impossible. Just complicated.
Finn Brooks: And look — I want to be honest about the objection to my own read here, because I think it's real.
Finn Brooks: More mathematicians with access genuinely does mean more eyes on the formalization gap — more chances someone spots whether the non-sofic group construction or the Connes disproof actually captured what the community was asking, or whether the encoding quietly shifted the goalposts.
Finn Brooks: That's a real upside.
Finn Brooks: But the Lean 4 certificate already did its job — it confirmed the logic is sound.
Finn Brooks: The question it cannot answer is what was proved.
