Zero Sorries, One Big Asterisk: Inside OpenAI's Ten Machine-Checked Proofs

Column Overview

On August 1, 2026, OpenAI put out a manuscript that would have sounded like fiction two years ago: an unreleased internal build of its next model, code-named Astra, generated fully machine-checked Lean 4 proofs for ten problems that had sat unsolved for a decade or more, spanning group theory, operator algebras, sphere packing, circuit complexity, and Ramsey theory. The 249-page document ships with a GitHub repository of proof certificates carrying an Apache 2.0 license, and the detail everyone is repeating is that the "sorry" count in the Lean code is zero — meaning no step was left as an unproven placeholder, no gap was papered over with an assumption. Every line was checked by a proof assistant that does not care about reputation, hype cycles, or who wrote the paper.

What Was Actually Solved

The headline result is an explicit construction of a non-sofic group, closing a question Mikhail Gromov posed when he defined soficity back in 1999. That single item would already be a career-defining result for a human mathematician. Alongside it sits a counterexample to Connes's rigidity conjecture on group von Neumann algebras, a subfield where progress has historically come in decades, not months. The manuscript also reports improved asymptotic upper bounds for sphere packing in high dimensions — a problem with a strange public profile, since the eight- and twenty-four-dimensional cases became famous after Maryna Viazovska's celebrated work, while the general high-dimensional asymptotics have remained comparatively unglamorous and stubborn. Rounding out the ten are additional results in circuit complexity and Ramsey theory, two areas where combinatorial explosion has historically made computer-assisted search either infeasible or unconvincing when it worked.

What makes this list different from the usual "AI solves math problem" press cycle is the verification layer. Lean doesn't grade on a curve. A Lean-checked proof either compiles against the formal statement or it doesn't, and there's no reviewer to charm, no advisor to reassure, no committee to satisfy. That's a meaningfully different claim than "the model produced a proof sketch that experts found plausible," which is closer to what earlier AI math announcements amounted to.

The Reception Split

The response from working mathematicians has been genuinely divided, and not along the lines you'd expect from a typical AI-hype story. Timothy Gowers, who has a Fields Medal and a long track record of skepticism toward inflated AI claims, said he would recommend one of the ten proofs for a top journal without hesitation — a real endorsement from someone who does not hand those out casually. Thomas Bloom, who curates the Erdos problems catalogue at the University of Manchester and therefore has as good a sense as anyone of what "unsolved" actually means in practice, called the results "big news."

But enthusiasm isn't unanimous, and the skepticism is worth taking seriously rather than waving off as reflexive gatekeeping. Steven Miller at Yeshiva University has accused OpenAI of effectively drawing on his own published research without attribution in producing these results — a complaint that lands differently than generic "AI ate my content" grievances, because it's coming from inside the specific mathematical community whose decades of published lemmas and techniques are exactly the training substrate a model like this would need. And the timing isn't neutral: the Leiden Declaration, now endorsed by the International Mathematical Union, has already put a marker down warning that AI companies are drawing on published mathematical research without consent, sidestepping peer review, and putting pressure on the norms of attribution and proof that the field has run on for a very long time. This announcement arrives right into the middle of that argument, not before it.

The $2,000 Question

OpenAI's own framing leans hard on cost: roughly $2,000 in compute, at current API rates, to produce ten proofs that took the mathematical community collectively over a hundred person-years to fail to solve. It's a striking number, and it's designed to be striking. But the figure only covers the successful runs — it says nothing about how many attempts didn't pan out, how many candidate proofs were generated and discarded, or how much compute went into problems that never made it into the manuscript at all. Critics have been quick to point out that this makes $2,000 a cost of publication, not a cost of discovery. It's the equivalent of a company reporting only its profitable trades: true as far as it goes, but not the number you'd want if you were trying to estimate the actual economics of the process.

There's also a reproducibility wrinkle that's easy to miss amid the excitement. Nobody outside OpenAI can run Astra — it's an unreleased internal build, not a shipped product. So the proofs themselves are independently checkable, because Lean is open and the certificates are public. But the process that produced them is not independently repeatable by anyone else. That's an unusual epistemic position: total transparency at the level of the output, total opacity at the level of the method. It's worth sitting with that distinction, because "verified" and "reproducible" are doing different jobs here, and conflating them is exactly how an accurate press release turns into an inflated public narrative.

It's also notable what this announcement doesn't claim. Earlier OpenAI math announcements have sometimes come bundled with broad claims of external review or expert validation across the board; this one doesn't make that move. The endorsements here — Gowers, Bloom — read more like individual reactions than an orchestrated review process, and OpenAI seems to be leaning on Lean's formal verification to do the credibility work instead. That's arguably more honest, but it also means the community's informal vetting, the kind that normally happens through seminars, referee reports, and years of people trying to poke holes in a result, hasn't happened yet for the mathematical substance even though the formal logic has already passed its check.

Reading This Against the Rest of the Week

None of this happened in a vacuum. In the same stretch of days, OpenAI cut pricing on GPT-5.6 "Luna" by roughly 80%, down to $0.20 per million input tokens, and ChatGPT reportedly crossed something like a billion weekly active users. The EU AI Act's transparency and labeling requirements also took effect on August 2. Put those pieces next to the math announcement and a pattern comes into focus: a company simultaneously racing on price and scale for its consumer product, while trying to plant a flag on the hardest possible terrain — pure mathematics — where "hallucination" isn't just an annoyance but a category error, because a wrong proof simply fails to compile.

That's precisely why the Lean verification detail matters more than the $2,000 figure or the user-count milestone. Pricing wars and usage numbers are business metrics that can be spun in either direction. A zero "sorry" count in a formal proof checker is not spinnable in the same way — it's either true or the repository doesn't compile. The interesting tension is that this hardest, most verifiable claim is also the one arriving with the least reproducibility, wrapped in the most contested attribution questions, at the exact moment the mathematical community is organizing around the Leiden Declaration to push back on how AI labs use published research. Whether Astra represents a genuine phase change in mathematical discovery or a very well-chosen highlight reel from a much larger and messier search process is a question the next year of scrutiny, not this one manuscript, will have to answer.