OpenAI Astra Solves 10 Open Math Problems With Lean Proofs

OpenAI has put a new claim on the table for AI and mathematics. On August 1, 2026, the company published ten results in mathematics and theoretical computer science that it says were produced by an internal version of Astra — the model OpenAI describes as its next major release. Each result arrives with a Lean proof certificate, a machine-checkable file that anyone can verify with the open-source Lean 4 compiler, and OpenAI posted all ten to a public GitHub repository.

The framing is deliberate. Rather than lead with a benchmark score, OpenAI led with artifacts: new constructions, counterexamples, and bounds that, it says, working mathematicians had not been able to produce, each one paired with a formal proof that either compiles or doesn’t.

What Astra is said to have produced

According to OpenAI, every one of the ten problems had been open for at least a decade, and several for far longer. The headline result is the first explicit construction of a non-sofic group — a question that has stood in group theory since Mikhail Gromov introduced the notion of soficity in 1999, roughly 27 years ago. For most of that time, mathematicians could not even agree on whether non-sofic groups existed; producing one explicitly is the kind of result that, if it holds, rewrites a section of a textbook.

Two other results have drawn particular attention. One concerns the closest vector problem, a question that sits at the heart of lattice-based cryptography — the same mathematics underpinning several post-quantum cryptography schemes. Another improves a bound on sphere-packing density, pushing it down toward the Cohn–Elkies threshold, a well-studied limit in the geometry of packings. These are not toy problems dressed up for a press release; they are questions professional researchers recognize on sight.

OpenAI describes a specific workflow behind the results. The model first produced the mathematical arguments; humans then worked with the same model to turn those arguments into written manuscripts; and finally the model formalized each argument as a Lean certificate. Lean 4’s trusted kernel gives a binary verdict — the proof compiles and is therefore valid under its stated assumptions, or it fails. That is a meaningfully different bar from a model asserting an answer in prose that a reader has to check by hand.

Perhaps the most striking number is the price. OpenAI estimated the total token cost of finding all ten solutions at roughly $2,000 at its Sol API rates. If accurate, that figure reframes the economics of research-grade mathematics: the bottleneck OpenAI is implicitly claiming to have moved is not compute cost but capability.

Why the Lean angle matters

The choice to ship formal proofs is the story’s technical center of gravity. Large models have long been prone to confident, fluent, and wrong mathematical arguments — the hallucination problem in its most consequential form. A Lean certificate is designed to cut through exactly that. Because the compiler mechanically checks every inference against the axioms, a proof that compiles cannot contain a hidden logical gap of the sort that fools human referees.

That is genuinely valuable, and it is why the reaction was not immediate dismissal. Thomas Bloom, a University of Manchester mathematician who runs the erdosproblems.com catalog of open questions, called the results “big news” on X. Verification here does not require trusting OpenAI or holding a mathematics PhD; it requires installing Lean and running the files, which is precisely the point of publishing them.

This is not the first time OpenAI has pointed a model at hard mathematics — the company has been probing open problems with earlier systems — but the combination of decade-old questions, an explicit construction like the non-sofic group, and public formal certificates is a step up in ambition from anything it has shown before.

The caveats mathematicians are raising

The skepticism is specific, and it is worth taking seriously. The first objection is about what a Lean certificate actually guarantees. The compiler confirms that a proof of the formal statement written in Lean is valid. It does not confirm that the formal statement faithfully captures the open problem as the mathematical community understands it. If the Lean encoding quietly weakens a hypothesis or restates the question, a green checkmark can attach to a theorem that is true but not the theorem everyone cared about. Judging that alignment still requires a domain expert to read the formalization closely — and that work is only beginning.

The second objection is about disclosure. Critics have noted that the accompanying 249-page paper says little about how the model works, how the proofs were checked beyond the Lean pass, what role humans played at each step, or whether any of the model’s proposed proofs contained errors that were caught and discarded along the way. A selection effect — showing the ten that worked without saying how many were attempted — would change how impressive the results look.

The third is more philosophical. As several mathematicians have pointed out, some problems are open because they are genuinely hard, and others are open simply because they sit in a corner of the field that no one had a reason to visit. A decade without progress is consistent with both. Sorting the truly difficult results from the merely neglected ones will take time and expert attention. The AI researcher and critic Gary Marcus captured the split reaction in a post calling the release “amazing — but vastly oversold.”

There is also a naming question OpenAI has left open: it has not said whether Astra is GPT-6, a distinct product line, or a research system that may never ship in its current form. For now, “Astra” is a label attached to a set of proofs rather than a product anyone can use.

How this fits the broader model race

The release lands in a competitive environment where every frontier lab is trying to demonstrate not just fluent output but verifiable capability. The shift toward reasoning models — systems that trade speed for extended deliberation — has made benchmarks feel increasingly saturated and, to skeptics, gameable. Formal mathematics offers something benchmarks don’t: a result that is either correct or not, with no partial credit and no room for a leaderboard to be quietly overfit.

That is also why the release doubles as a statement about evaluation. If a lab can point to a machine-checked proof of a previously open problem, it sidesteps the usual argument that a model merely memorized its test set — you cannot memorize the answer to a question no one had solved. Expect rivals to feel pressure to produce comparable artifacts rather than headline numbers, and expect the community to scrutinize the Lean formalizations line by line before granting the claim.

What it means

If the ten results hold up under expert review — and that review will take weeks, not hours — this is one of the more consequential demonstrations yet that a large language model can contribute original, verifiable mathematics rather than merely reproduce known arguments. The non-sofic group construction alone, if it survives scrutiny, is the sort of result that gets its own seminar. The $2,000 compute figure is the part that should make researchers and lab budgets sit up: it implies that, for a growing class of problems, the scarce resource is no longer machine time.

But the caveats are not garnish. The gap between “the Lean file compiles” and “the model solved the problem the community posed” is exactly where this claim will be won or lost, and OpenAI’s thin disclosure about method and human involvement leaves that gap wider than it needs to be. The healthiest outcome is the one the Lean certificates invite: don’t take OpenAI’s word for it, and don’t take the critics’ word either — run the proofs, read the formalizations, and check whether the statements match the problems.

For the industry, the near-term signal is strategic. A lab that can turn a model loose on open research questions and return machine-checked answers has a new axis to compete on, one that is far harder to fake than a benchmark and far more legible to outsiders than a private eval. Whether Astra becomes a shipping product or stays a research demonstration, the bar for what “the next major model” is expected to do just moved — and it moved toward results that can be independently verified by anyone willing to run the compiler.