What OpenAI Astra is
Astra is the model OpenAI describes as its next major one. The results came from an internal version of it, which is not generally available. What was published is the output, not the model.
The announcement at a glance
Today, we are sharing a selection of ten results, each of which resolves or makes substantial progress on a long-standing open problem. ／ The results were achieved by an internal version of Astra, our next major model. The total number of tokens needed to find solutions to these problems would cost roughly $2,000 at Sol API rates. These arguments were then prepared into manuscripts by humans with the same model. Afterward, the model formalized each argument in a Lean certificate ／ We are also releasing for each solution a model's narration of its thinking process. — From the introduction and "The results" section, covering what was achieved, what the search cost, and what was released
The ten results cut across fields
The breakdown is not concentrated in one speciality. It spans high-dimensional geometry, coding theory, arithmetic circuit complexity, group theory, operator algebras, quantum complexity, lattice cryptography, and extremal combinatorics.
The ten results (condensed from the official list)
| Problem | Result obtained |
|---|---|
| High-dimensional sphere packing | Upper bound on packing density improved to the Cohn–Elkies threshold |
| Binary and spherical codes | Exponentially stronger upper bounds on maximum code size at a given minimum distance |
| Non-sofic groups | A construction establishing their existence, answering a central open question in group theory |
| Connes's rigidity conjecture | Disproof of the conjecture that certain groups are uniquely determined by their von Neumann algebras |
| Arithmetic circuit complexity | New lower bounds for computing the permanent; an n⁴/log n bound for arithmetic formulas |
| Quantum parallel repetition | An exponential parallel repetition theorem for general two-player quantum games |
| Closest vector problem | Hardness of approximation to within polynomial factors, on a lattice problem underpinning post-quantum cryptography |
| Ehrhart's volume conjecture | Maximum volume of a convex body whose only interior lattice point is its centroid, settled in all dimensions |
| Multicolour Ramsey numbers | A superexponential lower bound for multicolour triangle Ramsey numbers (resolving Erdős problem 183) |
| Extremal number conjectures | The compactness and degeneracy conjectures in extremal graph theory (resolving Erdős problems 146 and 180) |
These problems span high-dimensional geometry, coding theory, arithmetic circuit complexity, group theory, operator algebras, quantum complexity, lattice cryptography and extremal combinatorics. ／ Non-sofic groups. A construction establishing the existence of non-sofic groups, addressing a central open question in group theory. ／ Connes's rigidity conjecture. Disproof of a longstanding conjecture that certain groups are uniquely determined by their von Neumann algebras. ／ Multicolor Ramsey numbers. A superexponential lower bound for multicolor triangle Ramsey numbers, resolving Erdős problem 183. ／ Extremal number conjectures. Results on the compactness and degeneracy conjectures in extremal graph theory, resolving Erdős problems 146 and 180. — From the list of fields and four of the entries in the table above; the remaining six appear in the same list
It follows on from the May disproof
This did not arrive out of nowhere. In May 2026 OpenAI published a disproof of the Erdős unit-distance conjecture, discovered while evaluating an unreleased model. The company says that work has already inspired further developments in mathematics and theoretical computer science, and its footnote names five follow-on papers.
Wider access for researchers sits in the background. Under ChatGPT for Academic Researchers, OpenAI is giving 100,000 scientists and mathematicians free access to its best models, and it says it continues to evaluate models on open research problems during development. We cover that programme in OpenAI's free access for 100,000 researchers.
That is why we recently announced ChatGPT for Academic Researchers, an initiative providing 100,000 scientists and mathematicians with free access to our best ChatGPT models. We also continue to evaluate our models on open research problems during development. ／ In May, we shared an AI-generated disproof of the Erdős unit-distance conjecture, discovered while evaluating an unreleased model. This work has already inspired further developments in mathematics and theoretical computer science ／ Subsequent research includes Bloom, Sawin, Schildkraut, and Zhelezov, "The sum-product conjecture is false for real numbers"; Pohoata, "Split primes and the Elekes-Rónyai problem"; Saha, Xu, and Ye, "Furthest Pair Requires Quadratic Time in Superconstant Dimension under SETH"; Goh and Hatami, "Communication complexity of point-line incidences over the reals"; and Lee, Pohoata, and Zhu, "The Minkowski grid has robustly many repeated distances." — From the researcher programme, the line from the May disproof, and the five follow-on papers listed in the footnote
The proofs are machine-checked in Lean
What separates this announcement from earlier ones is that the claims come with proofs in a machine-checkable form. That part can actually be inspected, so we pulled the published material and looked.
Measured from the published Lean proofs (August 4, 2026 — main branch at 94bc0fe, counted by our editorial team)
| Item | Measured |
|---|---|
| Main formalisation files | 10 (one per result) |
| Total lines across those 10 | 548,215 |
Unproved holes (sorry) in those 10 | 0 |
Added axioms (axiom) in those 10 | 0 |
| Licence | Apache License 2.0 |
| Toolchain | Lean 4.32.0 / mathlib (Lean's standard mathematics library) / Lake (Lean's build tool) |
This repository contains Lean 4 formalizations of the results presented in ／ The project uses Lean 4.32.0, mathlib, and Lake. ／ Apache License — From the repository's stated purpose, its build environment, and its licence
No unproved holes and no added axioms
Reading a formalised proof, two things get checked first. Whether any sorry — the marker for an unproved step — remains, and whether any axiom, a statement assumed without proof, has been quietly added. Park the inconvenient part as an axiom and you can "prove" anything.
Cloning the repository and counting mechanically across the ten main files, there was not a single sorry or axiom. The line count totals 548,215. That is not a volume a person writes.
One caution. It is not true that the repository as a whole has zero sorry. A separate directory holding challenge files for independent verification contains 42 of them. Those are not gaps left unfilled; they are deliberate, for the third-party checking described below. Read a secondary summary saying only "the repository has no sorry" and you lose that distinction.
Challenges for independent verification ship with it
The other thing not to overlook is that a route for cross-checking against a different implementation is built in from the start. The repository includes challenge files for a checking tool called Comparator, and its instructions tell you to install landrun, lean4export, and nanoda_bin.
That nanoda is an independent checker written in Rust, separate from Lean's official kernel. OpenAI is, in other words, supplying the path for someone else's implementation to verify its proofs. And in the same week, a bug in nanoda itself became an issue. See why an AI "disproof" of the Collatz conjecture was invalid for that. If you want to check a published proof file against your own copy, laying the differences side by side is where to start.
For instructions on checking the formalizations with Comparator, see the…／Install `landrun`, `lean4export`, and `nanoda_bin`, and make them available on `PATH`. — From the independent-checking instructions and the Comparator challenge readme
nanoda is an independent kernel (aka proof/type checker) for Lean implemented in Rust by Chris Bailey . ／ The original Collatz repository also passed a week-old version of nanoda , the main external checker. — Cited for nanoda being an independent Rust implementation, and for the trouble on nanoda's own side during the same period
OpenAI's position on authorship
Alongside the technical result, what drew attention was the statement about whose achievement this is. It answers head-on whether an AI-generated proof can be published as a human paper.
No claim of human authorship
The position is unambiguous. Claiming human authorship for a proof generated entirely by an AI system would misrepresent both the system's contribution and the nature of genuine human intellectual work. The principle is that attribution should honestly reflect how a result was actually produced.
The division of labour is then spelled out. OpenAI helped prepare the manuscripts and formalise the proofs in Lean, and takes responsibility for their correctness, while the mathematical arguments themselves were generated by the system. As a template for how to write up AI-assisted work, it is a reference point.
We believe attribution should honestly reflect how a result was produced: claiming human authorship for a proof generated entirely by an AI system would misrepresent both the system's contribution and the nature of genuine human intellectual work. We helped prepare the manuscripts and formalize the proofs in Lean, and we take responsibility for their correctness, while the mathematical arguments themselves were generated by our system. — From the passage on authorship and where responsibility sits
It also addresses those who are worried
OpenAI writes that systems capable of contributing to mathematical research raise questions that cannot be answered by a technology company alone. It goes on to express deep respect and understanding for those concerned with the impact, including the signers of the Leiden declaration on AI and Mathematics.
To the mathematical community it puts a request: engage deeply with the results, place them in context, and bring the ideas behind them to life through new research. Not treating the results as something to throw over the wall is written into the announcement itself.
The emergence of systems capable of contributing to mathematical research raises questions that cannot be answered by a technology company alone. There are many views as to the role of AI in mathematics, and we have deep respect and understanding for those concerned with its impact, including the signers of the Leiden declaration on AI and Mathematics ／ We hope the mathematical community will engage deeply with these results, place them in context, and bring the ideas behind them to life through new research and discovery. — From the passage on how the company positions itself toward the mathematical community
Conclusion: how to read "machine-checked"
Three things stand out in this announcement: the scale — ten open problems, publication in a verifiable form as Lean proof files, and a statement on authorship that does not misrepresent the AI's contribution. The second in particular differs in kind from previous "AI solved X" announcements. Anyone can put the claim through a machine on their own hardware.
But reading "machine-checked" as a blanket guarantee would be premature. The guarantee rides on the quality of the checker's implementation. In the same week as the Astra announcement, a soundness bug in Lean's own kernel was reported and fixed, and it emerged that an invalid "proof" exploiting it had been accepted at one point. Which is exactly why shipping a route to cross-check with an independent implementation matters.
A soundness bug in the Lean kernel ( #14576 ) was reported and fixed during the week of July 27. ／ On July 25, Ramana Kumar published a repository containing a sorry -free "disproof" of the Collatz conjecture, produced with AI assistance. It is not a valid proof because it exploits a bug in the kernel's handling of nested inductive types. — Cited for the kernel soundness bug reported and fixed the same week, and for the invalid "proof" that exploited it
The practical reading is this. This result builds trust through procedure, not through a number. Fetch it, run it in your own environment, put it through a second implementation as well. Only then have you checked anything. AI moving into mathematics is a line that runs on from Fable 5's attempt at the Jacobian conjecture, and for the foreseeable future the discipline of verification looks like the more interesting argument than the claims themselves.
