What the Astra model is and why it has no release date
Astra is the name OpenAI uses for its next major model, and the announcement of August 1, 2026 names no release date. The ten results came from an internal version of it. OpenAI published what the model produced: manuscripts, Lean certificates (proofs a computer can check) and a narration of the model's thinking for each solution.
What the announcement says about Astra
Astra is not the first step. In May 2026 OpenAI shared a disproof of the Erdős unit-distance conjecture that an AI system produced while OpenAI was testing an unreleased model. OpenAI says the disproof has already prompted further work, and the page's footnote lists five follow-on papers.
The same page mentions a program that gives 100,000 scientists and mathematicians free access to OpenAI's best models. ChatGPT for Academic Researchers: Free for 100,000 covers that program.
The results were achieved by an internal version of Astra, our next major model. ／ We are also releasing for each solution a model’s narration of its thinking process. ／ of the Erdős unit-distance conjecture, discovered while evaluating an unreleased model. ／ an initiative providing 100,000 scientists and mathematicians with free access to our best ChatGPT models. — From "The results" section and the introductionView official source →
Astra model math breakthrough: ten problems in eight fields
The ten problems fall in eight fields, and OpenAI says each result settles or greatly advances a problem that has stood open for years. The fields are coding theory, group theory, operator algebras, extremal combinatorics, quantum complexity, lattice cryptography, arithmetic circuit complexity and geometry in high dimensions. The table below lists the results in the order OpenAI gives them.
The ten results, as OpenAI lists them
| Problem | Result |
|---|---|
| High-dimensional sphere packing | Tighter upper limits on how densely spheres can be packed, reaching the Cohn–Elkies threshold |
| Binary and spherical codes | Exponentially better limits on how large a binary code can be for a given minimum distance, plus analogous results for spherical codes in high dimensions |
| Non-sofic groups | Shows that non-sofic groups exist, which answers a central open question in group theory |
| Connes's rigidity conjecture | Disproves the old conjecture that certain groups are pinned down by their von Neumann algebras |
| Arithmetic circuit complexity | New lower bounds on the cost of computing the permanent with circuits and formulas, among them a formula bound of order n^4/log n |
| Quantum parallel repetition | An exponential repetition theorem that covers two-player quantum games in general |
| Closest vector problem | Approximating it within polynomial factors is hard; it is a basic lattice question tied to post-quantum cryptography |
| Ehrhart's volume conjecture | The largest volume, in any dimension, of a convex body whose only interior lattice point is its centroid |
| Multicolor Ramsey numbers | A lower bound that grows faster than any exponential, which settles Erdős problem 183 |
| Extremal number conjectures | Work on the compactness and degeneracy conjectures from extremal graph theory, closing Erdős problems 146 and 180 |
Two entries are tied to named Erdős problems: Ramsey numbers settle problem 183, and the extremal number conjectures settle problems 146 and 180. OpenAI also says all ten problems are of substantial interest to their own mathematical communities, and several to mathematics as a whole.
These problems span high-dimensional geometry, coding theory, arithmetic circuit complexity, group theory, operator algebras, quantum complexity, lattice cryptography and extremal combinatorics. ／ A superexponential lower bound for multicolor triangle Ramsey numbers, resolving Erdős problem 183. ／ Results on the compactness and degeneracy conjectures in extremal graph theory, resolving Erdős problems 146 and 180. — From the introduction and two entries of the list in the table aboveView official source →
Ten proofs in Lean 4: what the math files contain
A Lean certificate is a proof written so strictly that a program called a proof checker can confirm every step. If the checker accepts the file, every step has been checked, whoever or whatever wrote it. OpenAI says its team helped prepare the manuscripts and formalize the proofs this way, while the arguments themselves came from the model.
The files are public in the openai/ten-proofs repository, licensed Apache-2.0. The project builds with Lean 4.32.0, mathlib (Lean's shared library of mathematics) and Lake (Lean's build tool).
Each result has its own file, named in the README beside the result. Examples are SpherePacking, ConnesRigidity, Permanent and GapCVP. With elan, the Lean installer, in place, two commands rebuild all ten proofs:
lake exe cache get
lake build All
A single result builds by passing its module name, for example lake build SpherePacking. For independent checking, the README has an "Independent proof checking" section that points to the ComparatorChallenges folder and its instructions for using Comparator. The repository page shows no counts of lines or unproved steps and no third-party verification results.
The project uses Lean 4.32.0, mathlib, and Lake. ／ To build an individual formalization, pass its module name to Lake ／ For instructions on checking the formalizations with Comparator, see the — From the build instructions and the independent-checking section of the READMEView official source →
Astra model cost: $2,000 in tokens and no benchmark scores
OpenAI puts the cost of the tokens used to find the solutions at roughly $2,000 at Sol API rates, and the announcement gives no benchmark scores. The figure is conditional: it prices the search at a stated rate, not money actually spent and not the cost of the whole research effort. The page does not say what Sol is.
The announcement page reports results on named open problems and no benchmark scores. It therefore has no leaderboard figure for comparing Astra with other models, and it gives no price for a future release.
The total number of tokens needed to find solutions to these problems would cost roughly $2,000 at Sol API rates. — From "The results" sectionView official source →
Astra model authorship: who gets credit, per OpenAI
OpenAI says credit should follow how a result was produced, so it will not claim human authorship for a proof its system generated. Its stated reason is that a human name on such a proof would misstate both the system's part and what real human intellectual work is. It divides the work this way:
- OpenAI helped prepare the manuscripts.
- OpenAI helped formalize the proofs in Lean and takes responsibility for their correctness.
- The mathematical arguments themselves came from OpenAI's system.
OpenAI also says that systems able to contribute to mathematical research raise questions a technology company cannot answer alone. It expresses respect and understanding for those concerned about AI's role in mathematics, including the signers of the Leiden declaration on AI and Mathematics. It hopes the mathematical community will engage with the results, place them in context and build new research on them.
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. ／ The emergence of systems capable of contributing to mathematical research raises questions that cannot be answered by a technology company alone. ／ 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 authorship and responsibilityView official source →
What the ten Astra proofs show, and what they leave open
The claim in this announcement is different in kind from a leaderboard number: the ten proofs can be rebuilt by anyone from a public repository. A Lean checker either accepts them or it does not. Astra itself is not among what was published, but its output is.
Three things are still missing from the announcement and repository pages: a release date, benchmark scores and verification results from outside parties. Until those appear, the ten results are best read as ten checkable proofs from an internal version of the model, not as a measure of how Astra compares with others. OpenAI's page also lists a related post dated Sep 8, 2026, titled "An OpenAI model proposes a solution to the Navier–Stokes problem".
FAQ
- What is OpenAI Astra, and when is it released?
- Astra is the name OpenAI gives its next major model. The ten results came from an internal version of it. The announcement is dated August 1, 2026 and names no release date. What OpenAI published is the output (manuscripts, Lean certificates and narrations of the model's thinking), not the model.
- How much did the Astra solutions cost?
- OpenAI says the tokens needed to find the solutions would cost roughly $2,000 at Sol API rates. That is a conditional price for the search itself, not the cost of the whole research effort, and the page does not say what Sol is.
- How do we know the Astra proofs are correct?
- Each argument comes with a Lean certificate, and the Lean 4 files are public in the openai/ten-proofs repository, where anyone can rebuild them. OpenAI says it takes responsibility for their correctness. The announcement does not report results from outside verifiers.
- Who is the author of a proof that an AI system generated?
- OpenAI says attribution should reflect how a result was produced, and that putting a human name on a proof generated entirely by an AI system would misstate both the system's contribution and genuine human intellectual work.
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