Is Astra available to the public?

Astra is described as OpenAI's next flagship model but is not yet publicly available. Only the ten formal proofs, along with Lean 4 certificates and chain-of-thought walkthroughs, are shipping in this release. The proofs live in the openai/ten-proofs repository under the Apache-2.0 license.

How can readers verify the Astra proofs?

Each of the ten Astra proofs ships as a Lean 4 file that anyone can machine-check against Lean's kernel. If the file compiles clean, the mathematical result is formally verified. That removes the usual 'AI might be hallucinating' worry — a broken step in the argument would fail the Lean typecheck.

What kinds of open problems did Astra solve?

The ten Astra targets span high-dimensional geometry (sharper sphere-packing upper bounds meeting the Cohn-Elkies threshold), coding theory (stronger bounds on binary and spherical codes), operator algebras (a disproof of Connes' rigidity conjecture), group theory (a non-sofic group construction), plus results in arithmetic circuit complexity, lattice cryptography, and extremal combinatorics.

How much did it cost to generate the ten proofs?

OpenAI researcher Sebastien Bubeck says the total token cost to solve all ten Astra problems was under $2,000 at Sol API prices. That figure is cheaper than a single human research month and is one of the reasons the release is being framed as a preview of what Astra can do for research.

OpenAI · 2026-08-01 · major

OpenAI Astra cracks 10 open math problems — Lean proofs on GitHub

OpenAI's Astra model solved 10 open problems in math and theoretical CS — from sharper sphere-packing bounds to a disproof of Connes' rigidity conjecture. Lean 4 proofs shipped on GitHub, generated for under $2,000 in Sol API tokens.

GitHub social card for the openai/ten-proofs repository — Lean certificates for ten Astra proofs

Astra, OpenAI's next flagship model, cracked 10 open math and CS problems and shipped Lean-verified proofs for each.

Quick facts

Maker	OpenAI
Model	Astra (internal preview)
Format	Blog announcement + Lean 4 certificates on GitHub
License	Apache-2.0 (proofs repo)
Proofs shipped	10
Total cost	Under $2,000 at Sol API prices
Availability	Astra not yet released; proofs are public

What is it?

Astra is OpenAI's next flagship model, first shown publicly by proving 10 problems in math and theoretical computer science that have stood open for at least a decade — most for far longer. The targets span high-dimensional geometry, coding theory, arithmetic circuit complexity, group theory, operator algebras, quantum complexity, lattice cryptography, and extremal combinatorics.

How does it work?

For each problem, the model writes a full mathematical argument then formalizes every step in Lean 4. The release ships every proof as a checkable Lean certificate plus a chain-of-thought walkthrough, so mathematicians can verify each step against Lean's kernel instead of taking the model's word for it. OpenAI researcher Sebastien Bubeck says the entire batch cost under $2,000 in Sol API tokens.

Why does it matter?

Formal-proof AI cracking questions like non-sofic groups or the Cohn-Elkies sphere-packing threshold is a first for a general-purpose model — these are results human researchers have wrestled with for years. Because every proof compiles under Lean, an auditor can reject the entire chain if a single line lies. That short-circuits the usual 'the AI might be hallucinating' worry for mathematics research.

Who is it for?

mathematicians, formal-verification researchers, and AI reasoning teams

Frequently asked questions

Is Astra available to the public?: Astra is described as OpenAI's next flagship model but is not yet publicly available. Only the ten formal proofs, along with Lean 4 certificates and chain-of-thought walkthroughs, are shipping in this release. The proofs live in the openai/ten-proofs repository under the Apache-2.0 license.
How can readers verify the Astra proofs?: Each of the ten Astra proofs ships as a Lean 4 file that anyone can machine-check against Lean's kernel. If the file compiles clean, the mathematical result is formally verified. That removes the usual 'AI might be hallucinating' worry — a broken step in the argument would fail the Lean typecheck.
What kinds of open problems did Astra solve?: The ten Astra targets span high-dimensional geometry (sharper sphere-packing upper bounds meeting the Cohn-Elkies threshold), coding theory (stronger bounds on binary and spherical codes), operator algebras (a disproof of Connes' rigidity conjecture), group theory (a non-sofic group construction), plus results in arithmetic circuit complexity, lattice cryptography, and extremal combinatorics.
How much did it cost to generate the ten proofs?: OpenAI researcher Sebastien Bubeck says the total token cost to solve all ten Astra problems was under $2,000 at Sol API prices. That figure is cheaper than a single human research month and is one of the reasons the release is being framed as a preview of what Astra can do for research.

Try it

git clone https://github.com/openai/ten-proofs