OpenAI's Astra model solves ten decades‑old math problems

OpenAI Astra math problems Lean formal proofs $2000 token cost

OpenAI announced that an internal version of its next‑generation model, tentatively named Astra, produced solutions to ten long‑standing open problems in mathematics and theoretical computer science. The problems, which had seen no progress for at least ten years, cover areas such as high‑dimensional geometry, coding theory, group theory, operator algebras, quantum complexity, lattice cryptography and extremal combinatorics. Astra generated the core arguments; human researchers turned the output into manuscripts and then formalised the proofs in the Lean 4 theorem‑proving system. The Lean certificates and the 249‑page manuscript were released publicly on GitHub, showing zero “sorry” placeholders. OpenAI estimated the token usage for the ten solutions at roughly $2,000 under its Sol API pricing. The announcement was made in early August 2026, and the model’s final name and commercial release plan remain undecided, with possibilities including GPT‑5.6, GPT‑6 or another designation.

Entities

Astra · Greg Brockman · Lean · Lean proof assistant · Noam Brown · OpenAI · Sam Altman · Sol API · U.S. Federal Government

Claims

What the coverage asserts, and how many sources carry each claim.

[● 16 SOURCES] The proofs were formalized in Lean 4 and released on GitHub with zero “sorry” placeholders. garymarcus.substack.com · www.merkur.de · ai.cnmo.com · www.kurierverlag.de · mashdigi.com · +11 more
[● 16 SOURCES] OpenAI published a 249‑page manuscript describing the ten advances on August 1‑3, 2026. garymarcus.substack.com · www.merkur.de · ai.cnmo.com · www.kurierverlag.de · mashdigi.com · +11 more
[● 14 SOURCES] The token cost to solve the ten problems via OpenAI's Sol API was about $2,000. garymarcus.substack.com · ai.cnmo.com · mashdigi.com · www.techritual.com · 36kr.com · +9 more
[● 14 SOURCES] Astra generated the core mathematical arguments, which human researchers turned into manuscripts. garymarcus.substack.com · ai.cnmo.com · mashdigi.com · www.techritual.com · 36kr.com · +9 more
[● 17 SOURCES] An internal version of OpenAI's next‑generation model Astra solved ten open problems in mathematics and theoretical computer science. garymarcus.substack.com · www.merkur.de · ai.cnmo.com · www.kurierverlag.de · mashdigi.com · +12 more
[● 15 SOURCES] The problems span high‑dimensional geometry, coding theory, group theory, operator algebras, quantum complexity, lattice cryptography and extremal combinatorics. www.merkur.de · ai.cnmo.com · www.kurierverlag.de · mashdigi.com · www.techritual.com · +10 more
[● 15 SOURCES] The ten problems had been open for at least a decade, with some remaining unsolved for much longer. www.merkur.de · ai.cnmo.com · www.kurierverlag.de · mashdigi.com · www.techritual.com · +10 more
[● 14 SOURCES] The model name Astra is tentative and may be released as GPT‑5.6, GPT‑6 or under another designation. garymarcus.substack.com · ai.cnmo.com · mashdigi.com · www.techritual.com · 36kr.com · +9 more