# OpenAI's Astra Solves 10 Decades-Old Math Problems

> OpenAI says an internal model it calls Astra produced proofs for more than ten long-standing open problems in mathematics and theoretical computer science, including a non-sofic group construction open since 1999, the Connes rigidity conjecture, the Ehrhart volume conjecture, and three problems posed by Paul Erdos. The results span high-dimensional geometry, coding theory, complexity, group theory, quantum complexity, and lattice cryptography. The claim that matters most is verification. Every proof was formally checked in Lean 4 with zero sorry statements, meaning nothing was left unproven and hand-waved, and the work was written up in a 249-page manuscript. OpenAI also put a price on it, roughly $2,000 of compute per problem at its API rates. Formal verification is what separates this from the familiar story of a model confidently asserting nonsense, because Lean either accepts a proof or it does not. If these hold up under outside scrutiny, the combination of genuine results and a low compute bill is the striking part, hinting that AI-assisted mathematics may be crossing from curiosity into a working research tool. The mathematics community's independent review will be the real verdict.

_Section: [Daily AI Updates](https://www.wortins.com/daily-ai) · Source: OpenAI · Published Tuesday, August 11, 2026_

## Wortins' read

OpenAI says an internal model it calls Astra produced proofs for more than ten long-standing open problems in mathematics and theoretical computer science, including a non-sofic group construction open since 1999, the Connes rigidity conjecture, the Ehrhart volume conjecture, and three problems posed by Paul Erdos. The results span high-dimensional geometry, coding theory, complexity, group theory, quantum complexity, and lattice cryptography. The claim that matters most is verification. Every proof was formally checked in Lean 4 with zero sorry statements, meaning nothing was left unproven and hand-waved, and the work was written up in a 249-page manuscript. OpenAI also put a price on it, roughly $2,000 of compute per problem at its API rates. Formal verification is what separates this from the familiar story of a model confidently asserting nonsense, because Lean either accepts a proof or it does not. If these hold up under outside scrutiny, the combination of genuine results and a low compute bill is the striking part, hinting that AI-assisted mathematics may be crossing from curiosity into a working research tool. The mathematics community's independent review will be the real verdict.

## Source

[Read the full story at OpenAI](https://openai.com/index/ten-advances-in-mathematics/)

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