OpenAI says it directed an internal model they refer to as “Astra, our next major model” to find solutions for ten mathematical problems that, according to the company, had seen no main‑result progress for at least a decade. OpenAI states the expenditure per problem was under $2,000 when calculated at GPT‑5.6 Sol token prices.
What was released
OpenAI published an openai/ten-proofs repository containing Lean 4 formalizations of their results. They also released a paper describing the proposed solutions and an additional PDF generated by a large language model in which the model "reconstructs how the proof came together" based on unpublished internal reasoning traces.
Together these materials provide a notable degree of transparency: formal proofs in Lean 4, an explanatory paper, and an LLM‑produced reconstruction of the proof process.
Recent context: Anthropic and broader reactions
The announcement follows reports from a few days earlier by Anthropic, who said they had discovered cryptographic weaknesses with the Claude model in Mythos Preview, claiming about $100,000 in token usage and prompts targeted at deep research rather than "low hanging fruit."
OpenAI’s update has prompted comparisons among mathematicians to landmark AI moments such as Deep Blue, with many observers noting the potential shift in how hard mathematical work is done.
Terence Tao on “big mathematics”
The development echoes Terence Tao’s description in IEEE Spectrum (June) of a transition toward what he calls "big mathematics." Tao is neither dismissive nor fearful of AI; he frames it as a catalyst for large‑scale, decentralized collaborations between humans and machines, where humans handle creative, conceptual pieces and AI takes on much of the technical, repetitive work.
Open questions
OpenAI’s communication does not specify how many attempts exceeded the stated per‑problem cost before reaching successful solutions, nor does it publish the exact prompts or the full internal model configuration. Members of the mathematics community have requested the prompts and more reproducibility details.
Summary
According to OpenAI, an internal model produced solutions and Lean 4 formal proofs for ten mathematical problems that had been stagnant for at least ten years. The company has released a repository with the formalizations, a descriptive paper, and an LLM‑generated reconstruction of the proofs, while some technical and reproducibility details remain undisclosed.
