OpenAI has once again put artificial intelligence in the spotlight with an announcement that has shaken the mathematical community. Its latest model, Astra, has solved ten open problems that had remained unsolved for decades, using a method that allows for the automatic verification of each step. This is no small feat: we are talking about questions that have withstood the attempts of the best specialists for years, and which have now been solved thanks to a system trained to reason autonomously.
The company has published the results in a collection of manuscripts that includes 249 pages of arguments and formal certificates generated with Lean, software that verifies the validity of proofs step by step. Furthermore, the computational cost of these findings has been surprisingly low: less than $2.000 in API tokens. But what does this advance really mean? Are we witnessing a turning point in mathematical research, or should we approach it with caution?
Ten decades-old mathematical problems solved
The results span areas as diverse as high-dimensional geometry, coding theory, the complexity of arithmetic circuits, group theory, quantum complexity, lattice cryptography, and extremal combinatorics. Among the most significant achievements are the refutation of Connes' rigidity conjecture , a long-standing problem in operator algebras, and the construction of non-Sophic groups, a question that had preoccupied specialists for years. The upper bound for the packing density of spheres in high dimensions, a limit that had remained unchanged since 1978, has also been improved.
Furthermore, Astra has solved three problems from Paul Erdős's catalog, including the well-known Problem 21 , which mathematician Thomas Bloom has described as "the most surprising of the entire list." In total, these are ten problems that had remained unsolved for at least a decade, and in most cases, much longer. The mathematical community has received the news with a mixture of astonishment and skepticism, as was to be expected.
Automatic verification with Lean
What makes this announcement different from others like it is the verification process. Each demonstration has been formalized in Lean , a wizard that automatically checks each logical step. The certificates are publicly available on GitHub, and anyone can download them and run the verifier. If a single step isn't correctly deduced, the software rejects it. This eliminates the need to take OpenAI's word for it: the validity of the results can be independently verified.
This method contrasts with the traditional way of validating mathematical proofs, which requires a peer-review process that can last for months. With Lean, verification is reduced to a few minutes. The researchers have emphasized that this is the first time an AI system has presented results with such a high level of transparency. However, some experts, such as Bharath Ramsundar, caution that "we are rapidly building a tower of 'maybe true' AI results" and call for human mathematicians to review the findings before integrating them into the established body of knowledge.
The cost and selection of problems
Another striking detail is the cost. OpenAI claims that the total number of tokens needed to find solutions to these problems would cost approximately $2.000, based on its API fees. But this figure only reflects successful executions, not failed attempts. As one expert points out, "It's like winning the lottery and saying that I spent €20 on a ticket and won €20.000, but I don't count the €5.000 I spent on previous tickets." The true cost of the discovery process is much higher.
Furthermore, OpenAI has carefully selected the problems it chose to publish. It's unknown how many problems Astra attempted and failed to solve. If it selected 10 successes out of 1.000 attempts, the interpretation is very different than if it were 10 out of 12. Until the denominator is known , the "Deep Blue moment of mathematics" remains more of a headline than a conclusion. Even so, the results are verifiable, and that's a significant improvement over previous announcements.
Reactions from the mathematical community
Fields Medal winner Tim Gowers has said he would have wholeheartedly recommended publishing the proof in a leading mathematics journal. A team of nine mathematicians, including Gowers and Noga Alon, has published a companion paper explaining the proof in a more accessible way. Thomas Bloom, who manages the Erdős problem catalog, called the results "major news" and said they are even more significant than Astra's solution to the unit distance conjecture in May.
But there are also critical voices. Gary Marcus , a well-known AI skeptic, called the announcement "amazing" but believes its scope has been exaggerated. Some experts think only a few of the ten demonstrations will be considered truly astonishing, while the others address problems that were within the community's reach, even if no one had bothered to tackle them. Despite the criticism, the fact that the results are automatically verifiable marks a fundamental difference from other AI announcements.
In short, OpenAI's announcement with Astra represents a milestone in the application of artificial intelligence to mathematical research. For the first time, an AI system has solved long-standing open problems and provided machine-verifiable proofs. While questions remain about problem selection and the true cost, the transparency of the process and the ability to verify each step open the door to a new era in which AI will collaborate with human mathematicians. The scientific community will have to adapt to this new reality, but what is clear is that the frontier of knowledge has shifted, and Astra is the tool that has made it happen.
