OpenAI shares ten advances in mathematics and theoretical computer science
OpenAI announced ten new results from its Astra model addressing long-standing open problems
OpenAI released a blog post describing ten advances in mathematics and theoretical computer science that were produced by an internal version of its next major model, Astra. The results resolve or make substantial progress on long-standing open problems in areas such as high-dimensional geometry, coding theory, group theory, operator algebras, quantum complexity, lattice cryptography and extremal combinatorics. The total token cost to find the solutions was roughly $2,000 at Sol API rates, and the arguments were prepared into manuscripts by humans and then formalized in Lean certificates.
High-dimensional sphere packing
The advance provides new upper bounds on sphere-packing density down to the Cohn–Elkies threshold.
Binary and spherical codes
The work yields exponentially improved bounds on the maximum size of binary codes at any prescribed minimum distance, with analogous results for high-dimensional spherical codes.
Non-sofic groups
A construction establishes the existence of non-sofic groups, addressing a central open question in group theory.
Connes’s rigidity conjecture
The conjecture that certain groups are uniquely determined by their von Neumann algebras is disproved.
Arithmetic circuit complexity
New lower bounds are given for computing the permanent using arithmetic circuits and formulas, including an arithmetic-formula lower bound of order n 4⁄log n.
Quantum parallel repetition
An exponential parallel repetition theorem is proved for general two-player quantum games, extending a foundational principle from classical complexity theory.
Closest vector problem
Polynomial-factor hardness of approximation is shown for the closest vector problem, a foundational lattice question related to post-quantum cryptography.
Ehrhart’s volume conjecture
The maximum possible volume of a convex body whose centroid is its only interior lattice point is determined in every dimension.
Multicolor Ramsey numbers
A superexponential lower bound is obtained for multicolor triangle Ramsey numbers, resolving Erdős problem 183.
Extremal number conjectures
Results are presented on the compactness and degeneracy conjectures in extremal graph theory, resolving Erdős problems 146 and 180.
OpenAI stresses honest attribution and shared responsibility for AI‑generated mathematics
OpenAI states that attribution should honestly reflect how a result was produced, noting that claiming human authorship for a proof generated entirely by an AI system would misrepresent both the system’s contribution and genuine human intellectual work. The company helped prepare the manuscripts and formalize the proofs in Lean, takes responsibility for their correctness, while the mathematical arguments themselves were generated by the system. It hopes the mathematical community will engage deeply with the results, place them in context, and bring the ideas behind them to life through new research and discovery.
Hacker News commenters reacted with a mixture of excitement, concern, and skepticism
Commenters expressed a range of views on the announcement. Some highlighted the rapid pace of progress, noting that the results appear to follow an exponential trend in AI capability. Others raised concerns about the impact on academia and the potential for AI to reshape traditional research processes. A few commenters questioned the novelty or significance of the advances, while others emphasized the need for practical applications beyond pure mathematics. Selected remarks include:
"People argue whether we are at y-5, y, or y+5, meanwhile we seem to be on a y=2^x exponential that keeps delivering more and more impressive results." – @sothatsit
"I feel increasingly anxious reading this. Machine research shouldn’t be merged into mainline of human knowledge." – @cwiz
"Crazy progress. I wonder how institutional academia would adjust with this. Now its more apparent than ever that the prestige and honour system in academia is having shaky foundations" – @dipanshuhappy
"Any computable problem will eventually fall to computers. LLMs have made math proofs more computable, in the sense that a computer can both generate potential solutions and check the validity of its solutions on its own, with a reasonable chance of converging on something correct." – @plaidfuji
"While these advances are genuinely impressive, I'm curious when we will see practical implications for this work. For example, will we see advances in material science, medical cures, etc?" – @10dpd
"Wow, here are solutions to 10 problems that we spent millions of dollars on out of the 100s/1000s of other problems that we tried and failed to solve." – @pbkompasz
"This is impressive. The real game-changer will be when AI creates an entirely new, significant branch of mathematics." – @nnm
These comments illustrate both enthusiasm for the technical achievements and broader reflections on the role of AI in mathematical research.
The results demonstrate AI’s ability to tackle open problems, but practical impact and scope remain uncertain per community feedback
The blog post shows that an internal Astra model can produce proofs for a variety of long-standing mathematical questions, with the work subsequently checked and formalized in Lean. However, commenters on Hacker News noted that the translation of such theoretical advances into practical applications—such as material science or medicine—has not yet been demonstrated, and they questioned whether the current approach will scale to the most challenging problems like the Millennium Prize questions. The discussion also touched on concerns about attribution, the potential disruption to academic incentives, and the need for the mathematical community to critically evaluate and build upon the AI‑generated work.