Ten advances in mathematics and theoretical computer science
Quick Answer
OpenAI's Astra model has generated ten significant mathematical results, including disproofs and new bounds in various fields, empowering 100,000 researchers with free access to ChatGPT.
Quick Take
These results address long-standing open problems in areas like group theory and quantum complexity, costing approximately $2,000 in API usage.
Key Points
- New upper bounds on high-dimensional sphere packing density achieved.
- Disproof of Connes's rigidity conjecture established by Astra model.
- Polynomial-factor hardness for the closest vector problem demonstrated.
- Superexponential lower bounds for multicolor triangle Ramsey numbers resolved.
- AI-generated proofs formalized in Lean, ensuring correctness and accountability.
DeepSignal Analysis
What happened
OpenAI's Astra model has produced ten significant mathematical results, including disproofs and new bounds in various fields. These results were generated while providing 100,000 researchers with free access to ChatGPT, costing around $2,000 in API usage.
Key evidence
- OpenAI announced that its Astra model generated ten results addressing open problems in mathematics, including group theory and quantum complexity.
- The results were achieved using an internal version of Astra, with a total cost of approximately $2,000 for the API usage to find solutions.
- The mathematical arguments were formalized in Lean certificates, and OpenAI prepared the manuscripts while taking responsibility for their correctness.
Why it matters
These advancements could significantly impact various fields within mathematics and theoretical computer science, potentially accelerating research and discovery. By providing free access to powerful AI tools, OpenAI aims to democratize access to advanced mathematical insights, fostering collaboration and innovation among researchers.
What to watch
📖 Reader Mode
~4 min readWe want to empower scientists and mathematicians with tools that accelerate discovery. That is why we recently announced ChatGPT for Academic Researchers, an initiative providing 100,000 scientists and mathematicians with free access to our best ChatGPT models. We also continue to evaluate our models on open research problems during development.
In May, we shared an AI-generated disproof of the Erdős unit-distance conjecture, discovered while evaluating an unreleased model. This work has already inspired further developments in mathematics and theoretical computer science1. Today, we are sharing a selection of ten results to problems that have been open and have seen no progress on the main result for at least a decade, and in most cases much longer. These problems span high-dimensional geometry, coding theory, arithmetic circuit complexity, group theory, operator algebras, quantum complexity, lattice cryptography and extremal combinatorics. All of these problems are of substantial interest to their respective mathematical communities, and several are of broad interest across mathematics as a whole.
The results
We provide new results for the following problems. The results were achieved by an internal version of Astra, our next major model. The total number of tokens needed to find solutions to these problems would cost roughly $2,000 at Sol API rates. These arguments were then prepared into manuscripts by humans with the same model. Afterward, the model formalized each argument in a Lean certificate(opens in a new window). We are also releasing for each solution a model’s narration of its thinking process.
- High-dimensional sphere packing. New upper bounds on sphere-packing density down to the Cohn–Elkies threshold.
- Binary and spherical codes: 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 establishing the existence of non-sofic groups, addressing a central open question in group theory.
- Connes’s rigidity conjecture. Disproof of a longstanding conjecture that certain groups are uniquely determined by their von Neumann algebras
- Arithmetic circuit complexity. New lower bounds for computing the permanent using arithmetic circuits and formulas, including an arithmetic-formula lower bound of order n4/log n.
- Quantum parallel repetition. An exponential parallel repetition theorem for general two-player quantum games, extending a foundational principle from classical complexity theory.
- Closest vector problem. Polynomial-factor hardness of approximation for the closest vector problem, a foundational lattice question related to post-quantum cryptography.
- Ehrhart’s volume conjecture. Determining, in every dimension, the maximum possible volume of a convex body whose centroid is its only interior lattice point
- Multicolor Ramsey numbers. A superexponential lower bound for multicolor triangle Ramsey numbers, resolving Erdős problem 183.
- Extremal number conjectures. Results on the compactness and degeneracy conjectures in extremal graph theory, resolving Erdős problems 146 and 180.
Responsibility to the mathematical community
The emergence of systems capable of contributing to mathematical research raises questions that cannot be answered by a technology company alone. There are many views as to the role of AI in mathematics, and we have deep respect and understanding for those concerned with its impact, including the signers of the Leiden declaration on AI and Mathematics(opens in a new window). We believe attribution should honestly reflect how a result was produced: claiming human authorship for a proof generated entirely by an AI system would misrepresent both the system’s contribution and the nature of genuine human intellectual work. We helped prepare the manuscripts and formalize the proofs in Lean, and we take responsibility for their correctness, while the mathematical arguments themselves were generated by our system. We hope the mathematical community will engage deeply with these results, place them in context, and bring the ideas behind them to life through new research and discovery.
As AI systems evolve into more sophisticated research collaborators, ensuring widespread access is fundamental to supporting scientists and mathematicians as they navigate and define the future of their disciplines during this transformative era.
Footnote
- 1
Subsequent research includes Bloom, Sawin, Schildkraut, and Zhelezov, “The sum-product conjecture is false for real numbers(opens in a new window)”; Pohoata, “Split primes and the Elekes-Rónyai problem(opens in a new window)”; Saha, Xu, and Ye, “Furthest Pair Requires Quadratic Time in Superconstant Dimension under SETH(opens in a new window)”; Goh and Hatami, “Communication complexity of point-line incidences over the reals(opens in a new window)”; and Lee, Pohoata, and Zhu, “The Minkowski grid has robustly many repeated distances(opens in a new window).”
— Originally published at openai.com
Want this in your inbox every morning?
Daily brief at your local 8am — bilingual EN/中文, free.
More from OpenAI Blog
See more →Scientific computing in the age of agentic AI
AI agents are transforming scientific computing by streamlining software development, enabling researchers to focus on discovery. Projects using Codex and Claude Code report accelerated development and improved maintenance, though challenges in validating AI outputs remain. Long-term stewardship of research software is crucial to ensure reliability and reproducibility.