OpenAI (via Future Tools)

Ten advances in mathematics and theoretical computer science

2026-08-01 · 00:00 UTC ·3 min read

Brief

OpenAI announced that an internal model called Astra produced proofs resolving ten longstanding problems across mathematics and theoretical computer science, with the announcement dated 2026-08-01 (created 2026-03-11, last updated 2026-04-04). The claimed advances cover high-dimensional sphere packing (bounds down to the Cohn–Elkies threshold), exponentially improved binary and spherical code bounds, construction of non-sofic groups, a disproof of Connes’s rigidity conjecture, new permanent lower bounds (including an arithmetic-formula bound quoted as “of order n- 4/log n”), an exponential quantum parallel-repetition theorem, polynomial-factor hardness for CVP, a full resolution of Ehrhart’s volume conjecture, a superexponential lower bound for multicolor triangle Ramsey numbers (Erdős problem 183), and results on Erdős problems 146 and 180. OpenAI says manuscripts were human-prepared with the model, all proofs were formalized in Lean (repo: github.com/openai/ten-proofs), model narrations were released, the token search cost about $2,000, and the work follows an earlier May AI-generated disproof of the Erdős unit-distance conjecture.

Why it matters

OpenAI says an internal model, Astra, produced proofs for ten major results (announcement dated 2026-08-01) including new upper bounds on high-dimensional sphere-packing density down to the Cohn–Elkies threshold and exponentially improved bounds on the maximum size of binary and high-dimensional spherical codes.

Key details

  • Astra-generated work reportedly establishes the existence of non-sofic groups and gives a disproof of Connes’s rigidity conjecture; each argument was prepared into manuscripts by humans, formalized as Lean certificates, and published in the repository github.com/openai/ten-proofs.
  • In complexity theory and cryptography, the results include new lower bounds for computing the permanent (including an arithmetic-formula lower bound described as “of order n- 4/log n”), an exponential parallel-repetition theorem for general two-player quantum games, and a claimed polynomial-factor hardness of approximation for the closest vector problem (CVP).
  • OpenAI notes these solutions span geometry, coding theory, group theory, operator algebras, quantum complexity, lattice cryptography, and extremal combinatorics (e.g., resolution of Ehrhart’s volume conjecture in every dimension, a superexponential lower bound for multicolor triangle Ramsey numbers resolving Erdős problem 183, and progress on Erdős problems 146 and 180); the search reportedly required roughly $2,000 in tokens and accompanies the ChatGPT for Academic Researchers program granting free access to 100,000 scientists.
Source evidence

Ten advances in mathematics and theoretical computer science

We 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 science 1. 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.

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 n- 4/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.

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

  • 1Subsequent research includes Bloom, Sawin, Schildkraut, and Zhelezov, “ The sum-product conjecture is false for real numbers(opens in a new window)Split primes and the Elekes-Rónyai problem(opens in a new window)Furthest Pair Requires Quadratic Time in Superconstant Dimension under SETH(opens in a new window)Communication complexity of point-line incidences over the reals(opens in a new window)The Minkowski grid has robustly many repeated distances(opens in a new window)

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