OpenAI's New Model Solves 10 Math Problems in a Row, Turning Math into a Click Game for $2000

In August 2026, OpenAI released research results showing that the internal test version of its next-generation flagship model Astra solved 10 long-standing open problems in mathematics and theoretical computer science in one go, covering high-dimensional geometry, coding theory, group theory, operator algebras, quantum complexity, and extremal combinatorics. All proofs were machine-verified using the Lean formal verification tool, and OpenAI also released the complete reasoning process text for each problem for mathematicians to review.

Thomas Bloom, a mathematician at the University of Manchester, believes that the overall academic value of these ten results far exceeds the single achievement five months ago when OpenAI disproved Erdős's unit distance conjecture. Among them, the construction of a non-sofic group is considered by many mathematicians to be the most significant result, with a Caltech math PhD calling it 'Fields Medal-level stuff.' The problem was posed by Abel Prize laureate Mikhail Gromov in 1999, asking whether all countable groups are sofic. For 27 years, countless top mathematicians attempted to construct a counterexample but failed. Astra forced a decisive contradiction by combining the unit group of the binary Leavitt algebra, Kun-Thom's extended graph theory, and Thompson's group V.

Another major result is the disproof of the rigidity conjecture of Alain Connes, a Fields Medalist from 1982. Connes had asserted that certain groups are uniquely determined by their von Neumann algebras. Astra constructed a countable infinite family of groups that are pairwise non-isomorphic but have identical von Neumann algebras. The key to the construction was Astra's active distinction between measurable conjugacy and algebraic conjugacy, two easily confused relations.

The high-dimensional sphere packing problem had seen no improvement since the Kabatiansky-Levenshtein bound in 1978, a 46-year stagnation. Astra precisely computed the exponential decay rate of the Cohn-Elkies linear programming bound, achieving the first breakthrough over the KL bound. It initially attempted a global norm estimate but quickly rejected it, reasoning that the global norm forgets where negative mass is concentrated, so it turned to local mass exclusion inequalities, using harmonic measure and the maximum modulus principle to pin down the lower bound.

For binary codes and spherical codes, Astra provided exponential improvements to the bounds, refreshing the classical MRRW bound and updating the theoretical limits for error-correcting codes and signal transmission in communications. In arithmetic circuits, Astra gave a lower bound of order n⁴/log n for the permanent, using rectangular matching polynomials to solve the counting failure of traditional derivations, and employing Möbius transformations to handle division complexity uniformly.

On the quantum front, Astra proved the exponential parallel repetition theorem for general two-player games, filling a gap in quantum parallel repetition theory. In lattice-based cryptography, Astra completed the proof of the polynomial approximation hardness of CVP, upon which the security of post-quantum encryption partly rests. Additionally, Astra solved three classic open problems of Erdős, including a super-exponential lower bound for the 183rd problem on multicolor Ramsey numbers, and the tightness and degeneracy conjectures for extremal graphs in problems 146 and 180.

Astra's problem-solving process is roughly divided into four stages: First, the model autonomously reasons around the open problem, generating a complete argument and core proof idea. Second, researchers reuse the same Astra model to assist in organizing the manuscript, adapting it to the general standards for reading, review, and citation in the mathematical community. Third, the model formalizes the proof into a Lean proof, where Lean turns every mathematical step into a computer-verifiable logical expression, and any gap fails verification, effectively adding an extra layer of review. Finally, OpenAI releases the complete AI reasoning narrative text for each problem, preserving the entire process of trial and error, tool changes, and self-revisions, allowing mathematicians worldwide to review and trace the origins.

This process demonstrates a new paradigm for AI in mathematical discovery: it not only provides results but also provides verifiable proofs and transparent reasoning processes. AI pioneer Hinton once predicted that within 10 to 20 years, AI might create new mathematics that humans cannot understand. Astra's performance seems to bring that future closer.