A Single AI Model Just Solved 10 Math Problems That Stumped Experts for Decades

Astra Model Solves Ten Decade-Old Math Problems in a Single Release

OpenAI today published a sweeping set of ten new results in mathematics and theoretical computer science, all generated by an internal version of its forthcoming Astra model.

The problems span high-dimensional geometry, quantum complexity, lattice cryptography, and extremal combinatorics, and had seen no progress on their main results for at least a decade, in most cases much longer.

The total compute cost to find the solutions was roughly $2,000 at Sol API rates, a figure that upends assumptions about the economics of frontier mathematical research.

From Sphere Packing to Ramsey Numbers: The Ten Breakthroughs

According to OpenAI’s announcement,

these breakthrough results—achieved using an internal version of their next major model, Astra—span ten distinct open problems. Each solution was prepared into a manuscript by human researchers collaborating with Astra and later formalized in the Lean proof assistant, complete with model-generated narrations of its thinking process.

1. High-Dimensional Sphere Packing: New upper bounds on sphere-packing density push performance down to the Cohn–Elkies threshold, sharpening our understanding of how tightly spheres can fill space in high dimensions.

2. Binary and Spherical Codes: In coding theory, exponentially stronger bounds now exist for the maximum size of binary and spherical codes at any prescribed minimum distance.

3. Non-Sofic Groups: Group theory received a definitive construction proving that non-sofic groups exist, settling a central open question that had long divided researchers.

4. Connes’s Rigidity Conjecture: The model produced a disproof of Connes’s rigidity conjecture, demonstrating that certain groups are not uniquely determined by their von Neumann algebras.

5. Arithmetic Circuit Complexity: In computational complexity, novel lower bounds for computing the permanent were established, including an arithmetic-formula bound on the order of $n^4 / \log n$.

6. Quantum Parallel Repetition: Quantum complexity saw an exponential parallel repetition theorem for general two-player quantum games, extending a foundational classical principle into the quantum domain.

7. Closest Vector Problem: The closest vector problem—a bedrock of post-quantum cryptography—was shown to be NP-hard to approximate within polynomial factors, tightening core security assumptions.

8. Ehrhart’s Volume Conjecture: Ehrhart’s volume conjecture was fully resolved by determining, in every dimension, the maximum volume of a convex body whose centroid is its only interior lattice point.

9. Multicolor Ramsey Numbers: In extremal combinatorics, a superexponential lower bound for multicolor triangle Ramsey numbers resolved Erdős problem 183.

10. Extremal Number Conjectures: Separate results in extremal graph theory settled the compactness and degeneracy conjectures, closing Erdős problems 146 and 180.

The total token cost to generate these proofs amounted to roughly $2,000 at Sol API rates, a figure that dramatically lowers the barrier to computer-assisted mathematical discovery.

AI’s Entry Into Peer-Reviewed Mathematics and Community Response

The Conversation recently observed that the combination of AI with formal proof assistants like Lean is accelerating discovery and could herald a new golden age for mathematics.

That pattern was already visible in May, when OpenAI’s model disproved the unit distance conjecture, and human mathematicians rapidly adapted the core technique to topple the sum‑product conjecture.

In the weeks leading up to this announcement, other AI systems found counterexamples to a 60‑year‑old question of Grothendieck and the century‑old Jacobian Conjecture, each subsequently verified in Lean.

Such speed is reshaping the human side of the discipline: some researchers are leaving academic posts for frontier AI labs, while communities grapple with anxiety about authorship and career displacement.

OpenAI addressed the tension by openly stating that the model generated the mathematical arguments, while humans curated the manuscripts and formalized the proofs — a stance aligned with the Leiden declaration on AI and mathematics.

Yet The Conversation also cautions that many fields and the remaining Millennium Prize problems remain out of reach for current systems, grounding the excitement in realistic limits.

A New Research Paradigm Where AI and Humans Co-Author Discovery

The ten results are not merely a list of solved puzzles; they signal a structural shift in how mathematical knowledge is created.

When a model can independently surface proofs for problems that stalled progress for decades, the human role evolves from sole explorer to curator, contextualizer, and extender of machine-generated insight.

This hybrid workflow — search, formal verification, human refinement — promises to compress research cycles and open lines of inquiry that once seemed impossibly distant.

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