OpenAI Unveils Astra: AI System Solves 10 Stagnant Open Math Problems for €1,840

For decades, progress in fundamental mathematics and theoretical computer science relied almost exclusively on years of manual labor by elite academic minds. On August 1, 2026, OpenAI published complete research papers, model reasoning transcripts, and formal computer-verified proof files demonstrating that frontier AI models can resolve stagnant mathematical conjectures in bulk.

This disclosure follows an earlier milestone in May 2026, when an unreleased OpenAI system disproven Paul Erdős’s 80-year-old planar unit-distance conjecture. With today's release, OpenAI has formally branded this reasoning-focused architecture as the Astra model family, positioning it alongside current commercial models like GPT-5.6, Anthropic's Claude Opus 5, and DeepSeek-V4-Flash-0731.

The 10 Mathematical Breakthroughs at a Glance

The ten problems solved or significantly advanced by the internal Astra build span diverse domains in pure mathematics and computer science. In every case, the main result had seen no substantial progress for at least a decade, and in several cases for over half a century:

• Non-Sofic Groups: Astra successfully constructed an explicit mathematical object establishing the existence of a non-sofic group, resolving one of the most famous open questions in modern group theory.
• Connes’s Rigidity Conjecture: Disproved a long-standing conjecture in operator algebras which postulated that certain groups were uniquely determined by their von Neumann algebras.
• Three Erdős Conjectures Resolved: Astra resolved three distinct open problems posed by legendary mathematician Paul Erdős: Erdős Problem 183 (establishing a superexponential lower bound for multicolor triangle Ramsey numbers), alongside Erdős Problems 146 and 180 concerning compactness and degeneracy in extremal graph theory.
• Arithmetic Circuit Complexity: Established new lower bounds for computing the matrix permanent using arithmetic formulas, specifically proving a lower bound order of $n^4 / \log n$.
• Closest Vector Problem (CVP): Proven polynomial-factor hardness of approximation for the closest vector problem in lattices, providing vital theoretical grounding for post-quantum lattice-based cryptography.
• High-Dimensional Sphere Packing: Discovered tighter upper bounds on sphere-packing density approaching the Cohn–Elkies theoretical threshold.
• Binary and Spherical Codes: Delivered exponentially improved bounds on the maximum density and size of binary codes for any given minimum distance, with matching bounds for high-dimensional spherical codes.
• Quantum Parallel Repetition: Extended classical complexity principles by proving an exponential parallel repetition theorem for general two-player quantum games.
• Ehrhart’s Volume Conjecture: Fully determined, across all dimensions, the maximum volume of a convex body whose centroid is its sole interior lattice point.

The Economics of Automated Discovery: Cost & Token Breakdown

What makes Astra's achievement particularly noteworthy for European research institutions and enterprise AI labs is the sheer efficiency of the compute required. According to OpenAI's official report, the total API token expenditure needed to discover all ten mathematical solutions was approximately $2,000 USD (roughly €1,840 EUR) evaluated at Sol API rates.

To put this economic metric in perspective, traditional mathematical breakthroughs on major open conjectures typically require thousands of hours of effort by senior university professors and postdoctoral researchers—translating to academic budget expenditures exceeding €100,000 per paper in grants and salaries. Astra delivered ten major proofs at an average inference cost of €184 per breakthrough solution.

Lean Certificates and the European Regulatory Standard

A critical issue when using large language models for scientific research is hallucination—models generating plausible-sounding steps that contain hidden logical fallacies. To eliminate this vulnerability, OpenAI took every mathematical argument produced by Astra and converted it into formal Lean certificates.

Lean is an interactive theorem prover developed with strong roots in European academia and open-source mathematics. By running Astra's proofs through automated Lean logic checkers, the mathematical correctness of each generated paper is guaranteed deterministically by computer software rather than relying solely on human peer review.

This approach directly aligns with upcoming regulatory standards in the European Union. Under Article 50 of the EU AI Act—which becomes statutory law across all 27 EU member states tomorrow, August 2, 2026—providers of General-Purpose AI (GPAI) systems face strict mandatory requirements regarding technical documentation, traceability, and verifiable transparency for high-capability models.

Furthermore, OpenAI explicitly acknowledged the Leiden Declaration on AI and Mathematics, an initiative crafted by European researchers advocating for honest academic attribution. OpenAI confirmed that while humans assisted in formatting the paper manuscripts, the foundational reasoning and proofs were generated entirely by Astra, establishing a new model for transparent human-AI scientific collaboration in Europe and worldwide.