OpenAI's Astra Solves 10 Unsolved Math Problems, Publishing Formal Proofs for $2,000 in Compute

OpenAI has announced that an internal version of its forthcoming "Astra" artificial intelligence model successfully solved 10 long-standing open problems in mathematics and theoretical computer science. The resolved problems, which span highly complex fields including high-dimensional geometry, group theory, cryptography, and quantum complexity, had seen no substantial progress from human mathematicians for at least a decade, and in most cases, significantly longer. The announcement marks a major milestone in artificial intelligence, demonstrating that large language models can move beyond summarizing existing human knowledge to generating net-new, highly advanced scientific discoveries that have eluded the world's top domain experts.[1][5]

Rather than simply publishing a traditional academic paper claiming success, OpenAI released a comprehensive 249-page manuscript accompanied by machine-checkable "Lean 4" certificates for every single mathematical result. This rigorous formal verification approach directly addresses the most common and severe criticism of AI-generated mathematics: the tendency of language models to hallucinate plausible-looking logical steps that quietly fail upon rigorous human inspection. By providing the underlying code that proves the math, OpenAI has shifted the burden of trust from the AI's natural language output to a deterministic, mathematically sound compiler.[4][7]

The financial metrics behind this mathematical breakthrough are arguably as notable as the proofs themselves. OpenAI estimates that the total computational power required to generate all 10 solutions cost approximately $2,000 at current API rates for its GPT-5.6 Sol model. This remarkably low price point effectively commoditizes advanced theorem proving, shifting the primary bottleneck of mathematical discovery away from massive, multi-million-dollar compute budgets. Instead, the constraint now lies in prompt design, problem framing, and the formal verification of the outputs, putting world-class mathematical research within the financial reach of almost any funded laboratory.[2][3][6]

The headline mathematical result from the Astra model is the first-ever explicit construction of a non-sofic group. The highly abstract concept of soficity was originally introduced by mathematician Mikhail Gromov in 1999, and the question of whether non-sofic groups actually exist in reality has remained a central, unresolved mystery in group theory for 27 years. Human mathematicians had repeatedly failed to prove or disprove their existence, making Astra's definitive proof a historic moment for the field of algebra.[3][4]

Beyond group theory, the Astra model also successfully disproved Connes's rigidity conjecture regarding von Neumann algebras by constructing infinitely many nonisomorphic groups that share the exact same algebra. Additionally, the AI resolved three distinct problems from Hungarian mathematician Paul Erdős's famous catalog of open mathematical questions. This includes a definitive solution to problem number 183, which concerns the highly complex behavior of multicolored Ramsey numbers, a topic that has frustrated combinatorial mathematicians for decades.[1][3][4]

In the realm of high-dimensional geometry, Astra produced the first major improvement to the general upper bound on high-dimensional sphere-packing density since 1978, characterizing the absolute limits of a major proof method. In theoretical computer science, the model proved an exponential parallel repetition theorem for two-player quantum games, extending a foundational principle from classical complexity theory into the quantum realm. It also established entirely new lower bounds on the circuit complexity of computing the permanent.[1][3]

The workflow used to generate these historic results relied on a tightly integrated, hybrid human-AI pipeline. Astra generated the core mathematical arguments autonomously, exploring pathways that human researchers had previously abandoned. Human mathematicians then used the model to help format these raw arguments into a traditional, readable academic manuscript. Finally, Astra was tasked with translating each of its own natural-language arguments into Lean 4, a strict programming language used specifically for theorem verification.[1][5][7]

Lean 4 acts as an uncompromising computational referee. The language features a small, highly secure checking kernel that only accepts a theorem if every single logical step strictly follows from declared definitions, axioms, and previously proven results. By shipping a Lean certificate for each of the 10 problems, OpenAI provided an auditable, transparent artifact that anyone with a standard compiler can verify independently, entirely removing the need to blindly trust the AI company's corporate claims or the model's fluent but potentially flawed prose.[4][7]

The release of these proofs lands amid escalating tensions between the commercial technology sector and traditional academia. In June 2026, a coalition of mathematicians issued the "Leiden Declaration," a formal warning that AI companies were utilizing published mathematical research without consent and threatening the integrity of traditional peer review by flooding the zone with unverified, AI-generated claims. The declaration demanded greater transparency regarding when and how artificial intelligence is used in mathematical discoveries.[4]

In a direct nod to these academic concerns, OpenAI explicitly credited the AI system with the intellectual heavy lifting for the 10 proofs. The company stated that claiming human authorship for the discoveries would "misrepresent both the system's contribution and the nature of genuine human intellectual work." While OpenAI took full responsibility for the ultimate correctness of the Lean code and the manuscript preparation, it attributed the actual mathematical arguments and the creative leaps entirely to the Astra system.[1][5]

Despite the rigorous presence of Lean 4 certificates, the results are not entirely immune to academic scrutiny. A machine-checked argument is not identical to a peer-reviewed one, and the formalization process itself can introduce subtle errors in translation. The human mathematical community must still manually review the formalized Lean statements to ensure they accurately and faithfully represent the original, real-world conjectures they claim to solve, rather than a simplified version of the problem.[6][7]

Furthermore, Astra remains an unreleased, internal system, meaning its exact capabilities cannot be independently audited. Outside researchers cannot yet test the model to verify how much human "problem-shaping" or prompt engineering was required before the AI could successfully tackle the proofs. The exact ratio of human intuition to machine reasoning remains a corporate black box, leaving some skeptics to wonder if the AI merely executed the final steps of a path heavily guided by OpenAI's internal mathematics team.[6]

Astra is reportedly the internal codename for OpenAI's next major model family, designed specifically to handle long-running, highly complex tasks by coordinating multiple AI agents simultaneously. Unlike current commercial models that answer discrete prompts in a matter of seconds, Astra is built to work continuously and autonomously on complex problems for hours or even days at a time, testing hypotheses and iterating on failures without human intervention.[2]

The model is currently undergoing rigorous internal testing and is expected to be the first system evaluated under new U.S. government regulatory frameworks that require federal safety review before any public launch. OpenAI CEO Sam Altman recently demonstrated Astra's capabilities directly to policymakers and regulators in Washington, D.C., highlighting its potential to act as an autonomous researcher rather than a simple chatbot.[2][3]

The successful resolution of 10 decadal math problems for a mere $2,000 in compute signals a profound transition in the trajectory of artificial intelligence. Frontier models are rapidly evolving from tools that summarize existing knowledge into agentic systems capable of generating net-new, verifiable discoveries at the very edge of human understanding. As these systems become cheaper and more reliable, the fundamental nature of scientific and mathematical research is poised for a historic acceleration.[6][7]