On August 1, 2026, OpenAI published a paper titled "Ten advances in mathematics and theoretical computer science." The results were generated by an internal version of Astra, a model family the company has not yet released. Mathematicians had made no progress on any of these ten problems for at least a decade. Some had been open for much longer. The total token cost to find all ten solutions would run about $2,000 at OpenAI's Sol API rates.

That number deserves a second read. Problems that resisted the combined effort of professional mathematicians for years β€” some for decades β€” fell to a model running test-time compute for the price of a decent laptop.

What Astra actually solved

The ten results span a wide range of mathematics and theoretical computer science. Here are the highlights that matter most.

The biggest headline is probably the existence of non-sofic groups. This has been a central open question in group theory for years. A group is sofic if it can be approximated by finite symmetric groups in a specific technical sense. Most groups mathematicians encounter in practice are sofic, and there was genuine debate about whether non-sofic groups even existed. Astra settled it: they do.

Then there is Connes's rigidity conjecture, which proposed that certain groups are uniquely determined by their von Neumann algebras. Astra disproved it. That is not a small result β€” it overturns a longstanding assumption in operator algebras.

On the geometry side, the model produced new upper bounds on high-dimensional sphere-packing density down to the Cohn-Elkies threshold. It also resolved Ehrhart's volume conjecture, determining in every dimension the maximum possible volume of a convex body whose centroid is its only interior lattice point. Both of these had been stubbornly resistant to progress.

In coding theory, Astra delivered exponentially improved bounds on the maximum size of binary codes at any prescribed minimum distance, with analogous results for high-dimensional spherical codes. In arithmetic circuit complexity, it established new lower bounds for computing the permanent, including an arithmetic-formula lower bound of order n⁴/log n.

The list goes on. An exponential parallel repetition theorem for general two-player quantum games. Polynomial-factor hardness of approximation for the closest vector problem, which is foundational to lattice-based post-quantum cryptography. A superexponential lower bound for multicolor triangle Ramsey numbers, resolving ErdΕ‘s problem 183. And results on the compactness and degeneracy conjectures in extremal graph theory, resolving ErdΕ‘s problems 146 and 180.

Every one of these is a result that would normally be the headline of a career-defining paper.

The $2,000 question

The cost figure is the part that gets people's attention, and it should. OpenAI says the total tokens needed to find all ten solutions would cost roughly $2,000 at Sol API rates. That does not include the human work to prepare the manuscripts and formalize the proofs β€” OpenAI's researchers helped with that, and the proofs were checked in Lean, a formal verification language. But the mathematical arguments themselves came from the model.

Noam Brown, one of the researchers behind the test-time reasoning technology Astra uses, was candid about the limits. "Sadly, no Millennium Prize Problems (yet)," he wrote on X. "But also, we didn't spend a lot on each problem. It's possible to push test-time compute much further." He called Astra a "major step for scientific reasoning."

That framing matters. These ten results were achieved without dedicating massive compute budgets to each individual problem. The implication is that harder problems β€” potentially including some of the seven Millennium Prize Problems, each carrying a $1 million bounty β€” might be reachable if you throw more test-time compute at them. Nobody has done that yet. But the door is open.

The name and the roadmap

Astra is OpenAI's next major model family. The name had been rumored for weeks β€” Sam Altman demoed it to politicians and regulators in Washington, D.C. in late July. The Information reported that Astra would sit alongside OpenAI's existing Sol, Terra, and Luna families. Whether it ships as GPT-6 or as a variant within the GPT-5 line has not been decided. There is no release date.

What makes Astra different from previous models is its focus on long-running tasks. OpenAI's Chief Scientist Jakub Pachocki said last year that the company wants to build AI systems that can work on a problem for hours or days, not just respond in seconds. Astra appears to be the first concrete result of that ambition. The model coordinates multiple agents over extended periods, and the math results demonstrate that this approach works on genuinely hard problems.

The model is also expected to be the first to go through the Trump administration's new AI regulatory framework, which would require AI models to be submitted to the federal government before public release.

What the math community thinks

Thomas Bloom, a University of Manchester mathematician who runs erdosproblems.com, called the results "big news" on X β€” and he would know, given that three of the ten results resolve problems from his site. He considered them more significant than the counterexample to the unit-distance conjecture that OpenAI published in May, which was already a major headline.

Bloom also pushed back on the idea that AI is replacing mathematicians. "The claim makes little sense when the AI draws on more than a century of mathematical theory, was built by mathematicians, and was trained on everything mathematicians have ever wrote," he argued. That is a fair point. Astra did not invent mathematics. It synthesized and extended existing knowledge at a scale and speed that humans cannot match.

OpenAI itself took a notably careful approach to credit. The company said that 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. They published the model's reasoning walkthroughs alongside the formal papers, making the process transparent rather than treating the AI as a black box.

Why this matters beyond math

If you are not a mathematician, you might wonder why you should care. Here is the reason: these results are a proof of concept for something much bigger.

The same test-time compute approach that cracked ten open math problems is being built into the next generation of AI systems designed for long-running, multi-step tasks. Research, engineering, drug discovery, materials science β€” any field where progress depends on sustained reasoning over complex problems is now in play.

The $2,000 price tag also shifts the economics of AI-assisted research. If a model can produce results that would take a human researcher months or years, and it costs less than a conference registration fee, the bottleneck is no longer compute or money. It is knowing which problems to point the model at.

Astra is not publicly available yet. But the results are real, the proofs are formalized in Lean, and the reasoning walkthroughs are published. Whatever comes next, August 1, 2026 is the date the goalposts moved.

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