OpenAI Claims Internal Astra Model Solved 10 Open Problems in Mathematics for Around £1,600

OpenAI has posted that an internal version of Astra — described as its next major model family — produced ten new results on long-standing open problems in mathematics and theoretical computer science, at a total token cost of roughly $2,000 (around £1,600) at Sol API rates. That price tag, relative to the complexity of the problems involved, is the detail that has stopped researchers mid-scroll.

The announcement was framed not as a product launch but as a research release. OpenAI said it is publishing a 249-page manuscript collection, model-written reasoning walkthroughs, and Lean 4 formal certificates for all ten results — meaning the claims can, in principle, be machine-checked by anyone with the tools and inclination to do so.

What Problems Did Astra Actually Tackle?

The ten results span a range of areas that sit at the harder end of modern mathematics and computer science. OpenAI listed sphere packing, coding theory, group theory, quantum complexity, lattice cryptography, and extremal combinatorics among the topics covered. Two results were highlighted specifically: a claimed proof of the existence of non-sofic groups, a problem in operator algebras that has been open for decades, and progress on exponential bounds in high-dimensional sphere packing.

These aren’t exam questions. Non-sofic groups, for instance, sit within a corner of group theory that most professional mathematicians never need to visit. The sphere packing problem in high dimensions has implications that stretch into coding theory and the design of error-correcting codes used in data transmission.

The breadth matters. Ten results across seven distinct mathematical fields, if they hold up, would be a different kind of claim from a single isolated proof.

Why the Lean 4 Certificates Are the Key Detail

Machine-checkable proofs change the verification picture considerably. Standard AI research announcements often rely on the reader trusting that the model’s output is correct, or waiting for human experts to work through it. Lean 4 is a formal proof assistant — software that verifies logical steps mechanically, without human judgement calls. If a Lean 4 certificate exists for a result, it means the logical structure of the proof has passed an automated check.

That doesn’t mean the result is necessarily correct in every sense a mathematician would care about. Formal verification checks that the logic follows from the stated premises; it doesn’t independently confirm that the premises themselves capture the intended mathematical object. But it’s a much higher bar than a text summary, and it makes the claims inspectable in a way that matters to the research community.

The 249-page manuscript collection and reasoning walkthroughs add another layer. Mathematicians and theoretical computer scientists can read the model’s working, not just its conclusions.

The Cost Figure and What It Suggests

Around £1,600 to generate ten results on open mathematical problems is the kind of number that prompts a double-take. Whether that figure is genuinely meaningful depends on questions the available sources don’t fully answer — including how much human guidance shaped the model’s direction, how many token-hours were spent on approaches that didn’t work, and whether the $2,000 figure covers the full compute cost or only the final successful runs.

OpenAI’s framing presents it as the cost of finding the solutions, which is a specific claim worth treating carefully until independent researchers have had time to examine the materials. The company has not, based on available reporting, provided a full accounting of the total compute used in development.

None of that makes the announcement unimportant. But scepticism is reasonable. AI-generated mathematical results have been announced before and later found to contain errors that formal certificates didn’t catch — usually because the formalisation itself contained a subtle mistake. The research community will work through these manuscripts, and that process will take time.

Independent Verification Is Still Ahead

The results have not, as of this announcement, been through conventional peer review. OpenAI’s position is that the supporting materials have been released precisely so that independent examination can happen. Mathematicians and theoretical computer scientists working in the relevant fields — quantum games, arithmetic circuit complexity, lattice problems — are the people best placed to assess whether the results represent genuine advances or contain gaps.

SiliconANGLE, which covered the release, described the package as including machine-checkable proofs, and treated the announcement as a significant research development. Critical or sceptical voices from named researchers weren’t available in the sources at the time of writing.

The honest position is that the materials exist, the formal certificates exist, and the scrutiny is now underway.

What This Means for Kent Residents

The direct day-to-day impact on people in Kent is minimal for now — this is a global research story aimed squarely at mathematicians and computer scientists. But lattice cryptography, one of the fields covered in OpenAI’s release, underpins the next generation of encryption standards being adopted by governments and banks worldwide, so advances there eventually touch everyone who uses online banking or sends sensitive data. Academics at Kent’s universities with interests in mathematics, formal verification, or AI research may find the published proofs and Lean 4 certificates worth examining directly.