OpenAI’s Astra Solves Ten Open Math Problems for $2,000 — With Machine-Checkable Proofs

TL;DR

OpenAI says an internal version of its next major model, Astra, produced ten new results in mathematics and theoretical computer science — each problem open for at least a decade. The headline result is the first-ever explicit construction of a non-sofic group, a question standing since 1999. Every result ships with a machine-checkable Lean 4 certificate on GitHub, and OpenAI puts the total inference cost at roughly $2,000 at Sol API rates.

Introduction

On August 1, OpenAI published a 249-page manuscript alongside Lean 4 proof certificates for ten previously open problems. This is not a demo: correctness of each result is verified by a computer program, not by trust in the lab that produced it.

The announcement lands at a delicate moment for the relationship between AI labs and the mathematics community. In June, mathematicians issued the Leiden Declaration, endorsed by the International Mathematical Union, warning that AI companies are announcing results through press releases rather than peer-reviewed journals, using published research without consent, and threatening the integrity of proof and attribution. (Source : The Next Web — OpenAI says its next model, Astra, has solved ten open problems in mathematics)

OpenAI is leaning on the one thing that cuts through that debate: proofs a machine can check.

The Ten Results

The headline result is the first explicit construction of a non-sofic group, resolving a central question in group theory that has stood since Mikhail Gromov introduced the concept of soficity in 1999 — 27 years with no proof or disproof from human mathematicians. (Source : The Next Web — OpenAI says its next model, Astra, has solved ten open problems in mathematics)

The other results span several fields:

Result	Field
First explicit construction of a non-sofic group	Group theory (open since 1999)
Disproof of Connes’s rigidity conjecture	Von Neumann algebras
Proof of Ehrhart’s volume conjecture	Geometry of numbers
Three problems from Erdős’s catalogue, including #183 on multicoloured Ramsey numbers	Combinatorics
First improvement to the general upper bound on high-dimensional sphere-packing density since 1978	Discrete geometry
Parallel repetition theorem for two-player quantum games	Quantum information
New lower bounds on the circuit complexity of computing the permanent	Computational complexity

Each proof ships with a Lean certificate and a chain-of-thought walkthrough. OpenAI’s head of mathematics research, Sébastien Bubeck, confirmed the results on X, calling them “beautiful.” (Source : The Next Web — OpenAI says its next model, Astra, has solved ten open problems in mathematics)

Why the $2,000 Number Matters

The total compute cost for all ten solutions was roughly $2,000 at Sol API rates, according to OpenAI. That is the real story for builders: the marginal cost of frontier-grade mathematical reasoning has collapsed to the point where a lab can solve a century of open problems for less than the price of a mid-range laptop.

For comparison, this is the same long-horizon model family that in May disproved the Erdős unit distance conjecture, an 80-year-old problem in discrete geometry. Fields Medalist Tim Gowers said at the time that he would recommend that proof for publication in Annals of Mathematics without hesitation. (Source : The Next Web — OpenAI says its next model, Astra, has solved ten open problems in mathematics)

Thomas Bloom, who runs the erdosproblems website, called the ten new results “big news,” saying they are more significant than the unit distance counterexample. (Source : The Next Web — OpenAI says its next model, Astra, has solved ten open problems in mathematics)

The Verification Strategy: Lean Certificates

The Lean certificates address the key objection the mathematical community has raised about AI-generated proofs: that they are difficult to verify independently.

Machine-checkable proofs can be validated by anyone with the Lean compiler, without trusting the model or its operators. This is a structural shift in how AI research claims can be audited — the artifact itself carries its own verification, decoupled from the reputation of the producing organization. (Source : The Next Web — OpenAI says its next model, Astra, has solved ten open problems in mathematics)

Whether the broader community will accept results announced via blog post rather than peer review remains an open question — the Leiden Declaration was written precisely to answer it. But the verification burden has changed: skeptics can now check the proofs mechanically rather than argue about methodology.

What We Know (and Don’t) About Astra

OpenAI has not said when Astra will be released publicly, describing it only as its “next major model.” Some observers, including investor Mark Kretschmann, speculate Astra is the GPT-6 series. (Source : The Next Web — OpenAI says its next model, Astra, has solved ten open problems in mathematics)

The company is also giving 100,000 academic researchers free access to its frontier models through 2027, deepening its ties to the scientific community while concentrating research infrastructure on its own platform. (Source : The Next Web — OpenAI says its next model, Astra, has solved ten open problems in mathematics)

For AI agent builders, the signal is clear: long-horizon reasoning at this level is no longer theoretical. A model that can hold a multi-hour chain of thought across ten distinct mathematical research programs is a model that can plan, execute and verify complex agentic workflows — the same capabilities, applied to code, infrastructure and business logic.

The Bottom Line for Agent Builders

Three takeaways:

Verifiability is becoming a feature. Lean-certified outputs are the template for agent outputs that need to be trusted — the model’s claims carry their own proof, checkable by anyone.
Reasoning cost is collapsing. $2,000 for ten open-problem solutions redefines what “expensive reasoning” means. Long-horizon planning agents become economically viable for tasks far beyond math.
The academic trust gap is the next battleground. The Leiden Declaration, the press-release announcements, and the free research access are all part of the same negotiation over how AI-discovered knowledge enters the scientific record.

FAQ

Q: Is Astra publicly available? A: No. OpenAI describes it as its “next major model” and has not announced a release date. The results were produced by an internal version.

Q: What is a Lean certificate? A: Lean is an interactive theorem prover. A Lean certificate is a proof file that the Lean compiler can verify mechanically, so correctness does not depend on trusting the model or the lab.

Q: How much did it cost? A: OpenAI estimates roughly $2,000 at Sol API rates for all ten results combined.

Q: Is a non-sofic group construction important? A: Yes. It answers a central question in group theory open since 1999, when Mikhail Gromov introduced the concept of soficity. No human mathematician had proved or disproved the existence of non-sofic groups in 27 years.

Q: Will mathematicians accept these results? A: The verification is mechanical via Lean, which addresses the independence objection. But the Leiden Declaration warns that announcing via press release instead of peer review threatens the integrity of proof and attribution — so acceptance is not guaranteed.

OpenAI's Astra Solves Ten Open Math Problems for $2,000 — With Machine-Checkable Proofs