On the morning of August 2nd, OpenAI did something unusual in the AI industry: instead of releasing a new model first, it published the model's math exam results—an internal version of its next-generation model, Astra, which delivered 10 new results in mathematics and theoretical computer science, all accompanied by Lean(a system where a computer checks proofs line by line) formal proofs, with a total cost of about $2000 according to Sol API(the plug apps use to call the AI) pricing. While Astra is not yet accessible for external testing, its work can be examined through these results.
10 Decade-Old Mysteries, Solved at Once
The 10 results disclosed by OpenAI span high-dimensional geometry, coding theory, group theory, arithmetic circuit complexity, quantum complexity, lattice cryptography, and extremal combinatorics. They share one commonality: each problem has been open for at least 10 years, with some lasting several decades, and three are classic open problems posed by mathematician Paul Erdős.
These include new upper bounds for high-dimensional sphere packing density and coding theory, the construction of non-sofic groups (and the refutation of the Connes rigidity conjecture), a sharp bound for the Ehrhart volume conjecture, new lower bounds for arithmetic circuit complexity, the quantum parallel repetition theorem, super-exponential lower bounds for multicolor Ramsey numbers, and more. Some push known boundaries forward, some construct objects previously unknown to exist, and some directly refute or prove specific conjectures.
OpenAI simultaneously released three materials: a 249-page complete argument paper, a 62-page problem-solving explanation, and a set of Lean 4 formal proof certificates (open-sourced on GitHub). The process is: Astra generates mathematical arguments → Humans and the model collaboratively organize them into papers → The model writes Lean proofs.
$2000: The First Clear Price Tag on Research
OpenAI provided a very tangible number: according to Sol API rates, the total token(the units AI bill by) cost for Astra to find these 10 solutions is approximately $2000 (about RMB 13,500). This number may not seem astronomical, but it is the first time that "original research" has been turned into a quantifiable bill.
For comparison: A team of top human mathematicians might spend years and millions of dollars to solve any one of these problems. 10 problems, $2000—this is not just "a bit cheaper," it's a difference of orders of magnitude. Researcher Noam Brown's comment is quite candid: "Unfortunately, the Millennium Prize Problems(seven famed puzzles, each worth $1 million) have not yet been solved. But we didn't spend much on each problem, and there is still a lot of room for advancement in the calculations."
The significance of $2000 is not about "how much money is saved," but that it proves "AI doing original research" is a reusable, quantifiable process, not a one-time mystical demonstration. When costs can be estimated, investment decisions can be made—the logic of research infrastructure has changed from "maintaining a team" to "receiving a cloud bill."
| Dimension | Top Human Team | Astra |
|---|---|---|
| Time per problem | Months to years | Batch completed 10 problems |
| Cost per problem | Hundreds of thousands to millions in funding | Average about $200 per problem |
| Proof format | Human-readable paper | Paper + Lean machine-verifiable certificate |
| Reproducibility | Depends on peer review | Proofs open source, anyone can verify |
Cost comparison is a rough order-of-magnitude estimate; human team costs vary by problem. Source: OpenAI official + ZhiDongXi, MeiJing cross-reporting.
The Key is Not "Calculating It Out," but "Being Able to Verify"
Many people, upon hearing "AI makes mathematical breakthrough," first reaction is skepticism: Is the AI just "guessing the answer," or even "fabricating the proof"? The way this announcement was made precisely addresses this question—it did not deliver an answer, but a proof chain that can be verified line by line by a machine.
Lean is a formal proof system: it translates human verbal "I think this proof is correct" into machine language, allowing the computer to automatically check each step of the derivation. Whether the proof is correct or not, the machine can judge objectively, leaving no room for ambiguity in textual debate. OpenAI has open-sourced all 10 proofs' Lean certificates to GitHub, meaning anyone with a computer can independently verify—no need to trust OpenAI's words, just trust the machine.
This is the essential difference between "AI doing research" and "AI answering questions": answering questions only requires providing an answer that looks correct, while research must provide verifiable evidence. When "proposing an answer" and "verifying the answer" are first possibly taken over by the same system on a large scale, AI truly steps into the realm of original research.
AI is Becoming Research Infrastructure
Taking a broader view. In the same announcement, OpenAI also launched a program: to 100,000 scientists and mathematicians free access to its most powerful model. Connecting the two events makes it clear—on one hand, using Astra to demonstrate "AI can do original research," on the other hand, integrating the tool into researchers' workflows for free.
This is not the first time OpenAI has demonstrated AI's mathematical capabilities. In May, it publicly announced an AI-discovered counterexample to the Erdős unit distance conjecture, which has spurred several subsequent studies. But this is the first time it has systematically and on a large scale tackled cutting-edge unsolved problems, complete with verifiable evidence.
Of course, we must remain clear-headed: Astra is still an internal version, not yet open; some of the 10 results are "advancing boundaries" rather than "completely solving"; the Millennium Prize Problems still stand. Industry evaluations describe "any one of them as having the weight of a Fields Medal(the highest honor in mathematics)," but whether they can withstand the test of time depends on the follow-up review by mathematicians in each field. However, the direction is clear: this most expensive intellectual activity of humanity is transforming from "the privilege of a few" to "a purchasable service."
No Need to Wait for Model Release, You Can Verify Now
Although Astra itself is not yet accessible, the mathematical results it has produced are fully public. If you or someone you know is doing research, you can do three things right now:
Download the 249-page paper to see what specific problems Astra has solved and what methods it used.
Clone the Lean certificate repository on GitHub and use Lean 4 to independently run the verification—the machine will decide if the proof is correct.
Pay attention to OpenAI's scientist free access program; if your team is doing research, this is a real bonus of computing power(the computing resources needed to run models).
The key here is not "AI calculating the answer," but "AI delivering a proof that can be verified line by line by a machine." When proposing and verifying answers are first possibly taken over by the same system on a large scale, research is transforming from "the privilege of a few" to "a cloud bill that can be estimated." $2000, 10 decade-old mysteries—this number will be remembered for a long time.
The specific mathematical content of the 10 results is subject to the official OpenAI paper; "Fields Medal-level weight" is an industry evaluation, not an official conclusion; Astra's naming and release date are subject to OpenAI's subsequent announcements. Cost figures are calculated according to the Sol API rate and are from OpenAI's official sources.