On August 1, 2026, OpenAI confirmed the existence of its next major model in the third paragraph of a research post about mathematics. The sentence is short: "The results were achieved by an internal version of Astra, our next major model." There is no release date, no pricing, no availability, and no model card. The post is tagged as a publication, not a product.
The name is not the interesting part. The interesting part is the sentence immediately after it: "The total number of tokens needed to find solutions to these problems would cost roughly $2,000 at Sol API rates." OpenAI led with a dollar figure instead of a benchmark score, and then published Lean 4 proofs so anyone can check the claim by machine. Those two choices matter more than the mathematics, because they are the parts that other labs now have to match.
The unit of the frontier argument just changed
For three years the standard frontier claim has been a percentage on a benchmark. MMLU, then GPQA, then SWE-bench, then whichever eval has not yet saturated. Benchmarks are cheap to game, expensive to audit, and almost impossible for a buyer to translate into a decision.
OpenAI's math post does not lead with any of that. It leads with a price. Ten open problems, resolved or substantially advanced, for roughly $2,000 of tokens at published API rates. Greg Brockman put the same number on X the day of the release, framed as "a total cost of about $2000 at Sol API prices."
This is a different kind of claim. A benchmark score tells you where a model sits on a curve. A cost-per-result tells you what a unit of hard cognitive work costs to buy, which is the first frontier claim in a while that a buyer can act on.
It also landed two days after OpenAI's July 30 price cuts, where the headline was also money rather than capability: GPT-5.6 Luna down 80 percent to $0.20 and $1.20 per million input and output tokens, Terra down 20 percent to $2 and $12. Sol, the frontier tier, was left untouched at $5 and $30. Two announcements in three days, both denominated in dollars.
$2,000 is a floor, and a lot of coverage got the unit wrong
Read OpenAI's sentence carefully. "The total number of tokens needed to find solutions to these problems" is the total across all ten, not per problem. Brockman's post says "total cost." Simon Willison, whose link blog is one of the most-read secondary sources on this release, read it the other way and wrote that OpenAI spent "less than $2,000 at GPT-5.6 Sol token prices on each one." That is a tenfold difference in the headline number, sitting in the most-cited commentary on the story. If you are quoting the figure, quote it as roughly $2,000 for all ten.
More importantly, $2,000 is a floor, not a budget. Three things it does not appear to cover, none of them disclosed by OpenAI:
Failed attempts. Noam Brown, who worked on the reasoning methods behind Astra, said on X that OpenAI tried and failed on other problems: "Sadly, no Millennium Prize Problems (yet)." He added, "But also, we didn't spend a lot on each problem. It's possible to push test-time compute much further." Neither line actually says what the $2,000 covers, and OpenAI has not said either. Ernie Davis, in an email quoted by Gary Marcus, treats the exclusion as an inference rather than a disclosure: "It seems a safe bet that this includes only the conjectures where Astra succeeded, not the ones where it failed."
Human labour. Davis also asks what the mathematicians and computer scientists who worked on the project were paid: "I'd be astonished if it was less than $20,000 and would not be surprised if it was upward of $200,000." That is one to two orders of magnitude above the token bill.
Selection. Nobody outside OpenAI knows how many conjectures were attempted. Ten out of ten attempted is a very different result from ten out of ten thousand, and the post does not say.
What $2,000 of Sol actually buys
Sol's published rates are $5 per million input tokens, $0.50 per million cached input, and $30 per million output tokens, with a 1,050,000-token context window. Do the arithmetic on the stated figure.
Output is the expensive side. At $30 per million, $2,000 buys at most about 67 million output tokens across all ten problems combined, so the average is at most 6.7 million output tokens per problem. Input costs a sixth as much, so any realistic input-heavy mix buys more tokens for the same money. That makes 67 million a floor on total tokens and 6.7 million a ceiling on output tokens per problem.
One caveat runs the other way. The same model page states that "Prompts with >272K input tokens are priced at 2x input and 1.5x output for the full request." On million-token maths contexts that surcharge plausibly applied, which would pull both figures down. For scale, the finished manuscript runs about 122,000 words across 249 pages, so the generation-to-publication ratio is on the order of hundreds to one.
That is the number worth carrying around. Not "AI does maths now," but at most a few million generated tokens per hard problem, at a rate you can look up. If you are throwing a genuinely hard problem at a frontier model through ChatGPT or the API, this is the order of magnitude to budget: tens to hundreds of dollars for a serious attempt, not cents. That anchor is useful even though the model that produced the result is not one you can buy.
The Lean certificates are the part rivals now have to match
Every claim in the release is backed by a Lean 4 formalization in the public openai/ten-proofs repository, Apache-2.0 licensed, built against Lean 4.32.0 and mathlib. We read the repository's formalization.yaml manifest. It reports sorry_count: 0 overall and for each of the twelve named main results, and lists exactly three axioms per result: propext, Classical.choice, and Quot.sound. Those are the three standard axioms of Lean's core logic. No extra axioms were introduced to make the proofs go through.
The repository also ships twelve challenge configurations for Comparator, the Lean proof-checking tool, so a third party can re-verify each result independently rather than trusting OpenAI's own build. That is a falsifiability bar. If any of these proofs is wrong, a stranger can demonstrate it in an afternoon with a laptop.
Compare that to the other big "our AI did science" claim of the same fortnight. On July 28, Anthropic's Frontier Red Team published cryptanalysis results from Claude Mythos Preview: a structural weakness in HAWK, a NIST post-quantum signature candidate that had survived two rounds of expert review over two years, plus a 200-800x speedup on an attack against round-reduced AES. Anthropic disclosed responsibly, published papers and demonstration code, and stated that "each of the results cost roughly $100,000 in API cost to develop."
That is honest, careful work. It is also not machine-checkable. Cryptanalysis gets verified by human experts over months. Lean certificates get verified by a compiler in minutes. OpenAI picked a domain where the proof of the claim ships with the claim, which is either a smart choice of ground or a genuine standard.
| OpenAI, Aug 1 2026 | Anthropic, Jul 28 2026 | |
|---|---|---|
| Model | Internal version of Astra | Claude Mythos Preview |
| Domain | 10 open problems in maths and TCS | HAWK signature scheme, round-reduced AES |
| Stated cost | ~$2,000 total, all ten, at Sol rates | ~$100,000 per result, API cost |
| Sol/API rate cited | $5 in / $30 out per 1M tokens | Not broken out |
| Verification | Lean 4 certificates, 0 sorries, public repo | Papers plus demo code, expert review |
| Independent re-check | Yes, 12 Comparator configs shipped | Human review, CryptanalysisBench |
| Human role stated | Humans prepared manuscripts with same model | 1 researcher with Claude; autonomous scaffold for AES |
| Model availability | None. No date, no price | Mythos Preview available |
The 249-page paper never mentions the model
Here is the gap nobody is talking about. We downloaded the full 249-page manuscript and the 62-page reasoning walkthrough and searched both. The word "Astra" appears zero times. "Lean" appears zero times. "Token" appears zero times. "OpenAI" appears twice, once as the author line and once in the abstract's phrase "an internal OpenAI model." "GPT" appears once, in an acknowledgement noting concurrent independent work by Shuoxing Zhou on the Connes rigidity counterexample, "developed in part with the assistance of GPT-5.6 Sol."
So the model name, the cost figure and the verification method are all in the blog post and the repository metadata. None of it is in the paper. Gary Marcus's complaint is accurate on the facts: "not one page is about how the model works, how the proofs were verified, what role if any humans played, whether any of the proposed proofs had errors."
The repository manifest is more revealing than the paper. Under review it says status: "agent-reviewed". Under automation it lists method "agent", model "Astra (OpenAI)", framework "Codex", and wall time "one week". Agent-reviewed is not human-reviewed, and on the manifest's own account the Lean compiler is the guarantee rather than a mathematician's read. The blog post claims more than the manifest does: "We helped prepare the manuscripts and formalize the proofs in Lean, and we take responsibility for their correctness, while the mathematical arguments themselves were generated by our system." Corporate accountability and an agent-run review step are not the same guarantee, and the release never reconciles them. Either way the guarantee is narrower than the headlines imply: Lean certifies that the formal statement follows, not that the formal statement faithfully captures the informal conjecture. That gap is where formalization errors actually live.
What this means if you are picking a tool
Practically, almost nothing changes this week. Astra is not purchasable. Reporting from The Information, relayed by The Decoder, says Astra is expected to be the first model routed through a new US federal pre-release review framework, and that no decision has been made on whether it ships as GPT-6 or a GPT-5.x point release. We could not confirm that framework from a primary government source, so treat it as reporting.
What you can act on is the price anchor. The best model you can actually buy today is the one in the acknowledgement, not the one in the headline: a working mathematician reached a comparable Connes rigidity result concurrently with help from GPT-5.6 Sol, a model that has been generally available for weeks. The gap between "frontier demo" and "shipping tier" on this class of work is smaller than a $2,000 headline suggests.
If you are choosing between assistants for research-grade work, the practical comparison is still between shipping models, which we cover in ChatGPT vs Claude vs Gemini, and the tooling around them matters as much as the model, which is the argument in our AI research tools roundup. Claude currently has the stronger public track record on autonomous long-horizon security research, on the evidence of the Mythos cryptanalysis work. Neither lab has shipped its headline model.
What we could not verify
We read the OpenAI post, both PDFs, the GitHub repository and the pricing documentation directly. We did not verify the mathematics and did not build the Lean project. We could not verify how many problems were attempted and failed, what human labour cost, whether the $2,000 includes retries, or what the reasoning traces contain, since the prompts and traces are unpublished. Willison's request is the right one: "I want to see the prompts they used." Until those exist, the cost figure is a marketing number with a checkable artifact attached, which is better than most, and still not a reproducible result.
FAQ
What is OpenAI Astra? Astra is the name OpenAI gave its next major model, confirmed publicly for the first time on August 1, 2026, in a research post about mathematics. OpenAI described the system used for the results as "an internal version of Astra, our next major model." No release date, pricing, benchmark table or availability has been announced. It is not in ChatGPT or the API, and OpenAI has not said whether it will ship as GPT-6 or as a point release in the GPT-5 line.
Does the $2,000 figure cover all ten math problems or one? All ten. OpenAI's wording is "the total number of tokens needed to find solutions to these problems," and Greg Brockman described it on X as "a total cost of about $2000 at Sol API prices." Some coverage read it as $2,000 per problem, which is a tenfold overstatement. The figure is a token cost only, so human labour sits outside it, and OpenAI has not said whether failed attempts are counted.
What are Sol API rates? Sol is the frontier tier of OpenAI's GPT-5.6 family. Its published rates are $5 per million input tokens, $0.50 per million cached input tokens, and $30 per million output tokens, with a 1,050,000-token context window. Prompts above 272,000 input tokens are billed at 2x input and 1.5x output for the whole request. Those rates were unchanged in the July 30, 2026 price cut, which reduced Luna by 80 percent and Terra by 20 percent.
What are the Lean certificates and why do they matter? Lean 4 is a proof assistant. A Lean certificate is a formal proof a computer can check without trusting the author. OpenAI published one per result in the openai/ten-proofs repository under Apache-2.0, with a manifest reporting zero incomplete proofs and only Lean's three standard axioms. It also shipped Comparator configurations so third parties can re-verify independently. That makes the claims falsifiable in a way most AI research announcements are not.
Can I use Astra today? No. There is no public access, no API model ID, no waitlist and no announced date. The most capable OpenAI model you can actually use is GPT-5.6 Sol, available through the API, Codex and ChatGPT paid tiers. Notably, the paper's own acknowledgements credit concurrent independent work on one of the ten results developed with help from GPT-5.6 Sol, which suggests the shipping tier is already useful on this class of problem.
What were the ten results? High-dimensional sphere packing, binary and spherical codes, the existence of non-sofic groups, a counterexample to Connes's rigidity conjecture, arithmetic circuit and formula lower bounds for the permanent, exponential quantum parallel repetition, polynomial-factor hardness for the closest vector problem, Ehrhart's volume conjecture (formalized in the repository as a sharp Ehrhart volume inequality), multicolor triangle Ramsey numbers resolving Erdos problem 183, and results on the compactness and degeneracy conjectures in extremal graph theory.
How does this compare with Anthropic's cryptography results? Anthropic published cryptanalysis findings from Claude Mythos Preview on July 28, 2026, weakening the HAWK post-quantum signature candidate and speeding up an attack on round-reduced AES by 200 to 800 times, at roughly $100,000 in API cost per result. Anthropic's results are stronger evidence of autonomous open-ended research. OpenAI's are cheaper and machine-checkable. They are different claims, not directly comparable.
Does any of this affect encryption or security today? Not directly. OpenAI's closest vector problem result concerns the theoretical hardness of a lattice problem underpinning post-quantum cryptography, which is a complexity-theory advance rather than an attack on deployed systems. Anthropic stated explicitly that neither of its findings affects production software, since HAWK is an unstandardised candidate and the AES attack targets a reduced-round variant. No system needs patching because of either release.
Sources
- https://openai.com/index/ten-advances-in-mathematics/
- https://cdn.openai.com/pdf/ten-proofs-oai.pdf
- https://cdn.openai.com/pdf/reasoning-walkthroughs.pdf
- https://github.com/openai/ten-proofs
- https://raw.githubusercontent.com/openai/ten-proofs/main/formalization.yaml
- https://developers.openai.com/api/docs/models/gpt-5.6-sol
- https://openai.com/index/advancing-the-price-performance-frontier-with-gpt-5-6/
- https://simonwillison.net/2026/Aug/1/ten-advances-in-mathematics/
- https://www.anthropic.com/research/discovering-cryptographic-weaknesses
- https://garymarcus.substack.com/p/openais-amazing-but-vastly-oversold
- https://the-decoder.com/openai-announces-its-next-major-model-astra-by-dropping-ten-previously-unsolved-math-solutions/