Ten advances in mathematics and theoretical computer science
OpenAI reports that an internal, unreleased version of its next model, Astra, resolved or advanced ten long-standing open problems across mathematics and theoretical computer science.
A concrete, Lean-formalized example of a frontier lab's model producing checkable new mathematics, paired with an explicit public stance on how to attribute AI-generated proofs.
Two readings, equal authority
How to choose: The paper’s words is verbatim — use it when you need to quote, or to judge how they write. Plain language is a paraphrase written for comprehension — use it when you want the idea fast. Neither is a summary of the other; they are two doors into the same room.
This source carries no verbatim abstract.
OpenAI says an internal, unreleased version of its upcoming model Astra produced new results on ten open problems spanning geometry, coding theory, complexity theory, group theory, operator algebras, quantum complexity, lattice cryptography, and combinatorics. Humans worked with the same model to write the arguments up as manuscripts, the model formalized each proof in the Lean proof assistant, and OpenAI is releasing each solution along with the model's own narration of its reasoning.
What they actually did
Each step is a synthesis. Open any step to see the paper’s own sentence it was derived from, with its locator — so nothing here floats free of the source.
- Selected ten long-standing open problems spanning high-dimensional geometry, coding theory, arithmetic circuit complexity, group theory, operator algebras, quantum complexity, lattice cryptography, and extremal combinatorics.
Trace this step to the paper
“These problems span high-dimensional geometry, coding theory, arithmetic circuit complexity, group theory, operator algebras, quantum complexity, lattice cryptography and extremal combinatorics.”The results (intro)
- Had an internal version of Astra, OpenAI's next major model, generate the mathematical argument for each problem.
Trace this step to the paper
“The results were achieved by an internal version of Astra, our next major model.”The results
- Estimated the compute spend needed to find the ten solutions in dollar terms.
Trace this step to the paper
“The total number of tokens needed to find solutions to these problems would cost roughly $2,000 at Sol API rates.”The results
- Had humans work with the same model to turn its arguments into written manuscripts.
Trace this step to the paper
“These arguments were then prepared into manuscripts by humans with the same model.”The results
- Had the model formalize each argument as a machine-checked proof.
Trace this step to the paper
“Afterward, the model formalized each argument in a Lean certificate”The results
- Released, alongside each solution, the model's own narration of its thinking process.
Trace this step to the paper
“We are also releasing for each solution a model’s narration of its thinking process.”The results
Exactly what was run, and how
| Model | Developer | Temp | Effort / reasoning | Deployment | Other settings |
|---|---|---|---|---|---|
| Astra | OpenAI | not reported | not reported | unstated | The total number of tokens needed to find solutions to these problems would cost roughly $2,000 at Sol API rates. |
Source for Astra settings
“The results were achieved by an internal version of Astra, our next major model.”The results
What they reported — and what they left out
The post names the model (an internal, unreleased version of Astra, OpenAI's next major model) and gives a rough aggregate token-cost figure, but reports no temperature, sampling settings, reasoning-effort level, or deployment details beyond calling it an internal version.
The numbers they report
New upper bounds on sphere-packing density were established.
See it in the paper
“New upper bounds on sphere-packing density down to the Cohn–Elkies threshold.”The results
Exponentially improved bounds on the maximum size of binary codes at a given minimum distance, plus analogous results for spherical codes.
See it in the paper
“Exponentially improved bounds on the maximum size of binary codes at any prescribed minimum distance, with analogous results for high-dimensional spherical codes.”The results
A construction proving that non-sofic groups exist, settling a central open question in group theory.
See it in the paper
“A construction establishing the existence of non-sofic groups, addressing a central open question in group theory.”The results
Disproved Connes's rigidity conjecture, that certain groups are uniquely determined by their von Neumann algebras.
See it in the paper
“Disproof of a longstanding conjecture that certain groups are uniquely determined by their von Neumann algebras.”The results
New lower bounds for computing the permanent with arithmetic circuits and formulas.
arithmetic-formula lower bound of order n 4 /log n
See it in the paper
“New lower bounds for computing the permanent using arithmetic circuits and formulas, including an arithmetic-formula lower bound of order n 4 /log n.”The results
Proved an exponential parallel repetition theorem for general two-player quantum games.
See it in the paper
“An exponential parallel repetition theorem for general two-player quantum games, extending a foundational principle from classical complexity theory.”The results
Established polynomial-factor hardness of approximation for the closest vector problem, relevant to post-quantum cryptography.
See it in the paper
“Polynomial-factor hardness of approximation for the closest vector problem, a foundational lattice question related to post-quantum cryptography.”The results
Determined, in every dimension, the maximum volume of a convex body whose centroid is its only interior lattice point.
See it in the paper
“Determining, in every dimension, the maximum possible volume of a convex body whose centroid is its only interior lattice point.”The results
Proved a superexponential lower bound for multicolor triangle Ramsey numbers, resolving a named Erdős problem.
Erdős problem 183
See it in the paper
“A superexponential lower bound for multicolor triangle Ramsey numbers, resolving Erdős problem 183.”The results
Resolved the compactness and degeneracy conjectures in extremal graph theory, closing two named Erdős problems.
Erdős problems 146 and 180
See it in the paper
“Results on the compactness and degeneracy conjectures in extremal graph theory, resolving Erdős problems 146 and 180.”The results
What they assert, beside what they showed
Left is the claim in the paper’s own words. Right is the data offered for it. Where the two do not fully meet, a gold band names the distance.
The mathematical arguments behind each result were generated by the model itself; humans and the model together only handled writeup and formalization.
“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.”
“These arguments were then prepared into manuscripts by humans with the same model. Afterward, the model formalized each argument in a Lean certificate”
The resultsEach of the ten selected problems is a substantial, genuinely difficult open problem, not an easy or manufactured one.
“All of these problems are of substantial interest to their respective mathematical communities, and several are of broad interest across mathematics as a whole.”
“we are sharing a selection of ten results, each of which resolves or makes substantial progress on a long-standing open problem.”
The results (intro)OpenAI's earlier public math disclosure (the Erdős unit-distance disproof) has already produced real downstream research.
“This work has already inspired further developments in mathematics and theoretical computer science”
“Subsequent research includes Bloom, Sawin, Schildkraut, and Zhelezov,”
FootnoteClaiming human authorship for an entirely AI-generated proof would be a misrepresentation.
“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.”
“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.”
Responsibility to the mathematical communityHow they frame it, and what they want next
Their framing
OpenAI frames the release primarily as a responsibility and attribution story rather than a pure capabilities showcase: it opens with free access for mathematicians, states plainly that the model generated the arguments while humans only helped with writeup and formalization, and repeatedly acknowledges community concern about AI's role in mathematics.
Register: The post states its mathematical results plainly and without qualification ('the results were achieved by...'), reserving careful, hedged language for the attribution and responsibility discussion rather than for the results themselves.
Where they hedge
“There are many views as to the role of AI in mathematics, and we have deep respect and understanding for those concerned with its impact”Responsibility to the mathematical community
What they say it means
- By publicly committing to credit the system rather than claiming human authorship, OpenAI sets an attribution norm that other labs' AI-for-math announcements may be measured against.
the paper’s words
“We believe attribution should honestly reflect how a result was produced”Responsibility to the mathematical community
What they call for next
- Calls on the mathematical community to engage with, contextualize, and build new research on top of these results.
the paper’s words
“We hope the mathematical community will engage deeply with these results, place them in context, and bring the ideas behind them to life through new research and discovery.”Responsibility to the mathematical community
Moves worth stealing
Leads with a public-good access announcement before the results themselves, framing a capabilities showcase as a service to the field.
“That is why we recently announced ChatGPT for Academic Researchers”
States a hard normative rule about AI attribution inside a capabilities blog post, pre-empting criticism about credit-taking rather than leaving it to a footnote or FAQ.
“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.”
Translates raw compute into a plain dollar figure rather than a token count, making cost legible to a non-technical reader.
“The total number of tokens needed to find solutions to these problems would cost roughly $2,000 at Sol API rates.”
Where else this leads
Same territory
- Formalizing Fermat's Last Theorem Anthropic
lean mathematics - Learning more about Claude's mathematical capabilities Anthropic
mathematics
Published alongside it
The nearest publications in time, across all three labs.
- How enabling two settings tripled our scores on the ARC-AGI-3 benchmark OpenAI
2026-07-29 - A moral Turing test: How belief and source shape detection of and agreement with LLM judgments Google DeepMind
2026-08-05 - Discovering cryptographic weaknesses with Claude Anthropic
2026-07-28 - Visual prompt engineering for video models Google DeepMind
2026-07-28
What this page was built from
Working from OpenAI's blog/publication landing page text (intro paragraphs, the ten result bullets, the responsibility section, and the footnote) plus site navigation chrome; the linked full paper and 'reasoning walkthroughs' are not present in this source text, and no separate formal abstract is provided.