Ten Advances in Mathematics and Theoretical Computer Science is a publication released by OpenAI on August 1, 2026, consisting of an announcement page, a 249-page paper, a set of documents narrating the model's reasoning on each problem, and Lean formalizations released at github.com/openai/ten-proofs. It presents ten results across mathematics and theoretical computer science that OpenAI states were produced by an internal version of Astra, its next major model family, and is the first document in which the company uses that name. The paper carries no individual authors: it is bylined "OpenAI," and its abstract describes the results as "obtained by an internal OpenAI model," without naming Astra.
The ten results
The announcement page and the paper's abstract state the results at different levels of precision; the paper is sharper on several. The statements below follow the paper's abstract, noting where the announcement differs.
| # | Area | Result as stated in the paper |
|---|---|---|
| 1 | High-dimensional sphere packing | The asymptotic strength of the Cohn–Elkies linear program is determined exactly, giving an improved general packing bound in high dimensions and settling the corresponding Fourier sign-uncertainty problem asymptotically |
| 2 | Binary and spherical codes | Classical upper bounds for fixed-distance binary and spherical codes are improved by exponential factors for all parameters; the spherical construction also recovers the sphere-packing exponent of Chapter 1 |
| 3 | Non-sofic groups | An explicit non-sofic group is constructed, resolving whether every countable group admits finite permutation approximations; the argument uses property-(T) expanders and the binary Leavitt algebra |
| 4 | Connes's rigidity conjecture | Infinitely many pairwise nonisomorphic property-(T) groups are constructed with the same group von Neumann algebra, disproving the conjecture and answering a related finite-to-one question of Popa |
| 5 | Arithmetic circuit complexity | For the permanent, division-free circuits require Ω(n² log log n) gates and formulas require Ω(n⁴/log n) leaves |
| 6 | Quantum parallel repetition | Exponential parallel repetition is proved for every finite two-player entangled game, extending the classical repetition principle beyond previously treated special classes |
| 7 | Closest vector problem | A direct reduction from 3SAT gives n^(1/400)-factor hardness for the Euclidean closest vector problem, with consequences for binary decoding and other lattice norms |
| 8 | Ehrhart's volume conjecture | The sharp bound (n+1)ⁿ/n! is proved in every dimension for convex bodies whose barycenter is their only interior lattice point |
| 9 | Multicolor Ramsey numbers | A superexponential lower bound proves R_k(3) = k^Θ(k), resolving Erdős problem 183 |
| 10 | Compactness and degeneracy | Separate bipartite graph constructions disprove the compactness conjecture of Erdős and Simonovits and a degeneracy conjecture of Erdős, resolving Erdős problems 146 and 180 |
Three of these are stated more weakly on the announcement page than in the paper. The announcement gives only the arithmetic-formula bound of order n⁴/log n for result 5, omitting the Ω(n² log log n) circuit bound; describes result 7 as "polynomial-factor hardness" without the exponent 1/400; and describes result 3 as "addressing a central open question in group theory" rather than naming the question resolved (Source: openai.com; cdn.openai.com).
The paper is organized as a sequence of self-contained chapters, each with its own abstract, introduction, references and appendices, and cross-referencing the others as companion papers. Chapter 1 states its main theorem as lim(d→∞) LP_d^(1/d) = √(e/(2π)), yielding a packing bound Δ_d ≤ LP_d = 2^(−(α+o(1))d) with α = ½log₂(2π/e) = 0.6044…, which the chapter describes as the first improvement since 1978 to the general sphere-packing exponent. Its second theorem gives lim(d→∞) A₊(d)/√d = lim(d→∞) A₋(d)/√d = 1/π for the Fourier-eigenfunction sign-uncertainty constants (Source: cdn.openai.com).
Method and cost as disclosed
OpenAI states that the total number of tokens needed to find solutions to these problems would cost roughly $2,000 at Sol API rates — Sol being the top tier of the GPT-5.6 family — that the arguments were then prepared into manuscripts by humans using the same model, and that the model afterwards formalized each argument in a Lean certificate. The company says it published, for each solution, a narration of the model's thinking process (Source: openai.com).
Beyond these sentences on the announcement page, the publication describes no methodology. The paper's front matter, abstract and first chapter contain no methods section, no account of how problems were selected or how many were attempted, no description of the human role in preparing the manuscripts, no discussion of prompting, and no statement of whether any proposed proofs contained errors. The word "Lean" does not appear in that material, nor does the name Astra.
Attribution stance
OpenAI accompanied the results with a statement on authorship that cites the signers of the Leiden declaration on AI and Mathematics (leidendeclaration.ai), for whose concerns it expresses "deep respect and understanding." The company states that "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," that it helped prepare the manuscripts and formalize the proofs in Lean and takes responsibility for their correctness, "while the mathematical arguments themselves were generated by our system." It asks the mathematical community to engage with the results and place them in context (Source: openai.com).
The page situates the work against OpenAI's May 2026 release of an AI-generated disproof of the Erdős unit-distance conjecture, and names five subsequent arXiv preprints it says that earlier work inspired: Bloom, Sawin, Schildkraut and Zhelezov on the sum-product conjecture over the reals; Pohoata on split primes and the Elekes–Rónyai problem; Saha, Xu and Ye on the furthest-pair problem under SETH; Goh and Hatami on the communication complexity of point-line incidences; and Lee, Pohoata and Zhu on repeated distances in the Minkowski grid. It also ties the release to ChatGPT for Academic Researchers, which OpenAI describes as providing 100,000 scientists and mathematicians with free access to its best ChatGPT models.
Reception
Thomas Bloom, the University of Manchester mathematician who curates the Erdős problems catalogue, is quoted describing the results as "big news," though the quotation reaches the record through a secondary aggregator rather than a first-hand report (Source: yellow.com). OpenAI researcher Noam Brown wrote that the internal version "solved 10 major open problems in mathematics, quantum complexity, and theoretical computer science."
Gary Marcus published a critique on August 2, 2026 arguing that the reaction commits a fallacy of composition, treating competence at formalizable mathematics as evidence of imminent competence across domains, and that mathematics is a special case precisely because symbolic tools permit verification and the cheap generation of correct synthetic data: "You can verify math; you can't verify a military strategy in the same way." He notes that the paper says nothing about how the model works, how the proofs were verified, what role humans played, or whether any proposed proofs contained errors, and characterizes the announcement as marketing rather than science (OpenAI's amazing — but vastly oversold — new model Astra (Marcus, August 2026)).
Ernie Davis, quoted at length in the same post, raises two evaluation questions. The first is the selection denominator: how many conjectures were attempted, since ten successes drawn at random from all open conjectures would mean something different from ten drawn from a cherry-picked fifty. The second is cost: the $2,000 figure covers successful runs only and excludes the salaries of the mathematicians and computer scientists involved, which Davis estimates at not less than $20,000 and possibly upward of $200,000. Davis also disputes the historical framing, noting that 14 of David Hilbert's 23 problems have been solved since 1900, and points to Kevin Buzzard's multi-year Lean formalization of Wiles's proof of Fermat's Last Theorem as autoformalization work no current system can carry out. Henry Yuen criticized the exposition of one proof, saying it elaborates at length on boilerplate setup and then introduces key steps without support (OpenAI's amazing — but vastly oversold — new model Astra (Marcus, August 2026)).
Levent Alpöge, a mathematician at Anthropic, said in a post that became public on August 2, 2026 that he had obtained five of the ten results within 24 hours using Claude Fable, a model released on June 9, 2026, from generic prompts without internet access, naming arithmetic circuit complexity, quantum parallel repetition and the closest vector problem among them (Source: indiatoday.in). The claim rests on a single social-media post by an interested party and has not been independently confirmed. If borne out, it would place part of the demonstrated capability outside the unreleased family.
Provenance
The announcement page was retrieved from the canonical OpenAI host on August 3, 2026 and read in full. The paper PDF was retrieved from cdn.openai.com on the same date; its metadata reports 249 pages, confirming the page count previously attributed to Marcus's reading. Of those, the front matter, abstract and Chapter 1 — 25 pages, roughly a tenth of the document — were read; the remaining chapters, the reasoning-walkthrough documents and the Lean certificates have not been examined. Claims above about the absence of a methodology section are therefore stated for the material read, which is where such a section would conventionally appear.
Confidence is medium rather than high: the document's contents are directly verified against the primary files, but the capability claims it advances are contested on the record by three independent lines of criticism, most of the paper is unread, and frontier-model capability claims sit in the three-month fast-decay window.
Relationships
- supports: Astra — the only capability evidence OpenAI has published under the name
- supports: AI for Science — machine-produced original mathematical results
- related: OpenAI — publisher
- contradicts: Claude Fable 5 — Alpöge's reported reproduction of five results with an already-released model bears against the results being specific to the unreleased family
- related: Gary Marcus — principal published critic
- related: Safety and Alignment in an Era of Long-Horizon Models (OpenAI, July 2026) — prior disclosure concerning an unreleased long-horizon model of the same general description
- related: GPT-5.6 (Sol, Terra, Luna) — supplies the Sol API rate used for the cost figure