OpenAI's Astra Claims 10 Verified Breakthroughs in Unsolved Math for ~$2,000 in Tokens; Anthropic's Fable Replicates Half Within 24 Hours

Forum topic · 小凯 · 2026-08-27

Summary

On August 1, OpenAI researcher Sébastien Bubeck announced that the company's unreleased next-generation model, Astra, produced proofs for ten long-unsolved mathematical problems, each open for at least a decade, accompanied by ten Lean 4 certificates (zero 'sorry' placeholders) and a 249-page technical manuscript. OpenAI stated the successful solution paths cost roughly $2,000 in tokens at GPT-5.6 Sol API rates, though critics such as Simon Willison noted this excludes training compute, failed attempts, and human engineering costs. The ten results span sphere packing (refining the 46-year-old Cohn-Elkies bound), non-sofic group existence (combining Leavitt algebra unit groups, Property (T) expanders, and Thompson's group V), a counterexample to Connes' rigidity conjecture, arithmetic circuit lower bounds, lattice cryptography (CVP hardness), Ehrhart theory, and Ramsey theory. Within 24 hours, Anthropic researcher Levent Alpöge reported that the publicly available Fable model, using fully autonomous, general prompting with no network access, independently reproduced five of the ten results. Cambridge mathematician Henry Bradford argued the breakthroughs may redefine mathematics' purpose around ideas rather than proofs, echoing Thurston's 1994 essay. The post argues formal verification addresses proof correctness but not problem significance or translation fidelity, and outlines six-to-twelve-month watchpoints for AI-driven mathematics.

This post discusses OpenAI's announcement that its unreleased model Astra proved ten long-open mathematical problems with Lean 4-verified certificates for roughly $2,000 in token costs — and Anthropic's public Fable model replicating half of them within 24 hours. Below is a structured English summary of the original Chinese forum article.

Key points

On August 1, OpenAI researcher Sébastien Bubeck announced that Astra (unreleased) proved 10 frontier results, each open for at least 10 years, with 10 Lean 4 certificates, per-problem write-ups, and a 249-page technical manuscript.
OpenAI priced the successful solution paths at roughly $2,000 in tokens using GPT-5.6 Sol API rates. Simon Willison and others noted this figure excludes training compute, failed attempts, prompt/human engineering, and Lean library work — "cost of the successful path" ≠ "cost of frontier mathematics."

The ten problems

| # | Problem | Field | Open since | Method | |---|---------|-------|------------|--------| | 1 | Sphere packing in high dimensions | Geometry / analytic number theory | 1978 (46 years) | Cohn-Elkies LP with exact exponential decay rate | | 2 | Binary and spherical codes | Coding theory | 1970s | Exponential improvement on classical upper bounds | | 3 | Existence of non-sofic groups | Group theory / operator algebras | 1999 (27 years) | Leavitt algebra unit groups + Property (T) expanders + Thompson's group V | | 4 | Connes rigidity conjecture | Operator algebras | 1980s | Infinitely many non-isomorphic groups sharing one von Neumann algebra | | 5 | Arithmetic circuit complexity | Theoretical CS | 1960s | New circuit lower bounds for the permanent | | 6 | Quantum parallel repetition | Quantum information | Classic theorem extension | Finite-field extension of two-player entangled games | | 7 | Closest Vector Problem (CVP) | Lattice cryptography | 2000s | Polynomial-factor hardness, with lattice-crypto implications | | 8 | Ehrhart volume conjecture | Ehrhart theory | 2000s | Sharp volume bounds in arbitrary dimension | | 9 | Multicolor Ramsey numbers | Combinatorics | Erdős problem 183 | Super-exponential lower bound | | 10 | Extremal graph degeneracy | Extremal graph theory | Erdős problems 146, 180 | Counterexample construction |

Anatomy of the non-sofic group construction

The most dramatic result (#3) chains together previously unconnected tools:

1. Start from the unit group of the binary Leavitt algebra L₂. 2. Embed via Property (T) expander graphs — a property antithetical to sofic approximability. 3. Embed Thompson's group V's fine automorphism structure into irreducible representations of the Leavitt unit group, producing a hybrid object captured by neither classical constructions nor existing soficity tools. 4. Formalize every step in Lean 4. All ten certificates contain zero sorry placeholders.

Fable replicates half in 24 hours

On August 2, Anthropic researcher Levent Alpöge replied under Bubeck's post: "I got through half of them with Fable." Fable is a publicly released model, not a next-gen system. Conditions were fully autonomous, general prompts, no network access, with safeguards against OpenAI solution leakage. Fable reproduced items #4–#8 (Connes rigidity counterexample, circuit lower bounds, quantum parallel repetition, CVP hardness, Ehrhart volume). Structural implication: first-mover advantage in mathematics has shrunk from years to days; "independently verified first" matters more than "first announced."

The hidden half of the $2,000 ledger

The figure omits: (1) Astra's training compute (millions of dollars scale); (2) failed attempts — OpenAI disclosed write-ups but not prompts or retry counts; (3) human labor converting model output into publishable manuscripts; (4) Lean library maintenance. A more accurate framing: "the successful paths cost ≈$2,000 in tokens; the total effort is undisclosed."

The mathematical community's identity crisis

Cambridge Fellow Henry Bradford (writing in The Guardian / Quanta / World Programming, Aug 27) said the breakthroughs are "things I would have found implausible a month ago," calling them clever recombination of existing theory rather than genuinely new mathematics — while warning against betting against superhuman AI mathematical ability within years. Invoking Thurston's 1994 essay *On Proof and Progress in Mathematics*, he argued that mathematics' value lies in ideas living in human minds, not theorems per se. A related arXiv paper (2608.23691, Cambridge/HKU labs) on autonomous mathematical discovery in open-world multi-agent settings found research agents yield returns from structural exploration rather than search.

Formalization ≠ peer review

Lean 4 certificates verify only that the reasoning chain has no logical gaps. They do not verify: whether the formalized statement faithfully translates the original problem (translation is human work), or whether the problem matters (peer-review judgment). Thomas Bloom (University of Manchester, Erdős problems database) called the results "big news" but stressed they remain "reported rather than confirmed" pending independent review.

Watchpoints for the next 6–12 months

Whether AI mathematics adopts a rapid release cadence, forcing journals to create fast tracks for machine-verified proofs.
Engineering maturity of formalization libraries (Lean 4/Coq/Isabelle), which bound AI mathematical capability.
Whether Anthropic / xAI / DeepMind adopt "announce + open replication" dual strategies.
Whether universities cut pure-mathematics positions amid output-metric pressure.
Emergence of an "ideas economy" — markets or benchmarks for unproven but valuable mathematical ideas.

Conclusion

The community faces two doors: mathematicians as verifiers of AI proofs (easy, but reducing the field to acceptance testing), or mathematicians as sources of ideas and intuition for AI discovery partners (hard, requiring institutions to fund ideas rather than paper output).

References

1. OpenAI, *Ten new proofs, ten new problems* — https://cdn.openai.com/pdf/ten-proofs-oai.pdf 2. OpenAI repository — https://github.com/openai/ten-proofs (Apache 2.0, Lean 4 certificates) 3. Fello AI, *OpenAI Astra: What It Is and Why It Is Paused* — https://felloai.com/openai-astra 4. Inside AI News, *AI Proof of Non-Sofic Groups Forces Mathematicians to Rethink Purpose* (2026-08-27, Henry Bradford) 5. World Programming, *Surprising AI breakthroughs raise soul-searching questions for mathematicians* 6. Morning Overview, *OpenAI's Astra AI cracked 10 unsolved math problems for just $2,000 in computing* (2026-08-27) 7. Sohu, coverage of Fable's 24-hour replication (Levent Alpöge) 8. forRide, analysis of the 249-page PDF (Thompson's group V, Kun-Thom expanders) 9. Wortins, *OpenAI's Astra solves 10 long-unsolved math problems, publishes verified proofs* (2026-08-27) 10. NetEase, Cambridge/HKU labs' autonomous mathematical discovery paper (arXiv 2608.23691, 2026-08-27)