AI vs. AI: Inside the 2026 race where OpenAI's Astra and Anthropic's Claude Fable solve decades-old math problems for $2,000

The Jacobian conjecture, first posed by Ott-Heinrich Keller in 1939, is an open problem in algebraic geometry ranked #16 on Fields Medalist Stephen Smale's list of problems for the 21st century . It asks: if a polynomial map from C³ to C³ has a nonzero constant Jacobian determinant everywhere, must it be globally invertible? For 87 years, no one had found a counterexample — or a proof.

Alpöge, working with Anthropic's Claude Fable 5, found that the map F = (P, Q, R): C³ → C³ with Jacobian determinant −2 (a nonzero constant) sends three distinct inputs to the same output, directly disproving the conjecture . The result was formalized in Lean 4 and verified within a day by mathematicians including Kevin Buzzard at Imperial College London . It was widely described as the most difficult mathematical problem yet solved with AI .

OpenAI Astra: 10 problems for $2,000 (August 1, 2026)

On August 1, 2026, OpenAI published a research post revealing that an internal version of its next major model, Astra, produced new results on 10 open problems — each of which had seen little or no progress for at least a decade . OpenAI published the proofs on GitHub alongside machine-checkable Lean 4 certificates . The total compute cost for generating all 10 solutions was roughly $2,000 at API rates .

Here are the 10 problems Astra reportedly solved:

• Non-sofic group (group theory): First explicit construction of a non-sofic group, open since Mikhail Gromov introduced soficity in 1999 .
• Circuit lower bounds for the permanent (circuit complexity) .
• High-dimensional sphere packing: New upper bounds on sphere-packing density down to the Cohn–Elkies threshold .
• Binary and spherical codes: Exponential improvements to classical upper bounds .
• Operator algebras: Counterexample to Connes's rigidity conjecture on group von Neumann algebras .
• Quantum parallel repetition: Result in quantum parallel repetition for all entangled games .
• Lattice cryptography: New result on the closest vector problem .
• Extremal combinatorics: Resolution of Erdős problems #146, #180, and #183 .
• High-dimensional geometry: Novel geometric construction .
• Ehrhart's conjecture: Resolution on lattice point enumeration .

The 249-page manuscript was accompanied by model-written reasoning walkthroughs, and the GitHub repository reported a "sorry" count of zero — meaning every step across all ten formalized proofs was fully verified .

Anthropic's counterclaim: Claude Fable matches 5 of 10 (August 2, 2026)

Within 24 hours of OpenAI's announcement, Levent Alpöge responded on X, stating that he had used the publicly available Claude Fable (not an unreleased model) to independently solve 5 of the same 10 problems .

Key details:

• He used generic prompts with no internet access and safeguards ensuring Fable could not have seen the OpenAI solutions .
• Fable produced the results autonomously within hours .
• The solved problems spanned arithmetic circuit complexity, quantum parallel repetition, lattice cryptography, and the closest vector problem .
• Only one of the five used essentially the same argument; the other four may be independent proofs .
• Alpöge did not claim Fable had solved all 10 — only that it independently reproduced 5 of them with minimal human guidance .

The claim was intended to demonstrate that a publicly available model could match a significant fraction of Astra's results, challenging OpenAI's framing of Astra as uniquely capable .

The bigger picture: what this means for math and AI

Cost as a competitive dimension. The $2,000 figure from OpenAI draws a direct line to cost efficiency . If AI systems can now produce genuinely novel mathematical results at $200 per problem, the bottleneck shifts from discovery to verification and interpretation.

Machine-checkable proofs gain primacy. Both the Jacobian and Astra results were formalized in Lean 4, making them machine-checkable . The rapid verification (within hours/days instead of years) points to a future where Lean certificates become the gold standard, potentially displacing traditional peer review for certain types of results .

Authorship attribution in academic research. The Jacobian conjecture result raised immediate questions: who is the author — the mathematician who prompted the model, the model itself, or the company that built it? Alpöge credited both a human colleague and Claude Fable 5 on X . Academic journals have no standard policy for AI co-authorship, and the result was disseminated via social media and GitHub rather than traditional peer review .

The pace has been described by mathematicians as "very rapid and disorienting" .

Key uncertainties

The specific list of which 5 of the 10 problems Alpöge's Claude Fable reproduced has not been published in full detail . Reports say they spanned arithmetic circuit complexity, quantum parallel repetition, and lattice cryptography, but the exact mapping remains unconfirmed . Both claims should be viewed as company- and researcher-announced results that have not yet completed formal peer review — though the Lean-verified proofs provide a strong layer of machine-checkable confidence .