Meta Muse Spark Helps Solve Five Open Math Problems

Meta has published six mathematics papers co-authored by versions 1.1 and 1.2 of its Muse Spark reasoning model. Five of these papers resolve previously open research questions. Rather than using specialized agent harnesses, custom retrieval systems, or tool-calling pipelines connected to Mathematica or Lean, the collaborating mathematicians accessed the AI directly through the standard Meta.ai consumer chat interface using its Thinking Mode. The human researchers directed the exploration and performed final validation, while the papers explicitly label which sections were drafted by the AI.

The collaborations spanned several advanced fields. In group theory, Muse Spark generated search code for the GAP computer algebra system that successfully identified a 384-element counterexample to Kida's 2024 semiabelian conjecture. For differential equations, the team proved a finite-time blow-up in the mass-critical biharmonic nonlinear Schrödinger equation in two or more dimensions, settling a question open since 2015. In optimization, the model helped derive a criterion for when a cycle-based relaxation of a binary polynomial problem is exact, addressing a 2026 question from Del Pia and Khajavirad.

The remaining papers cover probability, arithmetic physics, and non-associative algebra. The probability paper identified a sharp threshold for fitting random Gaussian points to an ellipsoid in high dimensions, a result independently proven by three other teams in August. The arithmetic physics paper extended a 1980s proposal by Yuri Manin, proving a string-theory two-point function equals an arithmetic height function on p-adic curves. Finally, the algebra paper constructed a three-dimensional counterexample to a test for solvable evolution algebras.

While benchmarks like FrontierMath show AI models scoring between 10% and 30% on fixed, difficult math problems, Meta's real-world trial highlights how reasoning models can assist in active research. Muse Spark contributed by drafting technical prose, proposing proof strategies, and reformulating problems. However, Meta noted several limitations, including the lack of a controlled comparison to measure speed advantages, and the heavy reliance on human experts to guide the model and verify all results.