Meta publishes six maths papers written with Muse Spark models

Meta on 2 October published six mathematics papers produced by mathematicians working with its Muse Spark 1.1 and 1.2 models, five of which answer previously open research questions.

Meta said the researchers used the models in Thinking Mode through the regular meta.ai chat, with no custom research scaffold. All six papers were peer reviewed, and each marks which passages the AI drafted and which the researchers wrote, according to Meta's research blog.

Meta said the model developed proof strategies, worked through calculations, wrote search programs and drafted technical sections. Humans set the direction and verified the results.

The papers cover six fields. In probability, "The Strict Threshold for Gaussian Ellipsoid Fitting" gives a sharp threshold for fitting random Gaussian points to an ellipsoid in high dimensions. In differential equations, one paper addresses finite-time blow-up, or wave collapse, for radial negative-energy solutions of the mass-critical biharmonic nonlinear Schrödinger equation.

In group theory, "Semiabelian Groups Need Not Be Monomial" gives a 384-element counterexample that disproves a 2024 conjecture by Kida. In non-associative algebra, a three-dimensional counterexample disproves a conjecture by García-Martínez and Pérez-Rodríguez on solvable evolution algebras.

The remaining two papers deal with optimisation and arithmetic physics. The first concerns the tightness of the cycle-based relaxation for completed length-three alpha-cycles. The second shows that a string two-point function equals a height function on a curve, linking number theory and p-adic string theory.

"Making progress means trying and retrying ideas, making new mistakes and resolving them, and sometimes going back to the beginning," the Meta research blog said.

Muse Spark had earlier reached gold-medal level on five high-school olympiad competitions. Muse recently reached No. 1 on the App Store. OpenAI and Google DeepMind have also published AI-assisted mathematics results this year.