Meta Says Muse Spark Helped Crack Five Open Math Problems

Meta has published six mathematics papers that it says were produced by human mathematicians working with its Muse Spark model, and that five of them settle questions that were previously open. The papers, released on 2 October alongside a post on Meta’s research blog, cover probability, differential equations, group theory, optimisation, non-associative algebra and a corner of number theory that touches string physics. The model versions involved were Muse Spark 1.1 and 1.2 in Thinking Mode, used through the same meta.ai chat interface anyone can open, not a custom research harness.

That last detail is the part Meta is leaning on. When OpenAI claimed ten open problems for an internal Astra model in August, the proofs came with Lean certificates and the model was not public. Google DeepMind’s co-mathematician agent, which resolved a 1965 group theory question in May, is a purpose-built hierarchical system. Meta’s pitch is that a consumer chatbot, in the hands of working mathematicians, can now do research-grade work. Both versions it used are already a step behind: Muse Spark 1.3 shipped in early September.

The six papers

Field	Paper	What it does	Independent result?
Probability	The strict threshold for Gaussian ellipsoid fitting	Pins down the sharp threshold at which random Gaussian points in high dimensions can still be fitted by an ellipsoid	Yes, three concurrent solutions in August
Differential equations	Finite-time blow-up for the mass-critical biharmonic NLS	Proves symmetric negative-energy waves collapse in finite time, a question open since 2015 and predicted by 2002 simulations	None noted
Group theory	Semiabelian groups need not be monomial	Disproves a 2024 conjecture of M. Kida with a group of order 384	Yes, AI agent Nilradical, 16 September
Optimisation	Tightness of the cycle-based relaxation for alpha-cycles	Answers a 2026 question from Del Pia and Khajavirad on when a relaxation for binary polynomial optimisation is exact	None noted
Arithmetic physics	String two-point function = height function on a curve	Extends a known link between p-adic string amplitudes and height functions from the Tate curve to a wider class of curves	Extension, not an open problem
Non-associative algebra	Solvable evolution algebras and a conjecture of García-Martínez and Pérez-Rodríguez	Disproves the conjecture with a three-dimensional counterexample and offers a corrected characterisation	Yes, Hu and Wen

What the model actually did

Meta describes a division of labour rather than a machine working alone. Mathematicians chose the problems and steered the work; Muse Spark proposed and revised proof strategies, ground through calculations, drafted technical sections and suggested connections between fields. A second, separate group of mathematicians then reviewed each paper. Every paper marks which passages were drafted by the researchers and which by the model, an unusually explicit disclosure.

The group theory result shows the most concrete version of the workflow. M. Kida had conjectured in 2024 that every semiabelian group is monomial, a property about how a group’s representations can be built. Muse Spark wrote a search program for GAP, the standard computer algebra system for group theory, and the search turned up a counterexample: a group with 384 elements, catalogued in GAP’s library as SmallGroup(384, 20127). Joseph Phillip Brennan and Milana Golich, the paper’s authors, then verified it. A finite counterexample is the kind of result that is easy to check once found, which makes it among the most trustworthy items in the set.

The differential equations paper is a different kind of claim. It is a full proof that solutions of the mass-critical biharmonic nonlinear Schrödinger equation with negative energy and radial symmetry blow up in finite time, something numerical work had predicted in 2002 and analysts had left open since 2015. Proofs like this are long, and their correctness rests on expert reading rather than a quick check. Meta’s papers are not formalised in Lean, so for these the second-team review is the main safeguard.

Five new answers, or fewer?

The headline number drew pushback within a day. Some researchers on X argued that three of the six problems were already resolved, and Meta’s own blog supports the substance of that, if not the framing. For the Gaussian ellipsoid threshold it acknowledges three independent solutions that appeared in August. The 384-element counterexample was also reported on 16 September by Nilradical, another AI research agent. And the evolution algebra conjecture was independently disproved by Hu and Wen. Meta says it credits all of these in the papers.

Concurrent discovery is not the same as being scooped, and in mathematics two teams landing on the same answer within weeks is often a sign that a problem has become ripe. But it does change what the result says about AI. Three of the five open questions were evidently within reach of other methods at the same moment, and the sixth paper is an extension rather than an answer. That leaves two results, the blow-up proof and the optimisation question, that Meta can present as both open and uncontested. The company itself frames the goal more modestly than its headline: not to mass-produce papers, it says, but to help researchers develop insight.

None of the six has yet been through journal peer review, and Meta’s reviewing mathematicians, while independent of the authors, were recruited for the project. What the batch does establish is that Meta now sits alongside OpenAI and Google DeepMind as a lab putting its name to AI-assisted research mathematics, in a year that has also produced a Lean-checked GPT-6 Pro proof of Nivat’s conjecture, and that its entry came out of a chat window. The interesting test will be whether mathematicians outside Meta, using the public model, start publishing results of their own.