Muse Spark Helped Close 5 Open Mathematical Problems

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Meta published six mathematical papers on October 2, 2026, that people wrote together with the Muse Spark model - five of them solve problems previously considered open. The mathematicians didn’t use any special laboratory, just the regular meta.ai chat in Thinking mode.

The announcement appeared on the Meta Research blog under the title “Solving Open Research Problems Together”, and the list of papers was shared on X by Meta Superintelligence Labs head Alexandr Wang. The team used Muse Spark 1.1 and 1.2 in Thinking mode. Each paper indicates which parts were written by humans and which by the model, and the results were verified by a separate group of mathematicians.

Six Papers and What They Contain

• Fitting an ellipsoid to Gaussian points (Aykut Arslan) - established a sharp threshold: fitting succeeds with high probability below approximately d²/4 points, and fails above. Behavior exactly at the threshold remains open. Meta acknowledges that in August 2026, three independent teams reached similar results using different methods.
• Wave collapse in the biharmonic nonlinear Schrödinger equation (Leonard Dinh) - proof that symmetric solutions with negative energy collapse in finite time. The problem had been open since 2015, and confirms computer simulations from 2002.
• Semiabelian groups need not be monomial (Joseph Phillip Brennan, Milana Golich) - Muse Spark wrote a program in the GAP system that found a counterexample with 384 elements to M. Kida’s 2024 hypothesis. Another AI agent (Nilradical) independently found a different counterexample on September 16.
• Cycle-based relaxation (Aykut Arslan) - answer to a question posed in 2026 by Del Pia and Khajavirad about when approximations in polynomial optimization are exact.
• Two-point function of a string and height function on a curve (five authors) - work at the intersection of string theory and number theory. Does not close an open problem, but develops a known connection. The model generated candidate proofs and wrote drafts of three technical sections.
• Solvable evolution algebras (Andres Barei) - three-dimensional counterexample to the hypothesis of García-Martínez and Pérez-Rodríguez. Here too, other researchers (Hu and Wen) submitted counterexamples in parallel.

Limitations of This Result

I read Meta’s post and the comments, and I see three caveats that headlines don’t mention:

• The model didn’t choose the problems or verify the results itself. Mathematicians provided direction, and a second group of people verified the work.
• In three of the six cases, others reached similar results at the same time, including another AI agent. These problems were “ripe” for solving.
• No one measured how much time the model saved or how often its proposals were wrong. The AlphaSignal service notes that this is proof of useful contribution under close expert supervision, not autonomous discovery.

Nevertheless, the 384-element counterexample itself is a good example of what models are strong at today. They don’t “invent mathematics”, but quickly write search programs and propose arguments that humans can verify. OpenAI followed a similar path earlier, which I wrote about regarding mathematical problems solved with the Astra model.

What This Means for You

The most interesting thing is that the researchers used the regular meta.ai chat, not a tool for laboratories. I described Muse Spark in the guide Muse Spark 1.3 from Meta. If you’re learning mathematics, remember the lesson from these papers: the model is good at writing search code and sketching proofs, but every step must be verified. I checked which models make the fewest mistakes in school problems in a test of 9 models.

Sources: Meta Research, “Solving Open Research Problems Together” (October 2, 2026); Alexandr Wang’s post on X; AlphaSignal. Verified October 4, 2026.

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