Did Meta's Muse Spark Solve Open Math Problems? (2026)
The short answer
Yes, with the asterisk Meta itself attaches: on October 3, 2026 Meta AI published six mathematics papers produced by mathematicians working with Muse Spark 1.1 and 1.2 in Thinking Mode via the regular meta.ai chat, and five of them settle previously open questions — but humans chose the problems, steered the work, verified every argument, and a second group reviewed each paper. The results were open when the work began; three of them were independently resolved by other teams (one by another AI agent) before publication. Facts verified October 4, 2026 against Meta’s research blog.
The six papers
| Area | Result | What Muse Spark did | Concurrent work |
|---|---|---|---|
| Probability | Sharp threshold for fitting high-dimensional Gaussian points to an ellipsoid; behaviour exactly at the threshold still open | Developed and revised proof strategies under Aykut Arslan; four mathematicians checked | Misiakiewicz–Wen; De la Cerda–Potechin–Tulsiani–Xu; Koehler–Sohn (all Aug 2026) |
| Differential equations | Finite-time blow-up of radial negative-energy solutions for the mass-critical biharmonic NLS in 2+ dimensions; settles a 2015 question, confirms 2002 simulations | Worked calculations, tested arguments, revised the proof; Leonard Dinh chose the problem and key ideas | — |
| Group theory | Semiabelian groups need not be monomial — a 384-element counterexample disproves Kida’s 2024 conjecture | Generated the GAP search program that found the group; Golich and collaborators verified and completed the argument | AI agent Nilradical reported a different counterexample Sep 16, 2026 |
| Optimization | Tightness of the cycle-based relaxation for completed length-three alpha-cycles (question posed by Del Pia and Khajavirad, 2026) | Proof development with Arslan | — |
| Number theory / p-adic strings | String two-point function equals a height function on a curve, extending a Tate-curve result to Mumford curves (a direction Manin envisioned in the 1980s) | Helped identify the connection, generated candidate proofs, drafted three core technical sections that researchers corrected | — |
| Non-associative algebra | 3-dimensional counterexample to the García-Martínez–Pérez-Rodríguez test for solvable evolution algebras, plus a corrected subspace-based criterion | Generated the counterexample and proposed alternative characterisations; Barei checked and rewrote | Hu and Wen, independent counterexamples |
Five answer open questions; the p-adic string paper is a new connection rather than the resolution of a stated conjecture.
How the work was done
Meta’s process notes are the most useful part of the release for anyone trying to reproduce it:
- No custom scaffold. Muse Spark 1.1 and 1.2 in Thinking Mode through meta.ai — the same interface a consumer uses — not a research harness with Lean, retrieval or multi-agent search.
- Mathematicians led. A team chose each problem, directed exploration and developed the arguments with the model.
- Independent review. A second group of mathematicians reviewed each paper before release.
- Labelled drafting. Each paper marks which passages were primarily drafted by researchers and which by AI.
- Credit for priors. Each paper credits the earlier research it builds on and the concurrent independent results.
The group-theory paper is the cleanest illustration of the division of labour. Kida conjectured in 2024 that every finite semiabelian group is monomial. Disproving it needs one counterexample. Muse Spark wrote a search program in GAP (the standard computational group theory system); the program found a group of order 384; humans verified the group has the first property and lacks the second and wrote up the argument. The model did not “see” the counterexample — it built the tool that did.
The concurrent-results problem
Three of the five open questions were also resolved by others in the weeks before Meta published:
- The ellipsoid threshold by three independent human groups in August 2026.
- Kida’s conjecture by Nilradical, an AI agent working on the Kourovka notebook problems, with a different counterexample on September 16, 2026.
- The evolution algebra conjecture by Hu and Wen.
Meta discloses all of it. The honest reading is that these were ripe problems — the kind that fall when enough capable attention arrives — and that AI-assisted attention arrived from several directions at once. It is also a reminder that “first” is now a race measured in weeks, and that an AI agent (Nilradical) got to one of these before a frontier lab’s model did.
What this does and does not show
It shows that a consumer-grade frontier model, used by expert mathematicians through a chat box, can contribute to publishable research on genuinely open problems — including writing search code, generating counterexamples and drafting technical sections. It does not show autonomous mathematical discovery: every problem was selected, steered, verified and partly rewritten by humans. Meta’s own framing — “empower researchers,” “close collaboration between experts and AI” — is accurate, and more careful than most headlines about it.
For practitioners: the counterexample-search pattern (model writes the GAP/SageMath/Lean search, humans verify the hit) is the most transferable technique here, and it works with any model that can write correct code against a mathematical library. See how to verify an AI-generated math proof with Lean 4.
Related: best AI models for math 2026, GPT-5.6 Sol Ultra and the cycle double cover conjecture, Meta Muse Spark capabilities and access.
Last verified: October 4, 2026, against Meta AI’s research blog post and the linked paper pages.