On 2 October 2026, Meta AI Research published a blog post, "Solving Open Research Problems Together," sharing six research papers produced by mathematicians working with Muse Spark. Meta says five of them present answers to previously open research questions, across probability, differential equations, group theory, optimization, arithmetic physics, and non-associative algebra.
From competition problems to open questions
Meta notes that earlier this year its models achieved gold-medal-level performance across five high-school Olympiad competitions in mathematics, physics, and chemistry. Competition problems can be extremely hard, but they already have solutions. Open research has no answer key and no guarantee that an approach will work; progress means trying and retrying ideas, making and resolving mistakes, and sometimes starting over.
Over the past several months, mathematicians used Muse Spark 1.1 and 1.2 in Thinking Mode through the regular meta.ai chat interface, with no custom research scaffold. Meta says the goal was not to mass-produce papers, but to help researchers develop mathematical insights that others can understand and build on.
Four collaboration principles
The post states that every collaboration followed the same principles:
- A team of mathematicians guided the research and worked with Muse Spark to explore ideas and develop the arguments;
- A second group of mathematicians reviewed the work;
- Each paper clearly marks which passages were primarily drafted by researchers and which by AI;
- Each paper credits the earlier research and mathematical ideas it builds on.
Figure: Left to right — mathematicians set the direction, AI helps work through and draft, a second group of mathematicians reviews and finalises; colour-coded passage markers distinguish human- and AI-drafted sections (editorial illustration — no text or logos).
The six papers at a glance
| Field | Result (as described by Meta) | Muse Spark's role |
|---|---|---|
| Probability | A sharp threshold for when random Gaussian points in high dimensions can be fitted to an ellipsoid | Helped develop and revise proof strategies |
| Differential equations | Proves certain radial negative-energy solutions must collapse in finite time | Worked through calculations, tested arguments, revised the proof |
| Group theory | Disproves a 2024 conjecture with a 384-element counterexample | Generated the GAP search program that found the counterexample |
| Optimization | A rule for when a relaxation loses nothing | Reframed the problem with probabilities, found a counterexample, developed the proof strategy |
| Arithmetic physics | Shows two calculations in different mathematical languages describe the same quantity | Helped identify the connection, generated candidate proofs, drafted three core technical sections |
| Non-associative algebra | Disproves a classification conjecture with a 3-D counterexample and gives an alternative rule | Generated the counterexample, proposed alternative characterizations and proofs |
Probability: the strict threshold for Gaussian ellipsoid fitting
Led by Aykut Arslan, reviewed by Babak Modami, Alexander Roitershtein, Mark Sepanski, and Grigory Sokolov. Picture dots scattered on a plane and the task of drawing an ellipse, centred at a fixed point, that passes through all of them. The paper identifies a sharp threshold on the number of points: below it, such an ellipsoid exists with high probability; above it, one almost certainly does not. Behaviour exactly at the threshold remains unresolved.
Differential equations: when wave collapse is unavoidable
Led by Leonard Dinh, reviewed by Fazel Hadadifard and Salem Selim. The problem comes from a model inspired by laser physics: one effect squeezes a wave inward while another spreads it out. For waves that are symmetric around a centre, have negative energy, and live in two or more dimensions, the paper proves collapse must happen within finite time. Meta says this settles a question left open in 2015 and confirms, in this setting, a prediction from computer simulations in 2002. Dinh chose the problem and the key proof ideas.
Group theory: one black swan
Led by Joseph Phillip Brennan and Milana Golich, reviewed by Andres Barei and John Portin. A conjecture first proposed by M. Kida in 2024 stated that every finite group with the property called "semiabelian" must also be "monomial." The team found an exception in a group with 384 elements. Muse Spark generated the search program in GAP, a mathematical software system, that found the counterexample; the researchers verified the result and completed the argument.
Optimization: when a relaxation loses nothing
Led by Aykut Arslan, reviewed by Kien Trung Le. The paper answers a question posed by Del Pia and Khajavirad about when a relaxation of a binary polynomial optimization problem captures the original exactly. Using the image of three overlapping circles in a Venn diagram, for the family studied the approximation is exact when each region shared by two circles (but not the third) contains exactly one decision; if any such region holds more than one, a gap appears.
Arithmetic physics: two calculations, one quantity
By Anindya Dey, Gabriel Herczeg, An Huang, Nicolas Jaramillo Torres, and Jacob H. Swenberg. Following a direction envisioned by Yuri Manin in the 1980s, the work connects number theory and p-adic string theory: starting from a connection already known for the Tate curve, it extends to a much broader class of curves. In simpler cases, the calculation comes down to counting how many initial digits the coordinates of two points share in base p. Beyond helping identify the connection, Muse Spark generated candidate proofs and drafted three core technical sections, which the researchers then checked, corrected, and refined.
Non-associative algebra: a counterexample, then a better rule
Led by Andres Barei, reviewed by Nicolás Jaramillo Torres. García-Martínez and Pérez-Rodríguez had suggested a test for identifying "solvable" evolution algebras. The team found a small three-dimensional example that passes the test but does not belong to that class. The paper goes further, establishing an alternative rule that considers whole subspaces rather than individual elements. Working from the researchers' prompts, Muse Spark generated the counterexample and proposed alternative characterizations and proofs, which Barei checked, refined, and rewrote.
Independent concurrent results, acknowledged
Meta says that after completing the work, it learned that teams outside Meta had independently announced solutions to some of the same problems using different approaches, and the papers acknowledge them:
- Probability: three independent concurrent works posted in August 2026 — Misiakiewicz and Wen independently proved the Gaussian threshold; De la Cerda, Potechin, Tulsiani, and Xu established it up to a vanishing multiplicative factor; Koehler and Sohn obtained a broader universality result that includes it as a special case;
- Group theory: the AI agent Nilradical reported a different counterexample to the same conjecture on 16 September 2026;
- Non-associative algebra: Hu and Wen independently reported counterexamples to the same conjecture.
What can be checked
On Meta's primary material, what can be filed is the field and result of each of the six papers, who led and who reviewed each one, Muse Spark's specific role in each, the practice of marking human- versus AI-drafted passages, and the acknowledgement of independent concurrent work. Meta closes by saying it believes progress will require close collaboration between experts and AI, with results that are carefully verified, transparent in how they were produced, and clearly communicated to the research community.
Primary source:
- Meta AI Research, "Solving Open Research Problems Together," 2026-10-02, https://research.meta.ai/blog/solving-open-research-problems-together
