The company published six mathematics papers on October 2 covering probability, differential equations, group theory, optimisation, arithmetic physics and non-associative algebra. Five present answers to questions Meta described as previously open, while the sixth develops a connection between number theory and p-adic string theory.
Researchers used Muse Spark 1.1 and 1.2 in Thinking Mode through meta. ai, without a custom research scaffold or specialised agent framework. Meta said mathematicians selected and guided the research, explored ideas with the models and developed the arguments, while separate groups of mathematicians reviewed the work.
The papers also distinguish passages drafted primarily by researchers from those produced mainly by AI, an effort to make the model’s contribution traceable. Meta said researchers checked, corrected and refined model-generated material rather than treating its output as automatically valid.
Meta’s announcement does not state that the six papers completed journal peer review. Its described checks were conducted by second groups of mathematicians in the collaborations, leaving the findings open to scrutiny by the wider mathematical community.
One paper, led by Aykut Arslan, addresses the threshold for fitting random Gaussian points in high dimensions to a centred ellipsoid. The work identifies a sharp transition: below the threshold an ellipsoid exists with high probability, while above it one almost certainly does not. Meta said behaviour exactly at the threshold remains unresolved.
That result also illustrates an important qualification surrounding claims of solving open problems. Meta acknowledged three independent works posted in August that addressed the Gaussian threshold, including one that independently proved it and another establishing a broader result containing it as a special case. The company said the approaches were developed independently.
A differential-equations paper led by Leonard Dinh concerns finite-time collapse in the mass-critical biharmonic nonlinear Schrödinger equation. For radial negative-energy solutions in two or more dimensions, the paper proves finite-time blow-up, addressing a question left open in 2015. Muse Spark was used to work through calculations, test arguments and revise the proof, while Dinh supplied the problem and key proof ideas.
The model took a more computational role in group theory. Joseph Phillip Brennan and Milana Golich worked with Muse Spark on a conjecture proposed in 2024 that every finite semiabelian group must be monomial. Muse Spark generated a search program using the GAP mathematical software system that found a counterexample with 384 elements. The researchers verified the result and completed the argument.
Meta also acknowledged that the separate AI agent Nilradical reported a different counterexample to the same conjecture on September 16, saying its team’s result was developed independently. Such overlaps mean the timing and status of individual open questions require care when assessing the broader claim.
For an optimisation problem, Muse Spark helped reframe the question probabilistically, identify a counterexample and develop a proof strategy. The researchers then checked the arguments, corrected gaps and refined the proof. The work concerns when a relaxation of a binary polynomial optimisation problem captures the original problem exactly.
Another paper connects calculations in number theory and p-adic string theory. Meta said Muse Spark generated candidate proofs and drafted three core technical sections, which researchers subsequently checked, corrected and refined. Unlike the five papers framed as answering open questions, this work establishes a mathematical connection across two fields.
The non-associative algebra collaboration produced a three-dimensional counterexample to a proposed test for identifying solvable evolution algebras. Muse Spark generated the counterexample and suggested alternative characterisations and proofs, while mathematician Andres Barei checked, refined and rewrote the material. Meta also credited independent work by researchers Hu and Wen that reported counterexamples to the same conjecture.
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