Discovering a Zeta Map Algorithm on Dyck Paths via Mechanistic Interpretability
This work provides a controlled example of AI-assisted mathematical discovery, demonstrating how mechanistic interpretability can translate model behavior into human-verifiable combinatorial algorithms for mathematicians.
This paper explores the use of a small transformer model to learn the zeta map on Dyck paths. Through mechanistic interpretability, the authors translated the model's learned computation into a new, explicit combinatorial algorithm called the scaffolding map, which they proved to be equivalent to the zeta map.
Machine learning is increasingly used in mathematical discovery, but in mathematics the desired output is often not a prediction itself, but an explicit construction that can be checked independently. We study this setting through the zeta map on Dyck paths, a classical bijection in the combinatorics of the q,t-Catalan numbers. We train a deliberately small one-layer, one-head encoder-decoder transformer on this map and analyze its learned computation using mechanistic interpretability tools, including decoder cross-attention analysis, linear probing, and causal intervention. The analysis reveals a level-based mechanism: encoder representations make path levels linearly accessible, while the decoder selects and traverses input positions in a structured way. Translating these signals into combinatorics leads to the scaffolding map, an explicit peak-centered traversal algorithm for Dyck paths. We prove that this algorithm agrees with the zeta map, modulo a reversal convention in the labeling. This gives a controlled example of AI-assisted mathematical discovery in which mechanistic interpretability turns model behavior into a precise, human-verifiable combinatorial algorithm.