Discovering a Zeta Map Algorithm on Dyck Paths via Mechanistic Interpretability

• URL: http://arxiv.org/abs/2605.30482v1
• Date: Thu, 28 May 2026 18:59:30 GMT
• Title: Discovering a Zeta Map Algorithm on Dyck Paths via Mechanistic Interpretability
• Abstract summary: We train a small one-layer, one-head encoder-decoder transformer on a map of Dyck paths.<n>We analyze its learned using mechanistic interpretability tools, including decoder cross-attention analysis.<n>We prove that this algorithm agrees with the zeta map, modulo a reversal convention in the labeling.
• Score: 2.3895981099137535
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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.

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