Discovering a Zeta Map Algorithm on Dyck Paths via Mechanistic Interpretability
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866913170997641216 |
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| author | Huang, Xiaoyu Jackson, Blake Lee, Kyu-Hwan |
| author_facet | Huang, Xiaoyu Jackson, Blake Lee, Kyu-Hwan |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_30482 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Discovering a Zeta Map Algorithm on Dyck Paths via Mechanistic Interpretability Huang, Xiaoyu Jackson, Blake Lee, Kyu-Hwan Machine Learning 05A19, 68T07 G.2.1; I.2.6 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. |
| title | Discovering a Zeta Map Algorithm on Dyck Paths via Mechanistic Interpretability |
| topic | Machine Learning 05A19, 68T07 G.2.1; I.2.6 |
| url | https://arxiv.org/abs/2605.30482 |