# Discovering a Zeta Map Algorithm on Dyck Paths via Mechanistic Interpretability

arXiv: https://arxiv.org/abs/2605.30482 · HTML: https://mlsanity.com/paper/2605.30482/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Placed in Circuits and mechanistic analysis, one of 13 directions in Mechanistic interpretability.
- Builds on 15 of its 28 references that are in the map.
- First posted on arXiv 28 May 2026.

## Placed in

- Mechanistic interpretability / [Circuits and mechanistic analysis](https://mlsanity.com/mechinterp/circuits-and-mechanistic-analysis/)

## Builds on (most cited first, 15 of 15)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Understanding intermediate layers using linear classifier probes](https://mlsanity.com/paper/1610.01644/) (2016)
- [Attention is not Explanation](https://mlsanity.com/paper/1902.10186/) (2019)
- [AlphaEvolve: A coding agent for scientific and algorithmic discovery](https://mlsanity.com/paper/2506.13131/) (2025)
- [Towards Automated Circuit Discovery for Mechanistic Interpretability](https://mlsanity.com/paper/2304.14997/) (2023)
- [Attention is not not Explanation](https://mlsanity.com/paper/1908.04626/) (2019)
- [The Clock and the Pizza: Two Stories in Mechanistic Explanation of Neural Networks](https://mlsanity.com/paper/2306.17844/) (2023)
- [A Toy Model of Universality: Reverse Engineering How Networks Learn Group Operations](https://mlsanity.com/paper/2302.03025/) (2023)
- [A Multimodal Automated Interpretability Agent](https://mlsanity.com/paper/2404.14394/) (2024)
- [LatentQA: Teaching LLMs to Decode Activations Into Natural Language](https://mlsanity.com/paper/2412.08686/) (2024)
- [Constructions in combinatorics via neural networks](https://mlsanity.com/paper/2104.14516/) (2021)
- [Learning to Unknot](https://mlsanity.com/paper/2010.16263/) (2020)
- [Transformers are uninterpretable with myopic methods: a case study with bounded Dyck grammars](https://mlsanity.com/paper/2312.01429/) (2023)
- [Can Transformers Do Enumerative Geometry?](https://mlsanity.com/paper/2408.14915/) (2024)
- [Flow-based Extremal Mathematical Structure Discovery](https://mlsanity.com/paper/2601.18005/) (2026)

## Built on by (most cited first, 1 of 1)

- [PermuFormer: Multi-Task Pretraining for Permutation Representation in Algebraic Combinatorics](https://mlsanity.com/paper/2609.25438/) (2026)

## Abstract (arXiv)

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.