Discovering a Zeta Map Algorithm on Dyck Paths via Mechanistic Interpretability

2026-05-28

AI Summary

This work investigates how to extract verifiable mathematical constructions from machine learning models rather than merely obtaining predictive outputs. By training a minimal single-layer, single-head encoder-decoder Transformer to learn the zeta map on Dyck paths, the study employs interpretability techniques—including cross-attention analysis, linear probing, and causal interventions—to uncover the model’s internal computational mechanisms. Building on these insights, the authors formally introduce and rigorously prove the equivalence between the scaffolding map algorithm and the zeta map under reversed labeling conventions. This result constitutes the first instance of AI-driven, verifiable discovery in enumerative combinatorics and establishes a novel paradigm wherein mechanistic interpretability serves as a foundation for automated mathematical theorem discovery.

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.

Problem

• zeta map
• Dyck paths
• mathematical discovery
• combinatorics
• AI-assisted reasoning

Innovation

• mechanistic interpretability
• zeta map
• Dyck paths
• scaffolding map
• AI-assisted discovery