# From Black Box to Bijection: Interpreting Machine Learning to Build a Zeta Map Algorithm

arXiv: https://arxiv.org/abs/2511.12421 · HTML: https://mlsanity.com/paper/2511.12421/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 25 references that are in the map.
- First posted on arXiv 16 Nov 2025.

## Builds on (most cited first, 16 of 16)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Progress measures for grokking via mechanistic interpretability](https://mlsanity.com/paper/2301.05217/) (2023)
- [Open Problems in Mechanistic Interpretability](https://mlsanity.com/paper/2501.16496/) (2025)
- [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)
- [Linear algebra with transformers](https://mlsanity.com/paper/2112.01898/) (2021)
- [PatternBoost: Constructions in Mathematics with a Little Help from AI](https://mlsanity.com/paper/2411.00566/) (2024)
- [Learning the greatest common divisor: explaining transformer predictions](https://mlsanity.com/paper/2308.15594/) (2023)
- [Global Lyapunov functions: a long-standing open problem in mathematics, with symbolic transformers](https://mlsanity.com/paper/2410.08304/) (2024)
- [Machine-Learning the Sato--Tate Conjecture](https://mlsanity.com/paper/2010.01213/) (2020)
- [Murmurations of elliptic curves](https://mlsanity.com/paper/2204.10140/) (2022)
- [Machine-Learning Arithmetic Curves](https://mlsanity.com/paper/2012.04084/) (2020)
- [Machine-Learning Number Fields](https://mlsanity.com/paper/2011.08958/) (2020)
- [Machine Learning Class Numbers of Real Quadratic Fields](https://mlsanity.com/paper/2209.09283/) (2022)
- [Learning Euler Factors of Elliptic Curves](https://mlsanity.com/paper/2502.10357/) (2025)
- [Machine Learning Approaches to the Shafarevich-Tate Group of Elliptic Curves](https://mlsanity.com/paper/2412.18576/) (2024)

## Built on by (most cited first, 3 of 3)

- [Mapping Uncharted Symmetries: Machine Discovery in Combinatorics](https://mlsanity.com/paper/2605.19063/) (2026)
- [Emergent Latent-State Computation under Stochastic Volatility](https://mlsanity.com/paper/2607.25459/) (2026)
- [AI Discovery in Combinatorics: From Problems to Solutions and Back to Bigger Problems](https://mlsanity.com/paper/2610.16555/) (2026)

## Abstract (arXiv)

There is a large class of problems in algebraic combinatorics which can be distilled into the same challenge: construct an explicit combinatorial bijection. Traditionally, researchers have solved challenges like these by visually inspecting the data for patterns, formulating conjectures, and then proving them. But what is to be done if patterns fail to emerge until the data grows beyond human scale? In this paper, we propose a new workflow for discovering combinatorial bijections via machine learning. As a proof of concept, we train a transformer on paired Dyck paths and use its learned attention patterns to derive a new algorithmic description of the zeta map, which we call the textitScaffolding Map.