# Adjusted Cup-Product Neural Layer

arXiv: https://arxiv.org/abs/2606.13568 · HTML: https://mlsanity.com/paper/2606.13568/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 15 of its 21 references that are in the map.
- First posted on arXiv 11 Jun 2026.

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

- [Gauge Equivariant Convolutional Networks and the Icosahedral CNN](https://mlsanity.com/paper/1902.04615/) (2019)
- [A General Theory of Equivariant CNNs on Homogeneous Spaces](https://mlsanity.com/paper/1811.02017/) (2018)
- [Weisfeiler and Lehman Go Topological: Message Passing Simplicial Networks](https://mlsanity.com/paper/2103.03212/) (2021)
- [Simplicial Neural Networks](https://mlsanity.com/paper/2010.03633/) (2020)
- [Principled Simplicial Neural Networks for Trajectory Prediction](https://mlsanity.com/paper/2102.10058/) (2021)
- [Lattice gauge equivariant convolutional neural networks](https://mlsanity.com/paper/2012.12901/) (2020)
- [E(n) Equivariant Topological Neural Networks](https://mlsanity.com/paper/2405.15429/) (2024)
- [Machine Learning Topological Invariants with Neural Networks](https://mlsanity.com/paper/1708.09401/) (2017)
- [Deep Learning Topological Invariants of Band Insulators](https://mlsanity.com/paper/1805.10503/) (2018)
- [Gauge-Equivariant Graph Neural Networks for Lattice Gauge Theories](https://mlsanity.com/paper/2604.20797/) (2026)
- [Persistent Cup-Length](https://mlsanity.com/paper/2107.01553/) (2021)
- [Higher Order Gauge Equivariant CNNs on Riemannian Manifolds and Applications](https://mlsanity.com/paper/2305.16657/) (2023)
- [Learning Chern Numbers of Topological Insulators with Gauge Equivariant Neural Networks](https://mlsanity.com/paper/2502.15376/) (2025)
- [Cup Product Persistence and Its Efficient Computation](https://mlsanity.com/paper/2212.01633/) (2022)
- [Haldane Bundles: A Dataset for Learning to Predict the Chern Number of Line Bundles on the Torus](https://mlsanity.com/paper/2312.04600/) (2023)

## Abstract (arXiv)

Many important observables in physics and geometry are cup products of cochains. The adjusted cup product neural layer has been introduced in this paper. It is a neural primitive that hard wires the cup product with an adjustment term from higher gauge theory. This creates a readout that is gauge invariant by design. Their main theoretical result shows that on a closed cycle the output relies entirely on the adjustment coefficient. Setting this coefficient to zero removes the output completely regardless of other parameters. Thus the adjustment is the only source of gauge invariant signal. They prove this observable is a nonzero quadratic form and is exactly invariant under one and two gauge transformations.