# HodgePhaseNet: Discrete Hodge Decomposition of Semantic Flows in Transformer Attention

arXiv: https://arxiv.org/abs/2610.14971 · HTML: https://mlsanity.com/paper/2610.14971/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 20 of its 29 references that are in the map.
- First posted on arXiv 10 Oct 2026.

## Builds on (most cited first, 20 of 20)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Retrieval-Augmented Generation for Knowledge-Intensive NLP Tasks](https://mlsanity.com/paper/2005.11401/) (2020)
- [On Calibration of Modern Neural Networks](https://mlsanity.com/paper/1706.04599/) (2017)
- [TruthfulQA: Measuring How Models Mimic Human Falsehoods](https://mlsanity.com/paper/2109.07958/) (2021)
- [FEVER: a large-scale dataset for Fact Extraction and VERification](https://mlsanity.com/paper/1803.05355/) (2018)
- [FActScore: Fine-grained Atomic Evaluation of Factual Precision in Long Form Text Generation](https://mlsanity.com/paper/2305.14251/) (2023)
- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [Adversarial NLI: A New Benchmark for Natural Language Understanding](https://mlsanity.com/paper/1910.14599/) (2019)
- [Poincar'e Embeddings for Learning Hierarchical Representations](https://mlsanity.com/paper/1705.08039/) (2017)
- [FNet: Mixing Tokens with Fourier Transforms](https://mlsanity.com/paper/2105.03824/) (2021)
- [Weisfeiler and Lehman Go Topological: Message Passing Simplicial Networks](https://mlsanity.com/paper/2103.03212/) (2021)
- [Order-Embeddings of Images and Language](https://mlsanity.com/paper/1511.06361/) (2015)
- [Simplicial Neural Networks](https://mlsanity.com/paper/2010.03633/) (2020)
- [Topological Signal Processing over Simplicial Complexes](https://mlsanity.com/paper/1907.11577/) (2019)
- [Random Walks on Simplicial Complexes and the normalized Hodge 1-Laplacian](https://mlsanity.com/paper/1807.05044/) (2018)
- [HyperLex: A Large-Scale Evaluation of Graded Lexical Entailment](https://mlsanity.com/paper/1608.02117/) (2016)
- [Hodgelets: Localized Spectral Representations of Flows on Simplicial Complexes](https://mlsanity.com/paper/2109.08728/) (2021)
- [WavePhaseNet: A DFT-Based Method for Constructing Semantic Conceptual Hierarchy Structures (SCHS)](https://mlsanity.com/paper/2602.14419/) (2026)
- [QuantumPhaseNet: A Gauge-Covariant Geometric and Quantum-Spectral Theory of Semantic Concept Hierarchies with Prototype Validation of a Classical Quantum-Inspired Model](https://mlsanity.com/paper/2608.15820/) (2026)
- [ModularPhaseNet: Finite-Cyclic Phase Geometry for Computable Semantic Hierarchy, Direction, and Context Consistency in Standard Transformers](https://mlsanity.com/paper/2609.06000/) (2026)

## Abstract (arXiv)

WavePhaseNet introduced frequency-resolved semantic representations, QuantumPhaseNet introduced gauge-covariant phase connections, and ModularPhaseNet made a finite cyclic phase channel and its cycle holonomy computable in an ordinary Transformer. These constructions are complementary, yet they do not by themselves separate an observed collection of pairwise relations into a globally consistent direction, circulation around local compositions, and circulation that persists around a larger unfilled cycle. HodgePhaseNet addresses this separation by representing semantic transitions as an antisymmetric real-valued 1-cochain on a finite oriented simplicial complex. With vertex-edge incidence matrix B1 and edge-face incidence matrix B2, every such field has a unique orthogonal decomposition Omega = B1TPhi + B2Psi + H, where B1H=0 and B2TH=0. The three component fields are unique even when their potentials are not. We give explicit projectors, a sparse implementation, integrability and gauge results, a verified numerical example, and a redesigned attention interface. The attention bias is defined on allowed ordered query-key pairs, remains consistent under arbitrary choices of edge orientation, and preserves a decoder's causal information boundary by constructing its complex from the visible prefix. Gauge invariance of the attention output is conditional on using invariant or appropriately compensated features. A simple difference of vertex embeddings has no coexact or harmonic component; a genuine relation encoder is therefore indispensable. Local circulation is not automatically contradiction, and harmonic flow is not automatically missing knowledge. These interpretations and any reduction in factual error require controlled empirical evaluation. This paper presents a mathematical and architectural proposal, not external natural-language performance results.