# The Geometry of Meaning: Perfect Spacetime Representations of Hierarchical Structures

arXiv: https://arxiv.org/abs/2505.08795 · HTML: https://mlsanity.com/paper/2505.08795/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 5 of its 8 references that are in the map.
- First posted on arXiv 7 May 2025.

## Builds on (most cited first, 5 of 5)

- [Poincar'e Embeddings for Learning Hierarchical Representations](https://mlsanity.com/paper/1705.08039/) (2017)
- [Learning Continuous Hierarchies in the Lorentz Model of Hyperbolic Geometry](https://mlsanity.com/paper/1806.03417/) (2018)
- [The Geometry of Categorical and Hierarchical Concepts in Large Language Models](https://mlsanity.com/paper/2406.01506/) (2024)
- [Language Models as Hierarchy Encoders](https://mlsanity.com/paper/2401.11374/) (2024)
- [Neural Spacetimes for DAG Representation Learning](https://mlsanity.com/paper/2408.13885/) (2024)

## Abstract (arXiv)

We show that there is a fast algorithm that embeds hierarchical structures in three-dimensional Minkowski spacetime. The correlation of data ends up purely encoded in the causal structure. Our model relies solely on oriented token pairs -- local hierarchical signals -- with no access to global symbolic structure. We apply our method to the corpus of textitWordNet. We provide a perfect embedding of the mammal sub-tree including ambiguities (more than one hierarchy per node) in such a way that the hierarchical structures get completely codified in the geometry and exactly reproduce the ground-truth. We extend this to a perfect embedding of the maximal unambiguous subset of the textitWordNet with 82,115 noun tokens and a single hierarchy per token. We introduce a novel retrieval mechanism in which causality, not distance, governs hierarchical access. Our results seem to indicate that all discrete data has a perfect geometrical representation that is three-dimensional. The resulting embeddings are nearly conformally invariant, indicating deep connections with general relativity and field theory. These results suggest that concepts, categories, and their interrelations, namely hierarchical meaning itself, is geometric.