Persistent Topological Structures in LLM Embedding Spaces: From Geometric Analysis to Controllability A Tier-I Study within the Meaning Unification Framework

This work investigates the geometric and topological structure of embedding spaces in transformer-based language models, with a particular focus on persistent H₁ cycles, intrinsic dimensionality, and the spectral properties of attention matrices.

Using empirical tools such as TwoNN intrinsic-dimension estimation, spectral analysis, and persistent homology, the study demonstrates that LLM embedding manifolds exhibit:

• non-uniform low-dimensional structure,

• stable H₁ loops that persist under perturbation, and

• attention matrices behaving as near-stochastic operators with meaningful eigenspectra.

These findings support the broader view that semantic organization in LLMs reflects deeper mathematical regularities in the representation space.

A key contribution of this work is the proposal that persistent H₁ loops can serve as topologically stable subspaces, offering a potential mathematical basis for controllability and alignment preservation in large-scale models.
We outline how such stable cycles may function as low-energy pathways for conceptual transitions, providing robustness advantages over existing activation-steering methods.

All synthetic illustrations and reproducibility code are included in the appendix.
The aim of this study is to provide a rigorous foundation for the emerging connection between topology, geometry, and semantic stability in AI systems, and to encourage further work toward mathematically grounded control mechanisms for advanced models.

For readers seeking the broader theoretical foundations and long-term research direction, two companion documents offer an integrated view of the overarching framework:

1. Meaning Unification Framework: A Dual-Tier Whitepaper Connecting Quantum–Geometric Structures and AI Semantic Stability

DOI: 10.5281/zenodo.17686790
URL: https://zenodo.org/records/17686790

2. Meaning Unification Framework (Full Whitepaper)

URL: https://zenodo.org/records/17650669

These documents situate the present paper within Tier I, linking formal mathematical structures—geometry, topology, intrinsic dimensionality—to later applications in alignment, semantic stability, and internal phenomenology developed in Tier II.