Mathematical Foundations of Alignment: Formalizing "Understanding" in Large Language Models Using Topology or Measure Theory

This research paper introduces a novel, rigorous mathematical framework that formalizes the concept of "understanding" in Large Language Models (LLMs). Moving away from purely behavioral and empirical benchmarks, the framework unifies three distinct mathematical disciplines to analyze the internal representations of neural networks: Topology: Utilizes Topological Data Analysis (TDA), specifically persistent homology and zigzag persistence, to model knowledge states and track the structural stability of geometric features across transformer layers. Measure Theory: Establishes a semantic \sigma-algebra generated by equivalence classes alongside a probability measure to quantify semantic consistency and meaning preservation under perturbations. Dynamical Systems: Models the reasoning process as discrete trajectories, evaluating local stability through Jacobian-based constraints. Theoretical claims are backed by empirical experiments on transformer-based models, demonstrating that correct reasoning and alignment strongly correlate with topological and dynamical stability, while conceptual failures or hallucinations correspond to structural collapse. This work bridges the gap between geometric deep learning, safety alignment, and theoretical computer science