A Free Probabilistic Framework for Analyzing the Transformer-based Language Models

We present a formal operator-theoretic framework for analyzing Transformer-based language models using free probability theory. This leads to a spectral dynamic system interpretation of deep Transformers. We derive entropy-based generalization bounds under freeness assumptions and provide insight into positional encoding, spectral evolution, and representational complexity. This work offers a principled, though theoretical, perspective on structural dynamics in large language models Keywords: Transformers, Free Probability, Spectral Theory, Non-Commutative Random Variables, Language Models1. Introduction Large Language Models (LLMs) [1], particularly those based on Transformer architectures, are generative probabilistic models defined over sequences of discrete symbols.