Exploring Hilbert Space Characteristics in Large Language Models: Toward a Theory of Prompt Algebra

This paper proposes an experimental framework for investigating the latent structure of Large Language Models (LLMs), particularly GPT-based architectures, through the lens of Hilbert Space concepts. Specifically, the work hypothesizes that prompt engineering operates as an implicit form of inner-product manipulation within a high-dimensional learned vector space. It introduces the idea of \textit{Prompt Algebra} as an emergent methodology for mapping, exploring, and controlling LLM behavior.