This paper provides a first-time ever unification of information theory, semi-group theory with the theory of uncertain reasoning, through functional perspective. Fundamentally, the threshold theorems for the Inference Functional (IF) were devised. Furthermore, numerical experiments are illustrated. Some information-theoretic applications to Human Computer Interaction (HCI) are provided. The paper ends with concluding remarks, open problems, and future research pathways. The analytical derivations and numerical evidence collectively demonstrate that the proposed inference functional exhibits stable and predictable behavior under varying entropic parameters, offering a mathematically rigorous bridge between uncertainty modeling and algebraic structures. This reinforces the reliability of the unified framework for advanced information-theoretic reasoning and related applications.