We introduce LARS (Live Adaptive Reasoning System), a formal framework for interactive AI systems that maintain an explicit, continuously evolving reasoning state under user-driven interruptions. Unlike conventional stateless request-response paradigms, LARS models interaction as a continuous state transition process: S(t+1) = f(S(t), \Delta U(t)), where S(t) represents the internal reasoning state and \Delta U(t) represents structured user interventions during execution. We further propose a structured state decomposition, an intent-aware update mechanism, and a quantitative metric, Reasoning Preservation Rate (RPR), to measure stability and continuity of reasoning under interruption.