This work investigates thermodynamic constraints on task-weighted mutual information in open nonequilibrium systems. Building on the free energy–Kullback–Leibler divergence relation and entropy production theory in stochastic thermodynamics, we propose a conditional lower bound relating entropy production to the growth rate of task-relevant information under an alignment regime assumption. The framework distinguishes total statistical dependence from functionally weighted information and formulates a dynamical inequality connecting semantic growth to physical dissipation. The results are derived under standard assumptions (Markovian dynamics, canonical reference distribution, open systems) and are presented as conditional bounds rather than universal physical laws.