Spending Operators and Weak Power-Port Decompositions for Path-Dependent Entropic Lagrangians

Path-dependent entropic Lagrangians formulate irreversible evolution through terminal variations of accumulated process power. This paper establishes a mathematical basis for calibrated terminal spending and weak diffusion power-port decompositions. Endpoint calibration and cocycle additivity determine a unique oriented spending increment, while local selector fields resolve its channel contributions. The diffusion power-port decomposition extends from Sobolev fields with normal-trace boundary power to finite-energy fields through a distributional balance component. These constructions are combined in a synchronized mixed thermo-diffusion functional. A Cahn-Hilliard specialization illustrates the weak channel structure and the associated energy balance.