Path-Dependent Entropic Lagrangian for Probability Flows: Balance-Entropy Routing and Composable Information Potentials

Abstract

Probability distributions are central to information theory, statistical inference, and modern probabilistic learning. Maximum entropy selects a probability state under prescribed constraints, but it does not specify how that state is reached, how probability is transported, or how dissipation and external information exchange are accounted for along the path. We develop a path-dependent entropic Lagrangian calculus that extends static state selection to probability-path evolution through restricted generators, upper-limit history terms, and explicit balance--entropy port routing. The construction yields the thermal state relation, conservative probability balance, and nonnegative production under standard mobility closure. Its KL/Shannon sector recovers maximum-entropy and Bayesian laws as stationary no-flux states, while time-dependent information potentials separate internal dissipation from supplied information power. Composable information and structural potentials control tails, sparsity, robustness, regularization, and nonlocal multimodality without changing the accounting architecture. Two numerical examples verify mass conservation, energy decomposition, and the total free-energy ledger.