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

History-dependent entropic formulations require a precise meaning for channelwise terminal variation and a spatial power split that remains valid for weak diffusion fields.

This paper establishes both structures and places them in a single Path-Dependent Entropic Lagrangian construction.

Endpoint calibration and cocycle additivity uniquely determine the oriented spending increment, while a channelwise extension of a decomposed power port yields explicit terminal differentials for scalar dissipation and potential-weighted fluxes.

The diffusion split is formulated at Sobolev, finite-energy distributional, and bounded-variation/divergence-measure regularity.

A natural-dual-space counterexample proves that chemical-potential-weighted residuals do not imply conservation, and an independent multiplier restores the unweighted balance.

These ingredients are assembled in one augmented thermo-diffusion PDEL functional on a synchronized product space.

Its admissible directional stationarity produces energetic and thermal conjugacy, species balance, flux closure, entropy production, and the two terminal power-routing channels.

The same construction admits a finite-energy weak form and specializes to the Cahn--Hilliard equation.