Entropy Control from Lopsided Dependency: Profinite Filtrations and Finite Information Extraction

We establish a general entropy–summability principle for refinement protocols evolving under a filtration on a profinite probability space. For stagewise families of finite-alphabet observables, we show that total conditional entropy extraction is finite whenever the total queried entropy is finite. Lopsided dependency and bounded degree provide a local-to-global mechanism, via Shearer-type inequalities, for verifying such entropy budgets from structural constraints. The results are arithmetic-free, filtration-native, and designed as a reusable interface for impossibility arguments based on information accumulation. In particular, this paper replaces informal “dependency-control” assumptions by explicit, checkable entropy conditions in applications such as covering system problems.