Localization, Local--Global Transfer, and Hull Theory for $C{4}^{\ast }$-Modules over Commutative Rings

Let $R$ be a commutative ring and $M$ an $R$-module. We develop a localization and local-global theory for $C4$-modules, $C{4}^{\ast }$-modules, strongly $C{4}^{\ast }$-modules, $C4$-hulls, and pseudo-continuous hulls over commutative rings. The problem is structural: these notions are defined through decompositions, summand conditions, and minimal extensions, while localization changes decomposition data, support, and hull minimality. We prove forward localization theorems for the $C4$, $C{4}^{\ast }$, and strongly $C{4}^{\ast }$ conditions under exact lifting hypotheses formulated through decomposition lifting, morphism lifting, and submodule lifting. We also prove converse local-global theorems under descent and patching hypotheses, showing when primewise or maximal-local $C{4}^{\ast }$ behavior implies global $C{4}^{\ast }$ behavior. In addition, we establish obstruction results showing that no unrestricted local-global principle can hold. We compare the localization of a global $C4$-hull or pseudo-continuous hull with the hull formed after localization. We show that hull commutation requires both localization stability of the hull class and envelope-type axioms for hull minimality and uniqueness, and we prove conditional patching theorems for reconstructing global hulls from compatible local hulls. Our method is purely algebraic and support-theoretic, based on summand descent, patching of local witnesses, support control, and dimension-stratified transfer on $\mathrm{Spec}R$. As applications, we show that for commutative artinian rings these properties are detected exactly on the local factors, and that for finitely generated torsion modules over a Dedekind domain they are detected exactly on the primary components, equivalently on the localizations at maximal ideals in the support.