Graded Apophatic Algebras: Negation Hierarchies as Modules over the Progenitor

Building on the Universal Apophatic Progenitor A0A_0A0 introduced in Betzer (2025), we construct a purely algebraic realization of its internal structure. Starting from a single generator representing the void and using only a degree-shifting negation operator, we define a graded structure whose presheaf topos realizes the apophatic foundation. The category of presheaves over the negation hierarchy forms an elementary topos equipped with exponentials and a subobject classifier. Iterating the power object functor yields a transfinite cumulative hierarchy modeling intuitionistic higher-order logic internally, while determinization via double-negation sheafification produces classical structure. This algebraic modeling demonstrates the progenitor’s generative power without reliance on modal indexing, illustrating its framework-independent nature and the necessity that all classical structure arises from negation-rich indeterminacy through collapse.