Depth-Preserving Functors and the Categorical Structure of Layered Linear Systems

This paper develops a categorical framework for depth-structured linear systems, modeled as product categories of matrix algebras. While the algebraic structure of these systems corresponds to classical direct product rings, the categorical organization of depth, functorial constraints, and monoidal natural transformations introduces new structural properties not addressed in the classical literature. Depth-preserving functors enforce strict factorization, the determinant becomes a monoidal natural transformation, and the resulting system forms a strict, non-symmetric monoidal category suitable for modeling layered architectures in AI, optics, solar concentration systems, and morph-model dynamics. This positions the depth-matrix formalism as a natural isomorphism–analogue similar to categorical treatments of the Chinese Remainder Theorem.