Morita Invariance, Categorical Obstructions, and Dimension Transfer for \texorpdfstring{$C4$}{C4}, \texorpdfstring{$C4^{\ast}$}{C4*}, Strongly \texorpdfstring{$C4^{\ast}$}{C4*}, and Semi-Weak-CS Modules

Abstract

Let $R$ and $S$ be rings with equivalent module categories. We study the Morita behavior of the conditions $C4$, $C4^{\ast}$, strongly $C4^{\ast}$, and semi-weak-CS. The point is categorical. These conditions are expressed through direct summands, subobjects, essentiality, and finite decomposition data. Their Morita status must therefore be determined at the level of transported witness structure. We prove that the four classical conditions are Morita invariant. The $C4$ condition is treated through finite summand witness schemes. The $C4^{\ast}$ condition is treated through the absence of subobject-level $C4$ defects. The semi-weak-CS condition is treated through the absence of admissible semisimple obstruction pairs. The strongly $C4^{\ast}$ condition is then recovered as the simultaneous vanishing of the two corresponding defect types. From this point we derive ring-level characterizations, together with matrix and full-corner criteria. We also isolate obstruction and impossibility statements showing that the strong theory does not collapse into the pure $C4^{\ast}$ theory, and that the semi-weak-CS layer cannot be read off from ideal-theoretic $C4$ data alone. We then introduce finite depth and finite arity extensions of the $C4$ framework and prove their Morita invariance in the same witness-theoretic form. Finally, we formulate a categorical reconstruction principle showing that the $C4$-type theory of a module is determined by its transported defect geometry, and we indicate the conditional semiring path suggested by this formalism. The paper is purely algebraic. No empirical input is used. The proofs rest on categorical transport of finite witness data and on the separation between local $C4$ defects and semisimple essentiality obstructions.