Category Theory as Representational Artifact of Operational Structure: A Structural Theorem from Operatiology and Noology

This paper establishes that category theory is not a foundational layer of any operational system but a representational artifact: the minimal morphism-based formal language encoding the operational obstruction structure ℐ/∼ derived from Operatiology. The Unbounded Index Obstruction criterion classifies core categorical notions individually. Arbitrary categories, functors, and natural transformations require certification over non-finitely-exhaustible index families. Limits and colimits belong to the power-set type of ℐ/∼, the categorical analogue of the Power Set axiom in ZFC. Adjunctions belong to the unrestricted type, parallel to the Axiom of Replacement. The Yoneda lemma occupies the extremal position within the power-set type: the entire relational complexity of Nat (hA, F) collapses to a single evaluation, constituting maximum compression efficiency within the morphism-based language. The minimal categorical structures retaining executive correlates in C⁽³⁾_Πd are finite acyclic free categories and the single-object category BM₃ (ℂ). A fidelity ordering among categorical, set-theoretic, and type-theoretic foundations is established as a structural taxonomy. Category theory and ZFC are dual ℐ/∼ compression languages differing only in compression axis: membership versus morphism. Their foundational rivalry is a competition internal to the projective layer. The classification criterion (operational necessity, grounded in Axiom 2) and the classification function (Unbounded Index Obstruction) are both derived from the executive layer; no external foundation is structurally required or presupposed.