On the Non-Existence of a Unified Equality Across Descriptive Levels

This paper examines the role of equality as an identification mechanism across multiple descriptive levels. While equality is commonly treated as a single relation, it is routinely used to perform distinct operations: syntactic equivalence within formal systems, equivalence induced by invariant coincidence, and identification tied to admissible realization. We show that no single equality relation can simultaneously capture all three modes. The incompatibility is structural and arises from refinement by admissible second-order variation, which generically separates objects indistinguishable at lower resolutions. Syntactic equality and invariant-induced equivalence remain valid within their respective frameworks, but neither can be extended to encode admissible realization without loss of structural resolution. The result establishes a non-existence theorem for unified equality relations and clarifies the resolution-dependent nature of identification in analytic and structural settings.