Structural Distinction and Irreversible Identification: A Pre-Dynamical Set-Theoretic Framework

We develop a purely set-theoretic framework for structural distinction, identification, and irreversible loss. Starting from a single existence axiom, we characterize howrelations distinguish elements through incidence, how equivalence relations induce noninvertible quotient maps, and how structural information is lost under identification.All results are derived without reference to time, dynamics, probability, metrics, orsemantics. The framework does not model physical irreversibility. Instead, it isolatesstructural features that must be present in any theory admitting irreversible identification. Persistence, memory, order, and loss are defined as invariance and monotonicityproperties under quotient refinement. This work constitutes a pre-dynamical structuralsubstrate intended for extension in higher layers.