Determinable Identity Under Real Transformation: On the Structural Exclusion of Complete Pair-Collapseability

This work investigates the minimal structural conditions of determinable identity under real transformation.The starting point is neither a comprehensive ontology nor a physical model, but a deliberately reduced formal framework: a non-empty set of states, a set of realizable transformations, and a state-based identity mapping. Under these assumptions it is shown that non-trivial identity, insofar as it is to remain determinable under real dynamics, is incompatible with complete pair-collapseability. Total dynamic mixing of all states is structurally incompatible with well-defined identity. Furthermore, the algebraic structure of the identity mapping is made explicit: it induces a congruence relation of transformational action, enables a factorization of the dynamics on the quotient space, and generates—via strongly connected components—a partial order structure. Under additional intra-class dynamics, a minimal functional three-part structure arises from identity structure, variation, and merge-exclusion. In the further course, the ontological consequence is formulated: determinable existence under real transformation implies selective stabilization. The resulting directed order structure is interpreted as structural time and serves as the basis for a further independent paper. The claim is strictly conditional and purely structural. No derivation of physical laws is claimed; rather, a necessary condition for any form of determinable existence under change is formulated.