Closure-Continuity and the Trajectory Ontology

Identity-persistence has been treated, across philosophy, cognitive science, and theoretical biology, as a question about the relations between states. A system is identical to itself across time when its states at those times stand in some specifiable relation: continuity, similarity, causal connection, or psychological linkage. This paper argues that the approach cannot succeed. State-based ontologies cannot in principle ground identity-persistence. The strategies they have available — continuity, similarity, and stipulated rules connecting states-at-times — either fail outright or collapse into a trajectory condition under a different description. No state-based ontology produces persistence as the structural feature it is committed to explaining. The paper develops the alternative: identity is the persistence of a recursive maintenance regime, and the regime continues or fails. Persistence is the continuation. The Invariance Maintenance Condition specifies the structural requirement; closure is satisfaction of the condition. The cryptobiotic tardigrade is the operational test. Substrate is preserved across cryptobiosis; lineage is unbroken; no recursive maintenance occurs. The trajectory ontology returns identity-interrupted, against substrate-continuity intuition and against causal-historical theories of identity. The position is falsifiable: it would be defeated by a system whose identity-continuity is structurally preserved across genuine gaps in maintenance, or by a system that fails maintenance while maintaining identity-continuity by some non-question-begging structural criterion. The inversion is structural rather than preferential. State-based ontologies do not produce persistence less well than the trajectory ontology; they do not produce it at all.