Retention Without Activation: A Minimal Relational System of Failure Under Persistence

This preprint introduces a minimal discrete-time relational system in which structural persistence and functional activation are formally separated. Failure is not defined as disappearance, disconnection, or transition to an external state, but as the inability of a retained relation to activate. The model consists of nodes, edges, non-negative weights, and a history-dependent threshold mechanism. A non-deletion update rule guarantees structural persistence, while activation depends on a local balance condition between density and threshold. Under bounded density and a mean-field approximation, it is shown that configurations exist in which relations are retained while activation globally ceases after a finite stage. This establishes a minimal formal distinction between existence and functioning within a single relational system, providing a structural account of failure without loss of existence.