This preprint derives a minimal architectural condition for the preservation of functional continuity in distributed systems operating under scale, uncertainty, and adversarial disturbance. Continuity is defined as the maintenance of bounded operational cost and decision latency independent of system size and environmental hostility. Collapse is shown to arise endogenously when coordination cost, authority, or state reconciliation become non-local.A necessary and sufficient continuity condition is formalized: all state transitions and coordination operations must remain locality-bounded and system history must be retained in an externalized substrate. This condition decomposes into five required functions — local gradient navigation, substrate-retained history, bounded local verification, relay-based propagation, and fail-closed state ejection — and the removal of any single function is shown to produce systemic collapse under adversarial load, establishing the architecture as a minimal completeness set.The framework unifies previously constructed locality-preserving implementations into a single topological continuity invariant and reframes reliability, alignment, and scalability as structural properties rather than optimization outcomes. The result is substrate-independent and applies to distributed databases, multi-agent artificial intelligence, edge robotics, and biological regulatory systems.This Version 4.0 preprint constitutes the program-level unification of the locality-bounded continuity architecture.