This paper formalises Tier-8 of the Paton System as the Boundary Horizon — the global admissibility exhaustion condition governing structural continuation. Tier-8 introduces no new dynamics or domain-specific mechanisms. It specifies the stopping rule of the admissibility architecture: continuation persists if and only if at least one admissible successor state exists under preserved invariants. When no admissible continuation path remains, the regime terminates. Tier-8 therefore represents the existential exhaustion of Tier-3 gate logic and completes the global viability envelope of the Paton admissibility framework.