Constraint Degeneracy and the Inevitability of Cosmic Cycles under the Global Zero Constraint

Abstract This paper examines the dynamical characteristics of the 'Single Structural Black Hole (Single SBH)', a topological limit state reached by a relational universe evolving under the Global Zero Constraint. While previous research (Paper IV) demonstrated the local stability of structural prime states, this study traces the macroscopic degeneration process of the constraint mechanism occurring as global relaxation of the entire system intensifies. We mathematically prove that as the internal boundary vanishes in the global topological structure, the Jacobian matrix of the constraint function undergoes a Rank-drop. This causes the collapse of the Lagrange multiplier system that defined the system's restoration forces. In this process, it is revealed that the Single SBH functions not as a stable terminal point but as a 'Topological Saddle Point' where the gradient of the constraint vanishes. Consequently, the universe does not remain in the limit state of a Single SBH but repeats an 'Inevitable Cycle (Structural Respiration)' initiating new topological differentiation, catalyzed by the structural instability caused by constraint degeneracy. This study suggests that the Global Zero Constraint theory is not merely a proposal for a static universe image but an integrated framework encompassing both dynamical irreversibility and global cyclicity. Key Highlights Mathematical Proof of Instability: Demonstrates that the Jacobian Rank-drop at the limit of boundary vanishing turns the global attractor into a degenerate critical point (saddle point). Structural Respiration: Proposes a recursive dynamic model: Differentiation -> Stabilization -> Global Relaxation -> Approach to Single SBH -> Re-differentiation. Thermodynamic Consistency: Clarifies that this cyclicity does not violate the Second Law of Thermodynamics, as the entropy definition itself evolves with the changing topological structure of the state space. Local Sub-cycles: Discusses the emergence of "Effective Black Holes (EBH)" and local sub-universes within the metastable intervals of the global grand cycle. Context within Series This paper serves as the dynamical conclusion to the "Relational Physics under Global Zero Constraint" series. It builds upon the existence of structural primes established in Paper IV and proves the impossibility of their eternal permanence.