Structural Primeness under Global Constraint: The Stabilization Mechanism of the Universe and Its Spectral Signatures

Why does the universe possess a stable geometric structure instead of dispersing like dust? This paper answers this fundamental question by establishing a Structural Selection Principle under the framework of the 'Global Zero Constraint ($\sum U_i = 0$)'. Moving beyond the Anthropic Principle, we demonstrate that the stability of the physical world is a mathematical inevitability derived from spectral graph theory. 1. Core Argument: Survival of the "Structurally Prime" We mathematically prove that any 'Composite Structure'—a system loosely coupled enough to be decomposed—inevitably possesses internal Soft Modes (zero-energy deformation channels) due to the Cheeger Inequality. Under the differentiation pressure of the universe, these composite structures are dynamically unstable and destined to collapse. Consequently, the differentiation only stops when the system reaches "Structural Primes": irreducible topological states that possess Spectral Rigidity. 2. Physical Realization of Topological Stability The paper identifies the observable signatures of these surviving structures: Microscopic Stability: The topological gap of structural primes manifests as the Mass Gap and the Pauli Exclusion Principle (level repulsion). Macroscopic Separation: Dark Matter is redefined not as a new particle, but as a topological sector spectrally orthogonal ($\mathcal{O} \ll 1$) to the observer, sharing only global metric constraints. Black Hole Duality: The framework distinguishes Ringdown as a structural relaxation toward topological stability, and Hawking Radiation as spectral leakage from an incompletely closed boundary. Conclusion This work unifies ontological constraints and physical reality, proposing that the universe we observe is the unique stable solution remaining after the elimination of all unstable possibilities.