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

Abstract This paper addresses the fundamental cosmological question: "Why does the universe exist in a stable geometric form instead of collapsing into formless dust? " By establishing the Global Zero Constraint (Uᵢ = 0) framework, we propose a Structural Selection Principle governing the evolution of the relational universe. We mathematically demonstrate that the differentiation process of the universal network inevitably eliminates unstable 'Composite Structures' through Cheeger Instability, converging solely to 'Structural Primes'—irreducible topological states that cannot be further decomposed. This study unifies the microscopic stability of matter (such as the mass gap and exclusion principle) and the macroscopic structure of the cosmos (such as the dark sector and cosmic web) into a single coherent logical structure. It argues that the stability of the universe is not an anthropic coincidence or the result of fine-tuning, but a geometric inevitability remaining after all dynamically unstable trajectories have been eliminated. Core Theoretical Contributions 1. The Stabilization Mechanism We provide a rigorous graph-theoretical proof that any composite structure with internal bottlenecks inherently possesses Soft Modes, making it dynamically unstable under the global constraint. Consequently, the universe must evolve toward Structural Primeness, characterized by spectral rigidity and a non-vanishing spectral gap. 2. The Geometric Origin of Quantum Phenomena We demonstrate that key quantum phenomena are not independent laws but spectral signatures of Structural Primeness. Mass Gap & Vacuum Stability: The non-vanishing spectral gap (₁ > 0) of structural primes manifests as the physical mass gap and the stability of the vacuum. Particle Sharpness: The suppression of composite cycles leads to Spectral Purification, explaining the sharp spectral lines and distinct identity of elementary particles. Exclusion Principle: Strong internal connectivity induces Level Repulsion (Wigner-Dyson statistics), providing a geometric basis for the Pauli Exclusion Principle and the volume of matter. 3. 3-Axis Topological Classification of Physical Reality We identify that Structural Primeness is realized in three distinct physical forms based on their topological relationship with the observer. We introduce a 3-Axis Discrete Classification Space defined by Basis Alignment (O), Dynamical Connectivity (C), and Gravitational Background (G): Visible Sector (O=1, C=1): Shares both gauge basis and connectivity (Standard Model particles). Dark Sector (O=0, C=1): Shares connectivity and gravity but possesses an orthogonal spectral basis, rendering it electromagnetically invisible. Black Hole State (C=0): A topological limit where information connectivity is severed, leaving only a gravitational boundary. Observational Predictions & Verification 1. Topological Defocusing and Hubble Tension The paper reinterprets cosmic Voids not as empty spaces but as regions of sparse connectivity. We propose that this structural sparsity causes Topological Defocusing, which could lead to a statistical violation of the distance-duality relation. This provides a novel structural perspective that may contribute to explaining the Hubble Tension (the discrepancy in H₀ measurements) via distance overestimation in void-dominated late-universe observations. 2. The Micro-Macro Consistency Test We propose an integrated verification strategy connecting cosmology and quantum physics. The theory predicts that the stochastic gravitational wave background detected by Pulsar Timing Arrays (PTA) and the microscopic quantum noise in Atomic Clocks share the same decay correlation time (ₓₑ₀₍ₒ). We present a Hierarchical Bayesian Model to verify this intrinsic correlation and distinguish it from standard astrophysical noise. 3. Dynamical Trajectories We redefine high-energy phenomena such as Quasars and Black Hole Ringdowns not as static states, but as 'Collapse Trajectories' of transitional layers stabilizing toward structural primes. Their statistical extinction and spectral signatures serve as direct evidence of the universe's structural stabilization. Conclusion This work shifts the paradigm of physics from a taxonomy of "what exists" to a structural evolutionism of "what can survive. " It offers a mathematically consistent and observationally verifiable framework where Dark Matter, Black Holes, and the vacuum's stability are necessary consequences of the Global Zero Constraint.