[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 ($\sum U_i = 0$) 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 ($\lambda_1 > 0$) 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 ($\mathcal{O}$), Dynamical Connectivity ($\mathcal{C}$), and Gravitational Background ($\mathcal{G}$): Visible Sector ($\mathcal{O}=1, \mathcal{C}=1$): Shares both gauge basis and connectivity (Standard Model particles). Dark Sector ($\mathcal{O}=0, \mathcal{C}=1$): Shares connectivity and gravity but possesses an orthogonal spectral basis, rendering it electromagnetically invisible. Black Hole State ($\mathcal{C}=0$): 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 $H_0$ 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 ($\tau_{\mathrm{trans}}$). 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.

Version 2.0 Update Notes 1 Overview This notes summarizes the major changes from v1.1 to v2.0 of the paper ‘Structural Primeness under Global Constraint’. While v1.1 focused on establishing the Theoretical Framework of the structural selection principle and stabilization mechanism, v2.0 establishes a concrete Physical Classification based on this framework and significantly reinforces the Quantitative Verification methodology using observational data. The key changes can be summarized in the following three points: Introduction of 3-Axis Topological Classification: Formalization of physical reality into coordinates (O, C, G). Proposal of Integrated Verification Strategy: Addition of Bayesian verification models for macroscopic (PTA) and microscopic (Atomic Clocks) data. Redefinition of Dynamical Trajectories: Redefining phenomena such as quasars and black hole ringdowns not as states, but as trajectories. 2 Major Theoretical Expansions2.1 3-Axis Discrete Classification Space The distinction between the dark sector and black holes, which was only conceptually addressed in Section 5 (Discussion) of v1.1, has been systematized into a rigorous topological coordinate system in Section 5 of v2.0. v1.1: Defined the Dark Sector by ‘Spectral Orthogonality‘ and the Structural Black Hole (SBH) as a topological limit. v2.0: Classifies all physical realities by introducing three independent topological axes. 1st Axis (O): Basis Alignment (Visible vs. Dark) 2nd Axis (C): Dynamical Connectivity (Normal vs. Black Hole) 3rd Axis (G): Gravity (Common background shared by all states) This clearly distinguishes neutrinos (Visible Sector, weak coupling) from Dark Matter (Dark Sector, basis orthogonality) topologically. 2.2 Dynamical Trajectory vs. State v1.1: Described quasar extinction and black hole ringdowns in terms of Phenomenological Outlook. v2.0: Redefines these as ‘Collapse Trajectories‘ distinct from ‘Stable States (Vertices)‘. It specifies that continuity appears only in unstable trajectories (Transitional Layers, EBH) connecting discrete states. 3 Verification Methodology and Data Integration The most significant expansion in v2.0 is the creation of Section 6 (Integrated Verification) and Section 7 (Microscopic Implications), which contain specific observational verification strategies. 3.1 Macroscopic Verification: PTA and Triangular Correlation Added: Prediction of triangular correlation between Dark Matter (DM) structural fluctuations, Dark Energy (DE) tension, and Gravitational Wave Background (SGWB). Added: Presentation of a discrimination table (Table 1) to identify transitional layer decay signals in Pulsar Timing Array (PTA) data and a Hierarchical Bayesian Model. 3.2 Micro-Macro Consistency Test Added: Prediction that macroscopic gravitational waves (PTA) and microscopic quantum noise (Atomic Clocks) share the same correlation time ($\tau_{\mathrm{trans}}$). Established a cross-Bayesian testing strategy to verify this. 4 Structural Changes and Appendices4.1 Section Restructuring Expansion of Section 5: Section 5 (Discussion) of v1.1 has been significantly subdivided and expanded into Section 5 (Physical Realization), Section 6 (Integrated Verification), Section 7 (Microscopic Implications), and Section 8 (Discussion) in v2.0. Reinforcement of Void Interpretation: In Section 5.5 of v2.0, voids are concretized not as simple empty spaces but as regions generating ‘Topological Defocusing‘ due to sparse connectivity. 4.2 Addition of Appendices Appendix A & B: Derivation of level repulsion and proof of SBH existence are maintained and reinforced. Appendix C (New): Added a discussion on Robustness, reviewing alternative spectral models for transitional layer noise (such as cutoff power-law) and demonstrating that the existence of a characteristic timescale remains invariant regardless of model form. 5 Conclusion v2.0 maintains the theoretical skeleton of v1.1 while succeeding in concretizing abstract mathematical arguments into observable physical predictions. In particular, the ontological clarification through the 3-axis classification system and the securing of verifiability through Bayesian modeling represent a pivotal advancement that elevates this theory from a metaphysical hypothesis to a scientific theory.

Version 1.1 Update Notes Appendix B (Topological Necessity of Structural Black Holes) has been added.