Relational Reality under Global Zero Constraint : A Unified Interpretative Framework for Conundrums in Cosmology and Quantum Mechanics

This paper presents a unified quantitative framework that addresses the fundamental conundrums of modern physics—ranging from the Hubble Tension in cosmology to the Measurement Problem in quantum mechanics—through a single geometric axiom: "Relational Differentiation under Global Zero Constraint ($\sum U_i = 0$)." We rigoroulsy define the universe not as a set of particles moving on a spacetime background, but as the Spectral Boundary of a relational graph satisfying a global zero-sum condition. By applying Spectral Graph Theory and Stochastic Process Theory (Ornstein-Uhlenbeck process), we demonstrate that the laws of General Relativity and Quantum Mechanics are not fundamental axioms, but Structural Consequences that inevitably emerge when graph connectivity exceeds a critical threshold. Key Theoretical Contributions 1. Mathematical Foundation: The Geometry of Zero Global Zero Constraint: Proposes that for a closed system to exist without external reference, the sum of all relational values must be zero. This constraint acts as a strict topological limit on the allowed evolution of the universe. Spectral Gap Dynamics: Utilizing Cheeger’s Inequality, we prove that the differentiation of the relational network naturally induces the generation of a spectral gap ($\lambda_1$), which manifests physically as vacuum energy. 2. Macroscopic Reconstruction: Cosmology Inflation as Percolation: The Big Bang is modeled as a Geometric Percolation Phase Transition. The rapid formation of a "Giant Connected Component" naturally resolves the Horizon and Flatness problems without requiring acausal fine-tuning. Dark Energy & Hubble Tension: The cosmological constant $\Lambda$ is reinterpreted as the long-time average of the fluctuating spectral gap ($\Lambda \propto \langle \lambda_1 \rangle$). The observed accelerated expansion is explained as a result of Structural Saturation, where boundary tension dominates over the internal matter load. This model offers a verifiable solution to the Hubble Tension by predicting enhanced expansion rates in cosmic voids. Dark Matter as Ghost Graphs: Dark Matter is redefined as "Geometric Shadows"—inactive subgraphs that contribute to the global mass constraint but are spectrally decoupled from the electromagnetic interaction channel. 3. Microscopic Reconstruction: Quantum Mechanics Origin of Uncertainty: Heisenberg's Uncertainty Principle is derived as the Statistical Lower Bound of spectral gap fluctuations required to satisfy the global zero constraint in real-time. Derivation of the Born Rule: The probabilistic nature of measurement is derived as the Geometric Consistency Cost (Orthogonal Projection) required for a system to return to the constraint hyperplane. Particle Generations: The existence of three particle generations is explained as a topological necessity arising from the Minimal Genus ($g=3$) required for stable graph differentiation. 4. The Nature of Time and Black Holes Emergence of Time: Time is defined as the accumulation of information correlations. We introduce an effective dimension coefficient $\alpha$ to describe how the time dimension vanishes ($\alpha \to 0$) at singularities. Black Holes: Modeled as Topological Bottlenecks where the Cheeger constant approaches zero ($\Phi_G \to 0$), causing a "freezing" of information diffusion (scrambling) rather than information loss. Conclusion This study provides a new phenomenological map for Quantum Gravity. It suggests that the "dark sector" of the universe and the probabilistic nature of the quantum world are not mysterious entities or intrinsic randomness, but Geometric Inevitabilities required to maintain the existence of a relational universe under the strict constraint of Global Zero. Related Works: Paper I: Relational Differentiation under Global Zero Constraint (DOI: 10.5281/zenodo.18028564) Paper II: Geometry of the Differentiation Boundary under Global Zero Constraint (DOI: 10.5281/zenodo.18051033) Paper III: Local Observer and the Flow of Time (DOI: 10.5281/zenodo.18058187)