Theory of Hierarchical Dissipative Self-Organizing Binary Network Dynamics

This paper presents the Theory of Hierarchical Dissipative Self-Organizing Binary Network Dynamics, an evolutionary framework derived from Status-Relational Entropy (SRE) Dynamics. Operating entirely within a discrete space, the system defines the evolution of real symmetric binary matrices under strict spin constraints without continuous function truncation or zero-states. At each dimensional expansion, historical cells govern their activity via an endogenous binary random gate, establishing a global dynamic redshift field based on topological geodesic depth rheology. The non-occurrence probability of nodes is driven by a non-linear mapping equation integrating local frustration energy and the temporal scale factor. A global adaptive negative feedback damping mechanism regulated by the core diagonal corner element stabilizes the global net charge pool. To establish the mathematical validity of the framework, we execute multi-sample finite-size scaling numerical simulations. The results demonstrate that as the system expands toward the thermodynamic limit, the topological coherence order parameter of the ancient core exhibits strict asymptotic boundedness, converging to a stable, non-zero constant constant. Furthermore, multi-dimensional scaling (MDS) analysis verifies that the coherence length scales to zero, causing the underlying high-dimensional eigenvalue energy levels to collapse and spontaneously condense into a rigid, positive-definite three-dimensional manifold. This structural phase transition establishes that matter, distance-based space, and emergent geometric curvature can spontaneously arise from microscopic binary uncertain events and global algebraic continuous multiplication without hard-coded geometric design.