Discrete Saturation Topological Dynamics (DSTD): An Axiomatic Framework for Nonlinear Discrete Structural Evolution and Self-Regulated Complex Systems

This work introduces Discrete Saturation Topological Dynamics (DSTD), an axiomatic framework for modeling nonlinear structural evolution in complex discrete systems. DSTD formulates system evolution as a constrained nonlinear dynamical process governed by discrete state transitions, saturation-based regulation, global structural potential coordination, and self-adjusting feedback mechanisms. The framework defines a unified mathematical architecture consisting of a discrete state space, nonlinear evolution mapping, dynamic boundary constraints, structural potential functions, and a twelve-operator nonlinear functional family. The proposed framework integrates concepts from discrete dynamical systems, nonlinear saturation control, fixed-point theory, Lyapunov stability analysis, and topological state transition modeling. A global structural potential function is introduced using multiplicative aggregation to preserve sensitivity to individual structural components and enable coordinated regulation of complex system states. Theoretical propositions concerning fixed-point existence, contraction conditions, convergence behavior, and stability analysis are formulated within the DSTD framework. A supplementary computational realization is provided to demonstrate the implementability of the proposed nonlinear evolution architecture through scalar numerical simulation. This first release focuses on the formalization of the axiomatic framework and mathematical structure. Extended analytical proofs, high-dimensional implementations, and application-specific validations are reserved for subsequent versions.