Unified Gating: A General Mathematical Framework with Adaptive Operator-Based Discrete Dynamics

This paper introduces Unified Gating, a general mathematical framework for adaptive regulation in discrete dynamical systems. The central contribution is the definition of the Unified Gate Operator (UGO), a positive adaptive operator that transforms heterogeneous system descriptors into a dynamic regulation coefficient. Unlike conventional control approaches where regulation parameters are predefined, Unified Gating treats the regulation intensity itself as an evolving mathematical object. Based on the Unified Gate Operator, this work defines Unified Gating Dynamics (UGD), a constrained discrete-time dynamical model integrating adaptive transition, feedback regulation, drift coupling, and projection-based state preservation. The proposed framework consists of three hierarchical layers: (1) Unified Gate Operator (UGO), the fundamental mathematical primitive; (2) Unified Gating Dynamics (UGD), the discrete evolution mechanism; and (3) Unified Gating Framework (UGF), the application-oriented architecture. A general n-dimensional formulation is provided together with entropy-based feedback and bounded nonlinear gating examples. The framework is designed as an abstract mathematical tool for studying adaptive regulation mechanisms in complex discrete systems. This manuscript establishes the first public definition and timestamped research record of the Unified Gating terminology and its associated mathematical structure. Further theoretical analysis, stability proofs, and domain-specific implementations remain subjects for future research.