Mathematical Derivation and Computational Analysis Framework for Markets Dynamics Using Fuzzy Hala Discrete Chaotic Systems with an Extended Privacy Protocol

This research establishes a formal topological framework for managing non- stationary market assets in portfolios by synthesizing high-dimensional chaotic dy- namics with industrial quality control and cryptographic verification. We introduce the Hala Operator as a state-dependent regulator capable of inducing Successive Controlled Collapse (SCC)—a process that maps continuous chaotic flows onto discrete, stable fixed-point constellations. By utilizing Taguchi Design of Experiments (DoE) for off-market robustness and Zero-Knowledge SNARKs for execution privacy, we provide a mathematically rigorous solution to the "Newtonian Trap" of market unpredictability. Formal proofs of global stability, dimension collapse via divergence analysis, and the uniqueness of the discrete constellation are presented.