Definition of Calculus on Tsuifukusosu (Tsui-Complex Numbers)

This paper establishes the dynamic structure of Tsuifukusosu (Tsui-Complex Numbers),extending the algebraic and geometric framework previously introduced in\textit{Tsuifukusosu (Tsui-Complex Numbers): A Structural Extension of the Complex Field}.Tsuifukusosu (Tsui-Complex Numbers) are defined as ordered pairs of complex numbers$(Z_1, Z_2) \in \mathbb{C} \times \mathbb{C}$, where the first componentrepresents continuous variation and the second represents jump-like,discontinuous variation. Because of this asymmetry, the usual notion ofderivative does not apply. We introduce a dual differential structureconsisting of a continuous derivative and a jump derivative, define thecorresponding integral, prove a generalized fundamental theorem ofcalculus, and formulate the general form of Twin Ordinary DifferentialEquations (Tsui-ODEs). This completes the analytic foundation of theTsui-Complex system.