Existence of the mathematical prediction equation and its convergence condition and its simplified derivation

The mathematical prediction equation (MPE) offers a mathematical definition for wave-motion field prediction in physical systems. It applies to fluid mechanics and numerical weather forecasting models. MPE is an infinite series ensuring space-time accuracy in the Taylor expansion, and its convergence condition (convergence domain) must be established. The provided simplified derivation and inference of MPE include an illustrative Fourier wave-motion equation analysis. The derivation also establishes mathematical relationships between the Euler constant equation and Newton's displacement, velocity, and acceleration. MPE accurately predicts fluctuations of known variabilities for any order. It should serve as the mathematical foundation for fluid numerical prediction models. Moreover, combining MPE and its convergence condition with the cubic spline function yielded closed prediction equations for ideal gas pressure, temperature, and flow fields with second-order space-time accuracy. Further research directions include demonstrating the validity of MPE rigorously by the mathematical community and exploring MPE applications in quantum fields and discussing the mathematical definitions and differences between Newton and Einstein displacements.