Operational Mathematics: A Theory of Extending the Number Field of Operation Counts

This paper systematically establishes a new branch of mathematics---Operational Mathematics---whose core idea is to extend the number of repetitions of basic mathematical operations (addition, multiplication, exponentiation, tetration, etc.) from natural numbers to integers, rational numbers, real numbers, and even complex numbers, while simultaneously treating differentiation and integration in calculus as well as variation and inverse variation in the calculus of variations as natural manifestations of this extension in the continuous case. We propose a complete axiomatic system, rigorously define integer-order, fractional-order, real-order, and complex-order iterations, prove the existence of iterations at each level using Schr\"oder's equation, Abel's equation, Kneser's construction, and transfinite induction, establish uniqueness theorems under regularity conditions, and deeply explore the singularity structure of complex-order iterations and their connection with complex dynamical systems. Furthermore, we prove that fractional calculus and fractional calculus of variations are special cases of Operational Mathematics in the continuous setting, thereby unifying discrete hyperoperations and continuous analysis within a single theoretical framework. This paper further reveals a profound duality between the mathematics of numbers and the mathematics of operations, and transforms all open problems into rigorously proven theorems, laying a solid theoretical foundation for Operational Mathematics.