Operational Mathematics on Octonions: Extending the Number Field of Operation Counts to Octonions

This paper systematically extends the theoretical framework of Operational Mathematics from complex numbers to the octonion algebra. As the unique eightdimensional normed division algebra with non-associative and non-commutative properties, octonions introduce fundamentally new mathematical structures and challenges for extending operation counts. We establish a complete axiomatic system, rigorously define integer-order, fractional-order, real-order, and even complexorder iterations on octonions, prove the existence of iterations at each level using Schröder’s equation, Abel’s equation, Kneser’s construction, and transfinite induction, and establish uniqueness theorems under regularity conditions. We further prove that fractional calculus and fractional calculus of variations are special cases of Operational Mathematics in the continuous octonionic setting, thereby unifying discrete hyperoperations and continuous analysis within a single octonionic framework. This paper further reveals a profound duality between the mathematics of octonionic numbers and the mathematics of octonionic operations, and transforms all open problems into rigorously proven theorems, laying a solid theoretical foundation for Octonionic Operational Mathematics.