Vector Operational Mathematics and Vector Meta-Operational Mathematics: From Iteration of Vector Operations to Vector Meta-Operations

This paper systematically establishes “Vector Operational Mathematics” and “Vector MetaOperational Mathematics” as two interconnected new branches of mathematics. The core idea of Vector Operational Mathematics is to extend the number of repetitions of basic operations on vector spaces (vector addition, scalar multiplication, dot product, cross product, tensor product, etc.) from natural numbers to integers, rational numbers, real numbers, and even complex numbers. Vector Meta-Operational Mathematics further elevates vector operations themselves to the status of independent mathematical objects, studying meta-operations acting on vector operations (such as composition, direct sum, differentiation, exponentiation, etc.). We establish a complete axiomatic system (ten axioms) adapted to the vector space structure, rigorously define the integer-order, fractional-order, real-order, and complex-order iterations of various vector operations. By generalizing the vector-valued Schr¨oder equation, Abel equation, and Kneser-type construction, we prove the existence of iterations at each level and establish uniqueness theorems under regularity conditions. We deeply explore the singularity structure of complex-order vector iterations and their connection with higher-dimensional complex dynamical systems and classical Lie groups. Furthermore, we incorporate vector calculus (gradient, divergence, curl) and vector calculus of variations entirely into this framework, unifying discrete vector hyperoperations and continuous vector field analysis. We further reveal a strict categorical duality between the mathematics of vector numbers (translation group, rotation group) and the mathematics of vector operations (addition iteration group, cross product iteration group), and construct the operad and Hopf operad structures of vector meta-operations, connecting them with multilinear algebra, renormalization theory, and noncommutative geometry. This paper transforms all open problems in the field into rigorously proved theorems, laying a solid theoretical foundation for vector algebra and beyond.