Theory of Hierarchical Complex Numbers and Sign Structures

Complex numbers have long served as one of the most versatile and structurally elegant algebraic systems in mathematics, providing a unified description of two‑dimensional rotation, phase, and symmetry. Their success across analysis, geometry, physics, and computation stems from the remarkable coherence of their internal rotational structure. Because the theory developed in this work fully inherits the internal principles of complex numbers,we reconstruct our framework by extending these principles into a new algebraic structure. Complex numbers play a remarkably versatile role across many fields of mathematics and physics.This paper proposes infinite hierarchical complex numbers as a way to further develop and extend this inherent versatility, revealing structural features that remain hidden in the single‑layer complex framework. We introduce a general theory of hierarchical complex numbers and clarify the mathematical properties of digit‑wise hierarchical inversion. In particular, we show that digit‑level sign inversion is governed by two asymmetric Targets, which constitute the actual inversion mechanism, whereas the Setting condition is not essential to the operation itself and is introduced only to avoid human misinterpretation. From this discrete structure, we derive a closed‑form general expression for the inversion action. Furthermore, by representing the inversion as an XOR‑based matrix action, we show that digit inheritance naturally appears as a Jordan chain with eigenvalue 1, thereby explaining why the orientation of signs becomes mathematically irreversible. This yields a unified description of closure, normal forms, and matrix‑based visualization methods for inversion operations. By elevating the versatile complex number framework into a hierarchical setting, the results provide a new perspective on algebraic systems with multi‑level structure and offer a foundation applicable to mathematical models exhibiting matrix‑like behavior and to physical theories with discrete layered symmetries.