Noncommutative Interconversion

Complex numbers provide a fundamental and harmonious algebraic framework for describing two-dimensional rotations. Their internal structure, consisting of a single phase and a single rotational degree of freedom, has supported remarkably stable behavior across analysis, geometry, and physics. This paper introduces the Tsuifukusosu System, a new algebraic framework obtained by doubling the internal structure of complex numbers through the natural coupling of two complex planes. Unlike conventional approaches to four-dimensional rotations, this system requires no projection, coordinate decomposition, or quaternionic reinterpretation. Its internal structure directly encodes the geometry of four-dimensional rotations. Classical complex numbers appear naturally as a two-dimensional cross-section of the system, preserving full compatibility with complex analysis while enabling a unified treatment of four-dimensional rotation, twisting, and conjugation. The noncommutative product, defined by a conjugation-mediated asymmetric interaction between the two components, yields the noncommutativity characteristic of SO(4). The Tsuifukusosu System provides a minimal algebraic representation of four-dimensional geometry, offering computational advantages over matrix-based methods and enabling efficient four-dimensional calculations without projection.