Basic Angular Structures of Tsuifukusosu (Twin Complex Numbers)

This paper investigates the foundational layer of paired complex numbers under commutative multiplication and axiomatizes
six minimal structures that constitute this layer: the Real-Part Complex Plane, the Imaginary-Part Complex
Plane, the Real-Angle Plane, the Imaginary-Angle Plane, the Real Angle Difference, and the Imaginary Angle Difference.
These six structures provide a minimal vocabulary for describing the roles of “plane,” “angle,” and “angle
difference” within paired complex numbers, and they determine the fundamental behavior of the system in the commutative
setting.
Based on the geometric requirement that a plane is determined by three points, we clarify that angles span planes,
whereas angle differences are one-dimensional quantities that do not form planes. Consequently, the commutative
structure of paired complex numbers is naturally formulated as a symmetric system of two planes, two angles, and two
angle differences.
The framework developed here establishes the minimal commutative foundation of the theory and serves as a basis
for higher-order phenomena—such as non-commutative multiplication, twist structures, and multi-layered extensions
—which will be addressed in subsequent work.