This paper investigates the foundational layer of paired complex numbers under commutative multiplication and axiomatizessix minimal structures that constitute this layer: the Real-Part Complex Plane, the Imaginary-Part ComplexPlane, 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 “angledifference” within paired complex numbers, and they determine the fundamental behavior of the system in the commutativesetting.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 commutativestructure of paired complex numbers is naturally formulated as a symmetric system of two planes, two angles, and twoangle differences.The framework developed here establishes the minimal commutative foundation of the theory and serves as a basisfor higher-order phenomena—such as non-commutative multiplication, twist structures, and multi-layered extensions—which will be addressed in subsequent work.