Tsuifukusosu (Tsui-Complex Numbers): A Structural Extension of the Complex Field

Tsuifukusosu (Tsui-complex numbers) is introduced as a structural extension of the complex field that preserves commutativity, associativity, and distributivity while duplicating the internal form of complex numbers. Unlike dimensional extensions such as quaternions or octonions, which inevitably break commutativity or associativity, the twin complex system is constructed on $\mathbb{C} \times \mathbb{C}$ and equipped with two symmetric multiplicative operations: a complex-type product extending the usual complex multiplication, and a componentwise product acting independently on each coordinate. This double algebra structure arises naturally from the OS model, which requires two interacting complex states and both mixed and unmixed multiplicative behavior. We develop the algebraic foundations of tsuifukusosu (tsui-complex numbers), establish their fundamental properties, compare them with existing number systems, and show that they form the unique commutative double algebra satisfying these structural requirements. The twin complex framework provides a new direction for analysis, geometry, and mathematical modeling in systems where dual complex degrees of freedom naturally appear.