Foundational Structure of Tsuifukusosu(対複素数)

Tsuifukusosu (Tsui-complex numbers) is introduced as a structural extension of the complexfield that preserves commutativity, associativity, and distributivity whileduplicating the internal form of complex numbers. Unlike dimensionalextensions such as quaternions or octonions, which inevitably breakcommutativity or associativity, the twin complex system is constructed on$\mathbb{C} \times \mathbb{C}$ and equipped with two symmetric multiplicativeoperations: a complex-type product extending the usual complex multiplication,and a componentwise product acting independently on each coordinate. Thisdouble algebra structure arises naturally from the OS model, which requires twointeracting complex states and both mixed and unmixed multiplicative behavior.We develop the algebraic foundations of Tsuifukusosu (Tsui-complex numbers), establish theirfundamental properties, compare them with existing number systems, and showthat they form the unique commutative double algebra satisfying thesestructural requirements. The twin complex framework provides a new directionfor analysis, geometry, and mathematical modeling in systems where dual complexdegrees of freedom naturally appear.