Hypercomplex Binary Superposition Algebra: A Commutative and Associative Product

In this article, we introduce a hypercomplex algebra based on a binary superposition structure. Each algebraic unit is defined by a pair (ƒ; S) where ƒ ∈ {0; 1} encodes the logical presence of a base component, and S ∈ {−1; 1} encodes a geometric phase or orientation. This framework allows us to define an imaginary product that is both commutative and associative, properties rarely combined in higher-dimensional algebras. We demonstrate the consistency of this product through a binary and superposed formalism. This result provides a solid foundation for representing multi-level logic states, with potential applications in quantum computing processing.