Binary Superposition Algebra: A Hyper Complex, Commutative, and Associative Product

In this article, we present a novel hyper-complex algebra grounded in a binary superposition algebras. We demonstrate this consistency through a structured binary formalism. This new product framework. Each algebraic element is represented as a pair (f, S), where f ∈ {0,1} encodes logica presence and S ∈ {1, 1} captures a phase or orientation. This formuation enables the definition of an imaginary product that is both commutative and associative, a rare combination in high-dimensional supports the encoding of multi-level logic states and offers potential applications in quantum information theory and algebraic modeling.