cayley-dickson algebra-valued metric spaces and convergent fixed point theory for hypercomplex contractions

This paper introduces a novel theoretical framework for metric spaces valued in cayley-dickson algebras, extending beyond the limitations of bicomplex-valued b-metric spaces to encompass the entire hierarchy of hypercomplex number systems including quaternions, octonions, and sedenions. we establish a comprehensive theory of cayley-dickson algebra-valued metric spaces (cd-metric spaces) with generalized triangle inequalities parameterized by coefficients that account for the non-associative and non-commutative properties inherent in higher-dimensional cayley-dickson constructions. our principal contributions include: (1) the formulation of a consistent partial ordering relation across all finite-dimensional cayley-dickson algebras, (2) the development of convergence theory and completeness criteria for cd-metric spaces, (3) the establishment of fixed point theorems for contractive mappings under rational-type conditions involving hypercomplex control functions, and (4) novel applications to systems of hypercomplex integral equations.