Quadra-Line Functions — A mod-4 Euler-Type Basis Arising from Double-Square-Root Multivector Algebra —

In this paper, in the 4-dimensional Clifford algebra, we introduce the quadra-line functions (qdline functions), which naturally arise from the exponential expansion of a bivector satisfying these conditions. The quadra-line functions extend the classical cosine–sine basis (mod-2 structure) in Euler’s formula and form a new family of analytic functions possessing a mod-4 derivative cyclic structure. In this study, we provide closed forms by power series, trigonometric and hyperbolic functions, addition theorems, matrix representations, and numerical implementation methods, thereby establishing them as a systematic family of functions with both analytic and computational utility.We also touch upon possible applications to several fields such as geometric algebra, linear operator theory, and functional analysis.