Phase-Compactified Dimensions: An Algebraic Recasting of M-Theory

Abstract We propose an algebraic reformulation of M-theory based on the cyclic complex structures Cₙ defined by the generalized root condition jn=−1j^n = -1jn=−1. In this framework, higher-dimensional compactification arises not from spatial curling but from phase compactification—a closure of internal frequency channels within the cyclic algebra. Each Cₙ corresponds to an n-carrier phase domain whose visible projection defines the observable 3+1 spacetime, while the remaining n−kn-kn−k phase channels form closed oscillatory loops analogous to compactified dimensions. This algebraic formulation replaces differential geometric compactification with a self-consistent closure of cyclic phases, providing a simpler and more physically transparent language for supersymmetric and dual structures. Vacuum tension and dark-energy-like effects emerge as residual phase stresses, unifying the algebraic and physical pictures of dimensional compactness.