Phase–Compactified M–Theory: The C8 Algebra and the 4 + 7 Internal Geometry

Abstract We develop an algebraic formulation of M–theory based on the eightfold cyclic phase algebra C8, defined by k8=−1k^8 = -1k8=−1. This structure naturally reproduces the 4+7 dimensional hierarchy of eleven-dimensional supergravity within a compact algebraic framework. The four visible directions correspond to observable spacetime (one time and three space), while the remaining seven emerge as hidden phase channels that encode internal curvature and flux compactification. The model unifies supersymmetry, brane dynamics, and cosmological acceleration as manifestations of a single cyclic closure. In this framework, geometry arises not from spatial extension but from algebraic rotation in phase space, and the cosmological constant emerges from quantized torsional energy. The C8 system therefore provides a minimal, finite, and predictive realization of M–theory through purely algebraic compactification.