Real Phase Geometry on the C8 Algebra: A Compact 4 + 7 Dimensional Framework

Abstract We propose a compact algebraic framework based on the cyclic phase system C8 (j⁸ = −1), in which a single real phase axis represents time, three conjugate phase pairs generate the spatial directions, and the remaining four channels form hidden closed phase loops. When coupled with a secondary cyclic layer C3, this structure naturally yields an eleven-dimensional correspondence (4 + 7) without invoking external compactification. The C8 base preserves the curvature and variational properties of earlier C11 models while introducing a more stable real-time axis and a symmetric phase pairing that directly reproduces Lorentzian (1+3) geometry. In this formulation, time emerges as the only Hermitian phase, space as triplets of conjugate channels, and the internal M-theoretic manifold as composite phase loops of C8 ⊗ C3. This reformulation provides a unified algebraic origin for visible spacetime and hidden dimensions, offering a minimal and symmetric alternative to traditional differential compactification schemes.