From C8 Algebra to the E8 Phase Geometry: Hidden–Visible Closure in Eight Dimensions

Abstract This paper develops a geometric bridge between the algebraic structure of the C8 complex phase system and the exceptional E8 lattice. Starting from the closure condition j8=−1j^8 = -1j8=−1, the eight phase channels {1,j,j2,…,j7}\{1, j, j^2, \ldots, j^7\}{1,j,j2,…,j7} are organized into visible and hidden subspaces whose curvature balance reproduces the self-dual symmetry of the E8 lattice. The C8 algebra thereby provides an explicit computational and analytic realization of the E8 root geometry, where each phase direction corresponds to a discrete root vector. This approach transforms E8 from an abstract Lie algebra into a calculable phase–metric manifold, enabling linearized wave and curvature formulations directly over its root network. The result suggests that internal symmetry closure and vacuum tension in high-dimensional field theories can be understood as emergent properties of an eight-channel complex phase geometry.