Spectral Phase Geometry from Cyclotomic Roots: Algebraic Axis Decomposition Beyond Root Arithmetic

Abstract We introduce a phase-algebraic framework derived from the cubic relation j³ = −1, in which cyclotomic roots are not treated as isolated arithmetic objects but reorganized into collective algebraic axes. By defining symmetric and antisymmetric phase combinations, we show that the resulting algebra admits an intrinsic spectral decomposition governed by a nontrivial minimal polynomial. This structure induces a pair of idempotent projectors and yields a natural two-channel decomposition without reference to matrices, inner products, or probabilistic postulates. The resulting “root-axis geometry” offers an alternative perspective on cyclotomic structures by linking number theory, spectral algebra, and geometric decomposition within a single algebraic setting. The construction is independent of quantum-mechanical assumptions and suggests applications in matrix-free spectral theory, algebraic geometry, and phase-based decomposition of dynamical systems.