Spectral Decomposition without Matrices: Phase Algebra, Idempotent Projections, and Two-Channel Structures

Abstract We introduce a phase-based algebraic framework in which spectral decomposition, eigenvalue structure, and projector-based splitting arise intrinsically, without reference to matrices, determinants, or coordinate representations. Starting from a cyclic phase algebra generated by an element satisfying a cubic relation, we construct intrinsic phase combinations that generate a nontrivial minimal polynomial and an exact idempotent decomposition of the algebra. This yields a two-channel spectral structure governed by fixed eigenvalues, realized purely at the algebraic level. In contrast to standard linear algebra, where spectral theory is inseparable from matrix representations, the present approach demonstrates that spectrum is fundamentally a polynomial property of algebraic elements. The resulting decomposition precedes the introduction of norms, inner products, or probabilistic interpretation and therefore applies beyond the quantum-mechanical setting. We further show that the same algebraic structure underlies phase separation in dynamical systems, mode decoupling in linear flows, stratified components in algebraic geometry, and matrix-free operator theory. This work positions phase algebra as a unifying mathematical language connecting spectral theory, functional calculus, non-semisimple algebras, and multi-channel representations.