The ARE (Action, Rectification, and Structure) method reorganizes the Leibniz expansion of the determinant via the right action of the cyclic group Cₙ on Sₙ, partitioning permutations into orbital classes. This paper introduces the vector determinant D_φ (A) = (G₀,. . . , G₍-₁) ∈ ℂⁿ, where each mode Gₖ is the discrete Fourier transform of the orbital sums Λᵣ (A). The classicaldeterminant is recovered exactly as the fundamental mode G₀ (A) = det (A). The paper establishes multilinearity, Hermitian symmetry, orbital Parseval identity, vector Jacobi and Laplace formulas, vector Hadamard inequality, and a structural impossibility result (incompatibility of polynomial degrees) that prevents identification of orbital modes with circulant eigenvalues. A bilingual computational tool (Python/tkinter) is included.
Ramón Moya (Wed,) studied this question.