Vector Determinant in the ARE Framework: From Scalar to Vector-Valued

The ARE (Action, Rectification, and Structure) method reorganizes the Leibniz expansion of the determinant via the right action of the cyclic group C_n on S_n, partitioning permutations into orbital classes. This paper introduces the vector determinant D_φ(A) = (G_0,...,Gₙ₋₁) ∈ ℂⁿ, where each mode G_k is the discrete Fourier transform of the orbital sums Λ_r(A). The classicaldeterminant is recovered exactly as the fundamental mode G_0(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.