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

Abstract: The ARE (Action, Rectification, and Structure) method is presented as a framework for reorganizing the Leibniz expansion of the determinant through the action of the cyclic group C_n on S_n. This action partitions the permutations into orbital classes and naturally leads to a finite Fourier decomposition of the determinant structure. The central object of the work is the vector determinant, whose components are Fourier modes associated with orbital sums of Leibniz terms. The classical determinant appears exactly as the fundamental mode G_0(A)=det(A), while the remaining modes are multilinear spectral observables associated with the orbital organization of the expansion. The manuscript establishes the fundamental properties of this formalism, including exact determinant recovery through a linear readout functional, multilinearity, Hermitian symmetry for real matrices, vector Jacobi and Laplace formulas, orbital Parseval identities, a vector Hadamard inequality, and finite trigonometric interpolation. The framework extends naturally to continuous frequencies through an associated orbital polynomial. It does not reduce the factorial complexity of determinant computation, but reveals an orbital-Fourier structure not visible in the classical scalar determinant. The work also clarifies the relation between orbital modes and circulant structures, proving that the crude orbital modes cannot generally coincide with circulant eigenvalues because of polynomial-degree incompatibility.