Paratrophic determinants over Z/NZ encode multiplicative structure through matrix entries that depend on products of residues. Computing them directly is combinatorial. But the discrete Fourier transform turns multiplication into convolution, and convolution diagonalizes.
Liu applies discrete Fourier, cosine, and sine transforms to factor these determinants into products indexed by divisors of N. Each factor is a group determinant associated to the quotient group at that divisor. The factorization is explicit: the formulas express the determinant as a product over d | N of determinants of smaller matrices built from Fourier coefficients.
The method generates explicit formulas for multiple families. Determinants involving periodic Bernoulli functions — the sawtooth functions that appear throughout analytic number theory — factor cleanly. Determinants built from powers of the tangent function factor similarly. A corrected version of a conjecture by Sun Zhi-Wei falls out of the framework.
The structural point is algebraic: the multiplicative semigroup Z/NZ is too complicated to diagonalize directly, but the Fourier transform decomposes it along the divisor lattice of N. Each level of the lattice contributes one factor to the product. The arithmetic of N (its factorization, its divisors) directly controls the algebraic decomposition of the determinant.
The discrete Fourier transform doing what it always does — revealing multiplicative structure through additive means — applied to a new class of arithmetically defined matrices.