Spectral Sets Tile in Cyclic Groups of Square-Free Order

We prove the spectral-to-tiling implication in every finite cyclic group of square-free order. The main ingredient is an unconditional prime-step descent theorem. Let q ≥ 3 be prime, let (q, H) = 1, and let (A, Λ) be an m-point spectral pair in ZqH. If q ∤ m, then the reductions of A and Λ modulo H are injective and form an m-point spectral pair in ZH. To prove this, we decompose the prime-level Fourier blocks into local intersection spaces and copies of the regular representation of Zq. On every surviving regular summand an unnormalized block Gram matrix has the rational eigenvalue m/q. Its entries are cyclotomic integers, so algebraic integrality forces q | m, a contradiction. The regular summand therefore vanishes, and the two prime-level decompositions commute. A strong induction on the square-free modulus then proves that every spectral set tiles. Together with the known reverse implication, this gives the finite-cyclic Fuglede equivalence for all square-free orders.