Unsupported Cyclotomic Divisors in Three-Prime Integer Tilings

Cyclotomic divisibility imposes strong prime-power structure on integer tiles. We study unsupported cyclotomic divisors: mixed-order divisors for which none of the prime-power components of the order divides the mask, although every prime in the order divides the tile cardinality. Kiss, Łaba, Marshall and Somlai asked whether such a phenomenon can occur in the three-prime setting. We prove that unsupported cyclotomic divisors already occur for periods with three distinct prime factors. For primes $p<q<r$, we characterize the square-period case: an unsupported factor ${\mathrm{\Phi }}_{pqr}$ occurs in a tiling of ${\text{ZZ}}_{(pqr{)}^2}$ if and only if $r\in ⟨p,q⟩$, and every such tile lies in a single residue class modulo $r$. Among cyclic tilings with the unsupported order dividing the specified modulus, the smallest modulus is $180$; if the order has three distinct prime factors, it is $900$. An Apéry-set construction gives examples for every triple at period $p^2{q}^2{r}^3$.The proof of our results combines Fourier rigidity, a three-cylinder decomposition, and an integer mass obstruction.