Nonnegative Cyclotomic Products and Square-Sided Dice

We determine when a family of products of cyclotomic polynomials has nonnegative coefficients. For distinct primes p, q and positive integers a, b, the product Phi_(q^b)(x)^2 Phi_(p^a q^b)(x) is nonnegative coefficientwise precisely when a = 1, or when a = 2, p = 2 and q is congruent to 1 modulo 4. The case a = 2, b = 1 proves a conjecture of Fiore, Nasr and Stone arising from relabeling dice with square numbers of sides. The proof uses the classical decomposition of nonnegative vanishing sums of roots of unity whose orders have two prime divisors. For products with nonnegative coefficients, this gives a sharp degree bound on integer multipliers whose coefficient sums are not divisible by a prescribed prime, together with all equality cases. At the exceptional boundary, a positive factorization gives the admissible dice polynomial directly. We also obtain a uniform obstruction for higher powers of the first cyclotomic factor.