A polynomial can count objects: its coefficients tell us how many objects have each possible size. Cyclotomic generating functions form a particularly structured family. Their roots are roots of unity, and many can be written as quotients of simple building blocks called q-integers.

Billey and Swanson conjectured that, when such a quotient has nonnegative coefficients, its numerator parameters should outweigh its denominator parameters from both ends. Sort the two lists. Compare their smallest entries, then their largest entries. The proposed inequalities should always point in the same direction.

This Evidence Press candidate breaks the second part. Every polynomial coefficient is strictly positive, but the largest five thousand five hundred and ninety-one numerator parameters sum to less than the matching denominator parameters.

The mechanism separates two jobs. A signed list of divisors produces a negative upper-tail comparison. Multiplication by a sufficiently large power of one plus q makes all coefficients positive. In the parameter lists, that multiplication adds only twos above and ones below. Those small entries cannot repair the defect far out in the upper tail.

The arithmetic uses a previously known negative value of a smoothed Möbius sum. The new connection is how that value becomes a counterexample to the parameter conjecture. The construction is enormous but specified exactly; the polynomial does not need to be expanded.

Two clarifications sharpen the result. Cancelling common parameters does not remove the failure. And this particular example satisfies every inequality comparing the smallest entries. The universal smallest-entry question remains unresolved.

The paper supplies a written positivity proof and replayable exact arithmetic. Internal checks are not external peer review or formal verification. The manuscript, certificate and review response are linked from the release page. This is an AI-generated voice, not additional mathematical evidence.