Combinatorial support criterion

Prove that the rational signed weight enumerator F_ω(t) is identically zero exactly when the composition ω is not anchored, where anchored means having one part or having both first and last parts odd.

Background

The paper introduces anchored compositions as those with either one part or odd extreme parts. The conjecture predicts that precisely these compositions can yield nonzero signed enumerators and, conditionally, nontrivial principal homology in some admissible degree.

The criterion was checked computationally for all compositions of norm at most 11 and admissible degrees through 40, but a general sign-reversing involution or noncancellation proof is not established.