Euler characteristic detects vanishing

Prove that for every admissible composition-degree pair (ω,d), the reduced Euler characteristic χ_d(ω) vanishes if and only if the reduced homology of the corresponding one-point compactification is zero.

Background

The signed cell count χd(ω) is computable from the rational enumerator Fω(t). If homology is free and concentrated in one degree, its absolute value determines the total Betti number.

The conjecture asserts that a zero Euler characteristic cannot arise from torsion-only homology or cancellation among nonzero Betti numbers in different degrees. The paper notes that only the forward implication is unresolved.