Single-period degree law

Prove that for an anchored composition with at least two parts and generic enumerator F_ω(t)=εt^{|ω|+p(ω)}/Φ_{p(ω)}(t), assuming concentration and Euler-characteristic detection, the nonzero homology at degrees d=d₀(ω)+kp(ω) is concentrated in degree m(ω)+1+2k, and determine the correct replacement in the multi-period case.

Background

For compositions whose enumerators have a single-period denominator, the observed homological degree varies affinely with the degree parameter. The conjecture gives the slope and initial value in terms of p(ω) and the number of odd parts.

The paper explicitly notes that the formula fails without the single-period hypothesis, with (3,1,1,5) as an example, and leaves the multi-period replacement unresolved.