Dead zones and phase reduction of coupled oscillators

The paper’s Proposition 3 states that if N(Ẑ) and N(ĝin) each contain an interval of lengths L1 and L2, and L1+L2>2π, then the averaged coupling h(ϑ) = (1/2π)∫ Ẑ(ϑ+s) ĝin(s) ds has a nontrivial dead zone of length at least L1+L2−2π; the proof sketch argues by shifting one zero-arc to cover the complement of the other so the integrand vanishes for all s and hence h(ϑ)=0 on an interval of ϑ (Proposition 3 and equation (13) in the uploaded PDF) . The candidate solution gives exactly this geometric-shift/convolution argument in full detail—including an explicit construction via a lift to R and a length calculation of the shift set J, yielding |J|=L1+L2−2π—thereby matching the paper’s claim and filling in the omitted details. No substantive contradiction found.