Determine boundary contact asymptotics at the top-to-fully-zipped transition

Determine the detailed asymptotic expansion of the lower-pair contact statistic m_b along the phase boundary separating the top-zipped and fully-zipped phases, where the paper establishes only an O(\sqrt{n}) leading scale.

Background

On the transition boundary t_t, the upper pair remains macroscopically zipped and the lower-pair contact statistic is reported to scale as O(\sqrt{n}) plus a smaller correction. The authors explicitly state that they could not obtain more detailed asymptotics, so the amplitude and subsequent terms remain unresolved.