Most mathematical models for interfaces and transition layers in materials science exhibit sharply localized and rapidly decaying transition profiles. We show that this behavior can largely change when non-local interactions dominate and internal length scales fail to be determined by dimensional analysis: we consider a reduced model for the micromagnetic N\'eel wall which is observed in thin films. The typical phenomenon associated with this wall type is the very long logarithmic tail of transition profiles. Logarithmic upper bounds were recently derived by the author. In the present article we prove that the latter result is indeed optimal. In particular, we show that N\'eel wall profiles are supported by explicitly known comparison profiles that minimize relaxed variational principles and exhibit logarithmic decay behavior. This lower bound is established by a comparison argument based on a global maximum principle for the non-local field operator and the qualitative decay behavior of comparison profiles.