  In [Pal et al., arXiv:2206.03552], the authors discuss how an unsupported
flat annulus contracted at its inner boundary by a factor $\Delta$, buckles
into a radial wrinkling pattern that is fully isometric and tension-free. What
selects the wavelength in such a pure-bending configuration, in the absence of
any competing sources of work? In this paper, with the support of numerical
simulations, we argue that competition between stretching and bending energies
at local, mesoscopic scales leads to the selection of a wavelength scale
$\lambda^*$ sensitive to both the width $w$ and thickness $t$ of the sheet:
$\lambda^* \sim w^{2/3} t^{1/3} \Delta^{-1/6}$. This scale $\lambda^*$
corresponds to an arrest criterion for wrinkle coarsening starting from any
wavelength $\lambda \lesssim \lambda^*$, which can be interpreted in terms of
both size and energetic barriers to further coarsening. However, the sheet can
support coarser wavelengths: $\lambda \gtrsim \lambda^*$, since there is no
penalty to their existence. Since this wavelength selection mechanism depends
on the value of $\lambda$ itself, it is hysteretic.
