  We analyze the buckling of a rigid thin membrane floating on a dense fluid
substrate. The interplay of curvature and substrate energy is known to create
wrinkling at a characteristic wavelength $\lambda$, which localizes into a fold
at sufficient buckling displacement $\Delta$. By analyzing the regime
$\Delta<<\lambda$, we show that wrinkles are unstable to localized folding for
{\em arbitrarily small} $\Delta$. After observing that evanescent waves at the
boundaries can be energetically favored over uniform wrinkles, we construct a
localized Ansatz state far from boundaries that is also energetically favored.
The resulting surface pressure $P$ in conventional units is
$2-(\pi^2/4)(\Delta/\lambda)^2$, in entire agreement with previous numerical
results. The decay length of the amplitude is
$\kappa^{-1}=(2/\pi^2)\lambda^2/\Delta$. This case illustrates how a
leading-order energy expression suggested by the infinitesimal displacement can
give a qualitatively wrong configuration.
