  We consider the mass preserving $L^2$-gradient flow of the strong scaling of
the functionalized Cahn Hilliard gradient flow and establish the nonlinear
stability of a manifold comprised of quasi-equilibrium bilayer \muckmucks up to
the manifold's boundary. In the limit of thin but non-zero interfacial width,
$\varepsilon\ll1,$ the bilayer manifold is parameterized by meandering modes
that describe the interfacial evolution and "pearling" modes that control the
structure of the profile near the interface. The pearling modes are weakly
damped and can lead to the dynamic rupture of the interface. Amphiphilic
interfaces can lengthen to decrease energy. We introduce an implicitly defined
parameterization of the interfacial shape that uncouples this growth from the
parameters describing the shape and introduce a nonlinear projection onto the
manifold from a surrounding neighborhood. The bilayer manifold has
asymptotically large but finite dimension tuned to maximize normal coercivity
while preserving the wave-number gap between the meandering and the pearling
modes. Modulo a pearling stability assumption, we show that the manifold
attracts nearby orbits into a tubular neighborhood about itself so long as the
interfacial shape remains sufficiently smooth and far from self-intersection.
In a companion paper, arXiv:1907.02196, we identify open sets of initial data
whose orbits converge to circular equilibrium after a significant transient,
and derive a singularly perturbed interfacial evolution comprised of motion
against curvature regularized by an asymptotically weak Willmore term.
