  We complete here a three-part study (see also arXiv:1506.08095 and
1508.00856) of how codimension-two objects back-react gravitationally with
their environment, with particular interest in situations where the transverse
`bulk' is stabilized by the interplay between gravity and flux-quantization in
a dilaton-Maxwell-Einstein system such as commonly appears in
higher-dimensional supergravity and is used in the Supersymmetric Large Extra
Dimensions (SLED) program. Such systems enjoy a classical flat direction that
can be lifted by interactions with the branes, giving a mass to the would-be
modulus that is smaller than the KK scale. We construct the effective
low-energy 4D description appropriate below the KK scale once the transverse
extra dimensions are integrated out, and show that it reproduces the
predictions of the full UV theory for how the vacuum energy and modulus mass
depend on the properties of the branes and stabilizing fluxes. In particular we
show how this 4D theory learns the news of flux quantization through the
existence of a space-filling four-form potential that descends from the
higher-dimensional Maxwell field. We find a scalar potential consistent with
general constraints, like the runaway dictated by Weinberg's theorem. We show
how scale-breaking brane interactions can give this potential minima for which
the extra-dimensional size, $\ell$, is exponentially large relative to
underlying physics scales, $r_B$, with $\ell^2 = r_B^2 e^{- \varphi}$ where
$-\varphi \gg 1$ can be arranged with a small hierarchy between fundamental
parameters. We identify circumstances where the potential at the minimum can
(but need not) be parametrically suppressed relative to the tensions of the
branes, provide a preliminary discussion of the robustness of these results to
quantum corrections, and discuss the relation between what we find and earlier
papers in the SLED program.
