  More than twenty years ago a paradigm emerged according to which a
UV-insensitive Higgs mass $m_H$ and (more generally) a UV-insensitive Higgs
effective potential $V_{1l}(\phi)$ are obtained from higher-dimensional
theories with compact extra dimensions and Scherk-Schwarz supersymmetry
breaking. Since then, these ideas have been applied to different models of
phenomenological interest, including recent applications to the dark energy
problem. A thorough analysis of the framework on which such a paradigm is based
allows us to show that a source of strong UV sensitivity for $m_H$ and
$V_{1l}(\phi)$, intimately connected to the non-trivial topology of these
models' spacetime, was missed. The usual picture of the Scherk-Schwarz
mechanism and its physical consequences need to be seriously reconsidered.
