  Gouv\^ea-Mazur [GM] made a conjecture on the local constancy of slopes of
modular forms when the weight varies $p$-adically. Since one may decompose the
space of modular forms according to associated residual Galois representations,
the Gouv\^ea-Mazur conjecture makes sense for each such component. We prove the
localized Gouv\^ea-Mazur conjecture when the residual Galois representation is
irreducible and its restriction to
$\textrm{Gal}(\overline{\mathbb{Q}}_p/\mathbb{Q}_p)$ is reducible and very
generic.
