  We give a general description of gauge symmetry breaking using vertical and
remainder fluxes in 4D F-theory models. The fluxes can break a geometric gauge
group to a smaller group and induce chiral matter, even when the larger group
admits no chiral matter representations. We focus specifically on applications
to realizations of the Standard Model gauge group and chiral matter spectrum
through breaking of rigid exceptional gauge groups $E_7, E_6$, which are
ubiquitous in the 4D F-theory landscape. Supplemented by an intermediate
$\mathrm{SU}(5)$ group, these large classes of models give natural
constructions of Standard Model-like theories with small numbers of generations
of matter in F-theory.
