  We prove a multi-valued $C^{1,\alpha}$ regularity theorem for the varifolds
in the class $\mathcal{S}_2$ (i.e., stable codimension one stationary integral
$n$-varifolds admitting no triple junction classical singularities) which are
sufficiently close to a stationary integral cone comprised of 5
half-hyperplanes (counted with multiplicity) meeting along a common axis. Such
a result is the first of its kind for non-flat cones of higher (i.e. $>1$)
multiplicity when branch points are present in the nearby varifolds. For such
varifolds, this completes the analysis of the singular set in the region where
the density is $<3$, up to a set which is countably $(n-2)$-rectifiable.
  Our methods develop the blow-up arguments in \cite{simoncylindrical} and
\cite{wickstable}. One key new ingredient of our work is needing to inductively
perform successively finer blow-up procedures in order to show that a certain
$\epsilon$-regularity property holds at the blow-up level; this is then used to
prove a $C^{1,\alpha}$ boundary regularity theory for two-valued $C^{1,\alpha}$
harmonic functions which arise as blow-ups of sequences of such varifolds, the
argument for which is carried out in the accompanying work \cite{minter2021}.
