  Consider an area minimizing current modulo $p$ of dimension $m$ in a smooth
Riemannian manifold of dimension $m+1$. We prove that its interior singular set
is, up to a relatively closed set of dimension at most $m-2$, a $C^{1,\alpha}$
submanifold of dimension $m-1$ at which, locally, $N\leq p$ regular sheets of
the current join transversally, each sheet counted with a positive multiplicity
$k_i$ so that $\sum_i k_i = p$. This completes the analysis of the structure of
the singular set of area minimizing hypersurfaces modulo $p$, initiated by J.
Taylor for $m=2$ and $p=3$ and extended by the authors to arbitrary $m$ and all
odd $p$ in arXiv:2105.08135. We tackle the remaining case of even $p$ by
showing that the set of singular points admitting a flat blow-up is of
codimension at least two in the current. First, we prove a structural result
for the singularities of minimizers in the linearized problem, by combining an
epiperimetric inequality with an analysis of homogeneous minimizers to conclude
that the corresponding degrees of homogeneity are always integers; second, we
refine Almgren's blow-up procedure to prove that all flat singularities of the
current persist as singularities of the $\mathrm{Dir}$-minimizing limit. An
important ingredient of our analysis is the uniqueness of flat tangent cones at
singular points, recently established by Minter and Wickramasekera in
arXiv:2111.11202.
