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Consider an area minimizing current modulo $p$ of dimension $m$ in a smoothRiemannian manifold of dimension $m+1$. We prove that its interior singular setis, 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 ofthe 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 ofthe 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 allodd $p$ in arXiv:2105.08135. We tackle the remaining case of even $p$ byshowing that the set of singular points admitting a flat blow-up is ofcodimension at least two in the current. First, we prove a structural resultfor the singularities of minimizers in the linearized problem, by combining anepiperimetric inequality with an analysis of homogeneous minimizers to concludethat the corresponding degrees of homogeneity are always integers; second, werefine Almgren's blow-up procedure to prove that all flat singularities of thecurrent persist as singularities of the $\mathrm{Dir}$-minimizing limit. Animportant ingredient of our analysis is the uniqueness of flat tangent cones atsingular points, recently established by Minter and Wickramasekera inarXiv:2111.11202.

Fine structure of the singular set of area minimizing hypersurfaces modulo $p$