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We consider a class of multi-valued harmonic functions which locally minimize Dirichlet energy. Such functions are a generalization of classical single-valued harmonic functions and were introduced by Almgren as approximations of area minimizing submanifolds. It is well-known that the dimension of the singular set of a Dirichlet energy minimizing function on an $n$-dimensional domain is at most $n-2$. We show the uniqueness of homogeneous harmonic tangent functions at each branch point and rectifiability of the branch set. Our approach involves “blow up” method due to Leon Simon, which was originally applied to multiplicity one classes of minimal submanifolds. We adapt Simon’s method in the higher multiplicity setting of multivalued Dirichlet energy minimizing functions using new estimates and techniques from prior work of Neshan Wickramasekera. Joint work with Neshan Wickramasekera.