Constant frequency and the higher regularity of branch sets

We consider a two-valued function $u$ that is either Dirichlet energy minimizing, $C^{1,\mu}$ harmonic, or in $C^{1,\mu}$ with an area-stationary graph such that Almgren's frequency (restricted to the singular set) is continuous at a singular point $Y_0$. As a corollary of recent work of Wickramasekera and the author, if the frequency of $u$ at $Y_0$ equals $1/2+k$ for some integer $k \geq 0$, then the singular set of $u$ is a $C^{1,\tau}$ submanifold and we have estimates on the asymptotic behavior of $u$ at singular points. Using a nontrivial modification of the argument of Wickramasekera and author, we show that the frequency of $u$ at $Y_0$ cannot equal an integer and therefore must equal $1/2+k$ for some integer $k \geq 0$. We then use the asymptotic behavior of $u$ and partial Legendre-type transformations based on those of Kinderlehrer, Nirenberg, and Spruck to show that the singular set in this case is in fact real analytic.