Constant distance to the cone point makes the local frame orthonormal, in an apartment of the atlas

Let $M$ be a conical Euclidean building of type $W$ and rank $N$, with apartment atlas $\mathcal A$, and let $h:\mathbb C\to M$ be a nonconstant homogeneous harmonic map of order $\alpha\neq0$ about the origin whose unit circle stays at constant distance $L$ from the cone point. Then there is a $\delta>0$ such that around every base angle $\theta_0$ there are a chart $c\in\mathcal A$, a base point $p_0$ with $c(p_0)=h(0)$, and an orthonormal pair $\mathbf v_1,\mathbf v_2$ with

where $\gamma_{\mathbf v_1,\mathbf v_2}(s)=\cos s\,\mathbf v_1+\sin s\,\mathbf v_2$ is the unit-speed great circle spanned by the pair.

Role. This is the constant-distance branch of the local analysis of the circle, in the form in which the building axioms can act on it. In an apartment chart the arc is an ellipse $p_0+\cos(\alpha\theta)\mathbf a+\sin(\alpha\theta)\mathbf b$; the squared distance to the cone point is then

a first harmonic in $2\alpha\theta$. Requiring it to be the constant $L^2$ on an arc forces both oscillating coefficients to vanish, so $\|\mathbf a\|=\|\mathbf b\|=L$ and $\mathbf a\perp\mathbf b$: the ellipse is a genuine circle of radius $L$ traversed at angular rate $\alpha$. This rigidity is what turns the local picture into a spherical one, and it is the hypothesis from which the holonomy of the frames around the circle is computed.

Formalization note. This is HarmonicBuilding.frame_orthonormal_of_constantDistance with the chart required to lie in the building's atlas, the base point $p_0$ carried explicitly since the atlas need not contain a chart sending the origin to the cone point. Positivity of $L$ is not assumed: it follows from nonconstancy.