The circle of a homogeneous harmonic map is locally an ellipse in an apartment

Let $M$ be a conical Euclidean building of type $W$ and dimension $N$, and let $h:\mathbb C\to M$ be a nonconstant homogeneous harmonic map of order $\alpha$ about $0$. Then there is a $\delta>0$ such that around every $\theta_0$ the unit circle's image lies in a single apartment, on which it is an ellipse centred at the cone point: there are an isometric chart $\iota:\mathbb R^N\to M$ taking $0$ to $h(0)$ and vectors $\mathbf v_1,\mathbf v_2\in\mathbb R^N$ with

Role. This is the single place where the harmonic map equation is used in the analysis of the circle, and it is the common source of everything that is said about the circle afterwards. Homogeneity together with the regularity theorem for harmonic maps into buildings confines the singular set of $h$ to the origin, so every point of the circle has a neighbourhood mapping into one apartment $A$. Fixing coordinates on $A$ with the cone point at the origin and writing $h(r,\theta)=r^\alpha g(\theta)$, the harmonic map equation in polar coordinates,

becomes $g''+\alpha^2 g=0$, whose solutions are exactly $\mathbf v_1\cos(\alpha\theta)+\mathbf v_2\sin(\alpha\theta)$.

Two things follow with no further geometry. The squared distance to the cone point is $\|g(\theta)\|^2$, a first harmonic in $2\alpha\theta$ with coefficients $\tfrac12(\|\mathbf v_1\|^2+\|\mathbf v_2\|^2)$, $\tfrac12(\|\mathbf v_1\|^2-\|\mathbf v_2\|^2)$ and $\mathbf v_1\cdot\mathbf v_2$; and in the branch where that distance is a constant $L$, the coefficients of $\cos$ and $\sin$ must vanish, forcing $\mathbf v_1,\mathbf v_2$ orthogonal of length $L$, so that the circle is a genuine circle of radius $L$ traversed at angular rate $\alpha$ and the chord law $d=2L\lvert\sin(\alpha\Delta/2)\rvert$ holds.

Both the apartment and the vectors depend on $\theta_0$; the passage to global statements is a separate, elementary matter.

Formalization Note. Apartments are isometrically embedded copies of the model apartment $\mathbb R^N$, so the chart is recorded as an isometry $\iota:\mathbb R^N\to M$ rather than as a member of the building's atlas; the normalization $\iota(0)=h(0)$ places the cone point at the origin of the chart, which is the choice of coordinates the source makes.

Retired 2026-09-07 — disproved, false as formalized. Do not use as a dependency.

The defect is in the shared definition layer, not in the mathematics of Breiner--Dees. IsPlanarKSHarmonicOn (Def_frame_2026_harmonic_building_conical) is defined purely through Lebesgue integrals -- ksEnergy, ksApproxEnergy, IsKSSobolevOn, SameKSTraceOnCircle -- and, unlike the goal-level predicate IsKSHarmonic, it does not require ContinuousOn. An a.e.-constant map therefore qualifies as "harmonic", and altering a map on a Lebesgue-null, dilation-invariant set (a ray) preserves every hypothesis -- IsHomogeneousOfOrderOn and NonconstantOn included, both being pointwise -- while destroying the pointwise conclusion. The same gap admits order alpha = 0 for nonconstant maps, which the source excludes.

A faithful restatement needs Continuous h (or the conclusion attached to the continuous representative) together with 0 < alpha. No corrected replacement node exists yet.