The order in the constant-distance branch has denominator dividing the Weyl group

In the constant-distance branch of the dichotomy for homogeneous harmonic maps into a conical Euclidean building with directions, the order is rational with denominator dividing the order of the finite Weyl group: there are positive integers $m,k$ with $k\mid|W|$ and

Role. The dichotomy for a nonconstant homogeneous harmonic map $h$ from the plane into a conical building says that either $\alpha$ is a half-integer, or the image of the unit circle lies at constant distance from the cone point. This statement disposes of the second branch, and it is where the Weyl group enters the answer. The first branch contributes denominators $2$, which divides $|W|$ because $W$ contains a reflection; the second contributes exactly the denominators produced here.

The argument. Under the constant-distance hypothesis the unit circle unfolds into the model apartment as a closed billiards path: there are orthonormal $v_1,v_2$, an element $w\in W$ and $m\in\mathbb{N}$ with the great circle spanned by $v_1,v_2$ satisfying $\gamma(s+\lambda)=w\cdot\gamma(s)$ for $\lambda=2\pi\alpha/m$. The period of such a lift is constrained: iterating the relation $k$ times, where $k$ is the order of $w$, closes the great circle, so $k\lambda=2\pi j$ for some positive integer $j$, and $k$ divides $|W|$ by Lagrange's theorem. Substituting $\lambda=2\pi\alpha/m$ and cancelling $2\pi$ gives $\alpha=mj/k$, which is the claim with numerator $mj$. The order is at least $1$, which is what makes the numerator positive.

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.