The unit circle of a constant-distance homogeneous map lifts to a closed billiards path

Let $h$ be a nonconstant homogeneous harmonic map of order $\alpha$ from the plane into a conical Euclidean building carrying its $\Delta_{\mathrm{mod}}$-direction structure, and suppose the image of the unit circle lies at constant distance $L$ from the cone point. Then the circle lifts to a closed billiards path: there are orthonormal vectors $v_1,v_2$ of the model apartment, an element $w$ of the finite Weyl group and a positive integer $m$ such that the great circle they span satisfies

Role. This is the geometric heart of the classification of orders. In the constant-distance branch of the dichotomy for homogeneous maps, the image of the unit circle lies in the space of directions at the cone point; in a Euclidean building that space is a spherical building, and a locally geodesic closed curve in it unfolds, apartment by apartment, into a billiards path: a unit-speed path in the model sphere which is reflected by elements of the Weyl group each time it crosses a wall. Passing to the developed path in the model sphere turns a statement about the building into a statement about a great circle and a single Weyl element, and the period of that lift is what determines the order.

The direction structure is exactly what makes this possible. Angle rigidity — axiom EB2 — says that the Alexandrov angle between two segments issuing from a point is one of the finitely many distances between the $W$-orbits of their $\Delta_{\mathrm{mod}}$-directions; this is the discreteness condition under which the space of directions is a spherical building rather than merely a CAT(1) space, and it is the hypothesis under which the unfolding is defined.

Formalization Note. The developed path is written as the standard great circle $s\mapsto(\cos s)v_1+(\sin s)v_2$ in the model apartment. The shift by $2\pi\alpha/m$ records that the map from the circle wraps $m$ times before closing up, which is the source of the integer $m$ in the final answer $\alpha=mj/k$.

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.