The circle of a homogeneous harmonic map is locally an ellipse in an apartment of the atlas
DisprovedHarmonicBuilding.circleApartmentRepresentationInApartmentRetired 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.
import Definitions.Def_frame_2026_harmonic_building_conical
namespace HarmonicBuilding
universe v
theorem circleApartmentRepresentationInApartment
{N : ℕ} (C : EuclideanCoxeterData N) (M : ConicalBuildingModel.{v} N C)
(h : ℂ → M.carrier) (alpha : ℝ)
(hhom : IsHomogeneousOfOrderOn M Set.univ h 0 alpha)
(hharm : IsPlanarKSHarmonicOn Set.univ h)
(hnc : NonconstantOn h Set.univ) :
∃ delta : ℝ, 0 < delta ∧ ∀ theta0 : ℝ,
∃ c ∈ M.building.atlas, ∃ p0 v1 v2 : ModelEuclideanSpace N,
c p0 = h 0 ∧
∀ theta : ℝ, |theta - theta0| < delta →
h (circlePoint 0 1 theta)
= c (p0 + (Real.cos (alpha * theta) • v1
+ Real.sin (alpha * theta) • v2)) := by sorry
end HarmonicBuilding