In an apartment the circle satisfies the harmonic oscillator equation

Let $M$ be a conical Euclidean building of dimension $N$ and $h:\mathbb C\to M$ a nonconstant homogeneous harmonic map of order $\alpha$ about $0$. Then there is a $\delta>0$ such that around every $\theta_0$ the image of the unit circle lies in one apartment, where it is a solution of the harmonic oscillator equation: there are an isometric chart $\iota:\mathbb R^N\to M$ with $\iota(0)=h(0)$ and a twice differentiable curve $g:\mathbb R\to\mathbb R^N$ with

Role. This isolates the one genuinely analytic input in the whole study of the circle. Two things go into it and nothing else does. First, regularity: homogeneity together with the regularity theorem for harmonic maps into Euclidean buildings confines the singular set of $h$ to the origin, so each point of the circle has a neighbourhood whose image lies in a single apartment, and apartments are isometric copies of $\mathbb R^N$. Second, the equation: on such a neighbourhood $h$ is a regular harmonic map into a Euclidean space, so in polar coordinates centred at the cone point

and substituting the homogeneous form $w(r,\theta)=r^\alpha g(\theta)$ leaves precisely $g''+\alpha^2g=0$.

Everything downstream is then ODE theory and algebra: the solution is $\cos(\alpha\theta)\mathbf v_1+\sin(\alpha\theta)\mathbf v_2$, whence the squared distance to the cone point is a first harmonic in $2\alpha\theta$, and in the constant-distance branch the circle is a Euclidean circle of radius $L$ traversed at angular rate $\alpha$.

Formalization Note. The curve $g$ is asked for on all of $\mathbb R$ rather than only on the window. This is no strengthening: a solution of a linear constant-coefficient equation on an interval is the restriction of a unique solution on the line, given by the same closed formula. Stating it globally avoids carrying an interval through the differentiability hypotheses, and the agreement with $h$ is still asserted only on the window, where the geometry actually holds.

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.