Theorem 1.1 — Possible orders of harmonic maps into Euclidean buildings

For every natural number $N$, every Euclidean Coxeter datum $C$ of dimension $N$—in particular $N>0$, with a concretely represented affine-isometry subgroup generated by a nonempty set of affine reflections, a finite orthogonal subgroup $W$, and a surjective rotational-part homomorphism onto $W$ agreeing pointwise with affine linear parts—every complete metric space $X$ whose supplied measurable structure is its Borel structure, every Hausdorff second-countable smooth one-dimensional complex manifold $S$, every Euclidean-building datum $B$ on $X$ whose atlas has pair coverage, ray and line coverage, affine-Weyl-compatible overlaps, maximality among compatible charts, and the stated CAT(0) segment and comparison properties, every connected open domain $D\subseteq S$, every map $u:S\to X$, and every point $x_0\in D$, the following implication holds: if $u$ is continuous on $D$ and, at every point of $D$, its expression in the preferred extended complex chart minimizes the normalized KS liminf energy on some coordinate ball against every KS-Sobolev competitor having vanishing normalized shrinking-annulus trace distance, and if there exist $x,y\in D$ with $u(x)\ne u(y)$, then there is an $r_0>0$ such that for every $0<r<r_0$ the closed coordinate ball at $x_0$ lies inside the coordinate image of $D$, the KS energy on its open ball is finite, and the real conversion of the boundary moment is positive; moreover, there exist natural numbers $m,k$ with $m>0$, $k>0$, $k\le m$, and $k\mid |W|$ for which the real quotient of $r$ times that finite KS energy by the boundary moment tends to $m/k$ as $r\to0$ through positive values; and if $N=1$, there also exists a natural number $m'\ge2$ for which the same quotient tends to $m'/2$. Here the coordinate map is the total composition with the extended chart inverse, the KS energy is the positive-scale liminf of the nested normalized difference-quotient integrals, the boundary moment is $r$ times the angular integral of squared distance from the center value, and all divisions are total real divisions, although the OrderDefinedAt conclusion ensures finite energy and a positive finite denominator at sufficiently small positive radii. The building datum $B$ is required as an argument but disappears from the unfolded conclusion; the conclusion depends on $C$ through $N$ and $|W|$. If the harmonicity antecedent is false, or it is true but the explicit nonconstancy antecedent is false, the theorem’s asserted implication is true vacuously. The rank-one numerator $m'$ is existentially independent of the earlier $m,k$.