Orders of Harmonic Maps into Euclidean BuildingsResearch Paper
## Motivation
Harmonic maps into singular nonpositively curved spaces arise in geometric analysis, rigidity theory, and the study of group actions on buildings. Near a point in the domain, their infinitesimal growth is measured by an **order**, obtained from an Almgren-type frequency quotient. For smooth targets that order is tied to familiar Taylor expansion data. Euclidean buildings are instead assembled from Euclidean apartments along reflection walls, so a map can branch through a singular link and a priori might exhibit a much less controlled spectrum of homogeneities. Breiner and Dees prove that, for maps from surfaces, this spectrum is discrete and is governed by the finite rotational Weyl group of the building. The mission formalizes their headline classification theorem, Theorem 1.1 of [Breiner--Dees](https://doi.org/10.1007/s00526-026-03375-5).
The discreteness matters because frequency information is a basic input to stratification and regularity arguments for singular harmonic maps. A finite list of possible denominators prevents homogeneities from accumulating arbitrarily and isolates rank-one behavior. The formal target makes explicit the nonconstant condition used by the source paper's tangent-map reduction. Without it, the usual numerator and denominator of the frequency quotient both vanish for a constant map, so its order is not defined.
## Setting
A **Euclidean Coxeter complex** consists of Euclidean space together with an affine reflection group. Taking the linear parts of its affine isometries produces a finite rotational reflection group $W$. A **Euclidean building of type $W$** is a complete metric space covered by isometric Euclidean apartments whose overlaps are related by elements of the affine Weyl group; the atlas is required to contain the relevant geodesic segments, rays, and lines and to be maximal with these compatibility properties.
The domain is a connected open subset $D$ of a complex one-dimensional manifold, hence a Riemann surface domain. The formalization uses a concrete Korevaar--Schoen-style metric Sobolev energy built from normalized local difference quotients and Lebesgue area in charts. A map $u:D\to X$ is harmonic when it has finite local energy and minimizes that energy against competitors with the same trace. For $x_0\in D$ and small radii $r$, the energy and boundary moment determine a frequency quotient. When its limit exists with positive denominator, that limit is the order $\operatorname{Ord}_u(x_0)$.
## Formalization targets
### Main classification
For a nonconstant energy-minimizing harmonic map $u:D\to X$ and any $x_0\in D$, prove that the order is defined and that there are positive integers $m,k$ such that
$$
\operatorname{Ord}_u(x_0)=\frac{m}{k},
\qquad k\mid |W|.
$$
If the building has rank one, prove the sharper form
$$
\operatorname{Ord}_u(x_0)=\frac{m}{2}
\qquad\text{for some integer }m\ge 2.
$$
The same theorem also records the small-scale energy and positive-boundary-moment facts needed for the order to be meaningful; these are conclusions, not assumptions supplied by a solver.
## Significance
The result identifies a purely algebraic constraint on an analytic singularity invariant: every denominator divides the order of the finite rotational Weyl group. In rank one, where the target is a tree or an $\mathbb R$-tree, it recovers the half-integer spectrum and its lower bound. This converts an apparently continuous local invariant into a discrete one determined by the building type.
Formalizing the theorem requires reusable infrastructure that is largely absent from current Mathlib: concrete Euclidean-building atlases, metric-valued Sobolev energy, trace and boundary-moment constructions, harmonic energy minimization, frequency quotients, and homogeneous tangent-map interfaces. The paper theorem is proved in ordinary mathematics; the open task is to replace the single `sorry` in the target with a machine-checked Lean proof. A completed development would provide components useful for other singular-target harmonic-map and CAT(0) formalizations.
## Difficulty
The target is not a direct consequence of treating the building as a Euclidean vector space. A harmonic map can cross apartment walls, and a single chart need not contain the image of a punctured neighborhood. The local problem must respect both metric energy and Weyl-group compatibility. Moreover, the frequency quotient is defined through limiting analytic quantities, while the conclusion is an exact rational arithmetic classification. Bridging those levels requires controlling tangent maps and the geometry of directions in the building rather than merely proving monotonicity of the frequency.
The rank-one clause is not obtained by substituting $|W|=2$ into the general statement alone: it also asserts $m\ge2$. The formal proof therefore must preserve the nonconstant hypothesis and the positivity information that rules out the degenerate zero-order case.
## Formalization scope
The Lean bundle fixes a complex one-dimensional manifold model for the source, a genuine complete metric target, a finite affine reflection group acting by Euclidean isometries, and an explicit building atlas. The rotational group $W$ is the image of the affine group under taking linear parts, so $|W|$ is not an arbitrary external number. The domain carries a point $x_0$ and is nonempty by construction. The map is required to be nonconstant on the domain; this is the necessary explicit repair of the printed headline, whose later reduction theorem uses the same condition.
Energy, trace, boundary moment, frequency, and order are transparent definitions tied to the supplied geometry. In particular, the caller cannot choose a zero measure or an unrelated predicate to make the target vacuous. The theorem must establish finite small-scale energy, positivity of the boundary moment, existence of the frequency limit, and its classification. Solvers may contribute supporting files for metric Sobolev estimates, tangent-map compactness, homogeneous harmonic-map classification, or finite-reflection-group lemmas, provided they preserve the exact conventions in the definition bundle.
## Selected references
- Christine Breiner and Ben K. Dees, *On the Possible Orders of Harmonic Maps into Euclidean Buildings*, Calculus of Variations and Partial Differential Equations, 2026, Theorem 1.1 and Sections 2--4. [DOI](https://doi.org/10.1007/s00526-026-03375-5)
- Mikhail Gromov and Richard Schoen, *Harmonic Maps into Singular Spaces and p-adic Superrigidity for Lattices in Groups of Rank One*, Publications Mathématiques de l'IHÉS 76 (1992), 165--246. [EuDML](https://eudml.org/doc/104049)
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Vector Space Methods VII: Euler–Lagrange EquationsTextbook
## Motivation
The calculus of variations replaces optimization over finitely many coordinates by optimization over paths. Its necessary conditions underlie geodesics, minimum-energy curves, classical mechanics, and many optimal-control models. Chapter 7 of David G. Luenberger's [*Optimization by Vector Space Methods*](https://openlibrary.org/books/OL7612943M/Optimization_by_Vector_Space_Methods) presents this transition as an application of differentiation in normed vector spaces: a local extremum first forces every directional derivative to vanish, and the resulting integral identity forces a differential equation along the optimizing path. This mission formalizes the scalar, fixed-endpoint version in §§7.4–7.5. The target is intentionally the theorem actually isolated by the source, not a stronger modern Sobolev-space variant.
## Setting
Fix real numbers $a<b$. A **$C^1$ path on the segment** is represented in Lean by two functions, $x,\dot x:\mathbb R\to\mathbb R$. Both are continuous on $[a,b]$, and $x$ has derivative $\dot x(t)$ at every $t\in(a,b)$. Ordinary two-sided derivatives are not demanded at $a$ or $b$; this makes the formal endpoint convention match the one-sided role of endpoints in a closed interval.
Let $L(y,v,t)$ be a scalar Lagrangian. Along a candidate path, write
$$
L_x(t)=\frac{\partial L}{\partial y}(x(t),\dot x(t),t),\qquad
L_v(t)=\frac{\partial L}{\partial v}(x(t),\dot x(t),t).
$$
The Lean statement records these partial derivatives with `HasDerivAt` and assumes that $L_x$ and $L_v$ are continuous on $[a,b]$. A **fixed-endpoint variation** is another $C^1$ pair $(h,\dot h)$ with $h(a)=h(b)=0$. The first variation already computed from the action is
$$
\delta J(x;h)=\int_a^b\bigl(L_x(t)h(t)+L_v(t)\dot h(t)\bigr)\,dt.
$$
The main theorem begins from the stationarity identity $\delta J(x;h)=0$ for every such variation. It does not claim that the complete passage from a local extremum in Luenberger's $C^1$ norm to this integral formula has already been bundled into the root statement.
## Formalization targets
### Main goal: Euler–Lagrange equation
From the computed first-variation identity, prove that
$$
\frac{d}{dt}L_v(t)=L_x(t)\qquad(t\in(a,b)).
$$
The conclusion is expressed as `HasDerivAt Lv (Lx t) t`, so it asserts both differentiability of $L_v$ and the equality of its derivative with $L_x$. This is equation (2) and the conclusion reached on printed pages 180–181.
### Milestones
The first milestone formalizes §7.4, Theorem 1: a local minimum or maximum of a real functional has zero derivative along every direction whenever that scalar directional derivative exists. The remaining milestones are the three fixed-endpoint fundamental lemmas from §7.5. They respectively show that a continuous coefficient annihilating all variations is zero, that a continuous coefficient annihilating all variation derivatives is constant, and that an identity involving both $h$ and $\dot h$ forces the second coefficient to have derivative equal to the first. These are stated with the same $C^1$ variation class used by the goal.
## Significance
The result turns an infinite family of scalar integral equalities into a pointwise differential equation. Once available, the same interface can support standard variational examples by supplying a concrete $L$, its two partial derivatives, and a stationary path. It also provides the analytic core needed before treating natural boundary conditions, vector-valued paths, higher derivatives, or weak Euler–Lagrange equations.
The formalization adds reusable interval-sensitive infrastructure. In particular, `IsC1OnSegment` separates a path from its chosen continuous derivative and avoids silently imposing derivatives outside the optimization interval. The three fundamental lemmas are useful independently of the named Euler–Lagrange theorem: they are test-function principles for interval integrals and can serve later missions involving integration by parts or weak formulations. The mathematics is classical and proved in the cited text; the open work is a machine-checked Lean development of these exact statements in the pinned Mathlib environment.
## Difficulty
The source argument uses informal phrases such as “arbitrary $C^1$ function vanishing at the endpoints” and treats endpoint differentiation according to standard calculus convention. In Lean, those phrases must determine a precise domain, derivative witness, continuity requirement, and interval-integral orientation. Replacing $C^1$ variations by merely continuous functions would change Lemmas 2 and 3, while requiring `HasDerivAt` at the endpoints would add a hypothesis not present in the book.
Another tempting shortcut is to assume from the outset that $L_v$ is differentiable and then use integration by parts. That would trivialize the central regularity conclusion of Lemma 3: the book derives differentiability of $L_v$ from stationarity and continuity. The root therefore assumes only continuity of the two coefficient functions and concludes a `HasDerivAt` assertion on the open interval. Conversely, constructing the first variation from a local extremum of the action requires a separate differentiation-under-the-integral development and a topology on bundled $C^1$ paths; it is not hidden inside the main goal.
## Formalization scope
The scalar field, path values, time variable, and action values are all real. The interval is nondegenerate through the explicit hypothesis $a<b$. Integrals use Mathlib's oriented interval integral, but all principal statements are made in the forward orientation. Paths and variations are total functions on $\mathbb R$ whose relevant regularity is restricted to $[a,b]$. The Lagrangian is finite-valued. No measurability or integrability premise is omitted: continuity of the coefficient and variation factors on the compact interval supplies the intended finite integrals.
The goal starts from an already computed first-variation identity. Contributions connecting a genuine local extremum of the action in the norm $\max|x|+\max|\dot x|$ to that identity are welcome as a strengthening, but they must not be advertised as part of the present root theorem. Other welcome contributions include reusable continuous test-function constructions and endpoint-aware interval integration lemmas. Sobolev paths, vector-valued state spaces, free endpoints, and weak derivatives are outside this mission and should be proposed separately rather than obtained by weakening the stated hypotheses until the result becomes vacuous.
## Selected references
- David G. Luenberger, *Optimization by Vector Space Methods*, John Wiley & Sons, 1969, Chapter 7, §§7.4–7.5, pp. 178–181; definition of $D[a,b]$ on p. 23. [Open Library record](https://openlibrary.org/books/OL7612943M/Optimization_by_Vector_Space_Methods)