An expansive tangent obstruction to symmetry breaking

Definitions and mathematical statement

Let $G=(V,E)$ be a finite labelled graph with prescribed bar lengths, realized by placements $p:V\to{\mathbb{R}}^d$ satisfying $|p_i-p_j|=\ell_{ij}$ for $ij\in E$. A finite symmetry group consists of graph automorphisms $\Gamma\leq\operatorname{Aut}(G)$, preserving these lengths, together with ambient Euclidean isometries realizing the relabellings at an initial placement $p_0$. Let $\mathcal C$ be the configuration space modulo all ambient Euclidean congruences. For a path $q(t)=[p(t)]$, call the motion expansive if every labelled pairwise squared distance \[ D_{ij}(q(t))=|p_i(t)-p_j(t)|^2 \] is nondecreasing; bar distances remain constant. Schulze asks whether a symmetric framework with a one-degree-of-freedom expansive mechanism must preserve its symmetry.

The supplied research aid gives a regular-case statement. Suppose the connected component $C\subset\mathcal C$ containing $q_0=[p_0]$ is a smooth regular one-manifold, all its placements affinely span ${\mathbb{R}}^d$, and the finite relabelling action is smooth. Assume there is a nonzero tangent $0\ne\tau\in T_{q_0}C$ with \[ {\,\mathrm{d}} D_{ij}|_{q_0}(\tau)\geq0\qquad(i,j\in V). \] The aid reports that every $\gamma\in\Gamma$ acts as the identity on all of $C$. Thus every congruence class on the component is fixed by the original relabellings. In this statement symmetry is preserved through a possibly placement-dependent ambient realization, not necessarily through unchanged matrices in a fixed coordinate frame.

The reported obstruction is that a reversing action on $T_{q_0}C$ would send the nonnegative vector $({\,\mathrm{d}} D_{ij}(\tau))_{i<j}$, up to permutation, to its negative. This would force all its entries to vanish. Affine spanning and infinitesimal rigidity of the complete framework then contradict $\tau\ne0$ in the congruence quotient. A finite action on a connected regular one-manifold that fixes $q_0$ and preserves orientation is pointwise trivial.

For a reversing analytic branch germ, choose a parameter $t$ in which the symmetry reverses $t$ to $-t$. The aid further reports that if every labelled squared distance is expansive on the positive ray, then every nonconstant expansion \[ D_{ij}(t)=D_{ij}(0)+a_{ij}t^{k_{ij}}+\text{higher-order terms},\qquad a_{ij}\ne0, \] must have even leading order $k_{ij}$. Constant distances are excluded from this leading-order assertion. This gives a higher-order obstruction rather than a resolution of singular or higher-order mechanisms.

The source does not define one degree of freedom as regular local moduli dimension, infinitesimal mobility, or existence of a selected branch. It also does not fix the ambient dimension, permitted collisions or the precise sense of preserving symmetry. The regular theorem therefore addresses only the explicitly stated class, not every interpretation of the original question.

Short English statement

Must a one-degree-of-freedom expansive motion of a symmetric linkage preserve symmetry? The supplied aid answers under a regular, affinely spanning, one-dimensional moduli hypothesis with a nonzero expansive tangent. Singular branches and motions whose first distance changes occur only at higher order remain outside that conclusion.

Sources

• Schulze, problem extracted as item 4, AIM workshop Configuration spaces of linkages: https://aimath.org/pastworkshops/linkagesproblems.pdf. Canonical location: aim-geometry-notes.json, notes[200]. The extraction includes a broken word and a trailing page number.
• Local registry data/unsolvedmath/problems.json, permanent ID 20001863, code catalogue entry; updated 14 August 2026. The clean-statement field is null, statement status reconstructed unverified, and original literature field empty.
• Supplied research attempt 1: partial result, partially solved status, rigor label proved, novelty confidence low. This machine-generated research aid is not peer reviewed and requires independent verification before citation; no detailed external theorem reference is supplied.