Role-compatible symmetry reduction for persistent formations

Definitions and mathematical statement

The source asks whether configuration symmetry can reduce computations in persistence theory for a directed formation in which every agent has out-degree two except a leader and co-leader, and whether there is a body-CAD version. It does not state the exceptional degrees or a precise persistence model. The aid uses the planar leader--first-follower interpretation: a unique leader $L$ has out-degree zero, a unique first follower $F$ has out-degree one, and every other vertex has out-degree two. This convention is an explicit additional interpretation.

Let $G=(V,E)$ be such a directed graph and let $p:V\to{\mathbb{R}}^2$ be an injective placement, in particular $p_L\ne p_F$. A role-compatible geometric symmetry consists of a graph automorphism $\gamma$ preserving directed edges and a Euclidean isometry $\rho(\gamma)$ with \[ p_{\gamma v}=\rho(\gamma)p_v. \] Assume the geometric group action is faithful. The unique out-degree roles fix $L$ and $F$; consequently every geometric symmetry fixes both $p_L,p_F$. The resulting group is trivial or generated by reflection in their connecting line. This restriction need not hold if the exceptional roles are not unique or not preserved.

For distance constraints $c_{ij}(p)=\frac12|p_i-p_j|^2$, define the Jacobian \[ (J(p)u)_{ij}=(p_i-p_j)\cdot(u_i-u_j). \] Directed edges specify agent responsibility, although reversing a distance row does not change this infinitesimal equation. At a symmetric placement, the displacement representation $T_V$ and directed-constraint permutation representation $T_E$ satisfy \[ J(p)T_V(\gamma)=T_E(\gamma)J(p). \] For reflection, split both spaces into $+1$ and $-1$ eigenspaces, giving blocks $J_+$ and $J_-$. If $\mathcal R$ is the space of Euclidean infinitesimal rigid motions and $\mathcal R_\pm$ its sector decomposition, unrestricted infinitesimal rigidity requires \[ \ker J_+=\mathcal R_+\quad\text{and}\quad \ker J_-=\mathcal R_-. \] With two distinct fixed agents on the reflection axis, the expected dimensions are $\dim\mathcal R_+=1$ and $\dim\mathcal R_-=2$: translation along the axis is symmetric, while perpendicular translation and rotation are antisymmetric. Testing only the invariant orbit block $J_+$ does not certify unrestricted infinitesimal rigidity. Infinitesimal rigidity is also not, without the directed control hypotheses, a complete persistence theory.

The aid reports a five-agent directed graph that is generically minimally persistent but has a reflection-forced placement with $\ker J_+=\mathcal R_+$ and an additional nonrigid antisymmetric infinitesimal mode. The supplied record contains no edge list, coordinates or block matrices, so this reported example cannot be reconstructed uniquely here. Its claimed counterexample scope is only orbit-block-only certification of unrestricted infinitesimal rigidity at that special placement; it neither refutes generic persistence nor gives a negative answer to exploiting symmetry.

For body-CAD, let $c$ be a differentiable constraint map on a chosen body-configuration manifold. If a symmetry fixes the configuration and $c$ is equivariant, its differential intertwines the induced tangent and constraint representations. Decomposition into symmetry sectors then applies to this Jacobian as well. The actual body variables, directed responsibilities and persistence definition are not specified, so this linear reduction does not constitute a directed body-CAD persistence theorem.

Short English statement

Under the unique leader/first-follower convention, planar role-preserving geometric symmetry is at most a reflection. Symmetry splits the distance Jacobian into sectors, but all sectors must be checked for unrestricted rigidity. The aid's special five-agent example and the body-CAD extension need additional data beyond the invariant block.

Sources

• Permanent UnsolvedMath ID $20001865$, exact public code catalogue entry; local registry aim-geometry-notes.json, notes[202], canonical tag problem.
• AIM workshop "Configuration spaces of linkages," Geometry, in-text attribution St. John/Schulze: https://aimath.org/pastworkshops/linkagesproblems.pdf. The canonical extracted source item is 6; the supplied verification note reports that the original item is 16, the hyphen in "computa-tion" is a line wrap, and terminal 4 is a page number. These reported corrections preserve the permanent ID and public code. No PDF was accessed for this draft.
• Research attempt 1, "Role-compatible symmetry reduction for persistent formations," labelled partial_result, partially_solved, proved and low novelty confidence. It reports the five-agent example but supplies no defining graph or realization in this record. No full authoritative literature citation or year is supplied.