---

Source. The unnumbered linear-algebra observation on published p. 2 (canonical PDF); all arXiv versions have the same observation on p. 2. The proof below expands it. It is ancillary to the main simplex embedding construction.

Statement. If a nonempty finite Euclidean configuration $F$ admits a transitive abelian group of isometries, then $F$ is isometric to a subset of a regular polygonal torus. Conversely, every regular polygonal torus admits such a transitive abelian group. Thus subsets of finite transitive abelian configurations and subsets of polygonal tori give the same class up to isometry.

Proof. The converse is the independent coordinate-rotation action proved in the Ramsey-consequence page. For the forward implication, replace any ambient group by its image on $F$. The image is finite, abelian and transitive. Each distance-preserving permutation extends uniquely to an affine isometry of $\operatorname{aff}F$: the Gram-matrix construction in the canonical extension proof gives the extension, and agreement on an affine spanning set gives uniqueness. Consequently compositions and commutation are preserved on this affine hull.

Let $c=|F|^{-1}\sum_{x\in F}x$. Every extension fixes $c$, because it permutes the summands and is affine. On the real vector space $V=\operatorname{span}(F-c)$ the extensions are therefore commuting orthogonal linear maps. If $V=\{0\}$, then $F$ is a singleton and one polygon vertex suffices. Otherwise complexify $V$ to obtain commuting unitary maps on $\mathbb C^d$, where $d=\dim V$. A common orthonormal eigenbasis simultaneously diagonalizes them. This finite-dimensional fact follows by diagonalizing one unitary map, noting that all commuting maps preserve its eigenspaces, and continuing inside the common invariant eigenspaces.

In that basis write the diagonal entry of $g\in G$ in coordinate $a$ as $\chi_a(g)\in\mathbb C$, with $|\chi_a(g)|=1$. Matrix multiplication gives $\chi_a(gh)=\chi_a(g)\chi_a(h)$. Choose $x_0\in F-c$ and write its complex coordinates as $(z_a)_{a=1}^{d}$. Transitivity gives

$$F-c=Gx_0, \qquad (gx_0)_a=\chi_a(g)z_a.$$

Discard coordinates with $z_a=0$, since they vanish on the entire orbit. Any trivial character also has $z_a=0$: its coordinate is constant on the orbit, whose average is zero. For every remaining coordinate, $r_a=|z_a|>0$, and the image of $\chi_a$ is a finite nontrivial subgroup of the unit circle. It is cyclic: all its elements are roots of unity of order dividing $|G|$, and the $|G|$th roots form a cyclic group. Write its order as $m_a\ge2$. The possible values $\chi_a(g)z_a$ are precisely the vertices of a regular $m_a$-gon of radius $r_a$.

Translation by $-c$, a unitary change of coordinates, and removal of zero coordinates preserve distances. Regard each complex coordinate as an orthogonal real two-plane and rotate its polygon if necessary. The whole orbit therefore embeds into $\prod_a T_{m_a,r_a}$. At least one coordinate remains because $F$ is not a singleton. Restricting embeddings to subsets proves the final class equivalence. $\square$

Scope. The proof allows a selected abelian subgroup; the full isometry group need not be abelian. It proves no analogous enclosure for every finite transitive nonabelian group. The assertion does not identify the intrinsic circumradius of a later subset with that of its torus enclosure. The simultaneous unitary spectral theorem is a standard linear-algebra input, not a new Ramsey theorem.

Related. Theorem 2 supplies an abelian transitive enclosure even when the simplex itself has few symmetries.