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Source. Published p. 6, Definition 9 and the following unnumbered paragraph (canonical PDF). The arXiv versions use Definition 3.7. This page expands the source's geometric observation and supplies the convention used in Lemma 10 and Proposition 11.

Statement. Let $X=\{x_1,\ldots,x_n\}$ be a nonempty finite Euclidean configuration. A labeled configuration $Y=\{y_1,\ldots,y_n\}$ is a regular expansion of $X$ if, for some $\alpha>0$,

$$\|y_i-y_j\|^2=\|x_i-x_j\|^2+\alpha^2 \qquad(i\ne j). \tag{1}$$

This is equivalent to $Y$ being congruent, with the given labels, to $\{(x_i,z_i):1\le i\le n\}$, where $\{z_i\}$ is a regular simplex with edge length $\alpha$. Every such expansion is affinely independent, even when $X$ is affinely dependent. For $n=1$, the edge condition is vacuous and both configurations are singletons.

Proof. For $n\ge2$, choose a regular simplex $\{z_i\}$ of edge length $\alpha$. Product distances satisfy

$$\|(x_i,z_i)-(x_j,z_j)\|^2 =\|x_i-x_j\|^2+\alpha^2\qquad(i\ne j).$$

Thus (1) is precisely the assertion that matching the labels is an isometry from $Y$ to this diagonal subset. Conversely, any such diagonal subset satisfies (1).

To check affine independence directly, let $\sum_i c_i=0$. The squared-distance identity in the finite Gram criterion gives

$$\left\|\sum_i c_i y_i\right\|^2 =\left\|\sum_i c_i x_i\right\|^2 -\alpha^2\sum_{i<j}c_ic_j =\left\|\sum_i c_i x_i\right\|^2 +\frac{\alpha^2}{2}\sum_i c_i^2.$$

The right side is positive whenever $c$ is nonzero. Hence no nontrivial affine relation among the $y_i$ exists. A singleton is immediate. $\square$

Use. Lemma 10 proves that every simplex arises this way; Proposition 11 embeds every regular expansion in a polygonal torus. The additional positive term belongs only to off-diagonal squared distances.