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Source. Published pp. 6–7, Proposition 11 (canonical PDF); arXiv Proposition 3.9.

Statement. If $Y$ is a regular expansion of a finite Euclidean configuration $X$, then $Y$ embeds into an $m$-regular polygonal torus for some integer $m\ge2$. The radii of its factors need not be equal.

Proof. A singleton is immediate, so write $X=\{x_1,\ldots,x_n\}$ with $n\ge2$, and choose $\alpha>0$ such that

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

Put $\delta=\alpha^2/n^2$. By Proposition 8, there are $m\ge2$, $r>0$ and an injective $\delta$-embedding $f:X\to T_{m,r}^{k}$. Define the symmetric errors

$$e_{ij}=\|x_i-x_j\|^2-\|f(x_i)-f(x_j)\|^2, \qquad |e_{ij}|<\delta.$$

Set $a_{ii}=0$ and $a_{ij}=\sqrt{\alpha^2+e_{ij}}$ for $i\ne j$. These roots are positive because $\delta<\alpha^2$. To verify the hypothesis of Lemma 4, let $A^2=\max_{i<j}a_{ij}^2$. Then

$$A^2>\alpha^2-\delta, \qquad 0\le A^2-a_{ij}^2<2\delta,$$

and therefore

$$\begin{aligned} \sum_{i<j}(A^2-a_{ij}^2) &<n(n-1)\delta =\alpha^2\frac{n-1}{n}\\ &<\alpha^2\left(1-\frac1{n^2}\right) =\alpha^2-\delta<A^2. \end{aligned} \tag{1}$$

The middle strict inequality uses $n>1$. Lemma 4 constructs an almost-regular simplex $Z=\{z_1,\ldots,z_n\}$ with distances $a_{ij}$. Proposition 5 embeds $Z$ into an $m$-regular torus $T_0$, with the same $m$ already chosen above; write this isometry as $h$.

For $1\le i\le n$, put $y'_i=(f(x_i),h(z_i))$ in $T_{m,r}^{k}\times T_0$. This product is $m$-regular. For $i\ne j$,

$$\begin{aligned} \|y'_i-y'_j\|^2 &=\|f(x_i)-f(x_j)\|^2+a_{ij}^2\\ &=\|f(x_i)-f(x_j)\|^2+e_{ij}+\alpha^2\\ &=\|x_i-x_j\|^2+\alpha^2=\|y_i-y_j\|^2. \end{aligned}$$

For $i=j$ both sides are zero. The matching of labels is thus an isometric embedding of $Y$. $\square$

Source precision. The source calls the residual array almost regular without giving inequality (1); this is the full bound for its exact choice $\delta=\alpha^2/n^2$. Its final display is introduced for all $i,j$, but the added $\alpha^2$ applies only when $i\ne j$. The diagonal case is separated here. The corrected sufficient parameter choice in Lemma 7 changes neither the tolerance nor the conclusion of this argument. No Euclidean realization of the residual is assumed before Lemma 4 supplies it.

Use. Theorem 2.