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Source. Karamanlis, published p. 3, Lemma 3 (canonical PDF). The corresponding arXiv v1–v3 result is Lemma 3.1.

Statement. Let $\Delta$ be a regular simplex with $n\ge2$ vertices and edge length $a>0$. For every integer $m\ge2$, it embeds into $T_{m,r}^{n}$ for

$$r=\frac{a}{2\sqrt2\sin(\pi/m)}>0.$$

A singleton embeds into any nonempty regular polygon.

Proof. Adjacent vertices $p,p'$ of $T_{m,r}$ have distance $2r\sin(\pi/m)=a/\sqrt2$. This is also the distance between the two vertices when $m=2$. For $1\le i\le n$, define a point $v_i$ of $T_{m,r}^{n}$ by putting $p$ in coordinate $i$ and $p'$ in every other coordinate. If $i\ne j$, the two tuples differ in exactly coordinates $i$ and $j$. Hence

$$\|v_i-v_j\|^2=2\|p-p'\|^2=a^2.$$

Sending the labeled vertices of $\Delta$ to these tuples is the required isometric embedding. A singleton needs only the choice of one vertex. $\square$

Use. Proposition 5 applies this construction separately to finitely many regular-simplex factors. The construction prescribes $m$ and determines $r$; it does not preserve an independently prescribed sphere radius.