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Source. Published p. 6, Lemma 10 (canonical PDF); arXiv Lemma 3.8. Karamanlis gives a proof pointer to Schoenberg and Frankl–Rödl. The deduction below is expanded relative to the complete finite negative-type criterion already compiled with Frankl–Rödl's 1990 source. That finite Gram proof is not duplicated here.

Statement. Every nonempty simplex $Y=\{y_1,\ldots,y_n\}$ is a regular expansion of another simplex $X=\{x_1,\ldots,x_n\}$. For $n\ge2$, the amount subtracted from every off-diagonal squared distance may be chosen positive and sufficiently small.

Proof. Suppose $n\ge2$ and put $e_{ij}=\|y_i-y_j\|^2$. For every nonzero zero-sum vector $c$, affine independence gives

$$Q_e(c):=\sum_{i<j}c_ic_je_{ij} =-\left\|\sum_i c_i y_i\right\|^2<0.$$

The zero-sum unit sphere in $\mathbb R^n$ is compact. Consequently there is a $\gamma>0$ with $Q_e(c)\le-\gamma\|c\|^2$ on the whole zero-sum subspace. Choose $0<\alpha^2<2\gamma$, set $d_{ii}=0$, and set $d_{ij}=e_{ij}-\alpha^2$ for $i\ne j$. Since $\sum_{i<j}c_ic_j=-\|c\|^2/2$ on that subspace,

$$Q_d(c)=Q_e(c)+\frac{\alpha^2}{2}\|c\|^2 \le-\left(\gamma-\frac{\alpha^2}{2}\right)\|c\|^2<0$$

for every nonzero zero-sum $c$. The linked finite Gram criterion therefore realizes $(d_{ij})$ as the squared distances of an affinely independent set $X$ in $\mathbb R^{n-1}$. In particular these off-diagonal entries are positive; this also follows by applying the strict inequality to $c_i=1,c_j=-1$, with all other entries zero. We obtain $\|y_i-y_j\|^2=\|x_i-x_j\|^2+\alpha^2$ for $i\ne j$, as required. For $n=1$, choose any singleton $X$ and any $\alpha>0$; the off-diagonal condition has no instances. $\square$

Dependency scope. This proves the source's essential reduction, relative only to the precise elementary criterion linked above. It does not assume that an arbitrary difference of two Euclidean distance matrices is Euclidean. Neither a general spherical Ramsey theorem nor the Matoušek–Rödl spread-vector theorem is an input to this deduction.

Use. Theorem 2. The same contraction mechanism appears in Frankl–Rödl's original simplex proof, whose subsequent approximation and density argument are different.