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Source. Published p. 2, Theorem 2, proved through Section 3 on pp. 3–7 (canonical PDF). In all three arXiv versions the statement is Theorem 1.2.

Statement. Every finite affinely independent Euclidean configuration is isometric to a subset of a regular polygonal torus. More precisely, there are a common integer $m\ge2$, an integer $s\ge1$, and positive radii $r_1,\ldots,r_s$ such that the configuration embeds into $\prod_{a=1}^{s}T_{m,r_a}$.

Proof. Empty configurations and singletons embed trivially. For a simplex $Y$ with at least two vertices, Lemma 10 gives a simplex $X$ and a positive $\alpha$ for which $Y$ is a regular expansion of $X$. Proposition 11 embeds that expansion into an $m$-regular polygonal torus. These two deductions prove the stated conclusion with the same labels and exact distances. $\square$

Complete chain. The regular-simplex construction is Lemma 3. The pair-identification product in Lemma 4 and its application in Proposition 5 handle almost-regular residuals. The corrected line approximation in Lemma 7 gives the common-radius finite approximation. Proposition 11 adds the corrective factor to that approximation. Lemma 10 uses the canonical complete finite Gram criterion, not an unproved assertion about arbitrary distance-matrix differences.

The resulting torus has a transitive finite abelian group of isometries. Thus the theorem gives a subsoluble enclosure, and the separate Kříž deduction shows that every simplex is Ramsey for every finite number of colors. The enclosure argument itself does not use a Ramsey theorem.

Scope. Neither the factor orders, the number of factors, nor the radii are fixed in advance. The result does not establish the exponential density witness bound of Frankl–Rödl's 1990 proof, or force the simplex on every sphere of radius just above its own circumradius. It also does not classify all finite Ramsey sets or all spherical sets. Those questions must retain their separate evidence in Problem 174.