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Source. Published p. 228, Remark 3.8, and its use on pp. 229–230. (canonical PDF).

Fix $s,k$, the enumeration of $k$-sets, and the unit vector $a$ in lemma_3_5. If the ratio $b/l$ stays fixed as the admissible integers $b,l,n$ vary, the labeled configurations $(v_1,\ldots,v_r)$ are all congruent. Their common norms and all their pairwise squared distances are fixed.

Proof.

The norm formula in Lemma 3.5 is $1+(b/l)q^{r-1}$. For distinct rows its exact squared-distance formula is

$$\|y_i-y_h\|^2+ \frac{b}{l}q^{r-2}\sum_{j,j'=0}^k(a_j-a_{j'})^2.$$

All quantities on the right are fixed. If $r=1$, only the norm statement is needed. Thus the full Gram matrix is fixed as well, by $2\langle v_i,v_h\rangle=\|v_i\|^2+\|v_h\|^2-\|v_i-v_h\|^2$.

Two finite vector families with identical Gram matrices are isometric: the map sending each labeled vector to its counterpart extends linearly on their spans and is well defined because the norm of every linear combination is given by the same Gram quadratic form. It preserves inner products and hence distances. Consequently these changing ambient dimensions contain one fixed target congruence class, and every fixed labeled subfamily also stays congruent.

Source precision.

The source's last index range in Remark 3.8 reads $1\le i,i'\le k$; the construction has $r=\binom sk$ rows. The conclusion holds for all of those rows, as the exact formulas show. This step is required before the density theorem can be applied to a fixed target.

Bears on. #174.