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Use the definitions in L143 and write $\kappa=\kappa(A)$. Fix $S$ with exactly $\kappa$ split classes and orthonormal coordinates in which $S=\lambda\operatorname{diag}(1,-1)$, $\lambda\ne0$. Then $d^{\mathsf T}Sd=\lambda(d_x^2-d_y^2)$. In a class that is not $S$-split, $d_x^2-d_y^2$ and $d_x^2+d_y^2$ are both constant, so all its pairs have the same absolute displacements, say $(a,b)$.

If all points lie on one vertical line, every displacement is vertical. Each class is then a union of vertical paths with at most $n-1$ pairs, and the distances from the lowest point give $R(A)\ge n-1$, which implies the claim. Otherwise let the abscissae fill $[m,M]$ with $W=M-m>0$, let $c$ points lie on the middle line $x=(m+M)/2$, and choose $p_L,p_R\in A$ with abscissae $m$ and $M$.

An unsplit class with $a>W/2$ has at most $n$ pairs. A neighbor of $v$ in it lies at $v+(\pm a,\pm b)$, and the abscissae $x_v+a$ and $x_v-a$ cannot both lie in $[m,M]$ because they differ by $2a>W$. So $v$ has neighbors in one column only, at most two of them, and a graph of maximum degree two on $n$ vertices has at most $n$ edges. An unsplit class with $a=0$ consists of vertical pairs of one length. On each vertical line every point has at most one neighbor above and one below, so the class is a union of paths and has at most $n-1$ pairs.

For each point $z$ strictly right of the middle line, consider the class of $|z-p_L|$; for each point strictly left, the class of $|z-p_R|$. These pairs have $|d_x|>W/2$. If such a class is unsplit, it is rare by the preceding paragraph. If it is split, $z$ lies on one of at most $\kappa$ circles about $p_L$, or about $p_R$, with split radii, so at most $2\kappa M(A)$ of the $n-c$ off-middle points meet a split class. An unsplit class with absolute displacements $(a,b)$ is reached from $p_L$ only by $p_L+(a,\pm b)$ and from $p_R$ only by $p_R+(-a,\pm b)$, so by at most four of these points. This gives at least $(n-c-2\kappa M(A))/4$ rare classes, all with $a>W/2$.

If $c\ge2$, let $z_0$ be the lowest middle point. The $c-1$ vectors from $z_0$ to the other middle points are vertical with distinct lengths, and each split length is attained by only one of them. Their unsplit classes have $a=0$ and are rare, which gives at least $c-1-\kappa$ further rare classes. They differ from the classes of the previous paragraph, whose common $a$ is positive. Therefore

$$R(A)\ge\max\Bigl\{0,\frac{n-c-2\kappa M(A)}{4}\Bigr\}+\max\{0,c-1-\kappa\}.$$

For $c=0$ this implies the claim. For $c\ge1$ the right side is at least $(n+3c-4-2\kappa M(A)-4\kappa)/4\ge(n-1-2\kappa(M(A)+2))/4$. $\square$

The argument uses only the classes met from the two extreme anchors, so the same count holds with $2\kappa M(A)$ replaced by the number of off-middle points whose anchor class is $S$-split, for any one $S$.