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Claim. Let $A\subset\mathbb R^2$ be a set of $n\ge2$ points. A class is the set of unordered pairs of $A$ at one occurring distance, and $R(A)$ is the number of classes with at most $n$ pairs, that is, the number of distances which occur at least once but between at most $n$ pairs. For a nonzero real traceless symmetric $2\times2$ matrix $S$, a class is $S$-split if it contains two pairs with difference vectors $d,e$ such that $d^{\mathsf T}Sd\ne e^{\mathsf T}Se$. Let $\kappa(A)$ be the least number of $S$-split classes over all such $S$, and $M(A)$ the largest number of points of $A$ on one circle of positive radius. Then

$$R(A)\ge\frac{n-1-2\kappa(A)\bigl(M(A)+2\bigr)}{4}.$$

Covers. Both questions of the modified Statement of Problem 132, for the sets on which some quadratic form splits few classes. Whenever the right side exceeds $1$, the set has two such distances, and along any sequence of sets on which $n-2\kappa(A)(M(A)+2)$ tends to infinity, so does $R(A)$. In the case $\kappa(A)=0$, where some nonsimilarity affine map preserves every distance equality, the bound is $(n-1)/4$, so every such set with $n\ge6$ has two. The bound gives nothing when every form splits many classes, as for sets invariant under a rotation of order three, where every class is split by every form. Both questions remain open in general.

Standing. L143 is author-recorded at tier 0: a complete argument using no external theorem, without independent examination or a Lean proof. Catalog acceptance is pending.

Depends on. L143 (tier 0), which consumes no other native claim and no external theorem.