Remark: Family 167. Pinned distances and a power saving for planar unit distances

Proves the weak pinned Erdős distance conjecture: for every fixed $\epsilon >0$, all but $o(n)$ points of any $n$-point planar set determine at least $n^{1-\epsilon }$ distinct nonzero distances. A complementary theorem bounds the number of unit-distance pairs by $O(n^{4/3-\delta })$ for an absolute $\delta >0$.

The authors are OpenAI. This companion does not verify the proof. The claim is not a theorem of the Enter series and not a premise of any registered sheet. It does not close O.1. O.1 stays NOT_CLOSED. It is not a de Sitter state and not the QdS external-leg scheme. No quantum coefficient is admitted. No compilation is asserted.