Conjectured upper bound on how many pairs among $n$ points in the plane can be exactly one unit apart.

Sawin's Lemma 2: lattice geometry of unit distances (Sawin, arXiv:2605.20579).

Let $d \ge 1$, $R \ge 2$, and suppose $\Lambda \subset \mathbb{R}^{2d}$ is a lattice equipped with an additive embedding $\pi : \Lambda \to \mathbb{R}^2$. Suppose $S \subseteq \Lambda$ is a finite set of "matching" vectors satisfying $\|v\| \le 1$ and $\|\pi v\| = 1$ for every $v \in S$. Then there is a finite point set $U \subset \mathbb{R}^2$ with unit-distance density at least $(1 - 1/R)^{2d}\,\#S$, i.e. $(1-1/R)^{2d}\,\#S\,\#U \le \#\{\text{unit pairs in } U\}$.

This pure geometry-of-numbers reduction is the elementary heart of the disproof.