Remark: Family 166. The higher-dimensional Erd (o)s distinct-distances conjecture

For every fixed $d\ge 3$, any $n\ge 2$ distinct points in ${\mathbb{R}}^d$ determine at least $c_d{n}^{2/d}$ distinct distances, with $c_d>0$ depending only on dimension. This matches the integer-grid order and resolves the higher-dimensional Erdős distinct-distances conjecture with a constant-factor bound.

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.