Remark: Family 257. A hyperbolic group with no geometric CAT(0) action

Answers negatively whether every word-hyperbolic group is a CAT(0) group. Constructs one with a finite classifying space but no proper cocompact isometric action on any proper complete $\mathrm{CAT}(0)$ space, in any dimension.

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.