Remark: Family 165. Exact crossing numbers of complete and complete bipartite graphs

Resolves the Harary–Hill conjecture and Turán's brickyard problem in the Zarankiewicz formulation, determining the crossing numbers of every complete and complete bipartite graph. The result proves the optimality of the classical drawings among all plane drawings with continuous edge arcs.

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.