The crossing number of the complete graph minus an edge

We prove that for every integer n >= 2 and every edge e of the complete graph K_n, the ordinary crossing number is cr(K_n-e) = (1/4) floor((n+2)/2) floor((n-1)/2) floor((n-3)/2) floor((n-4)/2), which was conjectured by Zheng, Lin, Yang and Cui in 2007. The proof relies on the September 2026 OpenAI preprint proving the Harary–Hill formula for cr(K_n), whose stated results we use without re-proving them. Given that formula, the even orders follow from a theorem of Ouyang, Wang and Huang (2014), equivalently of Chia and Lee (2015), and the odd orders n <= 13 follow from their counting bound and integrality. The new content is the lower bound for odd n >= 15. We adapt the polynomial rank method of the OpenAI preprint to the missing edge. The adaptation restricts the test polynomials so that they put no weight on the missing pair. It pairs two different vanishing subspaces by mixed orthogonality. It also localizes, with a twisted detector that needs the full rectangular nonsingularity statement of the preprint, only at the vertices whose stars are complete. Via the product identity of Ouyang, Wang and Huang, the formula gives the crossing number of K_m square P_n conjectured by Zheng et al. for all m >= 4 and n >= 1, where P_n is the path with n edges. The new lower-bound input enters this consequence for odd m >= 13 and n >= 2; the literature comparison is bounded as stated in the novelty section of the paper.