Nine distinct distances among eighteen or nineteen planar points with no three collinear

Eighteen or nineteen points in the plane, no three on a line, determine at least nine distinct distances. For sixteen or seventeen points the corresponding minimum is eight. Regular polygons attain both bounds. We give a computer-assisted proof. An eighteen-point counterexample would contain thirteen points with at most seven distances and, among them, nine points with at most six. The reduction uses the diameter graph and an exclusion of fourteen-point subsets with at most seven distances. We check 635 inequivalent nine-point patterns against the complex-rank enumeration of Szöllősi and Östergård. Gluing pairs of nine-point patterns along common eight-point deletions gives all 228 ten-point candidates. Exact calculations of the possible extensions, including two one-parameter families, exclude every nine-point pattern in its required thirteen-point set. The accompanying archive contains the LaTeX source, the verification code and data, exact certificates and reproduction instructions. The published order-nine upper bound of 635 is an external input. The preprint includes a statement on the use of AI. The result has not been independently reviewed by a human reviewer.