The nonexistence of nine-point crescent configurations

Abstract

A planar crescent configuration consists of $n$ points, no three collinear and no four concyclic, whose distinct distances have multiplicities $1,2,\ldots,n-1$. We prove by exact computation that no such configuration exists for $n=9$. The three pairs carrying the two least frequent distances yield a six-point subset with at most six distances. Rational linear algebra on quartic polynomials separates the possible six-point distance patterns into those with certified finite lists of normalized distance tuples and a residual family. Compatibility of overlapping subsets forces at least one core into the first family, without classifying the residual realizations. The remaining three points are then obtained by rational constructions over the field of the squared core distances. The multiplication matrices used to enumerate the core distances have order at most twelve. Exhaustive label assignments and exact algebraic comparisons exclude all remaining extensions.