Counterexamples to GNS Conjecture 1.8

Abstract

We disprove both parts of Conjecture 1.8 of Gabrielov, Novikov, and Shapiro (GNS). For the Coulomb potential, a generic rational configuration of 24 unit charges in ${\mathbb{R}}^3$ has at least 18 nondegenerate critical points of Morse index one but exactly 14 effective Voronoi one-cells. A separate proper-line construction has at least two nondegenerate minima but only one relatively effective zero-dimensional cell intersection. For each positive subspace dimension, product suspension gives a proper-subspace counterexample.