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arXiv · 2609.09637

Counterexamples to GNS Conjecture 1.8

Xiuqing Duan
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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.

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Xiuqing Duan. 2026-09-09. Counterexamples to GNS Conjecture 1.8. https://arxiv.org/abs/2609.09637

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