A six-neuron counterexample to the target-free clique conjecture
Abstract
The target-free clique conjecture asserts that the supports of stable fixed points of a nondegenerate combinatorial threshold-linear network (CTLN) are exactly its target-free cliques: bidirected cliques for which no outside vertex receives an edge from every clique vertex. We give an explicit six-neuron counterexample. For every sufficiently small , the CTLN defined by one fixed graph at is nondegenerate and has a stable fixed point with nonclique full support. Its values of tend to . In the complementary direction, for any CTLN on vertices, we prove that in the parameter range no nonclique support can satisfy both the fixed-point positivity and linear stability conditions. Consequently, throughout this range, every nondegenerate CTLN has exactly its target-free cliques as supports of stable fixed points. In particular, this holds when .
Keywords
Cite
@article{arxiv.2607.21396,
title = {A six-neuron counterexample to the target-free clique conjecture},
author = {Jesse Geneson},
journal= {arXiv preprint arXiv:2607.21396},
year = {2026}
}