2026/07/23 by Jesse Geneson
#math.CO #cs.DM
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 ε>0, the CTLN defined by one fixed graph at δ=29ε/25 is nondegenerate and has a stable fixed point with nonclique full support. Its values of q=δ(1-ε)/ε tend to 29/25. In the complementary direction, for any CTLN on n≥3 vertices, we prove that in the parameter range q≥ n-2-(n-3)/(2)ε, 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 ε≤δ/(δ+n-2).