English

A six-neuron counterexample to the target-free clique conjecture

Combinatorics 2026-07-23 v1 Discrete Mathematics

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 ε>0\varepsilon>0, the CTLN defined by one fixed graph at δ=29ε/25\delta=29\varepsilon/25 is nondegenerate and has a stable fixed point with nonclique full support. Its values of q=δ(1ε)/εq=\delta(1-\varepsilon)/\varepsilon tend to 29/2529/25. In the complementary direction, for any CTLN on n3n\geq3 vertices, we prove that in the parameter range qn2n32ε, q\geq n-2-\frac{n-3}{2}\varepsilon, 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 εδ/(δ+n2)\varepsilon\leq\delta/(\delta+n-2).

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}
}