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

Disclosure

“tion give (33) as the smaller root. The larger root exceeds (δ + n − 2)/(n − 3) > 1, while legal ε satisfies 0 < ε < 1. Hence (33) implies (35). Finally, (34) is equivalent to q ≥ n − 2, which implies (25). Acknowledgments Codex with GPT-5.6 and Claude Code with Opus 4.8 and Fable 5 were used for proof exploration, proof criticism, exposition, and revision. 11”

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Classification

Proof ideas or individual proof-step assistance

Multiplier

8

Verified

2026-08-23T19:03:15.457518Z

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