In threshold-linear networks (TLNs), a fixed point is called minimal if no proper subset of its support is also a fixed point. Curto et al (Advances in Applied Mathematics, 2024) conjectured that every stable fixed point of any TLN must be a minimal fixed point. We provide a counterexample to this conjecture: an explicit competitive TLN on 3 neurons that exhibits a stable fixed point whose support is not minimal (it contains the support of another stable fixed point). We prove that there is no competitive TLN on 2 neurons which contains a stable non-minimal fixed point, so our 3-neuron construction is the smallest such example. By expanding our base example, we show for any positive integers $i, j$ with $i < j-1$ that there exists a competitive TLN with stable fixed point supports $\tau \subsetneq \sigma$ for which $|\tau| = i$ and $|\sigma| = j$. Using a different expansion of our base example, we also show that chains of nested stable fixed points in competitive TLNs can be made arbitrarily long.

翻译：在阈值线性网络中，若一个不动点的支撑集不存在真子集同样构成不动点，则称该不动点为最小不动点。Curto等人（《应用数学进展》，2024年）曾猜想：任何阈值线性网络的所有稳定不动点都必须是最小不动点。本文通过构造一个包含3个神经元的显式竞争性阈值线性网络作为反例，该网络存在一个支撑集非最小的稳定不动点（其支撑集包含另一个稳定不动点的支撑集）。我们证明了在2个神经元的竞争性阈值线性网络中不可能存在稳定的非最小不动点，因此本文的3神经元构造是最小反例。通过扩展基础示例，我们证明对于任意满足 $i < j-1$ 的正整数 $i, j$，均存在一个竞争性阈值线性网络，其稳定不动点的支撑集满足 $\tau \subsetneq \sigma$ 且 $|\tau| = i$，$|\sigma| = j$。此外，通过对基础示例进行另一种扩展，我们进一步证明竞争性阈值线性网络中嵌套稳定不动点的链可以任意延长。