{"ID":2850151,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2511.05517","arxiv_id":"2511.05517","title":"Stable non-minimal fixed points of threshold-linear networks","abstract":"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 $i \u003c j-1$ that there exists a competitive TLN with stable fixed point supports $τ\\subsetneq σ$ for which $|τ| = i$ and $|σ| = j$. 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.","short_abstract":"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 explic...","url_abs":"https://arxiv.org/abs/2511.05517","url_pdf":"https://arxiv.org/pdf/2511.05517v1","authors":"[\"Jesse Geneson\"]","published":"2025-10-26T21:19:38Z","proceeding":"q-bio.NC","tasks":"[\"q-bio.NC\",\"cs.DM\",\"math.CO\"]","methods":"[]","has_code":false}