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  "slug": "quadratic-equilibrium-sharpness",
  "title": "Five reactions, four equilibria",
  "shortTitle": "Five reactions, four equilibria",
  "url": "https://evidencepress.org/releases/quadratic-equilibrium-sharpness/",
  "oneLine": "An explicit mass-action realization reaches the positive-equilibrium bound, with a construction in every dimension.",
  "abstract": "This unrefereed candidate realizes the established Phillipson–Rojas polynomial chain as a quadratic mass-action network with n species, n+2 reactions, rank n and exactly n+1 positive nondegenerate equilibria for every integer n≥2. Integer products and positive rational rates attain the bound discussed by Banaji and Feliu. Two three-species examples provide an inflow-free realization with maximum product molecularity ten and an exact one-rate interval, and a reciprocal example with two sinks and two saddles. Quadratic restricts sources, not products; no experimental realization or global dynamical classification is claimed.",
  "datePublished": "2026-10-04",
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  "doi": "10.5281/zenodo.23146115",
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  "problem": {
    "name": "Can quadratic three-species, five-reaction networks attain four positive nondegenerate equilibria?",
    "url": "https://link.springer.com/article/10.1007/s00285-026-02429-8"
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  "keywords": [
    "mass-action kinetics",
    "chemical reaction networks",
    "positive equilibria",
    "multistationarity",
    "fewnomial systems",
    "quadratic networks",
    "bistability"
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    "For every integer n≥2, a quadratic (n,n+2,n) mass-action network attains exactly n+1 positive nondegenerate equilibria.",
    "An explicit chemical realization of the existing Phillipson–Rojas extremal polynomial chain, with integer products and positive rational rates.",
    "An inflow-free three-species, five-reaction example has maximum product molecularity ten and four positive nondegenerate equilibria precisely for 1568<L<117649/75 when its fourth rate is 1/L.",
    "The reciprocal N=7, K=145 example has two locally attracting equilibria and two saddles, each with one unstable direction."
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      "citation": "Banaji and Feliu (2026). Positive equilibria in mass action networks: geometry and bounds. Theorem 5.5 supplies the bound; Remark 5.8 asks whether quadratic networks attain it for n≥3.",
      "url": "https://link.springer.com/article/10.1007/s00285-026-02429-8",
      "doi": null
    },
    {
      "citation": "Phillipson and Rojas (2013). Fewnomial systems with many roots, and an Adelic Tau Conjecture. Theorem 1.6 provides the extremal polynomial chain realized chemically here; the root mechanism is prior work.",
      "url": "https://arxiv.org/pdf/1011.4128",
      "doi": null
    },
    {
      "citation": "Banaji, Boros and Hofbauer (2024). Bifurcations in planar, quadratic mass-action networks with few reactions and low molecularity. Remark 38 gives an earlier two-species, four-reaction example with three positive equilibria.",
      "url": "https://doi.org/10.1007/s11071-024-10068-1",
      "doi": "10.1007/s11071-024-10068-1"
    },
    {
      "citation": "Bihan (2015). Maximally positive polynomial systems supported on circuits. Earlier extremal polynomial-system framework.",
      "url": "https://doi.org/10.1016/j.jsc.2014.09.003",
      "doi": "10.1016/j.jsc.2014.09.003"
    },
    {
      "citation": "El Hilany (2017). Characterization of circuits supporting polynomial systems with the maximal number of positive solutions. Earlier support characterization.",
      "url": "https://doi.org/10.1007/s00454-017-9897-4",
      "doi": "10.1007/s00454-017-9897-4"
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    {
      "citation": "Bihan, Dickenstein and Forsgård (2021). Optimal Descartes’ rule of signs for systems supported on circuits. Established circuit root bounds.",
      "url": "https://doi.org/10.1007/s00208-021-02216-4",
      "doi": "10.1007/s00208-021-02216-4"
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      "citation": "Evidence Press (2026). Exactly five positive steady states in a two-ligand T-cell activation model. A related candidate using scalar reduction and exact root counting in a different model; not confirmation of this result.",
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