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  "slug": "tcell-exactly-five",
  "title": "Exactly five positive steady states in a two-ligand T-cell activation model",
  "shortTitle": "One model, five steady states",
  "url": "https://evidencepress.org/releases/tcell-exactly-five/",
  "oneLine": "A rational example and exact root-count certificate establish five positive equilibria in the full two-ligand model.",
  "abstract": "A mathematical model of early T-cell signalling can have exactly five positive steady states under the same ligand conditions. This unrefereed candidate supplies a rational N=100 example, a bijective scalar reduction and an exact degree-201 polynomial certificate. The count persists near the example in the twelve named model parameters. Stability and biological plausibility are not established.",
  "datePublished": "2026-09-20",
  "dateModified": "2026-09-20",
  "version": "0.1.0-candidate",
  "doi": "10.5281/zenodo.22861212",
  "doiUrl": "https://doi.org/10.5281/zenodo.22861212",
  "conceptDoi": "10.5281/zenodo.22861211",
  "pdfUrl": "https://github.com/ipitchford/tcell-exactly-five/releases/download/v0.1.0-candidate/paper.pdf",
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  "repoUrl": "https://github.com/ipitchford/tcell-exactly-five",
  "releaseUrl": "https://github.com/ipitchford/tcell-exactly-five/releases/tag/v0.1.0-candidate",
  "markdownUrl": "https://evidencepress.org/releases/tcell-exactly-five/index.md",
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  "authors": [
    "Anonymous"
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  "status": "unrefereed-candidate",
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    "independentlyReproduced": false,
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    "detail": "Unrefereed candidate. Exact multistationarity in the full two-ligand model at N=100. No dynamical stability, physiological plausibility, minimal chain length, agonist-only resolution or historical-priority claim."
  },
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      "note": "Public archive readback establishes availability, not correctness."
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      "question": "Does the producer’s own pipeline reproduce the stated result from the archived package?",
      "state": "passed",
      "note": "Exact finite certificate, original model residuals, normal/optimized replay and six rejected evidence corruptions. Analytic proof remains separate.",
      "evidenceUrl": "https://github.com/ipitchford/tcell-exactly-five/blob/v0.1.0-candidate/ASSURANCE.md"
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      "question": "Has someone else reached the result from an independent implementation?",
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      "note": "Internal model review and producer-coordinated checks do not establish this external assurance dimension."
    },
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      "dimension": "formalVerification",
      "label": "Formal verification",
      "question": "Is a formalised statement machine-checked, and over which trusted base?",
      "state": "not-assessed",
      "note": "Internal model review and producer-coordinated checks do not establish this external assurance dimension."
    },
    {
      "dimension": "specialistReview",
      "label": "Specialist review",
      "question": "Has a domain specialist assessed the argument?",
      "state": "not-assessed",
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    },
    {
      "dimension": "editorialPeerReview",
      "label": "Editorial peer review",
      "question": "Has a journal or venue run peer review to a decision?",
      "state": "not-assessed",
      "note": "Internal model review and producer-coordinated checks do not establish this external assurance dimension."
    },
    {
      "dimension": "dataEnvironmentReproducibility",
      "label": "Data and environment reproducibility",
      "question": "Are data and computational environment pinned well enough to rebuild?",
      "state": "partial",
      "evidenceUrl": "https://github.com/ipitchford/tcell-exactly-five/blob/v0.1.0-candidate/ASSURANCE.md",
      "note": "Standard-library Python checks; Linux CI plus macOS producer replay; no hermetic or bit-identical PDF build claim."
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  "provenance": {
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    "disclosure": "Internal review and implementation diversity do not establish external assurance. Supplied audits acknowledged without inferred redistribution rights."
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  "attributionBoundary": null,
  "problem": {
    "name": "Higher equilibrium multiplicity in the full two-ligand model",
    "url": "https://doi.org/10.1098/rsos.170821"
  },
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  "grandfatheredAtSchemaVersion": null,
  "keywords": [
    "T-cell activation",
    "multistationarity",
    "kinetic proofreading",
    "negative feedback",
    "exact arithmetic",
    "Descartes rule"
  ],
  "keyResults": [
    "Explicit positive rational N=100 parameters with exactly five physical equilibria in one conservation class.",
    "Five simple scalar roots isolated in exact intervals of width 10^-12.",
    "The count persists on an open neighborhood in the eight shared rates and four conserved totals."
  ],
  "reviews": [],
  "evidencePackage": "Written bijection and robustness proofs; 202 exact coefficients; six rational sign enclosures; five root brackets; 3232 original chain residuals; six rejected evidence corruptions; internal editorial role review.",
  "openProblems": [
    "Unaffiliated proof audit and reimplementation.",
    "Smallest chain length admitting five equilibria.",
    "Higher multiplicity in the agonist-only restriction.",
    "Stability in the complete 203-variable dynamics.",
    "Structural count bounds and a quantitative full-parameter robustness region."
  ],
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  "relatedWorks": [
    {
      "citation": "François et al. (2013). Phenotypic model for early T-cell activation displaying sensitivity, specificity, and antagonism. Original model.",
      "url": "https://doi.org/10.1073/pnas.1300752110",
      "doi": "10.1073/pnas.1300752110"
    },
    {
      "citation": "Rendall and Sontag (2017). Multiple steady states and the form of response functions to antigen in a model for the initiation of T-cell activation. Equations and antecedent equilibrium analysis.",
      "url": "https://doi.org/10.1098/rsos.170821",
      "doi": "10.1098/rsos.170821"
    },
    {
      "citation": "Rendall (2018). Initiation of T cell signalling, Oberwolfach Reports, pp. 487–489. Historical question with single-ligand context.",
      "url": "https://doi.org/10.4171/OWR/2018/8",
      "doi": "10.4171/OWR/2018/8"
    },
    {
      "citation": "Bali and Rendall (2025). A comparative mathematical modeling study of phenotypic approaches to T cell activation. Recent nine-model comparison.",
      "url": "https://doi.org/10.1038/s41598-025-32255-9",
      "doi": "10.1038/s41598-025-32255-9"
    }
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