{
  "abstract": "Going beyond the traditional Hubble-normalized framework, we formalize a unified dynamical system formulation for $f(R)$ and coincident gauge $f(Q)$ gravity using three distinct sets of dimensionless variables: (1) kinematical cosmographic parameters, (2) the dark energy equation of state parameter $w$ (with its $N=\\ln a$ derivatives) and matter density abundance $Ω_m$, and (3) theory space parameters $\\{m_i\\}$ characterizing the shape of the theory function. Our formulation is free of auxiliary variables, closing via three equivalent ways: (1) specifying a theory, (2) requiring a specific cosmological evolution, or (3) requiring a specific dark energy equation of state evolution. Each closure strategy is illustrated with explicit examples. The theory closure approach studies a case of Hu-Sawicki $f(R)$ and $f(Q)=-2Λ+Q+β\\sqrt{-Q}$ models, where traditional formalisms fail. A heteroclinic trajectory from a GR-matter-dominated epoch to a late-time de-Sitter epoch is general for $f(Q)$, but sensitive to initial conditions for $f(R)$. The cosmographic and equation-of-state closure approaches compare $f(R)$ and $f(Q)$ gravities kinematically equivalent to $Λ$CDM ($j(z)=1$) and dynamically equivalent to $Λ$CDM ($w(z)=-1$). In both, GR acts as a cosmological past attractor for $f(Q)$ gravity, but not for $f(R)$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.24733",
  "author_total": 1,
  "authors": [
    {
      "name": "Saikat Chakraborty"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/72hfrtsnht8ps.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.24733",
  "fulltext_url": null,
  "id": "72hfrtsnht8ps",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.24733",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "gr-qc"
    ],
    [
      "openalex",
      "3103|Cosmology and Gravitation Theories"
    ]
  ],
  "openalex": "W7214056418",
  "pdf_url": "https://arxiv.org/pdf/2609.24733",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/72hfrtsnht8ps.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:39Z",
      "native_id": "2609.24733",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.24733"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:gr-qc",
      "subfield": "gravitation"
    }
  ],
  "title": "Formalizing a unified dynamical system framework for $f(R)$ and $f(Q)$ gravity",
  "title_text": "Formalizing a unified dynamical system framework for f(R) and f(Q) gravity",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:39Z",
  "url": "https://research.draconitic.com/paper/72hfrtsnht8ps",
  "venue": "arXiv",
  "versions": []
}