{
  "abstract": "Let $\\mathbf{c}=(λ,α,μ,ν)\\in\\mathbb{R}^{4}$ denote the four running couplings of four-dimensional nonplanar Euclidean scalar $φ^{4}$ theory, and let $B(\\mathbf{c})=\\sum_{p\\ge2}\\sum_{|\\mathbf{m}|=p}β(\\mathbf{m})\\mathbf{c}^{\\mathbf{m}}$ be its formal beta-function series. Assuming the factorial coefficient bounds $|β(\\mathbf{m})|\\le (p-1)!C^{p-1}$ for $|\\mathbf{m}|=p$, we construct an explicit summation rule $\\mathcal{S}$, given by a cut Borel--Laplace transform whose cut is placed by a scale-covariant real-analytic gauge. We prove that it assigns a well-defined real value to $B$ on an explicit nonempty open domain, in fact on all of $\\mathbb{R}^{4}$. The summed function $\\mathcal{S}B$ is linear and real, real-analytic on $\\mathbb{R}^{4}\\setminus\\{0\\}$, $C^\\infty$ on $\\mathbb{R}^{4}$ with Taylor series $B$ at the origin, and Gevrey-$1$ asymptotic to $B$ to all orders with an explicit remainder bound. At optimal truncation the remainder is $O(e^{-η/(C\\|\\mathbf{c}\\|_\\infty)})$ for any $η<1$. We show that $\\mathcal{S}B$ reproduces the classical Borel sum and the ordinary sum of a convergent series up to errors of the same exponentially small order. Exact reproduction is impossible for any linear rule defined on the whole admissible class: regularity forces all weights to equal $1$, leading to divergence. Finally, we prove optimality under the stated hypothesis. Admissible series may diverge at every $\\mathbf{c}\\ne0$, and their Borel transforms may have the circle $|τ|=1/(C\\|\\mathbf{c}\\|_\\infty)$ as a natural boundary, so classical Borel-type summation need not be available on the class. Moreover, there exist two functions indistinguishable at the level of the coefficient bounds yet differing by exactly $(2π/C)e^{-1/(C\\|\\mathbf{c}\\|_\\infty)}$. All constants are explicit and all arguments are self-contained.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.29605",
  "author_total": 1,
  "authors": [
    {
      "name": "Puyu Zeng"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8tkffksrezjb2.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.29605",
  "fulltext_url": null,
  "id": "8tkffksrezjb2",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.29605",
  "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",
      "cond-mat.stat-mech"
    ],
    [
      "arxiv",
      "math-ph"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "openalex",
      "2610|advanced mathematical theories"
    ]
  ],
  "openalex": "W7214387855",
  "pdf_url": "https://arxiv.org/pdf/2609.29605",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8tkffksrezjb2.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:11Z",
      "native_id": "2609.29605",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.29605"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    },
    {
      "field": "physics",
      "origin": "arxiv:cond-mat.stat-mech",
      "subfield": "statistical-nonlinear"
    }
  ],
  "title": "A summation rule for the four-coupling beta function of four-dimensional nonplanar Euclidean scalar $φ^{4}$ theory under factorial coefficient bounds, and a domain on which the summed beta function is well defined",
  "title_text": "A summation rule for the four-coupling beta function of four-dimensional nonplanar Euclidean scalar φ^4 theory under factorial coefficient bounds, and a domain on which the summed beta function is well defined",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:15Z",
  "url": "https://research.draconitic.com/paper/8tkffksrezjb2",
  "venue": "arXiv",
  "versions": []
}