{
  "abstract": "We study zeros transported by the heat semigroup from stationary solutions of the sparse operators $L_n=c_{n,0}\\D^n+c_{0,n-1}x^{n-1}$. The starting point is the exact cubic boundary obtained by Hernández-del-Valle and Guerra-Polania~\\cite{HerGuerra}. We show that the hypergeometric function used there is the residue-$1$ component of a naturally graded stationary solution space, and that heat transport preserves this grading after the scaling $x=t^ny$. This identifies the cubic boundary as the transported zero of a specific stationary mode. We then study $L_4=c_{4,0}\\D^4+c_{0,3}x^3$. Its $6\\times6$ boundary-jet determinant leads to the universal scaling \\[ F(t)=ρ\\left(\\frac t2\\right)^4 Φ\\!\\left(ρ^2\\left(\\frac t2\\right)^7\\right), \\qquad ρ=\\frac{c_{0,3}}{c_{4,0}}. \\] The resulting nonlinear equation for $Φ$ is singular at the origin. After an explicit desingularization, its initial compatibility condition factors as \\[ (5Φ'(0)-1)(6Φ'(0)-1)=0. \\] Consequently there are exactly two normalized graded analytic boundary germs. We place the desingularized equation in Briot--Bouquet form, prove local existence and uniqueness of both branches, and give a triangular recursion for all their Taylor coefficients. Finally, we prove that the two branches are generated respectively by the residue-$1$ and residue-$3$ stationary solutions. The quadratic and cubic cases provide the established baseline; the fourth-order problem is the first point at which distinct stationary modes produce distinct analytic branches. We do not claim a general branch-count theorem.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.27914",
  "author_total": 1,
  "authors": [
    {
      "name": "Gerardo Hernández-del-Valle"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/16gmyaqykd2g9.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.27914",
  "fulltext_url": null,
  "id": "16gmyaqykd2g9",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.27914",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AP"
    ],
    [
      "openalex",
      "2608|Advanced Differential Equations and Dynamical Systems"
    ]
  ],
  "openalex": "W7214171370",
  "pdf_url": "https://arxiv.org/pdf/2609.27914",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/16gmyaqykd2g9.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:00Z",
      "native_id": "2609.27914",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.27914"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Stationary Grading and Heat-Transported Moving Boundaries: From the Cubic Case to Two Fourth-Order Branches",
  "title_text": "Stationary Grading and Heat-Transported Moving Boundaries: From the Cubic Case to Two Fourth-Order Branches",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:00Z",
  "url": "https://research.draconitic.com/paper/16gmyaqykd2g9",
  "venue": "arXiv",
  "versions": []
}