{
  "abstract": "Using the framework developed by OpenAI, we prove that finite Morse index implies $O(R^{n-1})$ energy growth for bounded entire solutions of the quartic Allen--Cahn equation in dimensions $4\\leq n\\leq9$, uniformly in the center of the ball. Combined with the cited finite-index classification results, this gives planar rigidity for $4\\leq n\\leq7$ and, for nonconstant solutions, multiplicity-one stable minimal cone blowdowns through dimension nine. Within the framework, we derive an additive adjacent-layer interaction estimate, a weighted curvature and defect estimate with exponent $p=9/4$, and the localization needed for exterior stability. The limiting interaction coefficient $48$ is sharp in the unrestricted low-defect class with sufficiently large logarithmic selection depth.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.28080",
  "author_total": 2,
  "authors": [
    {
      "name": "Zhouyu Long"
    },
    {
      "name": "Wenming Zou"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/d87n5z909vjh3.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.28080",
  "fulltext_url": null,
  "id": "d87n5z909vjh3",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.28080",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AP"
    ],
    [
      "openalex",
      "2604|Geometric Analysis and Curvature Flows"
    ]
  ],
  "openalex": "W7223117198",
  "pdf_url": "https://arxiv.org/pdf/2610.28080",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-23",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/d87n5z909vjh3.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:19Z",
      "native_id": "2610.28080",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.28080"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "On the Allen--Cahn equation",
  "title_text": "On the Allen--Cahn equation",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:19Z",
  "url": "https://research.draconitic.com/paper/d87n5z909vjh3",
  "venue": "arXiv",
  "versions": []
}