{
  "abstract": "We give a new proof that 1-extension preserves generic local rigidity of bar-joint frameworks in all dimensions. The proof does not use a special position argument, which makes the approach potentially useful for X-replacement in 3-space. Instead, the balancing conditions recently introduced by the first author use directed paths in the line graph, and by modifying the end of these paths, we transfer a self-stress on the 1-extended graph back to the original. Treating the new vertex as variable allows us to extract the balancing condition of the smaller graph as one coefficient of a homogeneous quadratic. A Lean 4 formalization of the main results accompanies the paper.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.31212",
  "author_total": 2,
  "authors": [
    {
      "name": "Alexander Heaton"
    },
    {
      "name": "Martin Winter"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/eha0m4q2q126t.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.31212",
  "fulltext_url": null,
  "id": "eha0m4q2q126t",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.31212",
  "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.CO"
    ],
    [
      "arxiv",
      "math.MG"
    ],
    [
      "openalex",
      "2205|Structural Analysis and Optimization"
    ]
  ],
  "openalex": "W7223300257",
  "pdf_url": "https://arxiv.org/pdf/2610.31212",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/eha0m4q2q126t.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:38Z",
      "native_id": "2610.31212",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.31212"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.MG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "An algebraic proof of 1-extension",
  "title_text": "An algebraic proof of 1-extension",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:38Z",
  "url": "https://research.draconitic.com/paper/eha0m4q2q126t",
  "venue": "arXiv",
  "versions": []
}