{
  "abstract": "An approximate action of the pair groupoid of a metric space determines an action of its Lipschitz-thin groupoid on complete extended metric fibers, uniquely characterized by its local sewing estimate. We prove this assertion under a superlinear three-point defect estimate and a Lipschitz bound for finite products. As a result, we obtain a positive proof of Curry and Manchon's sewing conjecture [2], with the necessary path-scaling correction to its estimate. Further restriction to isotropy constructs the based holonomy anticipated in their Remark 4.15 as a representation of based Lipschitz loops modulo thin equivalence. Rectangular comparison estimates give full relative Lipschitz-homotopy descent, in particular under their strong knitting hypothesis and under any three point estimate of total order greater than two. We prove the corresponding flatness and basepoint-covariance statements. An area model shows that the order threshold is sharp. We also separate the exponential estimate from the general knitting assumptions and exhibit a compact disk metric for which strong knitting does not imply invariance under merely continuous homotopy. As an application, for controlled fields on a metric space, we prove that the classical rough integral descends to the thin groupoid and admits substitution with a common rough controller.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.23831",
  "author_total": 1,
  "authors": [
    {
      "name": "Alexandre Reggiolli Teixeira"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/88kcaxrn9azsh.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.23831",
  "fulltext_url": null,
  "id": "88kcaxrn9azsh",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.23831",
  "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.FA"
    ],
    [
      "openalex",
      "2604|Geometric Analysis and Curvature Flows"
    ]
  ],
  "openalex": "W7214026060",
  "pdf_url": "https://arxiv.org/pdf/2609.23831",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-20",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/88kcaxrn9azsh.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2609.23831",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.23831"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Sewing on Thin Groupoids, Knitting, and Based Holonomy for Lipschitz Paths",
  "title_text": "Sewing on Thin Groupoids, Knitting, and Based Holonomy for Lipschitz Paths",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/88kcaxrn9azsh",
  "venue": "arXiv",
  "versions": []
}