{
  "abstract": "We define packaged structures through carriers, internal units, incidence, candidate spaces, and selected families. A realized packaging system additionally retains a context and formation datum related to its output by a possibly partial, multivalued realization relation. We establish comparison-relative recoverability and non-collapse criteria and an intrinsic refinement-completion theory. For atlas specializations, exact extension becomes a specialization-relative packaging completion problem. Under additional recovery and elimination axioms, we establish general globalization sufficiency criteria and sharp classifications for three classes of compatible finite atlases on a fixed cover $\\mathcal H$. A geometric obstruction rank $ν(\\mathcal H)$ yields thresholds $2k+1<ν(\\mathcal H)$ for empty-preserving closures of local Carathéodory number at most $k\\ge1$; $d<ν(\\mathcal H)$ for complete functional-dependency theories of local minimum Armstrong size at most $d\\ge3$; and $3<ν(\\mathcal H)$ for transitive relations, including equivalence relations. For nonempty finite label sets and empty-preserving closures, minimum commutant realization dimension equals minimum Armstrong size. We also determine intrinsic recovery ranks and construct canonical clique completions under bounded conformality without chordality. Explicit obstructions establish sharpness; a transport hypothesis is essential for the general recovery--obstruction converse. Complexity bounds apply locally, not necessarily to global extensions.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.33092",
  "author_total": 1,
  "authors": [
    {
      "name": "Rongchao Ma"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/61pn1gvagey57.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.33092",
  "fulltext_url": null,
  "id": "61pn1gvagey57",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.33092",
  "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",
      "math.GM"
    ],
    [
      "openalex",
      "2608|Advanced Topology and Set Theory"
    ]
  ],
  "openalex": "W7224009185",
  "pdf_url": "https://arxiv.org/pdf/2610.33092",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/61pn1gvagey57.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:40Z",
      "native_id": "2610.33092",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.33092"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "openalex:2608|Advanced Topology and Set Theory",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Packaging Systems: Intrinsic Structure, Recovery Ranks, and Globalization Theorems",
  "title_text": "Packaging Systems: Intrinsic Structure, Recovery Ranks, and Globalization Theorems",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:40Z",
  "url": "https://research.draconitic.com/paper/61pn1gvagey57",
  "venue": "arXiv",
  "versions": []
}