{
  "abstract": "For fixed nonempty words $u,v$, the minimum insertion degree of a shuffle is the least number of inserted-source runs among its representations. We prove that, after a block exchange from a minimum mask of degree $r$, the minimum over masks with commuting source projections is exactly $r-1$; a larger decrease occurs precisely when a noncommuting mask has fewer runs. If either source is unary, every such exchange is exact. We prove an explicit repair for an infinite periodic degree-three family. Computer-assisted results establish first-mask descent through combined length seventeen over every alphabet, degree-three first-mask descent through length thirty, and a uniform repair of all degree-three failures through length thirty, including all further letter identifications. Counterexamples to arbitrary, first, and last minimum-mask choices explain the need for alternative masks. Finally, we prove that minimum iteration depths have no gaps for arbitrary languages. General block-exchange descent and Singleton No-Gap remain open.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.39941",
  "author_total": 3,
  "authors": [
    {
      "name": "Haobo Ma"
    },
    {
      "name": "Wenlin Zhang"
    },
    {
      "name": "Charles E. Hughes"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2dk2bbq6w8kec.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.39941",
  "fulltext_url": null,
  "id": "2dk2bbq6w8kec",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.39941",
  "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",
      "cs.FL"
    ],
    [
      "arxiv",
      "math.CO"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.39941",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2dk2bbq6w8kec.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2610.39941",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.39941"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Block-exchange descent for insertion: commuting projections and minimum masks",
  "title_text": "Block-exchange descent for insertion: commuting projections and minimum masks",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:50Z",
  "url": "https://research.draconitic.com/paper/2dk2bbq6w8kec",
  "venue": "arXiv",
  "versions": []
}