{
  "abstract": "We classify the labelled final layer orders of ordinary orthogonal 2 x 2k maps whose mountain-valley assignments are periodic with period two in each crease family. For k >= 2, local flat-foldability leaves 16 of the 64 formal period words. Eight pleat words have a unique labelled state. The remaining eight words reduce, under labelled symmetries, to two constant-row families described by row pivots. For the canonical word MMMMVV, an ordered pivot pair (a,b) is realizable exactly when a = 1, a = m, or a = b, where m = 2k - 1. For MVMMMM, realizability is equivalent to BR(a) = BR(b). In both families each compatible ordered pivot pair determines a unique labelled state. The resulting state counts are 0, 1, 6k - 5, and 4k - 3. The case k = 1 is treated separately because two formal period bits have no physical crease carriers.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.16776",
  "author_total": 1,
  "authors": [
    {
      "name": "Michael Fofonov"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fckehpcpx88dq.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.16776",
  "fulltext_url": null,
  "id": "fckehpcpx88dq",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.16776",
  "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"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7213423392",
  "pdf_url": "https://arxiv.org/pdf/2609.16776",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fckehpcpx88dq.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:45Z",
      "native_id": "2609.16776",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.16776"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Exact Labelled Layer-Order Classification for Period-2 Mountain--Valley Assignments on 2 x 2k Maps",
  "title_text": "Exact Labelled Layer-Order Classification for Period-2 Mountain--Valley Assignments on 2 x 2k Maps",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:45Z",
  "url": "https://research.draconitic.com/paper/fckehpcpx88dq",
  "venue": "arXiv",
  "versions": []
}