{
  "abstract": "We study an integer-coordinate encoding of boundary incidences for finite subsets of the regular hexagonal tiling. Each selected cell contributes six ordered records with endpoints in two role-distinguished classes, point and tail. Identical records from neighboring cells cancel, and the surviving records form a disjoint union of cycles, including the exterior boundaries of disconnected components and the boundaries of holes. We derive the actual even- and odd-row record formulas from the A--B--C transforms used in Hexagones 2.0, prove exact shared-edge identities and the global role-qualified incidence structure, and thereby obtain an unconditional correctness theorem for the abstract coordinate construction. We compare the encoding with an independent exact cube-coordinate half-edge hashing implementation, proving an incidence isomorphism and checking 504 finite regions. We give an expected-linear-time indexed algorithm and distinguish it from the historical fixed-capacity C traversal, whose sentinel and closure conventions do not inherit the theorem automatically. The claim is a compact coordinate realization of a known planar boundary principle, not historical priority or a new three-axis lattice.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.25451",
  "author_total": 1,
  "authors": [
    {
      "name": "Philippe Mullhaupt"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/eqw49mzrga8kg.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.25451",
  "fulltext_url": null,
  "id": "eqw49mzrga8kg",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.25451",
  "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.CO"
    ],
    [
      "openalex",
      "1707|Digital Image Processing Techniques"
    ]
  ],
  "openalex": "W7223032296",
  "pdf_url": "https://arxiv.org/pdf/2610.25451",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-18",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/eqw49mzrga8kg.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2610.25451",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.25451"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "An A-B-C Coordinate Encoding for Exact Boundary-Cycle Extraction on Hexagonal Lattices",
  "title_text": "An A-B-C Coordinate Encoding for Exact Boundary-Cycle Extraction on Hexagonal Lattices",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:44Z",
  "url": "https://research.draconitic.com/paper/eqw49mzrga8kg",
  "venue": "arXiv",
  "versions": []
}