{
  "abstract": "We study a finite-domain event for the mirror model on the Manhattan lattice: every boundary-to-boundary trajectory crosses a marked connector an even number of times, while internal cycles are unrestricted. Our main theorem gives a finite-scale confinement criterion: if this event's probability reaches a universal threshold in one even two-square rectangle, then almost surely every trajectory in the plane is periodic. We construct an exactly normalized three-color tensor representation and prove deterministic bounds for approximate contractions using the signed partial-permutation structure of directed continuations. These tools turn numerical evaluations into rigorous finite-volume statements. At mirror density \\(p=2/5\\), we prove that the boundary-parity probability is nonmonotone in the rectangle width, with a decrease followed by a certified increase. These values remain below the sufficient confinement threshold and do not resolve planar localization at this density. Further experiments track how the measured finite-size minimum varies with density, compare matching-based and entrywise error bounds, and exhibit endpoint-sensitive and cooperative responses to local mirror changes.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.19498",
  "author_total": 2,
  "authors": [
    {
      "name": "Jian Gu"
    },
    {
      "name": "Qin Hao"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8gkhctd6pyd3f.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.19498",
  "fulltext_url": null,
  "id": "8gkhctd6pyd3f",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.19498",
  "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",
      "cs.NA"
    ],
    [
      "arxiv",
      "math-ph"
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      "arxiv",
      "math.MP"
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      "openalex",
      "2610|Stochastic processes and statistical mechanics"
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  "openalex": "W7213589102",
  "pdf_url": "https://arxiv.org/pdf/2609.19498",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8gkhctd6pyd3f.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:19Z",
      "native_id": "2609.19498",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.19498"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "Boundary parity in the mirror model on the Manhattan lattice",
  "title_text": "Boundary parity in the mirror model on the Manhattan lattice",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:23Z",
  "url": "https://research.draconitic.com/paper/8gkhctd6pyd3f",
  "venue": "arXiv",
  "versions": []
}