{
  "abstract": "For every uncountable regular cardinal $α$, $\\mathbb S^{\\ast}(α)$ is an explicitly typed four-coordinate forcing motivated by the ladder-system construction of Hayut and Magidor. The forcing is $σ$-closed and, after adjoining a formal maximum, $α$-strategically closed. For $α\\geqω_2$, every nonempty countable family of designated generic branches has a stationary and costationary common trace on the generic ladder-coordinate set $L_α$, while no countable family of cofinal branches generates $L_α$. These conclusions persist under a Kurepa-style level-size bound. In the unrestricted forcing, for every infinite cardinal $μ<α$ in the ground model, some level of the generic tree contains a copy of $({}^μ2)^V$. Consequently, the endpoint-corrected restriction family indexed by $\\mathcal P_{ω_2}α$ is too wide, whereas the scaled restriction system indexed by $\\mathcal P_αα$ has all levels of size less than $α$ exactly when $α$ is strongly inaccessible in the ground model. When these equivalent conditions hold, the branch-covering number of $L_α$ relative to the scaled system is at least $ω_1$. The low-cofinality empty-value convention also ensures that the set of domains of $L_α$ contains no club in $\\mathcal P_αα$. The unrestricted tree clause of the motivating presentation is retained, without asserting forcing equivalence.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.25033",
  "author_total": 1,
  "authors": [
    {
      "name": "Xing-Yu Hu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0y9hjbafjvkr3.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.25033",
  "fulltext_url": null,
  "id": "0y9hjbafjvkr3",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.25033",
  "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.LO"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2609.25033",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0y9hjbafjvkr3.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:40Z",
      "native_id": "2609.25033",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.25033"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.LO",
      "subfield": "logic-foundations"
    }
  ],
  "title": "Bistationary Traces, Wide Levels, and Branch-Cover Rigidity for an Unrestricted Typed Variant of the Hayut-Magidor Forcing",
  "title_text": "Bistationary Traces, Wide Levels, and Branch-Cover Rigidity for an Unrestricted Typed Variant of the Hayut-Magidor Forcing",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:40Z",
  "url": "https://research.draconitic.com/paper/0y9hjbafjvkr3",
  "venue": "arXiv",
  "versions": []
}