{
  "abstract": "GST v1.21 characterised, for a finite loop-free relation z, the family H_le(z) of sub-relations w of z that keep the reflexive transitive closure of z: w is strongly connected inside each strongly connected component of z and keeps at least one pair of z on each link of the transitive reduction of the condensation (v1.21 Theorem A2). This manuscript studies the smallest members of that family and keeps two notions apart: inclusion-minimal (no pair can be removed) and of minimum cardinality (fewest pairs).\n\nA pair is essential when it lies in every member. The family is an up-set, so its maximal proper members are the relations z - {e} with e not essential (Proposition B1), and a pair a>b is essential if and only if there is no path from a to b in z - {a>b} (Proposition B2). A pair inside a component is essential if and only if its removal breaks the strong connectivity of that component (these are the strong bridges), and a pair between components if and only if it is the only pair on a link of the transitive reduction (Proposition B3).\n\nThe inclusion-minimal members correspond one to one to the choices of an inclusion-minimal strongly connected spanning sub-relation for each component and of one pair for each link of the transitive reduction, which gives their number as a product (Theorem B4). The least number of pairs is the sum of the least numbers for the components plus the number of links of the transitive reduction; a member of minimum cardinality is inclusion-minimal, but not conversely in general (Proposition B5).\n\nFor the fixed cut z_t of twelve pairs, the essential pairs are five and do not themselves keep the closure; H_lt(z_t) has seven maximal members, H_le(z_t) has four inclusion-minimal members, of 7, 7, 8 and 8 pairs, and the minimum is 7; inside the component {0, 2, 3, 4} one inclusion-minimal sub-relation has six pairs while the minimum there is five. The stored answer y = z_t - {1>3} is one of the seven maximal proper members and is not inclusion-minimal; this places y in the structure and does not explain the procedure that stored it.\n\nThe general statements are related to known results of graph theory, and no novelty is claimed for B1-B5; the specific correspondence with the known decomposition by components is stated in §A8. All results are argued; no new machine enumeration or graph computation was made for v1.22. Nothing is claimed about other particular cuts, the member list of the class C evaluation, acceptance at a root, or why y was stored.",
  "abstract_shown": true,
  "affiliations": [
    "SD Lab LLC"
  ],
  "article_type": null,
  "arxiv": null,
  "author_total": 1,
  "authors": [
    {
      "affs": [
        0
      ],
      "name": "Koji Okino",
      "orcid": "0009-0003-9273-9813"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bs8qb3y1jj0hx.bib",
  "cited_by": 0,
  "doi": "10.5281/zenodo.23450445",
  "fulltext_url": null,
  "id": "bs8qb3y1jj0hx",
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  "landing_url": "https://zenodo.org/records/23450445",
  "language": "eng",
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": null,
  "native_subjects": [],
  "openalex": null,
  "pdf_url": "https://zenodo.org/api/records/23450445/files/GST_v1.22_paper.pdf/content",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-18",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bs8qb3y1jj0hx.ris",
  "sources": [
    {
      "first_seen": "2026-11-01T12:21:10Z",
      "native_id": "23450445",
      "source": "zenodo",
      "url": "https://zenodo.org/records/23450445"
    }
  ],
  "subjects": [],
  "title": "Generative Structural Theory v1.22 — Closure-Preserving Sub-relations: Essential Pairs, Inclusion-Minimal Members, and Minimum Cardinality",
  "title_text": "Generative Structural Theory v1.22 — Closure-Preserving Sub-relations: Essential Pairs, Inclusion-Minimal Members, and Minimum Cardinality",
  "type": "preprint",
  "updated_at": "2026-11-01T12:21:10Z",
  "url": "https://research.draconitic.com/paper/bs8qb3y1jj0hx",
  "venue": "Zenodo",
  "versions": []
}