{
  "abstract": "GST v1.20 explained where the number 59 reported for C_2 comes from. It characterised the set H_lt(z_t) of proper sub-relations of one cut, z_t, that keep its reflexive transitive closure, and counted 59 of them; the identification of that set with C_2 rests on the cited lemma D2 and condition C-cone. This manuscript states the characterisation for any finite loop-free relation z and separates what is general from what is particular to z_t.\n\nA sub-relation w of z has the closure of z if and only if w is strongly connected inside each strongly connected component of z and, for each link of the transitive reduction of the condensation of z, w keeps at least one pair of z along that link (Theorem A2). The two conditions fall on disjoint sets of pairs, which gives a product formula for the number of such sub-relations (Corollary A3); the factor for the inside of a component is left as it stands. No proper sub-relation of this kind is closed (Proposition A4).\n\nWith the code definitions used since v1.18, for loop-free relations on one node set with w in H_lt(z) and z a subset of z', the letter is letter(w, z') = [cl(z) != cl(z')] [h(z')] c3(z, z'), and [cl(z) != cl(z')] [h(z')] 0 when z is persistent (Proposition A5). For z_t the formula gives |H_le(z_t)| = 60 = 5 x 1 x 1 x 1 x 3 x 4 and |H_lt(z_t)| = 59, the factor 5 being argued inside the strongly connected part {0, 2, 3, 4}; it recovers the four groups of v1.20 and the letters 000 and 100.\n\nThe decomposition is related to the known reduction of the minimum equivalent digraph problem; no novelty is claimed. All results are argued; no computation was made. The general results are not applied to, or evaluated on, any other particular cut; nothing is claimed about the member list of the class C evaluation, acceptance at a root, or why the stored answer was chosen.",
  "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/bd1cfz5khc9ct.bib",
  "cited_by": 0,
  "doi": "10.5281/zenodo.23449320",
  "fulltext_url": null,
  "id": "bd1cfz5khc9ct",
  "issns": [],
  "landing_url": "https://zenodo.org/records/23449320",
  "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/23449320/files/GST_v1.21_paper.pdf/content",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-18",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bd1cfz5khc9ct.ris",
  "sources": [
    {
      "first_seen": "2026-11-01T12:21:12Z",
      "native_id": "23449320",
      "source": "zenodo",
      "url": "https://zenodo.org/records/23449320"
    }
  ],
  "subjects": [],
  "title": "Generative Structural Theory v1.21 — Closure-Preserving Sub-relations: Structural Conditions and Counting",
  "title_text": "Generative Structural Theory v1.21 — Closure-Preserving Sub-relations: Structural Conditions and Counting",
  "type": "preprint",
  "updated_at": "2026-11-01T12:21:12Z",
  "url": "https://research.draconitic.com/paper/bd1cfz5khc9ct",
  "venue": "Zenodo",
  "versions": []
}