{
  "abstract": "We give a reachable-state operator formulation of one-to-one deferred acceptance with strict, possibly incomplete preference lists. A local update operator is defined for each active proposer, while a scheduler selects which local update is applied. On the states reachable from the canonical empty initial state, three invariants are immediate: the set of proposed edges grows strictly, every proposer visits each acceptable receiver at most once, and every receiver holds its most-preferred proposal received so far. These invariants yield termination after at most |E| proposals and stability of every terminal reachable state. The classical rejection lemma then gives proposer optimality and schedule independence. We separate these trajectory statements from two logically different results: the distributive-lattice structure of the full stable-matching set and one-sided strategy-proofness. A diagnostic table records which proof obligation is lost when a model changes the bipartition, ordinal comparisons, proposal irreversibility, or receiver choice rule. The paper makes no new complexity claim; its purpose is a precise operator-level account of the classical proof architecture and of the limits of that architecture.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.19237",
  "author_total": 1,
  "authors": [
    {
      "name": "Yoshiteru Ishida"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/azn3s2dpbx6rv.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.19237",
  "fulltext_url": null,
  "id": "azn3s2dpbx6rv",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.19237",
  "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.DS"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2002|Game Theory and Voting Systems"
    ]
  ],
  "openalex": "W7213586506",
  "pdf_url": "https://arxiv.org/pdf/2609.19237",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/azn3s2dpbx6rv.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:23Z",
      "native_id": "2609.19237",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.19237"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "A Reachable-State Operator Formulation of Deferred Acceptance: Progress Invariants and Structural Diagnostics",
  "title_text": "A Reachable-State Operator Formulation of Deferred Acceptance: Progress Invariants and Structural Diagnostics",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:23Z",
  "url": "https://research.draconitic.com/paper/azn3s2dpbx6rv",
  "venue": "arXiv",
  "versions": []
}