{
  "format": "sumire-research-record",
  "version": 1,
  "exportedAt": "2026-11-04T08:45:49.195Z",
  "scope": "Reader-visible project fields and native live notes only; referenced notes in other projects, deleted notes and private AI context are excluded.",
  "submission": "not_submitted",
  "mathematicalCorrespondence": "not_automatically_checked",
  "project": {
    "id": "2610.00003",
    "url": "https://sumire.aiformath.jp/p/2610.00003",
    "title": "Games as recursive coalgebras",
    "tagline": "Recursive coalgebras, game values, and the categorical structure of combinatorial games.",
    "visibility": "public",
    "owner": {
      "name": "hora",
      "handle": "ryuya"
    },
    "origin": "",
    "source": "",
    "goal": "Explain how recursive game positions, game values, and operations such as Conway addition fit together, and identify what this explains about the Nim-sum.",
    "narrative": "The starting point is [Games as recursive coalgebras: A categorical view on the Nim-sum](https://arxiv.org/abs/2510.22886v3), available in [Main](https://sumire.aiformath.jp/p/2610.00003?view=main). The paper develops impartial games as recursive finite-powerset coalgebras, game values as recursive evaluations, and Bouton monoids as universal factorizations of game values. Conway addition is symmetric monoidal closed.\n\nThe next question is how much more explicit this framework can become. [The internal hom](https://sumire.aiformath.jp/p/2610.00003/n/4) has a proposed concrete cofree-subgame construction and proof in [note 29](https://sumire.aiformath.jp/p/2610.00003/n/29); its relationship to playing strategies remains separate. [Differential structures](https://sumire.aiformath.jp/p/2610.00003/n/6) asks for a common explanation of the Leibniz-type move formula and the Rota–Baxter-type law, with an explanation of Nim-sum as the target.\n\nTwo extensions have different starting points. [Partisan games](https://sumire.aiformath.jp/p/2610.00003/n/3) already recover short game forms, but values and strategies need additional structure. [A double category of games](https://sumire.aiformath.jp/p/2610.00003/n/1) still requires a precise strategy class and compatible squares; counterexamples for arbitrary relations do not settle that question.\n\nA settled boundary is that [Game itself is not a topos](https://sumire.aiformath.jp/p/2610.00003/n/16), by Remark B.27 of the paper. Questions about a topos completion are therefore distinct from that answered question.\n\nTo contribute, choose one of these linked questions and work against its stated remaining claim. For the internal hom, compute a small explicit example or clarify its game-theoretic meaning. For the double category, specify legal strategies and test compatibility with coalgebra maps before claiming a general construction or obstruction.",
    "history": "",
    "dates": {
      "inspiration": null,
      "sourceRecorded": null,
      "evidence": null,
      "registeredInSumire": "2026-10-07T16:29:39.051Z",
      "updatedInSumire": "2026-10-30T14:35:21.579Z",
      "legacyRecordedAt": "2026-10-07T16:29:39.050Z",
      "legacyUpdatedAt": "2026-10-30T14:35:21.579Z"
    },
    "goals": [
      {
        "id": "goal_71391ac0-c899-41e4-b6ed-e9d825d41001",
        "title": "To understand why Nim-sum works",
        "body": "",
        "noteIds": []
      },
      {
        "id": "goal_4d2683f0-360b-4edd-b553-ca70400f6943",
        "title": "Categorical unification of recursive, geometric and logical aspects of combinatorial game theory",
        "body": "",
        "noteIds": []
      }
    ],
    "goalDependencies": []
  },
  "notes": [
    {
      "id": "n_KwQkyfrnYxcZ",
      "number": 14,
      "url": "https://sumire.aiformath.jp/p/2610.00003/n/14",
      "title": "AI evaluation report (2026-08-15): A double category of games",
      "author": {
        "name": "hora",
        "handle": "ryuya"
      },
      "source": "",
      "body": "# claim a66892-state-objects の独立レビュー\n\n> この報告はcommit時点では独立reviewを経ていない。書誌情報と主張の検証状態は`research/runs/`のRunEventを正とし、人間の確認前の内容として読むこと。\n\n検討対象: 「Double category の縦圏(vertical 2-category)と横 bicategory は共通の対象族を持つため、候補構成には recursive-coalgebraic games と Joyal games の対象同定または共通部分への制限が必要である」。\n\n1) 数学的前提の検証: double category D は定義上ただ一つの対象類 Ob(D) を持ち、その上に縦射(vertical 1-cells)からなる2-圏と、横射(horizontal 1-cells)+方形2-cellからなる bicategory の両方が構成される（Ehresmann の原定義以来の標準事実）。検索で得た TAC 38-33「Double Categories of Relations」の要約記述および nLab「double category」の定義が明示的にこれを確認している：vertical 2-category は対象・縦射・縦セルから、horizontal bicategory は対象・横射・横セルから成り、対象は共通。この部分は独立に真であり、原典（Ehresmann/Grandis-Paré/Shulman系文献）でも一致する周知の事実であって novelty はない。\n\n2) 主張の論理構造: 「候補構成」が recursive-coalgebraic games の圏(縦)と Joyal games の(bi)圏(横)を組み合わせて一つの double category を作ろうとするものである限り、上記の定義上の制約から、両フレームワークの対象（=ゲーム）が一致しているか、少なくとも共通部分に制限しなければ double category として well-defined にならない。これは(1)からの直接帰結であり、reject する根拠は見当たらない。\n\n3) 個別ソースの直接検証結果:\n - arXiv:2510.22886「Games as Recursive Coalgebras」: 本文取得済み。recursive coalgebra による impartial games の圏 **Game** を構成（局所有限表示可能・対称モノイダル閉・Set上のcomonadic・subobject classifier を持つ）。本文検索で \"double category\" \"bicategory\" は一切出現せず、Joyal への言及は参考文献 joyal1977remarques の1箇所のみで、比較や対象同定の議論は無い。→ このソース単体は claim の対象同定の必要性を主張していないが、否定もしていない（単にその話題を扱っていない）。\n - TAC 12-08: 証明書エラー(www.tac.mta.ca のホスト名不一致)と https 版でのPDFバイナリ未展開により本文検証不可。検索でも当該号の表題を特定できず、二次情報のみ。\n - Joyal-games.pdf (uchicago, 別ミラー math.uchicago.edu/~may/IMA/JOYAL/Joyal.pdf も試行): PDFがストリーム圧縮のため機械読解不可、内容未検証。一般に Joyal 1977 の games/strategies 構成は(2-cellを持たない)通常の1-圏として知られており、bicategory として扱うには locally discrete bicategory 化などの追加操作が必要になりうる——この点は claim の「対象同定または制限が必要」という慎重な言い回しをむしろ補強する。\n\n結論: claim の主張内容自体は圏論的に正しく、90分の制限内でアクセスできた一次資料（2510.22886, および一般 double category 文献）と矛盾しない。ただし挙げられた3件の evidence_refs のうち2件(TAC 12-08, Joyal-games.pdf)は技術的理由で直接確認できておらず、2510.22886 はこの claim の内容を直接論じてはいない。support とするが、証拠の直接性には限界がある。",
      "dates": {
        "inspiration": null,
        "sourceRecorded": null,
        "evidence": null,
        "registeredInSumire": "2026-10-07T16:34:17.943Z",
        "updatedInSumire": "2026-10-10T08:50:49.812Z",
        "legacyRecordedAt": "2026-10-07T16:34:17.942Z",
        "legacyUpdatedAt": "2026-10-10T08:50:49.812Z"
      },
      "language": null,
      "projectLanguage": "ja",
      "languageSource": "not_recorded",
      "legacyStage": "note",
      "classification": "unclassified",
      "classificationStatus": "unclassified",
      "importance": "unspecified",
      "section": "unassigned",
      "problemState": "unknown",
      "resolution": "",
      "verificationEvidence": {
        "ai": "AI evaluation reportの独立reviewerが「claimの主張内容自体は圏論的に正しく」とsupport。double categoryの対象共有の必要性のみを対象とし、新規結果の主張ではない。",
        "lean": "",
        "human": ""
      },
      "aiClaimModel": "",
      "formalization": "unknown",
      "humanReview": "unreviewed",
      "verification": "unverified",
      "ai": {
        "use": "assisted",
        "tools": "Codex",
        "additions": ""
      },
      "originalQuestion": "",
      "references": []
    }
  ],
  "noteDependencies": [],
  "guides": {
    "arxiv": "https://info.arxiv.org/help/submit/index.html",
    "formal_conjectures": "https://github.com/google-deepmind/formal-conjectures/blob/main/CONTRIBUTING.md",
    "mathlib": "https://leanprover-community.github.io/contribute/index.html"
  },
  "search": {
    "arxiv": "https://arxiv.org/search/",
    "formal_conjectures": "https://google-deepmind.github.io/formal-conjectures/browse/",
    "mathlib": "https://leanprover-community.github.io/mathlib4_docs/"
  }
}