{
  "format": "sumire-research-record",
  "version": 1,
  "exportedAt": "2026-11-04T08:44:35.344Z",
  "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_vvbu-yN1SRqa",
      "number": 25,
      "url": "https://sumire.aiformath.jp/p/2610.00003/n/25",
      "title": "AI evaluation report (2026-08-21, #2): A double category of games",
      "author": {
        "name": "hora",
        "handle": "ryuya"
      },
      "source": "",
      "body": "# a66892-ulp-finite-strategy-preservation の独立レビュー: 証拠不十分\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\nclaimは「各長さの有限rooted path上で全単射を誘導するcoalgebra準同型について、有限経路strategyの直像はinteraction、hiding、および恒等strategyを保存する」というもの。\n\n1つ目の引用arXiv:2510.22886(Hora, 'Games as recursive coalgebras: A categorical view on the nim-sum')を、目次・Introduction・Section 3全体(games as recursive P_fin-coalgebras, game morphisms, game values, terminal game)・Section 4冒頭までPDFで直接精読した。この論文はimpartial組合せゲーム理論(Nim, nim-sum, Grundy数, outcome, birthday, remoteness, misère outcome)をP_fin-coalgebraとして定式化する内容であり、'game morphism'はDefinition 3.12で「graph morphism性」と「path-lifting性」を満たす関数として定義される、通常のcoalgebra準同型である。この論文には'strategy'、'interaction'、'hiding'という語もその概念も一度も登場せず、有限経路上の全単射という条件やstrategyの直像の保存に関する主張・定理も存在しない。したがってこの一次資料はclaimを述べても証明してもいない。\n\n2つ目の引用Joyal-games.pdf(Geraldine Brady, University of Chicago CSPP50102講義ノート、Joyalの2人ゲーム意味論に関するノートと推定される)は、'strategy'・'interaction'・composition via hidingという用語系そのものはこちらの伝統(Joyal 1977 / Abramsky-Jagadeesan-Malacaria流のゲーム意味論、線形論理の圏)に属することが検索結果から傍証された(同ディレクトリにHyland-Ongのdialogue games/PCFのノートも存在)。しかしこのPDFはTLS証明書検証エラーのため、直接fetch・Googleキャッシュ経由・web.archive.org経由の3通りすべてで内容取得に失敗し、独立に内容を確認できなかった。\n\n結論として、claimは2つの異なる「game」の圏論的伝統(Horaの単一グラフによる組合せゲーム論的coalgebraと、Joyal流の2人インタラクション・ゲーム意味論)の用語を混在させているように見える。1つ目の証拠はclaimと無関係であり、2つ目は検証不能である。反例も否定の証明も得ていないためrefuteではなく、insufficientと判定する。",
      "dates": {
        "inspiration": null,
        "sourceRecorded": null,
        "evidence": null,
        "registeredInSumire": "2026-10-07T16:36:13.164Z",
        "updatedInSumire": "2026-10-10T08:51:04.488Z",
        "legacyRecordedAt": "2026-10-07T16:36:13.164Z",
        "legacyUpdatedAt": "2026-10-10T08:51:04.488Z"
      },
      "language": null,
      "projectLanguage": "ja",
      "languageSource": "not_recorded",
      "legacyStage": "note",
      "classification": "unclassified",
      "classificationStatus": "unclassified",
      "importance": "unspecified",
      "section": "unassigned",
      "problemState": "unknown",
      "resolution": "",
      "verificationEvidence": {
        "ai": "",
        "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/"
  }
}