{
  "format": "sumire-research-record",
  "version": 1,
  "exportedAt": "2026-11-04T08:28:33.898Z",
  "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_owJazVkKk4Yt",
      "number": 22,
      "url": "https://sumire.aiformath.jp/p/2610.00003/n/22",
      "title": "AI evaluation report (2026-08-18): A double category of games",
      "author": {
        "name": "hora",
        "handle": "ryuya"
      },
      "source": "",
      "body": "# Review: 有限極性付きcoalgebraによるgameの表現とJoyal型dual/tensor（a66892-polarized-carrier）\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検証内容:\n\n1. 表現の妥当性: game G={G^L|G^R}をcoalgebra X→P_f(X)×P_f(X)の状態として見ると、0=(∅,∅)、1=({0},∅)、-1=(∅,{0})、*=({0},{0}) となり、4つの(L,R)ペアはすべて相異なる。distinctnessの主張は集合論的に自明かつ正しい。\n\n2. Joyal型dual/tensorの再帰式: web検索で得た複数の独立な二次資料（n-Category Café、Honsell–Lenisa–Pellarini論文の記述、Conway Games Coalgebraically論文）が一致して以下を報告している。\n   - disjunctive sum: G+H = {G^L+H, G+H^L | G^R+H, G+H^R}\n   - negation(conjugate): Ḡ = {Ḡ^R | Ḡ^L}（Left/Rightの役割を入れ替えて再帰）\n   これはJoyal(1977)がConway gameのcompact closed categoryを構成する際の定義そのものであり、claimの「左右成分ごとの再帰式によるJoyal型dual・tensor」の記述と完全に整合する。Joyal本人のPDF（uchicago掲載）は証明書エラーで直接fetchできなかったが、複数の独立した二次資料が一致した内容を報告しており、これはConwayのONAGの古典的定義とも一致するため、内容自体の信頼性は高い。\n\n3. 引用arXiv論文の関連性: arXiv:2510.22886(\"Games as Recursive Coalgebras\")のabstractを確認した。ゲームをrecursive coalgebraとして扱い、Game圏が\"symmetric monoidal closed\"かつ\"comonadic over Set\"であることを主張しており、これはclaimが言うtensor構造・(comonadから示唆される)双対的構造と整合する。ただし本文まで確認できておらず、claim文がこの論文からの直接引用か、それとも独立した再構成かは未確定（\"calc:minimal-polarized-games-and-tensor\"という内部計算参照が併記されていることからも、review側での自前の再導出だと推測される）。\n\n結論: 主張された数学的内容（distinctness、dual/tensorの再帰的定義）はConway/Joyalの古典理論と完全に一致し、正しい（support）。ただしこれは新規結果ではなく、既に複数の論文（Joyal 1977、Honsell–Lenisa–Pellarini 2014、Conway Games Coalgebraically、そして引用されたarXiv論文自体）で確立済みの内容の言い換えである。反例は見つからなかった。",
      "dates": {
        "inspiration": null,
        "sourceRecorded": null,
        "evidence": null,
        "registeredInSumire": "2026-10-07T16:35:54.738Z",
        "updatedInSumire": "2026-10-10T08:51:01.750Z",
        "legacyRecordedAt": "2026-10-07T16:35:54.737Z",
        "legacyUpdatedAt": "2026-10-10T08:51:01.750Z"
      },
      "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「主張された数学的内容」は正しいとsupport。0,1,-1,*の区別とdual/tensorの再帰式に限定し、一般double category構成の解決主張ではない。",
        "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/"
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}