{ "format": "sumire-research-record", "version": 1, "exportedAt": "2026-11-04T08:26:53.277Z", "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_7GqFSriXUoeY", "number": 13, "url": "https://sumire.aiformath.jp/p/2610.00003/n/13", "title": "AI research report (2026-08-14): A double category of games", "author": { "name": "hora", "handle": "ryuya" }, "source": "", "body": "# 一応答外側ゲームと二応答中間ゲームの全列挙\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## 対象\n一問一答ゲームを Q_n=(o^-;p_0^+,…,p_{n-1}^+) と書き、A=C=Q_1、X=Q_2 とした。合流写像 q:X→Q_1 は p_0,p_1 を同じ応答 p に送る。Joyal strategy は、Opponent の手に対して閉じ、Player の選択が決定的かつ全域である有限部分木として扱った。\n\n## 正確な主張\n強制される prefix を省略すると、合法な R:A→X は R_0（中間応答 p_0 を選ぶ）と R_1（p_1 を選ぶ）の2本だけである。合法な S:X→C は S_*（反転した X 側で到来し得る p_0,p_1 の双方を受け、唯一の C 応答へ返す）の1本だけである。したがって\nS_*∘R_0=S_*∘R_1=T,\nここで T:A→C は唯一の一応答 strategy である。さらに q_*R_0=q_*R_1、かつ q 後の中間ゲームを通る合成も T なので、interaction-and-hiding の結果は合流前後で一致する。\n\n## 根拠\nJoyal の符号付きゲームでは、射 A→B は反転した A と B 上の strategy として構成され、合成は共有ゲーム上の interaction から中間手を隠して得られる。この符号反転により、R で Player 選択だった X の二枝は S では Opponent 側の枝になる。Opponent 閉性は S に両枝への応答を要求するため、関係模型で必要だった「片方の中間枝だけを受理する S」は合法でない。定義の一次資料は https://people.cs.uchicago.edu/~brady/CSPP50102/notes/Joyal-games.pdf 。本問題の coalgebra 側の目的と既存設定は https://arxiv.org/abs/2510.22886 。\n\n## 失敗点\nこの列挙が排除するのは、A,C が各一応答、X が同一質問に対する二つの Player 応答を持ち、全域決定的 Joyal strategies を使う場合だけである。部分 strategy、非決定的 strategy、複数段の中間ゲーム、または枝が異なる極性・履歴に置かれる場合は検査していない。また、coalgebra morphism q が strategy の直接像を与える一般条件や、弱い・擬 double category の可能性は未解決である。\n\n## 次の決定的検査\n中間ゲームを二段 X=(o^-;p_i^+;o_i^-;p_i'^+) に拡張し、合流 q が最初の Player 枝 p_0,p_1 を同一視する一方、後続の Opponent 履歴を区別する場合について全 strategy を列挙する。符号反転後にも片枝選択を合法に保てる最小配置が存在するかを調べ、存在すれば interaction-and-hiding の可視 play 集合を q の前後で直接比較する。", "dates": { "inspiration": null, "sourceRecorded": null, "evidence": null, "registeredInSumire": "2026-10-07T16:34:12.312Z", "updatedInSumire": "2026-10-10T08:50:48.791Z", "legacyRecordedAt": "2026-10-07T16:34:12.311Z", "legacyUpdatedAt": "2026-10-10T08:50:48.791Z" }, "language": null, "projectLanguage": "ja", "languageSource": "not_recorded", "legacyStage": "note", "classification": "unclassified", "classificationStatus": "unclassified", "importance": "unspecified", "section": "unassigned", "problemState": "unknown", "resolution": "", "verificationEvidence": { "ai": "AI research reportの「正確な主張」と根拠が指定した有限ケースでの合成一致を主張。", "lean": "", "human": "" }, "aiClaimModel": "", "formalization": "unknown", "humanReview": "unreviewed", "verification": "unverified", "ai": { "use": "assisted", "tools": "Codex", "additions": "" }, "originalQuestion": "", "references": [ { "url": "https://arxiv.org/abs/2510.22886", "verification": "not_checked", "service": "arxiv", "identifier": "arXiv:2510.22886", "version": null, "pinned": false, "correspondence": { "status": "reader_memo_unverified", "assumptions": "", "conclusions": "", "evidence": "", "differences": "", "unchecked": "", "declaration": "" } } ] } ], "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/" } }