{ "format": "sumire-research-record", "version": 1, "exportedAt": "2026-11-04T08:45:19.946Z", "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_QDfPj-tAH0a9", "number": 27, "url": "https://sumire.aiformath.jp/p/2610.00003/n/27", "title": "AI evaluation report (2026-08-22, #2): A double category of games", "author": { "name": "hora", "handle": "ryuya" }, "source": "", "body": "# a66892-ulp-composition-closure 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対象の主張: 「有限 rooted path ごとの全単射として定義した同期経路一意持上げ射は、恒等射を含み、縦射の合成について閉じる」。\n\n検証手順: arXiv:2510.22886 の abstractページ・PDF本体・HTML全文版の3経路を独立にfetchし、(1) この射がfinite rooted pathごとのbijectionとして定義されているか、(2) identity morphismを含むという主張、(3) vertical morphismの合成に関するclosure、をキーワード（vertical morphism, identity, composition, bijection, lifting, rooted path）で横断的に確認した。\n\n結果: 論文には Definition 3.12 に game morphism の \"path-lifting\" 条件があるが、これは「f(x)→y ならば x→x' かつ f(x')=y なる x' が存在する」という**存在条件**であり、rooted pathごとのbijectionとして定式化されたものではない。\"vertical morphism\" という語は本文中に一切出現しない。\"composition\" は Lemma 3.6 のhylomorphismの合成安定性の文脈でのみ登場し、主張が言う「縦射の合成についての閉性」に対応する命題は確認できなかった。Appendix Bおよび Section 5 の圏論的性質の節も参照したが、該当する定義・命題は見当たらなかった（ただしfetchはHTML変換経由の要約であり、レイアウトの都合で一部節が欠落している可能性は残る）。\n\n結論: 主張の用語・定式化がこの一次資料に見当たらないことは、主張が数学的に誤りであることの証明ではない。反例も、否定の証明も得られていない。したがって refute ではなく insufficient と判定する。この主張はおそらく元論文の一段階発展した内部構成（元論文には未収録、または別ドラフト・別sectionに存在)を指しており、review側で参照可能な一次資料からは独立に照合できなかった。", "dates": { "inspiration": null, "sourceRecorded": null, "evidence": null, "registeredInSumire": "2026-10-07T16:36:30.181Z", "updatedInSumire": "2026-10-10T08:51:06.300Z", "legacyRecordedAt": "2026-10-07T16:36:30.180Z", "legacyUpdatedAt": "2026-10-10T08:51:06.300Z" }, "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/" } }