{ "format": "sumire-research-record", "version": 1, "exportedAt": "2026-11-04T08:12:11.981Z", "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_Lcr0JIcrqjrV", "number": 39, "url": "https://sumire.aiformath.jp/p/2610.00003/n/39", "title": "Grundy周期性のprofinite化", "author": { "name": "hora", "handle": "ryuya" }, "source": "", "body": "## Question\n\n有限値列 \\(g:\\mathbb N\\to A\\) の最終周期性は、各fiber \\(g^{-1}(a)\\) がunary regular languageであることと同値である。これは自由1生成 profinite monoid から有限離散空間 \\(A\\) への連続写像、すなわち有限 clopen 彩色と読める。\n\nこの言い換えを Grundy's game の価値列へ適用し、次を問う。\n\n1. Grundy数列が最終周期的であることを、自由profinite monoid上の連続有限彩色の存在として特徴づけられるか。\n2. mex再帰は、この連続彩色の空間上でどのような自己作用素または固定点方程式を定めるか。\n3. rare/common 分解による「rare values が最終的に消えれば周期性が従う」という既存ルートを、profinite空間上の境界または非連続点の有限性として書き直せるか。\n\n## Evidence boundary\n\nfinite subtraction game では過去の有限windowが状態になるため、最終周期性とunary Moore machineの対応は標準的である。しかし Grundy's game では同じ有限window論法を使えず、最終周期性自体が未解決である。\n\nprofinite化が新しい定理を与えることもまだ示していない。\n\n## Goal\n\nまず既知のrare/common定理と計算証拠を一次資料で固定する。次に、最終周期列の標準的profinite特徴のうち、mex再帰に特有の部分だけを切り出す。rare値有限性がその不変量で表せれば、計算実験と代数的形式言語理論を同じ証明義務へ収束させる。\n\n[Eventual affine mex則](https://sumire.aiformath.jp/p/2610.00003/n/38) は別の game operation における eventual affine mex の結果であり、このノートの派生元ではない。両者は「mex再帰が作る最終的な有限状態構造」を比較する実験対象として関連する。", "dates": { "inspiration": null, "sourceRecorded": null, "evidence": null, "registeredInSumire": "2026-10-07T16:38:24.840Z", "updatedInSumire": "2026-10-30T04:58:59.517Z", "legacyRecordedAt": "2026-10-07T16:38:24.839Z", "legacyUpdatedAt": "2026-10-30T04:58:59.517Z" }, "language": null, "projectLanguage": "ja", "languageSource": "not_recorded", "legacyStage": "conjecture", "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, Claude", "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/" } }