{ "format": "sumire-research-record", "version": 1, "exportedAt": "2026-11-04T08:44:11.765Z", "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_C6CeuRDOzCF0", "number": 12, "url": "https://sumire.aiformath.jp/p/2610.00003/n/12", "title": "word gameとstar height 1予想", "author": { "name": "hora", "handle": "ryuya" }, "source": "", "body": "Inoue の発表（2026-07-30）で聞いた「語の上の subtraction game」から。局面は \\(w \\in \\Sigma^*\\)、固定した \\(S \\subseteq \\Sigma^+\\) に対し合法手は接頭辞除去 \\(sx \\to x\\)（\\(s \\in S\\)）、normal play。P-局面全体 \\(P_S\\) は well-founded グラフの kernel であり、一意に\n\\[\n\\Sigma^* \\setminus P_S = S\\,P_S\n\\]\nを満たす。これは Larsson の invariant subtraction game の \\(\\star\\)-operator（\\(S^\\star := P_S \\setminus \\{\\varepsilon\\}\\)）を可換モノイド \\(\\mathbb{N}^d\\) から自由モノイド \\(\\Sigma^*\\) へ非可換化したもの。\n\n## Idea\n\n**予想 (FGS: finite game-star conjecture)**: \\(S\\) が有限なら \\(P_S\\) の generalized star height は 1 以下。\n\n分かっていること:\n\n- \\(S\\) 有限 \\(\\Rightarrow P_S\\) は正規。ただし像 \\(\\mathrm{KFin}_\\Sigma = \\{P_S : S \\text{ finite}\\}\\) は \\(\\mathrm{REG}\\) の真部分クラス（例: \\(P=\\{\\varepsilon\\}\\) は実現不能）。\n- \\(S\\) が prefix code の場合は証明済み: \\(T := \\Sigma^* \\setminus S\\Sigma^*\\) と置くと greedy 分解の一意性から\n \\[\n P_S = (S^2)^* T, \\qquad N_S = S (S^2)^* T\n \\]\n で、\\(S\\) 有限なら \\(S^2, T\\) は star-free なので \\(h_{\\mathrm{gen}} \\le 1\\)。\n- 一般の有限 \\(S\\) では元同士が接頭辞として重なりゲーム木が分岐し、\\(\\mathbf{1}_P(w) = \\neg \\bigvee_{w=sx,\\, s\\in S} \\mathbf{1}_P(x)\\) という bounded NOR 固定点になる。この分岐を一段の Kleene star に flatten できるかが本体。\n\n## Goal\n\n- (FGS) が正なら \\(\\mathrm{KFin}_\\Sigma \\subseteq \\mathcal{H}_1\\) という大域定理。\n- 逆に \\(h_{\\mathrm{gen}}(P_S) \\ge 2\\) となる有限 \\(S\\) が一つでもあれば、それ自体が Generalized Star Height 問題（高さ 2 以上の正規言語の存在）の否定的でない方向の解決例になる。どちらに転んでも価値がある。\n\n## Personal context\n\n- Generalized Star Height の既存ノートの prefix-code greedy 分解・一方向 flattening \\((X^*Y)^* = \\{\\varepsilon\\} \\cup (X \\cup Y)^* Y\\) がそのまま道具になる。\n- 関連未解決問題（発表者情報）: WordWythoff（\\(S = a^+ \\cup b^+ \\cup \\{|w|_a = |w|_b\\}\\)）の N-局面が context-free かは不明。\n\n## Machine evidence: S = {a, b, ab} は height ≤ 1（⚪COMPUTED, 2026-08-07）\n\n進化型探索ループ（検証器はゲームDP＋Brzozowski微分の式評価器、kernel方程式とprefix-code定理式による独立検証つき）が、重なりあり（a が ab の prefix）の最初のインスタンス S = {a, b, ab} で height-1 表現を発見した:\n\n P_S = (bb)* ( 1 | aa(aa)*(1 | bΣ*) | ba(aa)*(1 | bΣ*) )\n\nDSL原文 `(bb)*(1|aa(aa)*(1|b~0)|ba(aa)*(1|b~0))`。長さ≤12の全語（8191語）と乱択30000語（長さ≤300）でP-position判定と完全一致、star nesting depth は 1。**ラベルは⚪COMPUTED（証明ではない）**。証明化は未着手。", "dates": { "inspiration": null, "sourceRecorded": null, "evidence": null, "registeredInSumire": "2026-10-07T16:34:00.191Z", "updatedInSumire": "2026-10-30T04:54:35.666Z", "legacyRecordedAt": "2026-10-07T16:34:00.191Z", "legacyUpdatedAt": "2026-10-30T04:54:35.666Z" }, "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/" } }