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  "exportedAt": "2026-11-04T08:12:17.528Z",
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  "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,
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      "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"
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    "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": []
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    {
      "id": "n_SxlxO9L7g4no",
      "number": 2,
      "url": "https://sumire.aiformath.jp/p/2610.00003/n/2",
      "title": "The Bouton monoid",
      "author": {
        "name": "hora",
        "handle": "ryuya"
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      "source": "",
      "body": "## Idea\n\nWhat is the Bouton monoid \\(B_{\\vee,\\mathrm{Grundy}}\\)? Is it just the miniature monoid \\((H,\\vee_H)\\)?\n\nThis corresponds to Question 5.8 of the games-as-recursive-coalgebras paper.\n\n## Goal\n\n\\(B_{\\vee,\\mathrm{Grundy}}\\) を決定し、\\((H,\\vee_H)\\) と一致するかを判定する。\n\n## References\n\n- [Ryuya Hora, *Games as recursive coalgebras: A categorical view on the Nim-sum*](https://arxiv.org/abs/2510.22886)\n\n## 2026-08-26 AI研究監査：Bouton congruence が検出する第一層\n\nselective compound を $\\nabla$ とする．任意の short finite impartial game $A$ について\n\\[\n\\operatorname{Gr}(A\\nabla *k)=k+d(A)\n\\]\nが十分大きい $k$ で成り立つ（$d(A)$ は最大 play 長）．従って，すべての selective context に関して Grundy-equivalent な二つの game は同じ birthday を持ち，\n\\[\nd(A)=\\lim_{k\\to\\infty}(\\operatorname{Gr}(A\\nabla*k)-k)\n\\]\nで回収できる．したがって Bouton monoid から $(\\mathbb N,+)$ への非自明な quotient/invariant が得られる．\n\n一方，birthday と Grundy number のペアだけでは不十分である．\n\\[\nG=\\{*2\\},\\qquad H=\\{*1,*2\\}\n\\]\nはともに depth $3$，Grundy number $0$ だが，\n\\[\n\\operatorname{Gr}(G\\nabla*1)=1,\\qquad\n\\operatorname{Gr}(H\\nabla*1)=4.\n\\]\n\n従って残る核心は次の context-separation 問題である：$A\\ne B$ なら，ある有限 game $C$ が存在して\n\\[\n\\operatorname{Gr}(A\\nabla C)\\ne\\operatorname{Gr}(B\\nabla C)\n\\]\nとなるか．Yes なら $B_{\\nabla,\\mathrm{Grundy}}\\cong(H,\\nabla_H)$，No なら最小 collision を構造的に説明する必要がある．\n\n## 2026-08-27 ChatGPT研究継続：syntactic quotient theorem と selective audit\n\n任意の lax monoidal（unitなしなら lax semigroupal）tensor $\\otimes$ と game value $v$ に対し，terminal game $\\mathbb H$ 上の miniature multiplication を\n\\[\np\\star q:=\\xi_{\\mathbb H\\otimes\\mathbb H}\\bigl(\\mu_{\\mathbb H,\\mathbb H}(p,q)\\bigr)\n\\]\nで定める．$f=v_{\\mathbb H}$ として two-sided syntactic congruence\n\\[\np\\equiv_fq\n\\iff\nf(apb)=f(aqb)\\quad(\\forall a,b\\in\\mathbb H)\n\\]\n（semigroup の場合は unitization 内で quantification）を取ると，\n\\[\nB_{\\otimes,v}\\cong \\mathbb H/{\\equiv_{v_{\\mathbb H}}}.\n\\]\nこれは $v_{\\mathbb H}$ が factor する最大 monoid congruence，従って Hora の minimum monoid factorization の明示形である．commutative miniature monoid では one-sided contexts $v(p\\star c)$ だけでよい．形式言語の syntactic monoid，CGTの indistinguishability quotient／misère quotient と同じ contextual mechanism である．\n\nGenuinely lax な退化例は完全に計算できる．任意の value $v$ について，left projection tensor の Bouton semigroup は $\\operatorname{im}(v_{\\mathbb H})$ 上の left-zero multiplication，right projection は right-zero，pointed constant tensor は null multiplication になる．\n\nselective sum $\\nabla$ と Grundy value について，birthday と Grundy number の pair では contextual class を決められない明示例を得た．\n\\[\nP=\\{*1\\},\\quad Q=\\{P\\},\\quad D=\\{P,Q\\},\\quad C=\\{D\\},\n\\]\n\\[\nB=*3,\\qquad A=B\\cup\\{P\\}\n\\]\nとすると\n\\[\nh(A)=h(B)=3,\\qquad \\operatorname{Gr}(A)=\\operatorname{Gr}(B)=3,\n\\]\nだが\n\\[\n\\operatorname{Gr}(A\\nabla C)=8,\\qquad\n\\operatorname{Gr}(B\\nabla C)=5.\n\\]\nheight $\\le3$ の16 canonical games と height $\\le4$ の全65,536 canonical contexts の exhaustive audit では，selective-Grundy profile が一致する distinct pair はこの $(A,B)$ だけであり，height 5 の $C$ がそれを分離する．\n\n中心予想は次である：distinct canonical finite games $G\\ne H$ に対し，ある finite game $K$ が存在して\n\\[\n\\operatorname{Gr}(G\\nabla K)\\ne\\operatorname{Gr}(H\\nabla K)\n\\]\nとなる．Yes なら\n\\[\nB_{\\nabla,\\operatorname{Gr}}\n\\cong\n(\\mathbb H,\\nabla_{\\mathbb H})\n\\]\nであり，No なら最小 contextual collision の構造分類が必要である．有限 height truncation 上の congruence refinement は，stabilization certificate または separating context を返す計算手法になる．\n\n## 追記（2026-08-27・AI採掘）: Question 5.9 の一部と、残る具体化の問い\n\n**Question 5.9 の一部（Bouton monoid の構成）**（2026-03 の ChatGPT 会話）: Bouton monoid の構成に \\(P_{\\mathrm{fin}}\\) は不要で、「terminal object \\(H\\) をもつ monoidal category \\(\\mathcal R\\)＋lax monoidal functor \\(U:\\mathcal R\\to\\mathrm{Set}\\)＋value \\(v:U\\Rightarrow\\Delta_A\\)」だけから構成できる。証明が実際に使っているのは終対象性と lax monoidality のみで、論文 v3 Theorem 4.8 の証明がそのまま通る。検証状態: 会話中の証明であり、人間による検証は未了。Question 5.9 が問う枠組み全体の一般化は [[2610.00003/7]] に残る。\n\n**残る具体化の問い**（2025-10 の会話より）:\n\n- Bouton monoid は抽象構造しか与えられておらず、具体的構成が未達成で Nim 和の demystification として不満足。実際には多くの場合 \\(P_{\\mathrm{fin}}\\)-algebra \\(A\\) から誘導され、それが \\(G_b=\\mathrm{hylo}_A\\) の具体計算を可能にしている。Bouton function は常に \\(P_{\\mathrm{fin}}\\)-algebra 構造から誘導されるか（TAMS 版）。\n- Bouton monoid を計算する良い方法はあるか（Adámek 構成の類似。Nim-sum が Ackermann 解釈で対称差になることが手がかり。TAMS 版）。\n- monoidal 構造が Conway 和のとき、具体構造理解の鍵はゲームの持つ「微積構造」であろう（[Differential structures of games](https://sumire.aiformath.jp/p/2610.00003/n/6) と接続）。\n\n**CN(6,2) への動機**（論文 v3 §5.4.2）: circular nim CN(6,2)（Dufour–Heubach、未解決）の部分問題として \\(\\mathrm{Nim}_1+(\\mathrm{Nim}_1\\vee(\\mathrm{Nim}_1+\\mathrm{Nim}_1))\\) 型の必勝戦略理解があり、そのために nested Bouton monoid \\(B_{\\vee,\\mathcal G}\\) の決定——miniature monoid \\((\\mathbb H,\\vee_{\\mathbb H})\\) そのものか、真の商か——が必要になる。上の中心予想と同じ分岐である。",
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          "url": "https://arxiv.org/abs/2510.22886",
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          "identifier": "arXiv:2510.22886",
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