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  "exportedAt": "2026-11-04T08:29:33.678Z",
  "scope": "Reader-visible project fields and native live notes only; referenced notes in other projects, deleted notes and private AI context are excluded.",
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    "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",
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      {
        "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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  "notes": [
    {
      "id": "n_77SRNx-JErA5",
      "number": 38,
      "url": "https://sumire.aiformath.jp/p/2610.00003/n/38",
      "title": "Eventual affine mex則",
      "author": {
        "name": "hora",
        "handle": "ryuya"
      },
      "source": "",
      "body": "## Trigger\n\n[Multi-select sumと普遍monoid](https://sumire.aiformath.jp/p/2610.00003/n/19) と [The Bouton monoid](https://sumire.aiformath.jp/p/2610.00003/n/2) で得た selective sum の eventual birthday phenomenon を，[mex/rev-mex双対力学系](https://sumire.aiformath.jp/p/2610.00003/n/8) の mex 力学系も含む一般の well-founded game operation に抽象化した．\n\n## Result\n\n**定理（abstract eventual affine mex theorem）．** well-founded impartial games の二項 operation \\(\\diamond\\) と test games \\(H_k\\) があり，次を満たすとする．\n\n1. \\(\\operatorname{Gr}(H_k)=d(H_k)=k\\)．\n2. \\(A\\diamond H_k\\) とその全 options は \\(A\\diamond H_{k+1}\\) の options に含まれる．\n3. \\(d(A\\diamond H_k)\\le d(A)+k\\)．\n4. deepest option \\(A'\\) of \\(A\\) と任意の \\(j\\le k\\) に対し，\\(A'\\diamond H_j\\) は \\(A\\diamond H_k\\) の option である．\n\nこのとき任意の finite short impartial game \\(A\\) について\n\\[\n\\boxed{\n\\operatorname{Gr}(A\\diamond H_k)=k+d(A)\n\\qquad(k\\gg0)\n}\n\\]\nが成り立つ．ここで \\(d(A)\\) は最大 play length である．\n\n## Proof\n\n\\(f_A(k)=\\operatorname{Gr}(A\\diamond H_k)\\) と置く．仮定2より，\\(A\\diamond H_{k+1}\\) の options は \\(A\\diamond H_k\\) とその全 options を含む．mex の定義から後者の Grundy values は \\(0,\\dots,f_A(k)-1\\) を含み，前者自身が \\(f_A(k)\\) を与えるので\n\\[\nf_A(k+1)>f_A(k).\n\\]\n従って \\(\\delta_A(k)=f_A(k)-k\\) は非減少である．一方，Grundy number は depth 以下なので\n\\[\nf_A(k)\\le d(A\\diamond H_k)\\le d(A)+k,\n\\]\nよって \\(\\delta_A(k)\\le d(A)\\)．従って \\(\\delta_A(k)\\) は最終的に定数 \\(e\\) になる．\n\n\\(d(A)=d>0\\) とし，\\(d(A')=d-1\\) の option \\(A'\\) を取る．depth に関する帰納法により\n\\[\n\\operatorname{Gr}(A'\\diamond H_j)=j+d-1\\qquad(j\\gg0).\n\\]\nもし \\(e\\le d-1\\) なら\n\\[\nj=k+e-(d-1)\\le k\n\\]\nと取れる．仮定4より \\(A'\\diamond H_j\\) は \\(A\\diamond H_k\\) の option だが\n\\[\n\\operatorname{Gr}(A'\\diamond H_j)=j+d-1=k+e=f_A(k),\n\\]\nとなり，position とその option が同じ Grundy number を持つという矛盾を得る．従って \\(e=d\\)．\n\n## Selective-sum corollary\n\n\\(\\diamond=\\nabla\\) を selective sum，\\(H_k=*k\\) を Nim heap とすると仮定1--4が成立し，\n\\[\n\\operatorname{Gr}(A\\nabla*k)=k+d(A)\\qquad(k\\gg0).\n\\]\n特に\n\\[\n\\boxed{\nd(A)=\\lim_{k\\to\\infty}\\bigl(\\operatorname{Gr}(A\\nabla*k)-k\\bigr)\n}\n\\]\nであり，selective contexts は birthday/depth を復元する．\n\n## Interpretation\n\n大きな test object との compound が eventual affine になる機構は，個別の floor formula ではなく，\n\n- mex による strict growth；\n- depth による upper bound；\n- deepest option による lower forcing；\n\nの三点で説明される．\n\n## Status\n\n抽象定理と selective-sum specialization は証明済み．他の game operations・rev-mex dynamics・misère analogues が仮定を満たすかは未調査である．",
      "dates": {
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        "registeredInSumire": "2026-10-07T16:38:04.005Z",
        "updatedInSumire": "2026-10-30T04:49:28.660Z",
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        "legacyUpdatedAt": "2026-10-30T04:49:28.660Z"
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