{
  "format": "sumire-research-record",
  "version": 1,
  "exportedAt": "2026-11-04T08:29:35.278Z",
  "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": []
      }
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  "notes": [
    {
      "id": "n_OZtvO0vppO9o",
      "number": 19,
      "url": "https://sumire.aiformath.jp/p/2610.00003/n/19",
      "title": "Multi-select sumと普遍monoid",
      "author": {
        "name": "hora",
        "handle": "ryuya"
      },
      "source": "",
      "body": "## Idea\n\nGames $A_1,\\dots,A_n$ に対して、それらの **multi-select sum**（正の有限個の成分を選んでプレイする和）を考える。これは不偏ゲームの monoidal product で、Conway addition に似ている。\n\n**問い: この operation に関する普遍的な monoid 不変量（universal monoid invariant）は何か。**\n\n原メモの観察・部分結果:\n\n- LN(3,2) は 2山Nim と 1山Nim の multi-select sum なので、universal monoid invariant が分かれば LN(3,2) が解けるはず。\n- 予想（原メモ自身が曖昧と注記）: universal monoid invariant は Birthday と Grundy number のペアか？（「しなさそう．いや，するかも」）\n- 性質: $1\\mathrm{Nim}(n)+1\\mathrm{Nim}(m)=1\\mathrm{Nim}(n+m)$（一山）、$LN(3,2)(a,b,c)=2\\mathrm{Nim}(a,c)+1\\mathrm{Nim}(b)$、$X_1+\\dots+X_1=\\mathrm{Nim}_n$（$X_{n+1}=\\{X_n\\},\\ X_0=0$）。\n- Lemma: $A+B$ が P-position $\\iff$ $A$ も $B$ も P-position（LN(3,2) のときに観察済み）。よって universal monoid では P-position は全て単位元（後手必勝ゲーム全体のなす同値類）につぶれる。\n- universal monoid は群ではない: $\\mathrm{Nim}_n$（$n>0$）に何を multi-sum しても単位元になれない（Lemma より）。\n- $\\{P,N\\}$-universal monoid は $\\{P,N\\}$ 自体。だから本命は **Grundy-universal monoid** で、単位元も考え直す必要がある。Birthday が関わるかも。\n- 「Game Operation and Universal Monoids」という観点からは、パス付き三山ニムの未解決問題も関連しそう。\"Recursive aspect\" にも注目する。\n- 例の候補: Grundy数 / Remoteness / Selective sum + PN / 逆形ゲーム / Selective sum + Grundy / Selective and Conway sum（知りたい）/ パスつき3山Nim（\"dependent sum\"）。\n\nUniversal monoid・recursive coalgebras・monoidal structure を全部駆使したい方向（原メモの結び）。*Games as recursive coalgebras* 系の公開問題 [The Bouton monoid](https://sumire.aiformath.jp/p/2610.00003/n/2)（Bouton monoid）・[Closed structures on games](https://sumire.aiformath.jp/p/2610.00003/n/5)（monoidal closed structures）と近い場所にあるが、multi-select sum という別の operation の普遍不変量を問う点で独立した問い。\n\n## 2026-08-26 AI研究監査：eventual birthday theorem\n\nselective compound を $A\\nabla B$，Nim heap を $*k$，$d(A)$ を $A$ からの最大 play 長（birthday/depth）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が十分大きいすべての $k$ で成り立つ．\n\n証明：$f_A(k)=\\operatorname{Gr}(A\\nabla *k)$ と置く．$A\\nabla*(k+1)$ の option set は $A\\nabla*k$ とその全 option を含むので $f_A(k+1)>f_A(k)$．一方 Grundy number は depth 以下だから $f_A(k)\\le k+d(A)$．従って $\\delta_A(k)=f_A(k)-k$ は非減少かつ $d(A)$ 以下で，最終的に定数 $e$ となる．$d=d(A)>0$ とし depth $d-1$ の option $A'$ を取る．帰納法で $\\operatorname{Gr}(A'\\nabla *j)=j+d-1$（$j\\gg0$）．もし $e\\le d-1$ なら $j=k+e-(d-1)\\le k$ と置くと $A'\\nabla *j$ は $A\\nabla*k$ の option であり，その Grundy number は $k+e=f_A(k)$ となって mex に反する．よって $e=d$．\n\n従って selective contexts に関する universal monoid は birthday を検出する：\n\\[\nd(A)=\\lim_{k\\to\\infty}(\\operatorname{Gr}(A\\nabla*k)-k).\n\\]\nまた $(d,\\operatorname{Gr})$ は普遍不変量ではない．$G=\\{*2\\}$ と $H=\\{*1,*2\\}$ はともに $(d,\\operatorname{Gr})=(3,0)$ だが\n\\[\n\\operatorname{Gr}(G\\nabla*1)=1,\\qquad\n\\operatorname{Gr}(H\\nabla*1)=4.\n\\]\nBouton monoid 全体の決定には，異なる hereditarily finite games を有限 selective context が必ず分離するか，または collision があるかを決める必要がある．",
      "dates": {
        "inspiration": null,
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        "registeredInSumire": "2026-10-07T16:35:33.817Z",
        "updatedInSumire": "2026-10-30T14:34:00.934Z",
        "legacyRecordedAt": "2026-10-07T16:35:33.817Z",
        "legacyUpdatedAt": "2026-10-30T14:34:00.934Z"
      },
      "language": null,
      "projectLanguage": "ja",
      "languageSource": "not_recorded",
      "legacyStage": "conjecture",
      "classification": "unclassified",
      "classificationStatus": "unclassified",
      "importance": "unspecified",
      "section": "unassigned",
      "problemState": "withdrawn",
      "resolution": "multi-select sum（正の有限個の成分を選び，選んだ各成分で1手ずつ指す和）は論文 v3 Example 4.3 の selective sum \\(\\vee\\) であり，その普遍的な monoid 不変量は Bouton monoid \\(B_{\\vee,v}\\)（Definition 4.6，存在は Theorem 4.8）である．outcome については \\(B_{\\vee,\\mathrm{Outcome}}=\\{P,N\\}\\)（Example 4.15）で，原メモの補題（和が P ⇔ 各成分が P）はこの事実である．Grundy 数については Question 5.8 そのものなので [[2610.00003/2]] に統合する．selective context は birthday を回収し（\\(k\\gg0\\) で \\(\\operatorname{Gr}(A\\vee *k)=k+d(A)\\)：[[2610.00003/38]]），birthday と Grundy 数の組は完全な不変量でない（\\(\\{*2\\}\\) と \\(\\{*1,*2\\}\\) はともに \\((3,0)\\) だが，\\(*1\\) との selective sum の Grundy 数は 1 と 4）．原メモの LN(3,2) は \\(\\mathrm{Nim}_1\\vee(\\mathrm{Nim}_1+\\mathrm{Nim}_1)\\) で，論文 v3 §5.4.2 が CN(6,2) の中に見つけた部分ゲーム \\(\\mathrm{Nim}_1+(\\mathrm{Nim}_1\\vee(\\mathrm{Nim}_1+\\mathrm{Nim}_1))\\) の内側にあたる．",
      "verificationEvidence": {
        "ai": "「2026-08-26 AI研究監査：eventual birthday theorem」の証明と(d,Gr)反例。",
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      "aiClaimModel": "",
      "formalization": "unknown",
      "humanReview": "unreviewed",
      "verification": "unverified",
      "ai": {
        "use": "assisted",
        "tools": "Codex, Claude",
        "additions": ""
      },
      "originalQuestion": "",
      "references": []
    }
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