{
  "format": "sumire-research-record",
  "version": 1,
  "exportedAt": "2026-11-04T08:28:21.234Z",
  "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_dt9zllooENgg",
      "number": 40,
      "url": "https://sumire.aiformath.jp/p/2610.00003/n/40",
      "title": "AI evaluation report (2026-08-28, #2): A double category of games",
      "author": {
        "name": "hora",
        "handle": "ryuya"
      },
      "source": "",
      "body": "# 反対検討: reachable recursive game coalgebraにおけるpath-bijectiveだが非同型なcoalgebra準同型\n\n> この報告はcommit時点では独立reviewを経ていない。書誌情報と主張の検証状態は`research/runs/`のRunEventを正とし、人間の確認前の内容として読むこと。\n\n主張: reachable recursive game coalgebraにおいて、有限rooted pathの各長さで全単射を誘導するが同型でないcoalgebra準同型が存在する。具体例は、2つの枝の終点を分離した有限木から、それらを1つの終点へ合流させた有限DAGへの準同型。\n\n検証方法: 会話履歴・元の判定には依存せず、arXiv:2510.22886 (Ryuya Hora, \"Games as recursive coalgebras: A categorical view on the nim-sum\", v3, 2025-11-14)をPDFで直接取得し、以下の定義を確認した。\n- Definition 2.1 (game = finite-option + finite-time な有向グラフ)\n- Definition 3.1/3.2 (T-coalgebraとcoalgebra homomorphism: P_fin(f)∘θ_X = θ_Y∘f)\n- Definition 3.5 (recursive coalgebra)\n- Definition 3.12 (game morphism = graph morphism条件 + path-lifting条件。これはcoalgebra homomorphismと同値であることが本文で明示されている)\n\nこれらの定義に基づき、次の具体例を自分で構成し、直接計算で確認した。\n木X: root→a→leaf1, root→b→leaf2 (5頂点、根から全頂点へreachable)。\nDAG Y: root'→a'→leaf, root'→b'→leaf (4頂点、a',b'からの終点leafが共通)。\n写像f: root↦root', a↦a', b↦b', leaf1↦leaf, leaf2↦leaf。\n\n確認1 (coalgebra homomorphism): θ_X(root)={a,b}をfで送ると{a',b'}=θ_Y(root')。θ_X(a)={leaf1}→{leaf}=θ_Y(a')。θ_X(b)={leaf2}→{leaf}=θ_Y(b')。leaf1,leaf2はθ_X=∅でθ_Y(leaf)=∅とも整合。よって可換図式が成立しcoalgebra homomorphism(=game morphism)である。\n確認2 (path-lifting): root'→a'に対しroot→aが存在しf(a)=a'。root'→b'に対しroot→bが存在しf(b)=b'。a'→leafに対しa→leaf1でf(leaf1)=leaf。b'→leafに対しb→leaf2でf(leaf2)=leaf。全て満たされる。\n確認3 (path数の各長さでの全単射): 長さ0: {root}↔{root'} (1対1)。長さ1: {root→a, root→b}↔{root'→a', root'→b'} (2対2、かつfは経路を頂点列として送るので単射)。長さ2: {root→a→leaf1, root→b→leaf2}↔{root'→a'→leaf, root'→b'→leaf} (2対2、経路は中間頂点a'/b'で区別されるため終点が同じleafでも異なる経路として単射に対応)。長さ3以上: 両側とも0。よって各長さでpathの全単射が成立する。\n確認4 (非同型): fは頂点集合上|X|=5→|Y|=4で単射でない(leaf1,leaf2が同じleafに合流)ため、fは同型射ではない。\n\n結論: 主張が述べる現象(path-bijectiveだが非同型なcoalgebra準同型で、2分岐の終点合流という具体形)は、この論文自身の定義の枠内で完全に構成可能かつ正しいことを独立に確認した。これは標準的なcoalgebra理論の現象(bisimilarでない状態の合流がpath-count/trace同値を保ちつつ単射性を壊す)の一例であり、数学的内容として反論の余地はない。\n\n留保点: PDF 45ページのうち読んだ範囲(1–20ページ、28–31ページ)には「reachable」という語そのものの定義や「reachable recursive game coalgebra」という用語のverbatimな出現は確認できなかった。この語が論文の別部分(未読の21–27, 32–45ページ、または関連する共著予定の続編論文)由来である可能性は排除できていない。しかし、私が独立に構成した例は根からの到達可能性(reachability)を自明に満たす最小構成であり、用語の正確な出典に関わらず主張の数学的中身は成立する。ゆえにsupport(用語出典についてのみ未確認の留保あり)とする。",
      "dates": {
        "inspiration": null,
        "sourceRecorded": null,
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        "registeredInSumire": "2026-10-07T16:38:31.313Z",
        "updatedInSumire": "2026-10-10T08:51:25.472Z",
        "legacyRecordedAt": "2026-10-07T16:38:31.313Z",
        "legacyUpdatedAt": "2026-10-10T08:51:25.472Z"
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      "language": null,
      "projectLanguage": "ja",
      "languageSource": "not_recorded",
      "legacyStage": "note",
      "classification": "unclassified",
      "classificationStatus": "unclassified",
      "importance": "unspecified",
      "section": "unassigned",
      "problemState": "unknown",
      "resolution": "",
      "verificationEvidence": {
        "ai": "AI evaluation reportで具体的な5頂点木→4頂点DAGを構成し、四つの確認の後に主張の成立を述べる。",
        "lean": "",
        "human": ""
      },
      "aiClaimModel": "",
      "formalization": "unknown",
      "humanReview": "unreviewed",
      "verification": "unverified",
      "ai": {
        "use": "assisted",
        "tools": "Codex",
        "additions": ""
      },
      "originalQuestion": "",
      "references": []
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    "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"
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  "search": {
    "arxiv": "https://arxiv.org/search/",
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