{ "format": "sumire-research-record", "version": 1, "exportedAt": "2026-11-04T08:27:41.854Z", "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_hvtNuBNPGUxf", "number": 3, "url": "https://sumire.aiformath.jp/p/2610.00003/n/3", "title": "Partisan games via Pfin×Pfin", "author": { "name": "hora", "handle": "ryuya" }, "source": "", "body": "## Question\n\nHow much of partisan game theory is recovered by replacing \\(\\mathcal P_{\\mathrm{fin}}\\) with\n\\[\nF(X)=\\mathcal P_{\\mathrm{fin}}(X)\\times\\mathcal P_{\\mathrm{fin}}(X)?\n\\]\nThis is Question 5.3 of [Games as recursive coalgebras](https://arxiv.org/abs/2510.22886v3).\n\n## Established starting point: short game forms\n\nProposition 5.4 identifies the terminal recursive \\(F\\)-coalgebra with short partisan game forms. A position has a finite set of Left options and a finite set of Right options; recursive evaluation is well founded. This recovers forms such as\n\\[\n0=\\{\\mid\\},\\quad 1=\\{0\\mid\\},\\quad -1=\\{\\mid0\\},\\quad *=\\{0\\mid0\\}.\n\\]\n\nAn \\(F\\)-algebra gives a recursive evaluation. For normal-play outcomes, take \\(O=\\{0,1\\}^2\\), where the coordinates say whether Left wins when Left starts and when Right starts. Then\n\\[\n\\alpha(S_L,S_R)=\n\\left(\\bigvee_{(p,q)\\in S_L}q,\\;\n \\bigwedge_{(p,q)\\in S_R}p\\right),\n\\]\nwith empty join \\(0\\) and empty meet \\(1\\), computes the four outcome classes.\n\nThe familiar recursive definitions of sum and negation apply to forms:\n\\[\n(G+H)^L=(G^L+H)\\cup(G+H^L),\\qquad\n(G+H)^R=(G^R+H)\\cup(G+H^R),\n\\]\n\\[\n(-G)^L=-G^R,\\qquad(-G)^R=-G^L.\n\\]\n\n## What this does not yet supply\n\nExact option-preserving maps are not playing strategies. Normal-play equality, Conway order, simplification, and contextual quotients require their own definitions and arguments. Nor does the impartial internal-hom construction automatically prove closedness for this partisan category.\n\nThe proposed comparison with short Conway values and dyadic surreal numbers therefore belongs to a separate semantic layer. Full surreal numbers also require infinite option sets and transfinite birthdays, beyond this finite-powerset setting.\n\n## Next step\n\nFor the categorical extension, establish the colimit and Conway-sum properties of \\(\\operatorname{RecCoalg}_F\\), then prove an adjunction before asserting a partisan internal hom. For the semantic extension, state the contextual equivalence and prove that its quotient agrees with the intended game-value equivalence.\n\nThe two tasks should be kept distinct: Proposition 5.4 already provides the forms; it does not by itself establish the strategy category or the value quotient.\n\nSee also [the double-category question](https://sumire.aiformath.jp/p/2610.00003/n/1) and [the impartial internal hom](https://sumire.aiformath.jp/p/2610.00003/n/4).", "dates": { "inspiration": null, "sourceRecorded": null, "evidence": null, "registeredInSumire": "2026-10-07T16:30:19.301Z", "updatedInSumire": "2026-10-30T04:49:22.409Z", "legacyRecordedAt": "2026-10-07T16:30:19.300Z", "legacyUpdatedAt": "2026-10-30T04:49:22.409Z" }, "language": null, "projectLanguage": "ja", "languageSource": "not_recorded", "legacyStage": "conjecture", "classification": "unclassified", "classificationStatus": "unclassified", "importance": "unspecified", "section": "open-questions", "problemState": "open", "resolution": "", "verificationEvidence": { "ai": "", "lean": "", "human": "" }, "aiClaimModel": "", "formalization": "unknown", "humanReview": "unreviewed", "verification": "unverified", "ai": { "use": "assisted", "tools": "Codex", "additions": "" }, "originalQuestion": "", "references": [ { "url": "https://arxiv.org/abs/2510.22886v3", "verification": "not_checked", "service": "arxiv", "identifier": "arXiv:2510.22886v3", "version": "v3", "pinned": true, "correspondence": { "status": "reader_memo_unverified", "assumptions": "", "conclusions": "", "evidence": "", "differences": "", "unchecked": "", "declaration": "" } } ] } ], "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/" } }