{ "format": "sumire-research-record", "version": 1, "exportedAt": "2026-11-04T08:28:19.722Z", "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_-XO9VDEg_nSm", "number": 33, "url": "https://sumire.aiformath.jp/p/2610.00003/n/33", "title": "Wythoff Nim完全値の計算量", "author": { "name": "hora", "handle": "ryuya" }, "source": "", "body": "## Trigger\n\nWythoff Nim の特殊な色、特に `G(n,m)=1` の判定ではなく、完全な Sprague--Grundy 値\n`G(n,m)` を計算する時間計算量を上下から完全に決定したい。入力長を\n`L=Theta(log n+log m)` と見る単一問と、`0<=n,m<=N` の完全表を作る問題を区別する。\n\n## Goal\n\n完全表について既知の素朴な cubic 計算を破り、最終的には単一問を入力桁数の多項式時間で\n計算できるか、またはそのような算法が不可能であることを意味のある計算モデルで示す。\nWythoff の三方向を A2 正根系、Hall algebra、affine symmetric group と結び、単なる類似でなく\nmex の計算を実際に圧縮する構造を探す。\n\n## Current frontier\n\n- 三本の合法手を A2 の正根方向として、Boolean Hall support から厳密に復元できる。\n- 完全表には word-RAM で `O(N^3/log N)` の厳密な word-parallel 算法がある。ただし bit 計算量は\n まだ cubic であり、この評価の文献上の新規性も未確定である。\n- 各固定行は有限個の例外を除いて affine symmetric group の元になる。\n- 自然な Zeckendorf prefix、有限個の先頭hole、隣接行の短い Coxeter 更新は高速化に使えない。\n\n## Next theorem target\n\n色 `h` と行 `r` に対する三つの禁止集合 `D_h`, `B_h-r`, `L_r` の和集合が dyadic interval を\n完全に覆うかを、polylogarithmic size の affine cut-set で判定する。これができれば、最初の\n非被覆点を区間二分で求め、完全表を `O(N^2 polylog N)` で計算できる可能性がある。", "dates": { "inspiration": null, "sourceRecorded": null, "evidence": null, "registeredInSumire": "2026-10-07T16:37:28.216Z", "updatedInSumire": "2026-10-10T08:51:19.005Z", "legacyRecordedAt": "2026-10-07T16:37:28.216Z", "legacyUpdatedAt": "2026-10-10T08:51:19.005Z" }, "language": null, "projectLanguage": "ja", "languageSource": "not_recorded", "legacyStage": "conjecture", "classification": "unclassified", "classificationStatus": "unclassified", "importance": "unspecified", "section": "unassigned", "problemState": "unknown", "resolution": "", "verificationEvidence": { "ai": "", "lean": "", "human": "" }, "aiClaimModel": "", "formalization": "unknown", "humanReview": "unreviewed", "verification": "unverified", "ai": { "use": "assisted", "tools": "Codex", "additions": "" }, "originalQuestion": "", "references": [] } ], "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/" } }