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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"
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    "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.",
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        "title": "To understand why Nim-sum works",
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        "title": "Categorical unification of recursive, geometric and logical aspects of combinatorial game theory",
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      "id": "n_dzJDIVCOqoQY",
      "number": 8,
      "url": "https://sumire.aiformath.jp/p/2610.00003/n/8",
      "title": "mex and reverse-mex: a dynamical approach to sink–wall duality",
      "author": {
        "name": "hora",
        "handle": "ryuya"
      },
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      "body": "## Question\n\nCan sink–wall duality for additive subtraction games be explained by mex dynamics and reflection?\n\nLet\n\\[\nS=\\{s_1,s_2,s_1+s_2\\},\\qquad 1\\leq s_1<s_2.\n\\]\nThe formula below is the conjecture in Equation (3) of Bhagat–Larsson–Manabe–Yamashita, [Additive Sink Subtraction](https://arxiv.org/html/2601.18715v3). It is not a new conjecture of this project.\n\nThe wall convention forbids negative heaps. The sink convention makes nonpositive heaps terminal. For the periodic sink sequence, represent a residue in \\(\\{1,\\ldots,p\\}\\): residue zero means the value at \\(p\\), not the terminal heap \\(0\\). Wall representatives are in \\(\\{0,\\ldots,p-1\\}\\). Use the source's outcome period \\(p\\) and these aligned residue conventions.\n\n## Operators\n\nOn\n\\[\nX=\\{0,1,2,3\\}^{\\mathbb Z/p\\mathbb Z}\n\\]\ndefine\n\\[\n(M_Sf)(x)=\\operatorname{mex}\\{f(x-s):s\\in S\\},\n\\qquad\n(M_S^{\\mathrm{rev}}f)(x)=\\operatorname{mex}\\{f(x+s):s\\in S\\}.\n\\]\nThere are at most three distinct option values, so mex remains in \\(\\{0,1,2,3\\}\\). Here \\(\\mathrm{rev}\\) reverses the shifts; it does not assert that the operator is the inverse of \\(M_S\\).\n\nLet\n\\[\n\\sigma(0)=2,\\quad \\sigma(2)=0,\\quad\n\\sigma(1)=1,\\quad \\sigma(3)=3,\n\\qquad\n(J_bf)(x)=\\sigma(f(-b-x)).\n\\]\nThen \\(J_b\\) is an involution. The sink–wall conjecture is\n\\[\nv_{\\mathrm{wall}}([x]_p)\n=\\sigma\\!\\left(v_{\\mathrm{sink}}([-s_2-x]_p)\\right),\n\\]\nso the proposed relation uses \\(b=s_2\\).\n\n## What is already ruled out\n\nThe identity \\(M_SJ_b=J_bM_S^{\\mathrm{rev}}\\) does not hold throughout \\(X\\). For \\(f\\equiv3\\), its left side is constantly \\(0\\), while its right side is constantly \\(2\\).\n\nThere is a conditional identity. Suppose every successor-value set\n\\[\nA_f^{\\mathrm{rev}}(y)=\\{f(y+s):s\\in S\\}\n\\]\nis either a three-element subset of \\(\\{0,1,2,3\\}\\) or a two-element subset of \\(\\{0,1,2\\}\\). On these sets,\n\\(\\operatorname{mex}(\\sigma A)=\\sigma(\\operatorname{mex}A)\\).\nWith \\(y=-b-x\\),\n\\[\nM_S(J_bf)(x)\n=\\operatorname{mex}(\\sigma A_f^{\\mathrm{rev}}(y))\n=\\sigma(\\operatorname{mex}A_f^{\\mathrm{rev}}(y))\n=J_b(M_S^{\\mathrm{rev}}f)(x).\n\\]\nThe hypothesis concerns successor sets, not predecessor sets. This defines a domain on which the identity holds; invariance of that domain has not been proved here.\n\n## Remaining work\n\nProve that the relevant sink sequences satisfy the successor-set condition and the reverse-mex fixed-point equation. Finite checks alone do not prove either statement. Even then, being a fixed point does not identify the wall sequence: the boundary data must be matched over a full recurrence window of length \\(s_1+s_2\\), not at a single point.\n\n[Note 35](https://sumire.aiformath.jp/p/2610.00003/n/35) records the calculations and the distinctions between operator identities, finite evidence, and the conjecture. The connection with [differential structures of games](https://sumire.aiformath.jp/p/2610.00003/n/6) remains a motivation rather than an established categorical interpretation.",
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          "url": "https://arxiv.org/html/2601.18715v3",
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