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  "body": "\n## Definition\n\n**Repair radii and villainy** are defined for a finite graph $G$ as follows.\n\nUse [[words and profiles|28M8XD8EA9]] with palette size $\\chi(G)$. For a profile $\\mathbf m$, let $\\mathcal W_{\\mathbf m}(G)$ be the set of words on $G$ with profile $\\mathbf m$.\n\nLet $\\mathcal P_{\\mathbf m}(G)$ be the set of [[proper colourings|0874138YP6]] of $G$ with profile $\\mathbf m$. Let $\\mathcal P(G)$ be the set of all proper colourings of $G$ with this palette.\n\nFor two words $w$ and $y$ on $G$, the Hamming distance $d_{\\mathrm H}(w,y)$ is the number of vertices on which they differ.\n\n1. The **repair radius** of $\\mathbf m$ is $R_G(\\mathbf m)=\\max_{w\\in\\mathcal W_{\\mathbf m}(G)}\\min_{y\\in\\mathcal P_{\\mathbf m}(G)} d_{\\mathrm H}(w,y)$.\n\n2. The **weak repair radius** of $\\mathbf m$ is $R^{\\mathrm w}_G(\\mathbf m)=\\max_{w\\in\\mathcal W_{\\mathbf m}(G)}\\min_{y\\in\\mathcal P(G)} d_{\\mathrm H}(w,y)$.\n\n3. The **villainy** of $G$ is $B(G)=\\max R_G(\\mathbf m)$, taken over all feasible profiles $\\mathbf m$.\n\n4. The **weak villainy** of $G$ is $B_{\\mathrm w}(G)=\\max R^{\\mathrm w}_G(\\mathbf m)$, taken over all feasible profiles $\\mathbf m$.\n\nEach inner quantity is the [[distance from a point to a set|07TXYYVYZR]] for the Hamming distance. Both $B(G)$ and $B_{\\mathrm w}(G)$ are zero when $G$ is the empty graph.\n",
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      "notes": "The source is arXiv:2610.13556v1 (2026-10), by Jiang Jiang, Yuhan Jiang, Yiyang Li, Lujia Wang, Aitao Zhang. A paper also restates older results, so this is not the first statement until a person or a later pass confirms it."
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