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For adjacent vertices $x$ and $y$, we prove that the idleness function $\\alpha\\mapsto\\kappa_\\alpha^{\\mathcal H}(x,y)$ is piecewise affine and with no more than three affine pieces. A separate mass-balance argument gives linearity on $[1/2,1]$ for every locally finite simple hypergraph and, consequently, a limit-free expression for the Lin--Lu--Yau curvature. 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