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The full $N$-system is in a pure state, but the $L$-system is in a statistical mixture. As well known, for $L >>1$, the Boltzmann-Gibbs-von Neumann entropy violates thermodynamical extensivity, namely $S_{BG}(L) \\propto \\log L$, whereas the nonadditive entropy $S_q$ is extensive for $q=q_c=\\sqrt{37}-6 $, namely $S_{q_c}(L) \\propto L$. 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