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Statistical physics is unable to prove that $TdS=d\overline{E}$
I will pose $k_B=1$.
Suppose a system of statistical physics with the constraints:
$$
\begin{align}
1&=\sum_{q\in\mathbb{Q}}\rho(q)\\
\overline{E}(\beta)&=\sum_{q\in\mathbb{Q}} E(q)\exp(-\...