precise statement of  will involve upper and lower bounds, as well as an average over the translations, rotations and dilations of the partition itself. The lower bound was indeed already shown in [@LewLieSei-18] using the Graf-Schenker inequality [@GraSch-95]. The upper bound is the main new ingredient of our proof. The two bounds are derived in Sections \[sec:upper\_bound\] and \[sec:lower\_bound\]. This will allow us to provide a rather simple proof of Theorem \[thm:UEG\_thermo\_limit\_quantum\] in Section \[sec:proof\_thermo\_limit\], using the convergence for tetrahedra which will be studied in Section \[sec:CV\_rate\_tetrahedra\].

In Section \[sec:deviation\] we will estimate the deviation of the energy when we replace $\rho\chi_k$ by a constant function, say $\rho(x_k)\chi_k$ for some $x_k$ in the support of $\chi_k$. If $\rho$ is essentially constant in the corresponding region, the error will be small, but if $\rho$ is not constant we bound the energy using some gradient terms, utilizing our upper bound .

After showing the Lipschitz regularity of $e_{\rm UEG}$ in Section \[sec:Lipschitz\_UEG\], we will be able to conclude the proof of Theorem \[thm:LDA\] in Section \[sec:conclusion\]. We replace $E(\rho(x_k)\chi_k)$ by $\int_{{\mathbb{R}}^3}e_{\rm UEG}(\rho(x))\chi_k(x)\,dx$ when $\rho$ is large enough on ${\rm supp}(\chi_k)$, using some quantitative estimates derived for tetrahedra in Section \[sec:CV\_rate\_tetrahedra\].

In the 