\label{eq:resolution_identity}$$ Here $C_\ell=(-\ell/2,\ell/2)^3=\ell C_1$ is the cube of side length $\ell$. This is because for any $f\in L^1({\mathbb{R}}^3)$, $$\begin{aligned}
\int_{C_\ell}\sum_{z\in{\mathbb{Z}}^3}f(x-\ell z-\tau)\,d\tau&=\int_{C_\ell}\sum_{z\in \ell{\mathbb{Z}}^3}f(x-z-\tau)\,d\tau\\
&=\int_{{\mathbb{R}}^3}f(x-\tau)\,d\tau=\int_{{\mathbb{R}}^3}f(\tau)\,d\tau. \end{aligned}$$

The main result of this section is the following upper bound.

\[prop:upper\_bound\] There exists a universal constant $C$ such that for any $\sqrt{\rho}\in H^1({\mathbb{R}}^3)$, any $0<\delta<\ell/2$ and any $0<\alpha<1/2$, $$\begin{gathered}
 E(\rho){\leqslant}\left(\int_{1-\alpha}^{1+\alpha}\frac{ds}{s^4}\right)^{-1}\int_{1-\alpha}^{1+\alpha}\frac{dt}{t^4}\int_{SO(3)}dR\int_{C_{t\ell}}\frac{d\tau}{(t\ell)^3}\times\\
  \times \sum_{z\in{\mathbb{Z}}^3}\sum_{j=1}^{24} E\Big(\chi_{t\ell,t\delta,j}(R\,\cdot\,-t\ell z-\tau)\rho\Big)
 +C\delta^2\log(\alpha^{-1})\int_{{\mathbb{R}}^3}\rho^2.
 \label{eq:upper_bound_average}\end{gathered}$$ In particular, we can find $\ell'\in(\ell(1-\alpha),\ell(1+\alpha))$, $\delta'\in(\delta(1-\alpha),\delta(1+\alpha))$ and an isometry $(\tau,R)\in {\mathbb{R}}^3\times SO(3)$ such that $$E(\rho){\leqslant}\sum_{z\in{\mathbb{Z}}^3}\sum_{j=1}^{24} E\Big(\chi_{\ell',\delta',j}(R\,\cdot\,-\ell' z-\tau)\rho\Big)
 +C\delta^2\log(\alpha^{-1})\int_{{\mathbb{R}}^3}\rho^2.
 \label{eq:upper_bound}$$

The right side of  involves our incomplete partition of unity with holes, which is rotated (with the rotation $R$), translated (with the translation $\tau$) and dilated (with the dilation parameter $t$). The error is small only when $\delta$ (the size of the holes) is small. However we cannot take $\delta=0$ since that would make the gradient of the densities $\chi_{t\ell,t\delta,j}(R\,\cdot\,-t\ell z-\tau)\rho$ blow up. Nevertheless, the statement is that the energy decouples and the holes can be neglected, at the expense of an error of the order $\delta^2\int_{{\mathbb{R}}^3}\rho^2$. In  we use dilations for purely technical reasons, in order to better control error terms.

Using , we write our density $\rho$ as follows $$\rho(x)=\frac{1}{\ell^3}\int_{C_\ell}\sum_{z\in{\mathbb{Z}}^3}\sum_{j=1}^{24} \chi_{\ell,\delta,j}(x-\ell z-\tau)\rho(x)\,d\tau.$$ For every fixed $\tau\in C_\ell$, 