E$, and thus $$\begin{aligned}
&& 0={\mathcal{H}}_{E}^s(e_{n+1})
-{\mathcal{H}}_{F}^s(e_{n+1})
=-2\int_{{\ensuremath{\mathbb{R}}}^{n+1}}
\frac{\chi_{E\setminus F}(y)}{|e_{n+1}-y|^{n+1+s}}\,dy\\&&\qquad{\leqslant}-2
\int_{{\ensuremath{\mathbb{R}}}^{n-1}\times(-\infty,0)\times\left(-\frac1{100},0\right)}
\frac{dy}{|e_{n+1}-y|^{n+1+s}}<0.\end{aligned}$$ This contradiction proves .

Similarly, one proves that $-e_{n+1}\not\in
\overline{(\partial E)\cap\{x_n>0\}}$. Consequently, since $\partial E$ is a closed set, we obtain , as desired.

As a byproduct of Lemma \[3243-237gndlsm\], we obtain that the second blow-up of an $s$-minimal graph which is flat from one side is necessarily a graph as well (see e.g. Lemmata 2.2 and 2.3 in [@2019arXiv190405393D] for the basic properties of the second blow-up). The precise result goes as follows:

\[BOAmdfiUAMP\] Let $u$ be an $s$-minimal graph in $(-2,2)^{n-1}\times(0,4)$. Assume that there exists $h>0$ such that $$\label{568-029-3984112}
{\mbox{$u=0$ in~$ (-2,2)^{n-1}\times(-h,0)$.}}$$ Let $E_{00}$ be the second blow-up of $E_u$, being $E_u$ defined in . Then: $$\begin{aligned}
\label{POS-E001}
&& {\mbox{either $E_{00}\cap\{x_n>0\}=\varnothing$,}}
\\ \label{POS-E002}&& {\mbox{or $E_{00}\cap\{x_n>0\}=\{x_n>0\}$,}}\\&&
{\mbox{or $E_{00}$
has a graphical structure, namely}}\nonumber\\&&\label{POS-E003}
{\mbox{there exists~$u_{00}:{\ensuremath{\mathbb{R}}}^n\to{\ensuremath{\mathbb{R}}}$ such that~$E_{00}=E_{u_{00}}$}.}\end{aligned}$$ Furthermore, if  holds true, then $$\label{POS-E004}
\lim_{x\to0}u_{00}(x)=0.$$

We suppose that $$\label{454562382023}
{\mbox{\eqref{POS-E001} and~\eqref{POS-E002} do not hold,}}$$ and we aim at showing that  and  are satisfied. We start by proving . For this, we first observe that $E_{00}$ has a generalized “hypographical” structure, that is $$\label{657865-349}
{\mbox{if~$(y,y_{n+1})\in({\ensuremath{\mathbb{R}}}^n\times{\ensuremath{\mathbb{R}}})\cap E_{00}$,
then~$(y,\tau)\in E_{00}$ for every~$\tau{\leqslant}y_{n+1}$.}}$$ Indeed, each rescaling of $E_u$ has such property, and 