[proofplan]
The proof separates the rigidity and existence directions. For $n \leq 7$, we invoke the classical nonparametric Bernstein theorem in the sharp low-dimensional range: every entire $C^2$ solution of the minimal surface equation over $\mathbb{R}^n$ is affine. For $n \geq 8$, we start from the Bombieri-De Giorgi-Giusti non-affine entire smooth solution in dimension $8$ and extend it constantly in the remaining variables; the minimal surface equation is preserved because the added gradient components vanish and the divergence contains no new contribution.
[/proofplan]
[step:Apply the low-dimensional Bernstein theorem to force affine rigidity]
Assume $1 \leq n \leq 7$, and let
\begin{align*}
u: \mathbb{R}^n &\to \mathbb{R}
\end{align*}
be a function with $u \in C^2(\mathbb{R}^n)$ satisfying
\begin{align*}
\operatorname{div}\left(\frac{\nabla u}{\sqrt{1 + |\nabla u|^2}}\right) = 0
\end{align*}
on $\mathbb{R}^n$. The hypotheses match the classical nonparametric Bernstein theorem in dimensions $n \leq 7$, in the precise form asserting that every entire $C^2$ solution of the nonparametric minimal surface equation on $\mathbb{R}^n$ is affine for $1 \leq n \leq 7$: the domain is all of $\mathbb{R}^n$, the solution is $C^2$, the equation is the minimal surface equation for an entire graph, and the dimension lies in the low-dimensional range. Therefore $u$ is affine, meaning there exist $a \in \mathbb{R}^n$ and $b \in \mathbb{R}$ such that
\begin{align*}
u(x) = a \cdot x + b
\end{align*}
for every $x \in \mathbb{R}^n$.
[guided]
We prove the first assertion by using the sharp low-dimensional Bernstein theorem for entire minimal graphs, in the precise form that every entire $C^2$ solution of the nonparametric minimal surface equation on $\mathbb{R}^n$ is affine when $1 \leq n \leq 7$. The theorem applies to a map
\begin{align*}
u: \mathbb{R}^n &\to \mathbb{R}
\end{align*}
with $u \in C^2(\mathbb{R}^n)$ satisfying the minimal surface equation
\begin{align*}
\operatorname{div}\left(\frac{\nabla u}{\sqrt{1 + |\nabla u|^2}}\right) = 0
\end{align*}
on the whole domain $\mathbb{R}^n$, provided $1 \leq n \leq 7$. We verify these hypotheses one by one. The theorem statement gives that $u$ is $C^2$ on $\mathbb{R}^n$; it also gives that $u$ solves the minimal surface equation on $\mathbb{R}^n$; and in this step we are assuming $n \leq 7$. Thus the nonparametric Bernstein theorem yields that the graph of $u$ is a hyperplane. Equivalently, there are constants $a \in \mathbb{R}^n$ and $b \in \mathbb{R}$ such that
\begin{align*}
u(x) = a \cdot x + b
\end{align*}
for all $x \in \mathbb{R}^n$. This is precisely the assertion that $u$ is affine.
[/guided]
[/step]
[step:Use the Bombieri-De Giorgi-Giusti example in dimension $8$]
The Bombieri-De Giorgi-Giusti existence theorem gives a smooth non-affine entire graph in dimension $8$, in the precise form that there is a function $w \in C^\infty(\mathbb{R}^8)$ with domain $\mathbb{R}^8$ and codomain $\mathbb{R}$ satisfying
\begin{align*}
\operatorname{div}_{\mathbb{R}^8}\left(\frac{\nabla w}{\sqrt{1 + |\nabla w|^2}}\right) = 0
\end{align*}
on $\mathbb{R}^8$. This proves the existence assertion when $n = 8$.
[/step]
[step:Extend the dimension eight solution by ignoring the extra variables]
Let $n \geq 8$. Write each point $x \in \mathbb{R}^n$ as $x = (y,z)$ with $y \in \mathbb{R}^8$ and $z \in \mathbb{R}^{n-8}$. Define $v: \mathbb{R}^n \to \mathbb{R}$ by setting $v(y,z) = w(y)$ for every $(y,z) \in \mathbb{R}^8 \times \mathbb{R}^{n-8}$.
Since $w \in C^\infty(\mathbb{R}^8)$, the function $v$ belongs to $C^\infty(\mathbb{R}^n)$. Its gradient is
\begin{align*}
\nabla v(y,z) = (\nabla w(y),0),
\end{align*}
where $0 \in \mathbb{R}^{n-8}$ denotes the zero vector. Therefore
\begin{align*}
\frac{\nabla v(y,z)}{\sqrt{1 + |\nabla v(y,z)|^2}}
= \left(\frac{\nabla w(y)}{\sqrt{1 + |\nabla w(y)|^2}},0\right).
\end{align*}
Taking divergence in $\mathbb{R}^n$ gives
\begin{align*}
\operatorname{div}_{\mathbb{R}^n}\left(\frac{\nabla v}{\sqrt{1 + |\nabla v|^2}}\right)(y,z)
= \operatorname{div}_{\mathbb{R}^8}\left(\frac{\nabla w}{\sqrt{1 + |\nabla w|^2}}\right)(y)
= 0.
\end{align*}
Thus $v$ is an entire smooth solution of the minimal surface equation on $\mathbb{R}^n$. Since $w$ is non-affine, $v$ is non-affine: if $v$ were affine on $\mathbb{R}^n$, then its restriction to the affine subspace $\mathbb{R}^8 \times \{0\}$ would make $w$ affine on $\mathbb{R}^8$, contradicting the Bombieri-De Giorgi-Giusti example.
[/step]
[step:Combine the two directions to obtain the threshold]
The first step proves affine rigidity for every $1 \leq n \leq 7$. The preceding two steps prove the existence of non-affine entire smooth solutions for every $n \geq 8$. Hence dimension $8$ is exactly the threshold at which the higher-dimensional Bernstein conclusion fails.
[/step]
