\section{Proofs of Theorem~\ref*{thm:main} and Corollaries~\ref*{cor:minimiser} and~\ref*{cor:ID}}\label{sec:proofs}

Theorem~\ref{thm:main} is now a combination of Theorems~\ref{thm:smooth} and~\ref{thm:plane}. The variational
characterisation adds the ergodic decomposition and the strict convexity of the surface tension, and the
identification is the argument of \cite[Section~4]{OAI225} with the characterisation in place of the remark
it cites. Corollary~\ref{cor:domains} is proved in Section~\ref{sec:domains}.

\begin{proof}[Proof of Theorem~\ref{thm:main}]
Let $1<c\le2$ and let $\mu$ be a translation-ergodic gradient Gibbs measure of slope $0$. By
Theorem~\ref{thm:plane}, $\Pp_c$ is also one. By Lemma~\ref{lem:SFE}(iii) both are equilibrium states. If
$\mu\ne\Pp_c$, Theorem~\ref{thm:smooth} shows that $\Pp_c$ is smooth, which contradicts
Theorem~\ref{thm:plane}.
\end{proof}

\begin{remark}[The case $c=1$]\label{rem:c1}
At $c=1$ the model is a Lipschitz simply attractive potential (Proposition~\ref{prop:notNN}), so
\cite[Theorem~9.1.1]{Sheffield} gives at most one \emph{minimal gradient phase} of slope $0$, an interior
slope ($(\frac12,\frac12)$ in the coordinates $\psi$, inside $U_\Phi=(0,1)^2$), for the
translation group, that is, one ergodic gradient measure of slope $0$ that minimises the specific free
energy among measures of its slope. By \cite[Theorem~6.3.1]{Sheffield} every ergodic gradient Gibbs measure of slope $0$ is
such a phase, and $\Pp_1$ is an ergodic gradient Gibbs measure of slope $0$ by Theorem~\ref{thm:plane}. So the conclusion of
Theorem~\ref{thm:main} holds at $c=1$ as well.
\end{remark}

\begin{proof}[Proof of Corollary~\ref{cor:minimiser}]
By Theorem~\ref{thm:plane} and Lemma~\ref{lem:SFE}, $\Pp_c$ minimises the specific free energy. Let $\mu$ be a
translation-invariant measure that does. The specific free energy is affine on translation-invariant
measures and respects the ergodic decomposition \cite[Lemma~3.3.1 and Theorem~3.3.2]{Sheffield}, so
almost every ergodic component $\mu'$ of $\mu$ minimises it, that is $\SFE(\mu')=\sigma_c(0)$. By definition
of the surface tension, $\SFE(\mu')\ge\sigma_c(S(\mu'))$, where $S(\mu')$ is the slope.

The slopes of finite surface tension form the square $[-1,1]^2$ in height coordinates: random
height functions $h(x)=f_1(x_1)+f_2(x_2)$, with $f_i$ having independent unit steps of mean $u_i$,
realise every slope $u\in[-1,1]^2$ by translation-invariant measures, and every function on the faces with
unit steps between NN-adjacent faces is the height function of an arrow configuration. So $0$ is an
interior point. For $c\ge1$ the surface tension is strictly convex on the interior of the set of slopes
\cite[Theorem~13.16]{LammersTassy}; the condition there, $c_+c_-\ge\max\{a_+a_-,b_+b_-\}$ in their
notation, reads $c\ge1$ when $a=b=1$, and their surface tension agrees with $\sigma_c$ up to an additive
constant and the change of coordinates of Section~\ref{sec:gradient}, both being defined from the
energy $-\log w$ per vertex. The function $\sigma_c$ is even (Lemma~\ref{lem:SFE}). If it were minimal at
some $u\ne0$, it would be constant on the segment from $-u$ to $u$, contradicting strict convexity near
the interior point $0$. So $\sigma_c(u)>\sigma_c(0)$ for $u\ne0$, and $S(\mu')=0$.

A translation-invariant measure of finite slope that minimises the specific free energy among measures
of its slope is a gradient Gibbs measure \cite[Theorem~2.5.2]{Sheffield}, which applies by
Proposition~\ref{prop:LSAP}. So almost every ergodic component of $\mu$ is a translation-ergodic gradient
Gibbs measure of slope $0$, hence equal to $\Pp_c$ by Theorem~\ref{thm:main}. Therefore $\mu=\Pp_c$.
\end{proof}

\begin{proof}[Proof of Corollary~\ref{cor:ID}]
The sector event is invariant under the translations of the torus, so each $\Pp^{(n_L)}_{N,L}$ is, and so is
the local limit $\rho_c$, which exists by (O1). The functional minimised in (O2),
$\mu\mapsto-\mathsf h(\mu)-\E_\mu\log w_0$ over translation-invariant measures, is $-P(\mu)$, which differs
from $\SFE$ by a constant (Lemma~\ref{lem:SFE}). So $\rho_c$ is a translation-invariant minimiser of the
specific free energy, and $\rho_c=\Pp_c$ by Corollary~\ref{cor:minimiser}.
\end{proof}

We have used (O1) and (O2) as stated in \cite{OAI225} and have not checked their proofs. The Gibbs property of $\rho_c$, which is not needed here,
also follows directly: for a box $B$ and $N$ large, the sector event is determined by any column that
does not meet $B$, so conditionally on the configuration outside $B$ the measure $\Pp^{(n)}_{N,L}$ restricted
to $B$ is the six-vertex specification, and the property passes to the local limit.