\section{Swaps and the case distinction}\label{sec:swaps}

This section is the six-vertex form of Sheffield's cluster swapping \cite[Sections~8.1--8.5]{Sheffield}.
Subsection~\ref{subsec:localcomb} records what the difference of two height functions looks like around a
vertex. Subsection~\ref{subsec:FKswaps} defines the clusters that can be swapped and shows that swapping
them preserves the joint law (Figure~\ref{fig:swap}); the new ingredient is the role of the saddles.
Subsections~\ref{subsec:pairs} and~\ref{subsec:cases} turn this into identities for pairs of Gibbs
measures and use them to sort the possible patterns of infinite clusters: all but one pattern either
cannot occur or makes both measures smooth. The remaining pattern is treated in Section~\ref{sec:fold}.

\subsection{Local combinatorics}\label{subsec:localcomb}

\begin{lemma}\label{lem:lattice}
If $\psi,\psi'\in\Omega$, then $\psi\vee\psi'$ and $\psi\wedge\psi'$ are in $\Omega$.
\end{lemma}

\begin{proof}
For $\alpha,\beta\in\{0,1\}$, $\max(s+\alpha,t+\beta)-\max(s,t)\in\{0,1\}$, and likewise for the minimum.
\end{proof}

\begin{lemma}[Difference of two admissible functions]\label{lem:diff}\leavevmode
Let $\psi_1,\psi_2\in\Omega$ and $D=\psi_2-\psi_1$.
\begin{enumerate}
\item[(i)] $|D(x)-D(y)|\le1$ for NN-adjacent faces, and $\le2$ for the two faces of a diagonal.
\item[(ii)] Let $d=\{x,y\}$ be a diagonal, $\ell\in\Z$, and suppose $D=\ell$ on $d^\perp$. If
$D(x)=D(y)=\ell+s$ with $s\in\{+1,-1\}$ ($d$ is an \emph{$\ell$-saddle of sign $s$}), then
$\mathsf a(\psi_1;d)=\mathsf a(\psi_2;d)=1$. If $\{D(x),D(y)\}=\{\ell+1,\ell-1\}$ ($d$ is \emph{$\ell$-monotone}), then
$\mathsf a(\psi_1;d)=\mathsf a(\psi_2;d)=0$.
\end{enumerate}
\end{lemma}

\begin{proof}
(i) is immediate. (ii) Go around the block in the cyclic order $x,z,y,z'$ with $d^\perp=\{z,z'\}$. Let
$\varepsilon_1,\dots,\varepsilon_4\in\{\pm1\}$ be the steps of $h_1=2\psi_1-x_1-x_2$; the steps of $h_2$ are
$\varepsilon_i+2\delta_i\in\{\pm1\}$, where the $\delta_i$ are the steps of $D$. For a saddle,
$\delta=(-s,s,-s,s)$, which forces $\varepsilon=(s,-s,s,-s)$: both $h_1$ and $h_2$ alternate, so both are
equal across $d$. For the monotone pattern $D=(\ell+1,\ell,\ell-1,\ell)$ along $x,z,y,z'$,
$\delta=(-1,-1,1,1)$, which forces $\varepsilon=(1,1,-1,-1)$, so $h_1(y)=h_1(x)+2$ and $h_2(y)=h_2(x)-2$.
\end{proof}

For $\ell\in\Z$ and $A\subset F$ let $s^\ell_A(\psi_1,\psi_2)=(\psi_1',\psi_2')$ be given by
$(\psi_1',\psi_2')(x)=(\psi_2(x)-\ell,\psi_1(x)+\ell)$ for $x\in A$ and $(\psi_1',\psi_2')(x)=(\psi_1,\psi_2)(x)$
otherwise. Then $D'-\ell=-(D-\ell)$ on $A$, and $s^\ell_A$ acts trivially on faces where $D=\ell$. Call $A$
\emph{$\ell$-separated} if there is no NN-adjacent pair $x\in A$, $y\notin A$ with $D(x)\ne\ell\ne D(y)$.

\begin{lemma}[Admissibility of a swap]\label{lem:swapadm}\leavevmode
\begin{enumerate}
\item[(i)] $s^\ell_A(\psi_1,\psi_2)\in\Omega\times\Omega$ if and only if $A$ is $\ell$-separated.
\item[(ii)] If $A$ is $\ell$-separated, then for every diagonal $d$ the number
$\chi_d:=[\mathsf a(\psi_1';d)+\mathsf a(\psi_2';d)]-[\mathsf a(\psi_1;d)+\mathsf a(\psi_2;d)]$ equals $-2$ if $d$ is an $\ell$-saddle with
exactly one face in $A$, $+2$ if $d$ is $\ell$-monotone with exactly one face in $A$, and $0$
otherwise.
\end{enumerate}
\end{lemma}

\begin{proof}
Write $D_\ell=D-\ell$. (i) Let $u\in A$, $v\notin A$ be NN-adjacent with $v=u+e_i$ (the case $v=u-e_i$
is symmetric), and $\eta=\psi_1(v)-\psi_1(u)\in\{0,1\}$. Then $\psi_1'(v)-\psi_1'(u)=\eta-D_\ell(u)$ and
$\psi_2'(v)-\psi_2'(u)=\eta+D_\ell(v)$, while $\psi_2(v)-\psi_2(u)=\eta+D_\ell(v)-D_\ell(u)\in\{0,1\}$. If
$D_\ell(u)=0$ or $D_\ell(v)=0$, the new increments are the old ones, in the same or the exchanged order.
If both are nonzero, they have the same sign by Lemma~\ref{lem:diff}(i), so the two new increments
differ by at least $2$ and cannot both lie in $\{0,1\}$. Pairs inside $A$ or inside its complement keep
their increments up to exchange.
(ii) If both faces of $d$ are in $A$, the copies are exchanged on $d$ up to the common shift $\ell$; if
neither is, nothing changes; in both cases $\chi_d=0$. Let $x\in A$, $y\notin A$. If $D_\ell(x)=0$
nothing changes at $x$, and if $D_\ell(y)=0$ then $\psi_1'=\psi_2-\ell$ and $\psi_2'=\psi_1+\ell$ on both faces
of $d$; in both cases $\chi_d=0$. If both are nonzero, let $z\in d^\perp$. If $D_\ell(z)\ne0$, separation
applied to the NN pairs $\{x,z\}$ and $\{z,y\}$ puts $z$ and then $y$ in $A$, a contradiction. So
$D_\ell=0$ on $d^\perp$ and $|D_\ell(x)|=|D_\ell(y)|=1$: $d$ is an $\ell$-saddle or $\ell$-monotone, and the swap
at $x$ turns one into the other. Lemma~\ref{lem:diff}(ii) gives $\mathsf a$-sums $2$ and $0$ respectively.
\end{proof}

\subsection{FK clusters and swaps}\label{subsec:FKswaps}

For a triple $\om$ and $\ell\in\Z$ let $G_\ell(\om)$ be the graph whose vertices are the faces with
$D\ne\ell$, with an edge between NN-adjacent vertices (they then have the same sign of $D-\ell$), and
with an edge, an \emph{open bridge}, for every $\ell$-saddle $d$ with $n_d=0$. Its components are the
\emph{FK clusters at level $\ell$}, plus or minus according to the sign of $D-\ell$ on them. An $\ell$-saddle
with $n_d=0$ is \emph{open} and one with $n_d\ge1$ is \emph{closed}: by Lemma~\ref{lem:FKswap} below, a swap can
separate the two faces of a closed saddle, at the price of lowering $n_d$ by one, but not those of an open
one.
For a nearest-neighbour hard-constraint potential, such as the six-vertex model at $c=1$, Sheffield's
open clusters \cite[Section~8.1.3]{Sheffield} are the components of $\{D\ne\ell\}$; the open bridges are what
the diagonal interaction adds.

\input{fig-swap}

\begin{lemma}[FK swap]\label{lem:FKswap}
Let $A$ be a union of FK clusters at level $\ell$ of $\om$, and define
$\Sigma^\ell_A(\om):=(s^\ell_A(\psi_1,\psi_2),n')$ with $n'_d:=n_d+\chi_d/2$. Then:
\begin{enumerate}
\item[(i)] $\Sigma^\ell_A(\om)$ is a triple, that is $n'\ge0$, and $\tau_d$ is unchanged for every $d$;
\item[(ii)] $G_\ell(\Sigma^\ell_A\om)=G_\ell(\om)$ as graphs, with the sign reversed exactly on $A$;
\item[(iii)] $\Sigma^\ell_A$ is an involution on the set of triples for which $A$ is a union of FK
clusters at level $\ell$.
\end{enumerate}
\end{lemma}

\begin{proof}
$A$ is $\ell$-separated, so Lemma~\ref{lem:swapadm} applies. (i) $\chi_d=-2$ only at $\ell$-saddles with
exactly one face in $A$; such a saddle is not an open bridge, so $n_d\ge1$ and $n'_d=n_d-1\ge0$. The
definition of $n'$ makes $\tau$ invariant. (ii) The vertex set and the NN edges depend only on
$\{D=\ell\}$. An open bridge of $\om$ has both faces in $A$ or both outside, stays an $\ell$-saddle and
keeps $n=0$. A saddle with exactly one face in $A$ becomes monotone. A monotone diagonal with exactly one
face in $A$ becomes a saddle with $n'=n+1\ge1$, a closed one. All other saddles keep their $n$. (iii)
follows from (ii) and the formula for $n'$.
\end{proof}

The next lemma is the six-vertex form of \cite[Lemmas~8.1.5 and~8.1.6]{Sheffield}.

\begin{lemma}[Finite swaps preserve the kernel]\label{lem:finiteswap}
Let $\Lambda$ be finite, $\ell\in\Z$, and let $\om\mapsto A(\om)$ be a rule that selects a union of FK
clusters at level $\ell$ contained in $\Lambda$ and depends only on $G_\ell(\om)$ and on the restriction
of $\om$ to the complement of $(\Lambda,E(\Lambda))$. Then $\Theta(\om):=\Sigma^\ell_{A(\om)}(\om)$ is an
involution that changes $\om$ only on $(\Lambda,E(\Lambda))$, and
$\gb_\Lambda(\Theta^{-1}(\cdot)\mid\om)=\gb_\Lambda(\cdot\mid\om)$ for every $\om$.
\end{lemma}

\begin{proof}
By Lemma~\ref{lem:FKswap}, $A(\Theta\om)=A(\om)$, $\Theta$ is an involution of the set of triples that
agree with $\om$ off $(\Lambda,E(\Lambda))$, and it preserves \eqref{eq:density}.
\end{proof}

We use two rules: ``the FK cluster at level $\ell$ of a given face, if it is contained in $\Lambda$'',
and ``all FK clusters at level $\ell$ contained in $\Lambda$''.

\begin{lemma}[Monotonicity in the level]\label{lem:levels}
If $C$ is a plus FK cluster at level $\ell$ and $\ell'<\ell$, then $C$ is contained in one plus FK
cluster at level $\ell'$. If $C$ is a minus FK cluster at level $\ell$ and $\ell'>\ell$, then $C$ is
contained in one minus FK cluster at level $\ell'$.
\end{lemma}

\begin{proof}
Consider the minus case. Faces of $C$ have $D<\ell<\ell'$, and NN edges of $C$ are NN edges at level
$\ell'$. An open bridge $\{u,v\}$ of $C$ has $D=\ell$ on $d^\perp=\{z,z'\}$, and $D(z)=\ell<\ell'$, so
$u,z,v$ is an NN path in $G_{\ell'}$.
\end{proof}

\subsection{Pairs of Gibbs measures}\label{subsec:pairs}

Lemmas~\ref{lem:MM}--\ref{lem:exch} and Proposition~\ref{prop:onesided} below are the six-vertex forms of the
corresponding steps of \cite[Sections~8.4--8.5]{Sheffield}. Sheffield fixes the height of one copy; we
instead average over all relative heights of the two copies with the $\sigma$-finite measure $\MM$, which
makes the levels interchangeable.

In this subsection $\mu_1,\mu_2$ are translation-invariant gradient Gibbs measures and
$\mub=\mu_1\otimes\mu_2\otimes\pi$. For a gradient triple and $j\in\Z$ let $\om^{(j)}\in\Om$ be the triple
with $D(0)=j$, and let
\[
\MM:=\sum_{j\in\Z}\operatorname{law}(\om^{(j)}),
\]
a $\sigma$-finite measure on $\Om$ whose restriction to $\{D(0)=j\}$ is a copy of $\mub$ for every $j$.
Let $V_t$ be the vertical shift $(\psi_1,\psi_2,n)\mapsto(\psi_1,\psi_2+t,n)$. An event $E\subset\Om$ is
\emph{robust} if it is invariant under every $V_t$, under translations, and under every change of the
triple on finitely many faces and diagonals.

\begin{lemma}[The measure $\MM$]\label{lem:MM}\leavevmode
\begin{enumerate}
\item[(i)] $\MM$ is invariant under translations and under every $V_t$; for a $V$-invariant event $E$,
$\MM(E\cap\{D(0)=j\})=\mub(E)$ for every $j$.
\item[(ii)] $\MM\gb_\Lambda=\MM$ for every finite $\Lambda$.
\item[(iii)] If $\Theta:\Om\to\Om$ is measurable and $\gb_\Lambda(\Theta^{-1}(\cdot)\mid\om)=\gb_\Lambda(\cdot\mid\om)$
for every $\om$ and some finite $\Lambda$, then $\MM\circ\Theta^{-1}=\MM$.
\end{enumerate}
\end{lemma}

\begin{proof}
(i) $\MM=\sum_j\operatorname{law}(\text{lift with }D(x)=j)$ for any face $x$. (ii) Fix a face $b\notin\Lambda$
and write $\MM=\sum_j\MM_j$ with $\MM_j$ the law of the lift with $D(b)=j$. Resampling in $\Lambda$ does
not change $D(b)$, and $\mub$ is invariant under $\gb_\Lambda$. (iii)
$\MM(f\circ\Theta)=\int\MM(\dd\om)\,\gb_\Lambda(f\circ\Theta\mid\om)=\int\MM(\dd\om)\,\gb_\Lambda(f\mid\om)=\MM(f)$
for $f\ge0$.
\end{proof}

\begin{lemma}[Swap identity]\label{lem:swapid}
Let $\ell\in\Z$, $k\ge1$, $E$ robust, and let $C_\ell(0)$ be the FK cluster at level $\ell$ of the face
$0$. Then
\[
\MM\bigl(D(0)=\ell+k,\ |C_\ell(0)|<\infty,\ E\bigr)=\MM\bigl(D(0)=\ell-k,\ |C_\ell(0)|<\infty,\ E\bigr).
\]
\end{lemma}

\begin{proof}
Let $\Theta_N$ swap $C_\ell(0)$ at level $\ell$ if it is contained in $\Lambda_N:=[-N,N]^2$ and do nothing
otherwise. By Lemmas~\ref{lem:finiteswap} and~\ref{lem:MM}(iii), $\MM\circ\Theta_N^{-1}=\MM$. The map
$\Theta_N$ is an involution, preserves $E$ and the event $\{C_\ell(0)\subset\Lambda_N\}$
(Lemma~\ref{lem:FKswap}(ii)), and maps $D(0)-\ell$ to its negative on that event. Hence the identity
holds with $\{C_\ell(0)\subset\Lambda_N\}$ in place of $\{|C_\ell(0)|<\infty\}$; both sides are at most
$1$, and we let $N\to\infty$.
\end{proof}

\begin{lemma}[No infinite cluster implies exchangeability]\label{lem:exch}
Let $E$ be robust and suppose that $\MM$-almost everywhere on $E$ all FK clusters at level $0$ are
finite. Then $\mub(g\circ\mathrm{Ex};E)=\mub(g;E)$ for every bounded local function $g$ of the gradient pair, where
$\mathrm{Ex}$ exchanges the two copies. If moreover $\mu_1,\mu_2$ are ergodic and $\mub(E)>0$, then $\mu_1=\mu_2$.
\end{lemma}

\begin{proof}
Let $\Theta_N$ swap all FK clusters at level $0$ contained in $\Lambda_N$; then $\MM\circ\Theta_N^{-1}=\MM$ and
$\Theta_N$ preserves $E$. Fix $j_0$ and put $f:=g\cdot\ind{D(0)=j_0}$, so that $\MM(f\circ\Theta_N;E)=\MM(f;E)$.
For $\MM$-almost every $\om\in E$, the finitely many clusters that meet the window of $g$ or contain $0$
are finite, so for large $N$ the triple $\Theta_N(\om)$ coincides on that window with the full exchange
$(\psi_2,\psi_1,n)$; when every cluster is swapped, no diagonal has exactly one swapped face among two
faces with $D\ne0$, so $n$ is unchanged. Since $|f\circ\Theta_N|\le\|g\|\,\ind{|D(0)|=|j_0|}$, which is
$\MM$-integrable, dominated convergence gives $\MM(g\circ\mathrm{Ex}\cdot\ind{D(0)=-j_0};E)=\MM(g\cdot\ind{D(0)=j_0};E)$,
which by Lemma~\ref{lem:MM}(i) is the first claim. For the second take $g=g_1(\nabla\psi_1)$. Then
$\mub(g_1(\nabla\psi_1);E)=\int\mu_1(\dd\psi_1)\,g_1(\psi_1)\,u(\psi_1)$ with $u(\psi_1)$ the
$(\mu_2\otimes\pi)$-measure of the section of $E$, a translation-invariant function, hence $\mu_1$-almost
surely equal to $\mub(E)$; likewise $\mub(g_1(\nabla\psi_2);E)=\mub(E)\,\mu_2(g_1)$. So
$\mu_1(g_1)=\mu_2(g_1)$.
\end{proof}

\subsection{The case distinction}\label{subsec:cases}

The classification below follows \cite[Section~8.5]{Sheffield}; the measure $\MM$ of the previous
subsection, which makes the levels exchangeable, lets us state it without fixing a height.

For $\om\in\Om$ let $I^+(\om)$ (resp.\ $I^-(\om)$) be the set of levels $\ell$ at which $G_\ell(\om)$ has
an infinite plus (resp.\ minus) cluster. By Lemma~\ref{lem:levels}, $I^+$ is a down-set and $I^-$ an
up-set of $\Z$. Put
\[
\begin{aligned}
E_\emptyset&:=\{I^+=\emptyset=I^-\}, & E_{\mathrm{top}}&:=\{I^+\ne\emptyset,\ \sup I^+<\infty\}, &
E_{\mathrm{bot}}&:=\{I^-\ne\emptyset,\ \inf I^->-\infty\},\\
E_+&:=\{I^+=\Z,\ I^-=\emptyset\}, & E_-&:=\{I^+=\emptyset,\ I^-=\Z\}, & E_{\Z\Z}&:=\{I^+=\Z=I^-\}.
\end{aligned}
\]
These six events cover $\Om$.

\begin{lemma}\label{lem:robust}
If two triples agree off finitely many faces and diagonals, they have the same $I^+$ and $I^-$. In
particular the six events above are robust.
\end{lemma}

\begin{proof}
Let $S$ be the union of the blocks that contain a changed face or a changed diagonal, a finite set. Let
$C$ be an infinite plus cluster of $G_\ell(\om)$. The set $C\setminus S$ has finitely many components in
$G_\ell(\om)$, each containing a face adjacent to $S$, so one of them, $K$, is infinite. An edge of
$G_\ell(\om)$ between two faces outside $S$ is an edge of $G_\ell(\om')$: for an NN edge the two faces are
unchanged; for a bridge $d$, the block of $d$ contains a face outside $S$, hence no changed face, and $d$
is not a changed diagonal. So $K$ lies in an infinite plus cluster of $G_\ell(\om')$. Under $V_t$ and
translations, $I^\pm$ are shifted by $t$, respectively unchanged.
\end{proof}

\begin{proposition}[One-sided cases]\label{prop:onesided}
$\mub(E_+)=\mub(E_-)=0$.
\end{proposition}

\begin{proof}
On $E_+$ every minus cluster at every level is finite. By Lemmas~\ref{lem:swapid} and~\ref{lem:MM}(i),
for all $\ell$ and $k\ge1$,
\[
\begin{aligned}
\MM\bigl(D(0)=\ell+k,\ |C_\ell(0)|=\infty,\ E_+\bigr)
&=\mub(E_+)-\MM\bigl(D(0)=\ell+k,\ |C_\ell(0)|<\infty,\ E_+\bigr)\\
&=\mub(E_+)-\MM\bigl(D(0)=\ell-k,\ |C_\ell(0)|<\infty,\ E_+\bigr)\\
&=\mub(E_+)-\mub(E_+)=0 .
\end{aligned}
\]
So $\MM$-almost everywhere on $E_+$ the face $0$, and by translation invariance every face, lies in no
infinite plus cluster at any level. This contradicts $I^+=\Z$ unless $\mub(E_+)=0$. The case of $E_-$
is symmetric.
\end{proof}

A height that is determined by the pair up to finite modifications, in the way made precise below, makes
a measure smooth. This is \cite[Lemma~8.4.4]{Sheffield} in our setting.

\begin{lemma}[A canonical lift makes a marginal smooth]\label{lem:canonical}
Let $\varpi$ be a probability measure on a space $\mathcal U$ carrying a measurable action of $\Z^2$ that
preserves $\varpi$. Let $\mu_2$ be an ergodic gradient Gibbs measure, $\mu_1$ a translation-invariant
gradient measure, and $Q_0:=\mu_1\otimes\mu_2\otimes\varpi$. Let $\mathcal E$ be an event with
$Q_0(\mathcal E)>0$ and let $\mathfrak d$ be a measurable map that assigns to each
$(\nabla\psi_1,\nabla\psi_2,u)\in\mathcal E$ a function $\mathfrak d:F\to\Z$ with
$\nabla\mathfrak d=\nabla\psi_2-\nabla\psi_1$, such that
\begin{enumerate}
\item[(a)] $\mathcal E$ is invariant under the diagonal action of $\Z^2$, and
\item[(b)] if $(\nabla\psi_1,\nabla\psi_2,u)\in\mathcal E$ and $\tilde\psi_2\in\Omega$ differs from $\psi_2$ on
finitely many faces, then $(\nabla\psi_1,\nabla\tilde\psi_2,u)\in\mathcal E$ and
$\mathfrak d(\nabla\psi_1,\nabla\tilde\psi_2,u)=\mathfrak d(\nabla\psi_1,\nabla\psi_2,u)+(\tilde\psi_2-\psi_2)$.
\end{enumerate}
Then $\mu_2$ is smooth. The same holds with the roles of the two copies exchanged.
\end{lemma}

\begin{proof}
Let $Q:=Q_0(\cdot\mid\mathcal E)$. Its second marginal has density
$(\mu_1\otimes\varpi)(\text{section of }\mathcal E)/Q_0(\mathcal E)$ with respect to $\mu_2$; by~(a) this
density is translation-invariant, hence constant, so the second marginal is $\mu_2$. Conditionally on
$(\nabla\psi_1,u)$, the law of $\nabla\psi_2$ under $Q$ is $\mu_2$ conditioned on the section of
$\mathcal E$, which by~(b) is invariant under finite modifications; conditioning on such an event
preserves the DLR equations. Let $\psi_1^0$ be the representative of $\nabla\psi_1$ with $\psi_1^0(0)=0$
and $\hat\psi_2:=\psi_1^0+\mathfrak d$, an admissible function with gradient $\nabla\psi_2$. By~(b),
replacing $\psi_2$ by $\tilde\psi_2\sim\gamma_\Lambda(\cdot\mid\psi_2)$ replaces $\hat\psi_2$ by
$\hat\psi_2+(\tilde\psi_2-\psi_2)$, whose conditional law is $\gamma_\Lambda(\cdot\mid\hat\psi_2)$ because the
specification commutes with adding constants. Hence $\E_Q[f(\hat\psi_2)]=\E_Q[\gamma_\Lambda(f\mid\hat\psi_2)]$ for
bounded measurable $f$: the law of $\hat\psi_2$ is a Gibbs measure on heights with gradient law $\mu_2$.
\end{proof}

\begin{proposition}[A top or bottom level]\label{prop:topbot}
Let $\mu_1,\mu_2$ be ergodic. If $\mub(E_{\mathrm{top}}\cup E_{\mathrm{bot}})>0$, then $\mu_1$ and $\mu_2$
are both smooth.
\end{proposition}

\begin{proof}
On $E_{\mathrm{top}}$ let $\mathfrak d:=D-\max I^+$, the lift of $\nabla\psi_2-\nabla\psi_1$ for which
$\max I^+=0$; on $E_{\mathrm{bot}}$ use $\min I^-$ instead. Condition~(a) of Lemma~\ref{lem:canonical}
holds by robustness. For~(b), changing $\psi_2$ on a finite set changes the honest difference $D$ by
$\tilde\psi_2-\psi_2$ and does not change $I^+$ (Lemma~\ref{lem:robust}). Apply
Lemma~\ref{lem:canonical} with $\mathcal U$ the space of $n$ and $\varpi=\pi$, once as stated and once
with the copies exchanged (using $-\mathfrak d$).
\end{proof}

\begin{corollary}\label{cor:cases}
Let $\mu_1\ne\mu_2$ be ergodic translation-invariant gradient Gibbs measures. Then either both are
smooth, or $\mub(E_{\Z\Z})=1$.
\end{corollary}

\begin{proof}
$\mub(E_+\cup E_-)=0$ by Proposition~\ref{prop:onesided}; $\mub(E_\emptyset)=0$ by
Lemma~\ref{lem:exch}, since on $E_\emptyset$ all clusters at all levels are finite; and if
$\mub(E_{\mathrm{top}}\cup E_{\mathrm{bot}})>0$, both measures are smooth by
Proposition~\ref{prop:topbot}.
\end{proof}