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\hypersetup{pdftitle={Closing the identification step in the GFF limit: Uniqueness of the zero-slope six-vertex phase for -1 <= Delta < 1/2},pdfauthor={Yan Ru Pei},pdfkeywords={six-vertex model, height function, gradient Gibbs measures, uniqueness, cluster swapping, Gaussian free field}}
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\title{Closing the identification step in the GFF limit:\\
Uniqueness of the zero-slope six-vertex phase for $-1\le\Delta<\frac12$}
\author{Yan Ru Pei\\[0.3em]\small\href{mailto:yanrpei@gmail.com}{\texttt{yanrpei@gmail.com}}}
\date{November 2026\\[0.3em]\small\href{https://doi.org/10.5281/zenodo.24117031}{doi:10.5281/zenodo.24117031}}

\begin{document}
\maketitle

\begin{abstract}
For the six-vertex model with $a=b=1$ and $c\in(1,2]$, that is $-1\le\Delta<\frac12$, we prove that the
plane measure, the limit of balanced torus measures, is the only translation-ergodic gradient Gibbs
measure of slope zero, and hence the unique translation-invariant minimiser of the specific free energy.
Both statements had been asserted by reference to Sheffield's uniqueness theory, which is proved for
nearest-neighbour potentials; we show that for $c\ne1$ the six-vertex weights, which couple faces across
the diagonals of each vertex, are not a nearest-neighbour potential in height coordinates. The proof
extends Sheffield's cluster swapping to these diagonal interactions by means of residual energies on the
diagonals, shows with the Lanford--Ruelle theorem for subshifts of finite type and a Burton--Keane
argument that two distinct ergodic zero-slope phases are gradients of Gibbs measures on heights, and
shows from the spin representation in the form given by Duminil-Copin, Kozlowski, Lammers and
Manolescu that the plane measure is not.

This supplies a step that we found missing in the chain of recent results on the Gaussian free field
limit of the six-vertex height function. An unrefereed manuscript of OpenAI states the full-plane limit
for every $c\in(0,2]$; for $\sqrt2\le c<2$ it identifies a state built from nearly balanced sectors with the
plane measure through the characterisation above. Our result gives that identification, given two
properties of the state stated in the manuscript. We also check that the manuscript and the work of
Averous, Duminil-Copin, He, Lammers and Manolescu concern the same measure, so that their convergence in
domains holds, for $c\in[1,2]$ and with constants depending on $\Delta$, wherever the manuscript's
full-plane theorem holds.
\end{abstract}

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\input{sec1-intro}
\input{sec2-gradient}
\input{sec3-swaps}
\input{sec4-fold}
\input{sec5-plane}
\input{sec6-proofs}
\input{sec7-domains}
\appendix
\input{appA}

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\end{document}