Gaussian free field limits throughout the balanced six-vertex regime

The balanced square-lattice six-vertex height field with $a=b=1$ and $0\lt c\le2$ converges to a Gaussian free field, including at the endpoint c = 2. The plane state is defined by balanced-torus limits. For unit height increments and Green kernel $-(2\pi)^{-1}\log|x-y|$, the exact variance multiplier is $1/\arcsin(c/2)$.