Gaussian free field limit of the six-vertex height function throughout the regime a = b = 1, 0 < c <= 2

Theorem 1.1: for each $c\in(0,2]$ the balanced-torus plane law exists and $h(\cdot/\delta)$ converges to $\sigma(c)\Gamma$, $\sigma(c)^2=2/\arccos(1-c^2/2)=1/\arcsin(c/2)$, with Green kernel $-(2\pi)^{-1}\log|x-y|$, jointly on finite-energy mass-zero test measures, in $B^\alpha_{p,q}$ and $W^{\alpha,p}$ restriction spaces for $-1<\alpha<0$, and with Wick-type limits of increment moments. Priority: Duminil-Copin, Kozlowski, Lammers and Manolescu had proved this for $\sqrt3\le c\le2$. Not shown: other slopes or Gibbs states, anisotropic $a\ne b$, or the regime $c>2$.

Theorem 1.1: for each $c\in(0,2]$ the balanced-torus plane law exists and $h(\cdot/\delta)$ converges to $\sigma(c)\Gamma$, $\sigma(c)^2=2/\arccos(1-c^2/2)=1/\arcsin(c/2)$, with Green kernel $-(2\pi)^{-1}\log|x-y|$, jointly on finite-energy mass-zero test measures, in $B^\alpha_{p,q}$ and $W^{\alpha,p}$ restriction spaces for $-1<\alpha<0$, and with Wick-type limits of increment moments. Priority: Duminil-Copin, Kozlowski, Lammers and Manolescu had proved this for $\sqrt3\le c\le2$. Not shown: other slopes or Gibbs states, anisotropic $a\ne b$, or the regime $c>2$.