\begin{figure}[tp]
\centering
\newcommand*{\rmloc}[1]{\\[1pt]{\tiny\color{black!70}#1}}%
\begin{NoHyper}%
\begin{tikzpicture}[
  x=1cm, y=1cm, font=\scriptsize,
  >={Stealth[length=1.5mm,width=1.1mm]},
  thmlook/.style={fill=black!80, draw=black, line width=0.8pt},
  ourslook/.style={rounded corners=1.3mm, fill=black!14, draw=black, line width=0.6pt},
  repairlook/.style={rounded corners=1.3mm, fill=black!14, draw=black, line width=1.1pt, dash pattern=on 3pt off 1.6pt},
  knownlook/.style={fill=white, draw=black, line width=0.4pt},
  unreflook/.style={fill=white, draw=black, line width=0.6pt, dash pattern=on 2pt off 1.5pt},
  adaptlook/.style={fill=white, draw=black, line width=0.45pt,
    path picture={\draw[black, line width=0.45pt]
      ([shift={(1.6pt,1.6pt)}]path picture bounding box.south west) rectangle
      ([shift={(-1.6pt,-1.6pt)}]path picture bounding box.north east);}},
  box/.style={rectangle, align=flush center, text width=3.35cm, inner xsep=1.6pt, inner ysep=2pt, minimum height=0.95cm},
  wide/.style={text width=4.3cm},
  thm/.style={box, wide, thmlook, text=white, font=\scriptsize\bfseries},
  ours/.style={box, ourslook},
  repair/.style={box, repairlook},
  known/.style={box, knownlook},
  unref/.style={box, unreflook},
  adapt/.style={box, adaptlook, inner xsep=2.8pt, inner ysep=3pt},
  sw/.style={rectangle, minimum width=0.55cm, minimum height=0.32cm, inner sep=0pt},
  lg/.style={anchor=west, inner sep=2pt, text height=1.6ex, text depth=0.3ex},
  use/.style={->, line width=0.5pt, draw=black!80},
  rel/.style={use, dashed},
]
% central chain
\node[ours, wide]  (DOM) at (7.3,0)    {Gaussian free field in domains\\ for $-1\le\Delta\le\frac12$, conditional\rmloc{Corollary~\ref{cor:domains}}};
\node[unref, wide] (OAI) at (7.3,-1.75) {Full-plane Gaussian free field\\ limit for every $c\in(0,2]$\rmloc{\cite[Theorem~1.1(3)]{OAI225}, unrefereed}};
\node[repair, wide](ID)  at (7.3,-3.5)  {The identification $\rho_c=\Pp_c$\\ for $\sqrt2\le c<2$: the missing step\rmloc{Corollary~\ref{cor:ID}}};
\node[ours, wide]  (MIN) at (7.3,-5.25) {$\Pp_c$ is the unique minimiser of\\ the specific free energy, $1<c\le2$\rmloc{Corollary~\ref{cor:minimiser}}};
\node[thm]         (MAIN) at (7.3,-7.0) {Theorem~\ref{thm:main}\\ uniqueness of the zero-slope\\ phase, $1<c\le2$};
\node[adapt]       (TA)  at (4.6,-8.85) {Two distinct zero-slope\\ phases are smooth, $c>1$\rmloc{Theorem~\ref{thm:smooth}; Sections~\ref{sec:swaps}--\ref{sec:fold}}};
\node[ours]        (TB)  at (10.0,-8.85) {$\Pp_c$ is ergodic and rough,\\ $1\le c\le2$\rmloc{Theorem~\ref{thm:plane}; Section~\ref{sec:plane}}};
% inputs, bottom row
\node[known] (SH8) at (1.6,-10.7) {Cluster swapping for\\ nearest-neighbour potentials\rmloc{\cite[Chapter~8]{Sheffield}}};
\node[known] (LR)  at (5.4,-10.7) {Lanford--Ruelle for subshifts;\\ Burton--Keane\rmloc{\cite{BGMT,BurtonKeane}}};
\node[known] (SPIN)at (9.2,-10.7) {Spin representation,\\ circuits, Markov property\rmloc{\cite[Sections~14--18]{DKLM}}};
\node[known] (GL)  at (13.0,-10.7) {Ergodicity of the\\ spin measure\rmloc{\cite[Proposition~4.11]{GlazmanLammers}}};
% side inputs
\node[known] (VAR) at (13.0,-7.0) {Variational principle\rmloc{\cite[Theorems~2.5.2, 6.3.1]{Sheffield}}};
\node[known] (LT)  at (1.6,-5.25) {Strict convexity of\\ the surface tension\rmloc{\cite[Theorem~13.16]{LammersTassy}}};
\node[unref] (O12) at (1.6,-3.5) {$\rho_c$ exists and minimises\\ the free energy:\\ (O1), (O2)\rmloc{\cite[Sections~3--4]{OAI225}}};
\node[known] (DK)  at (1.6,-1.75) {Gaussian free field for\\ $-1\le\Delta\le-\frac12$\rmloc{\cite{DKLM}}};
\node[known] (AD)  at (13.0,-1.75) {Domains, given the\\ full-plane input ($\ICI$)\rmloc{\cite[Theorems~1.3, 1.4]{ADHLM}}};
\node[ours]  (FIT) at (13.0,0) {Same measure, membership,\\ discrete harmonicity\rmloc{Section~\ref{sec:domains}}};
\node[align=left, font=\scriptsize\itshape, text width=3.4cm] (GAP) at (13.1,-3.75)
  {Stated as a remark in \cite{DKLM}; sketched in \cite{Ryan} by applying Sheffield's nearest-neighbour theory, which does not cover the six-vertex model for $c\ne1$ (Proposition~\ref{prop:notNN}).};
% arrows
\draw[use] (SH8) -- (TA);
\draw[use] (LR) -- (TA);
\draw[use] (SPIN) -- (TB);
\draw[rel] (GL) -- (TB);
\draw[use] (TA) -- (MAIN);
\draw[use] (TB) -- (MAIN);
\draw[use] (VAR) -- (MAIN);
\draw[use] (VAR.north) |- (MIN.east);
\draw[use] (MAIN) -- (MIN);
\draw[use] (LT) -- (MIN);
\draw[use] (MIN) -- (ID);
\draw[use] (O12) -- (ID);
\draw[use] (ID) -- node[right, font=\tiny, align=left]{supplies the\\ identification step} (OAI);
\draw[use] (OAI) -- (DOM);
\draw[use] (AD.west) -- (DOM.south east);
\draw[use] (FIT) -- (DOM);
\draw[rel] (DK.north) |- (DOM.west);
\draw[rel, black!55] (GAP.west) -- (MIN.north east);
% legend
\node[sw, thmlook] (g1) at (0.35,-12.0) {}; \node[lg] at (g1.east) {main theorem};
\node[sw, ourslook, rounded corners=0.6mm] (g2) at (5.0,-12.0) {}; \node[lg] at (g2.east) {new here};
\node[sw, repairlook, rounded corners=0.6mm] (g3) at (9.8,-12.0) {}; \node[lg] at (g3.east) {missing step, supplied here};
\node[sw, knownlook] (g5) at (0.35,-12.55) {}; \node[lg] at (g5.east) {existing result};
\node[sw, adaptlook] (g4) at (5.0,-12.55) {}; \node[lg] at (g4.east) {existing argument, extended};
\node[sw, unreflook] (g6) at (9.8,-12.55) {}; \node[lg] at (g6.east) {unrefereed input (OpenAI)};
\draw[use] (0.1,-13.1) -- (0.6,-13.1) node[lg] {used in the proof};
\draw[rel] (4.75,-13.1) -- (5.25,-13.1) node[lg] {alternative source or context};
\end{tikzpicture}%
\end{NoHyper}
\caption{How the pieces fit. Read from the bottom up: an arrow means that its source is used to prove
its target; only the main dependencies are drawn. Box styles give provenance, and grey labels give
sources and where each item is proved. Here $c\in[1,2]$ and $\Delta=1-c^2/2$, so that $c=1,\sqrt2,\sqrt3,2$
correspond to $\Delta=\frac12,0,-\frac12,-1$; $\rho_c$ is the state constructed in \cite{OAI225} from nearly
balanced sectors, (O1) and (O2) are the two properties of it that we use (Section~\ref{subsec:gap}), and
a \emph{phase} is a translation-ergodic gradient Gibbs measure. The one step of the chain from the plane
measure to the Gaussian free field that we found not to be proved in the literature is the identification in the
dashed rounded box; it matters for $-\frac12<\Delta\le0$, and we supply it given (O1) and (O2). For
$-1\le\Delta\le-\frac12$ the full-plane input is the main theorem of \cite{DKLM}. Everything above the
identification is as strong as its inputs, some of which come from an unrefereed manuscript.}
\label{fig:roadmap}
\end{figure}