Half-circle lemma — Lemma 2.9

Because $V$ is finite and nonempty, choose $z$ maximizing $\mathrm{\mid }{\theta }_z\mathrm{\mid }$ and put $c=\mathrm{\mid }{\theta }_z\mathrm{\mid }$. Then $0\le c<\pi \mathrm{/}2$. At any vertex $y$ with $\mathrm{\mid }{\theta }_y\mathrm{\mid }=c$, [F4] gives $0\le {\sum }_x{A}_{x,y}\sin (\mathrm{\mid }{\theta }_x\mathrm{\mid }-c)$. Every difference $\mathrm{\mid }{\theta }_x\mathrm{\mid }-c$ lies in $(-\pi ,0]$; by [F5] its sine is nonpositive and is strictly negative unless $\mathrm{\mid }{\theta }_x\mathrm{\mid }=c$. Since the coefficients are nonnegative, every neighbor $x$ of $y$ has $\mathrm{\mid }{\theta }_x\mathrm{\mid }=c$. Repeating this along a finite path from $z$ to any vertex shows $\mathrm{\mid }{\theta }_x\mathrm{\mid }=c$ for all $x\in V$. Self-loops cause no exception.

By [F2], the local minimum is a critical point: apply the Euclidean extremum theorem in an angular chart. Thus [F3] gives the sine equilibrium identity at every vertex. If $c>0$ and both $S=\{x:{\theta }_x=c\}$ and $V\setminus S=\{x:{\theta }_x=-c\}$ are nonempty, connectedness gives an edge between them: a path between the two sets has a first crossing edge. Summing the equilibrium identities over $y\in V\setminus S$, terms with $x\in V\setminus S$ vanish and the remaining terms give $0=\sin (2c){\sum }_{y\in V\setminus S}{\sum }_{x\in S}{A}_{x,y}$. The double sum is positive and $0<2c<\pi$, contradicting [F5]. Hence, if $c>0$, all representatives have one common value $t\in \{c,-c\}$.

In the common-value case, the definition of ${\rho }_1$ gives ${\rho }_1=e^{it}$. Its being real makes $\sin t=0$ by [F5]. If $t\mathrm{\ne }0$, then $0<\mathrm{\mid }t\mathrm{\mid }<\pi \mathrm{/}2$, so [F5] gives $\sin \mathrm{\mid }t\mathrm{\mid }>0$ and oddness gives $\sin t\mathrm{\ne }0$, a contradiction. Therefore $c>0$ is impossible. With $c=0$, step 1.1 gives ${\theta }_x=0$ for every vertex, as required.