\section{The half-plane coboundary}
\label{sec:coboundary}

\begin{theorem}[Exact reverse-complement defect]
\label{thm:coboundary}
For every length-$k$ word $w$,
\begin{equation}
  m_0(\RC w)-m_0(w)
  =\chi(\tail(w))-\chi(\head(w))
  =(\nabla\chi)(w).
  \label{eq:coboundary}
\end{equation}
Equivalently, $\ind_{\RC(M_0)}-\ind_{M_0}=\nabla\chi$.
\end{theorem}

\begin{proof}
Put
$A=\operatorname{Im}P_s(w)$ and
$B=\operatorname{Im}P_s(Rw)$.  Lemma~\ref{lem:covariance} sends the ordered
imaginary coordinates of $\RC(w)$ to $(B,A)$.  If $A$ and $B$ are nonzero,
the selector rule gives
\[
  m_0(\RC w)-m_0(w)
  =\ind[B<0<A]-\ind[A<0<B]
  =\ind[A>0]-\ind[B>0].
\]

The ray cases fix the sign convention.  If $A=0$, write
$P_s(w)=r\in\mathbb R$.  Then
$B=-r\sin(2\pi/k)$: for $r<0$ both sides are $-1$, and for $r>0$ both are
zero.  If $B=0$, write $P_s(Rw)=r\in\mathbb R$: for $r>0$ both sides are
$+1$, and for $r<0$ both are zero.  When $k>2$, simultaneous vanishing of
$A$ and $B$ with $P_s(w)\neq0$ is impossible.  In the zero sector, the
chosen ties are equivariant, while both lower imaginary coordinates vanish.

Finally, Lemma~\ref{lem:covariance} identifies the sign difference with
$\chi(\tail(w))-\chi(\head(w))$ under convention~\eqref{eq:gradient}.
\end{proof}

The open upper half-plane and the negative-real-axis convention are
load-bearing.  Equation~\eqref{eq:coboundary} is a literal edgewise identity,
not an averaged symmetry statement.