\section{Supplementary proofs for standard results}\label{app:bruhat}
\subsection{A rank proof of the Bruhat transfer}
This appendix retains the proof of the standard lifting property
\cite[Theorem~2.1.5, Proposition~2.2.7]{coxeter} in the notation of
Lemma~\ref{lem:bruhat}.
\begin{proof}
Suppose $q$ has a descent at position $i$ and $p\preceq q$.
Sort that pair in both maps, skipping the exchange in $p$ if it has no
descent. Call the results $p',q'$. Only rank row $i$ can change.
For a threshold between $q(i+1)$ and $q(i)$ put
$A=\mathcal R_p(i-1,t)$, $B=\mathcal R_q(i-1,t)$,
$x=\ind{p(i)>t}$, and $y=\ind{p(i+1)>t}$.
Then $A\le B$ and $A+x+y\le B+1$.
The new row entries are $A+\min(x,y)$ for $p'$ and $B$ for $q'$.
If $A<B$, $A+\min(x,y)\le A+1\le B$;
if $A=B$, use $x+y\le1$.
Outside this threshold range the row for $q$ is unchanged and the row
for $p$ can only decrease. Hence $p'\preceq q'$.
Induction carries $p$ to the identity along with $q$.
Every performed exchange is a descent, so exactly $C(F)$ are performed.
Finally $p\preceq\mathrm{id}$ forces $p=\mathrm{id}$:
the identity has the smallest possible ranks $\max(0,i-t)$, and the
rank numbers determine the permutation.
\end{proof}

\subsection{Shallow involutions and noncrossing chords}\label{app:matching}
The classification used in Lemma~\ref{lem:invcap} is
\cite[Corollary~25]{berman-tenner}. Here is the retained direct calculation.
For an involution $\sigma$ with $h$ nontrivial pairs, let $J$ count unordered
pairs of chords whose four endpoints alternate in the linear order. Then
\[
 \Delta(\sigma)-I(\sigma)-h=2J.
\]
A single pair $(a,b)$ contributes $2(b-a)$ to $\Delta$ and $2(b-a)-1$
to $I$, hence one to $\Delta-I$. Contributions between two pairs are
computed by their four endpoints. Separate pairs $a<b<c<d$ paired as
$(a,b),(c,d)$ and nested pairs $(a,d),(b,c)$ give no correction;
alternating pairs $(a,c),(b,d)$ reduce the sum of the individual inversion
counts by two. Inserting a fixed position adds the same $2w$ to
$\Delta$ and $I$, where $w$ is the number of pairs crossing its cut.
Thus the formula also includes fixed points. Since $\rk(\sigma)=h$,
shallowness is equivalent to $J=0$.