\subsection{The two reservoir matchings share a cardinality constraint}
\label{sec:reservoir-rank-coupling}
The initial and enlarged reservoir constructions use related auxiliary
graphs. Comparing their maximum matching sizes yields an additional
constraint. Write $R=r+k$ for the enlarged eligible-pool mass,
$t\le r$ for the initial target mass, and $t'\le R$ for the enlarged target
mass. Let $b_3$ be the exceptional label-three pair mass. Both auxiliary
graphs include all $2b_3$ exceptional endpoints and start from their
old matching. The old maximum low-label matching covers $s$ target
$D$ vertices, and the enlarged one covers $s'$ target $D^*$ vertices.

Put $\delta_0=r-t$ and $\delta_1=R-t'$. The corresponding target
sets $I,I'$ satisfy
\[
 |I'\setminus I|/M\le k+\delta_0+o(1).
\]
Deleting the edges incident to these added vertices from the new
maximum matching leaves a matching in the old auxiliary graph.
Each deleted vertex destroys at most one matching edge. Because both
maximum matchings preserve every exceptional endpoint, this yields
\begin{equation}
 \frac{s'+2b_3}{2}\le\frac{s+2b_3}{2}+k+\delta_0+o(1),
 \qquad s'\le s+2k+2\delta_0+o(1).
 \label{eq:reservoir-rank-monotonicity}
\end{equation}
This argument compares ordinary maximum matching cardinalities; it
does not assume that the new matching contains the old matching's
edges. Starting with the exceptional pairs and augmenting guarantees
that a maximum matching preserving their covered vertices exists in
each graph.

The unmatched new target mass is $u'=t'-s'$. Before maximizing over
$u'$, the contribution of the low-label matching to the enlarged
critical-edge count is
\[
 u'\bigl(R-t'/2+b_3\bigr)-\frac{(u')^2}{2}.
\]
The unconstrained upper bound is $(R-t'/2+b_3)^2/2$. Completing the
square and using \eqref{eq:reservoir-rank-monotonicity} gives
an additional loss at least
\begin{equation}
 \frac12\left(\frac r2-s-\frac{3k}{2}-b_3
                   -2\delta_0-\frac32\delta_1\right)_+^2.
 \label{eq:reservoir-rank-loss-finite}
\end{equation}
Indeed $u'\ge t'-s-2k-2\delta_0$, and the unconstrained maximizing
value is $R-t'/2+b_3$; their difference is the expression inside the
positive part. All other terms in the enlarged count bound are
unchanged.

Condition on the old target and the selected swap links, so that
$r,s,k,\delta_0$ are fixed during the new top sampling. We have
\[
 \delta_0\le\frac{e_{24}}{2w}+o(1),\qquad
 \mathbb E\delta_1\le\frac{e_{24}}{2w}+o(1),\qquad
 b_3\le H/r+o(1).
\]
The function $x\mapsto(x_+)^2/2$ is convex. Jensen's inequality
applied to \eqref{eq:reservoir-rank-loss-finite} therefore proves that
one may subtract
\begin{equation}\label{eq:reservoir-rank-loss}
 \frac12\left(\frac r2-s-\frac{3k}{2}-\frac Hr
                         -\frac{7e_{24}}{4w}\right)_+^2
\end{equation}
from the right side of the enlarged-reservoir bound \eqref{eq:extended-reservoir}. The
coefficient $7/(4w)$ is the sum of the old-target contribution
$2/(2w)$ and the new-target contribution $(3/2)/(2w)$.
The bounds on the positive terms due to unused top vertices remain
valid, so they require no additional error term in
\eqref{eq:reservoir-rank-loss}. All errors are uniform in $N\ge2M$,
as both auxiliary graphs have only $O(M)$ vertices.

\paragraph{Quadratic forms for exact certificates.}
Let $g=r/2-s-3k/2=2r-s-3R/2$. The inequality
\[
 -\tfrac12(x_+)^2\le\tfrac12\tau^2-\tau x
 \qquad(\tau\ge0)
\]
provides useful rational quadratic bounds. Choosing $\tau=\eta r$
for any fixed rational $\eta\ge0$ adds to the right side of \eqref{eq:extended-reservoir}
\begin{equation}\label{eq:rank-tangent-r}
 \eta r(s-r/2+3k/2)+\frac{\eta^2r^2}{2}
       +\eta H+\frac{7\eta r}{4w}e_{24}.
\end{equation}
The last coefficient is at most $7\eta/2$ on $r\le2w$.
Writing the enlarged polynomial as $F(R)$, its modified part is
$F(R)+(3\eta/2)rR$. First taking the tangent at $R=q$, thus using
$\varepsilon=0$ in the preceding subsection, the two-orientation swap envelope uses
\[
 J_\eta=\frac{(7q/4+z-2y-3\eta r/2)_+}{2\sigma r},
 \qquad\sigma=s/r,
\]
and its main polynomial is
\[
 F(q)+\eta r\bigl(s-r/2+3(q-r)/2\bigr)+\eta^2r^2/2.
\]
Its $H$ coefficient increases by $\eta$, and its $e_{24}$ coefficient
increases by at most $7\eta/2$; the usual two-orientation defect
coefficients use $J_\eta$ in place of $J$.

Alternatively choose $\tau=\lambda q$ with fixed rational
$\lambda\ge0$. Using $q/r\le8/7$ and $q\le2w$, the new defect
costs are at most
\[
 \frac{8\lambda}{7}H+\frac{7\lambda}{2}e_{24}.
\]
In the envelope, the main polynomial changes by
\[
 \left(\frac{\lambda^2}{2}+\frac{3\lambda}{2}\right)q^2
       -\lambda(2-\sigma)qr,
\]
and
\[
 J_\lambda=
 \frac{((7/4-3\lambda/2)q+z-2y)_+}{2\sigma r}.
\]
These formulas introduce no new optimization coordinate: a rational
$\eta$ or $\lambda$ may be chosen separately on each ratio box.

A coarser consequence, useful when fixed coefficients on the deficits
are preferred, is the additional term
\[
 -\frac12g_+^2+\frac H2+\frac74e_{24}.
\]
To verify it, note that $g_+\le r/2$ and
$\tfrac12g_+^2-\tfrac12(g-D)_+^2\le(r/2)D$ for $D\ge0$.
Take $D=b_3+2\delta_0+3\delta_1/2$, average, and use
$r\le2w$. The certificate below uses the tangent form
\eqref{eq:rank-tangent-r}.

\paragraph{The envelope when the covered side is smaller.}
The full rectangular bound \eqref{eq:swap-coverage} gives
\[
 q-R\le(\delta-s)_++\frac{V+U+a+j}{2\max(s,\delta)}+o(1).
\]
After the rank tangent with parameter $\eta$ and the reservoir tangent
at $R=(1-\varepsilon)q$, the coefficient of $q-R$ is bounded by
\[
 \left[(7/4-23\varepsilon/4)q+z-2y-3\eta r/2\right]_+.
\]
To make this bound uniform on a parameter box, suppose
\[
 \sigma_0\le s/r\le\sigma_1,\qquad
 \rho_0\le r/q\le\rho_1,\qquad
 \zeta_0\le z/q\le\zeta_1,\qquad
 \upsilon_0\le y/q\le\upsilon_1.
\]
Put
\[
 D_{\rm box}=\max\{\sigma_0\rho_0,1-\rho_1\},\qquad
 A_{\rm box}=
 [7/4-23\varepsilon/4+\zeta_1-2\upsilon_0-3\eta\rho_0/2]_+.
\]
If $D_{\rm box}>0$, use
\begin{equation}\label{eq:partial-swap-envelope}
 J=\frac{A_{\rm box}}{2D_{\rm box}},\qquad
 E_{\rm box}=A_{\rm box}[1-(1+\sigma_0)\rho_0]_+.
\end{equation}
The resulting enlarged column has the same resource coefficients as
before, with this $J$, and the additional main term $E_{\rm box}q^2$.
Indeed the slope is at most $A_{\rm box}q$,
$\max(s,\delta)\ge D_{\rm box}q$, and
$(\delta-s)_+\le[1-(1+\sigma_0)\rho_0]_+q$.
If $D_{\rm box}=0$, give this comparison zero weight.
On a box satisfying $\sigma_0\rho_0\ge1-\rho_0$, the additional
term vanishes and the formula agrees with the envelope for complete
enlargement. Otherwise it accounts explicitly for the part of the
omitted set that the matching cannot absorb.