\section{Auxiliary matchings through good label-3 pairs}\label{sec:auxiliary-matchings}
This section concerns the orientation $d\ge q$. We construct a
bipartite matching that uses both blue $AD$ edges and certain edges
from $A$ to the $P$ endpoints of good label-3 pairs. Either this
matching has a prescribed size, giving a direct comparison, or its
minimum vertex cover yields another counting inequality. Put
$\kappa=1-y/z$ and $I_3=z^2/2-C_{33}\ge0$. The original, core, and reservoir
counts all retain an additional $-I_3$. The original and core matchings
leave the original label-3 edges unchanged; the joint reservoir may
rematch exceptional pairs, but their births only decrease, so the
original internal-count deficit is still available. Let $Z=z-h_3/r+o(1)$ be a chosen
mass of good pairs, discarding surplus good pairs when necessary.
We use the global internal estimate $E_{PP}\le I_3+o(1)$ proved in
\eqref{eq:PP-deficit}, and the retained-$C$ resource
\eqref{eq:TB-core}.

Call an original $A$ strong if it has at least two blue neighbors
among the initial retained $D$. Their mass is at least $d-a/r-o(1)$.
Any two strong rows can be placed in distinct $AD$ slots; keeping
the original $CC$ links gives an actual neutral core containing both
of their $B$ endpoints.
Build a bipartite graph from all $A$ to the retained $D$ and good $P$.
Allow every blue $AD$ edge; allow $AP$ precisely when $A$ is strong,
$AP=1$, and its corresponding $AQ=2$. Start with the actual initial
$AD$ matching and augment. Every initially covered $A$ and every retained
$D$ stays covered. A matching with $r+t$ edges consists of $r$ new
$AD$ links and $t$ compatible $AP$ links.

Keep the original blue $CC$ matching. Pair the new core's $B$ vertices
at label 4, and match the selected $B$ to the corresponding good $Q$
at label 4. The latter endpoints have exact outward label 4. Every new
core $A$ is insertable into the initial core, so its $B$ satisfies
$PB\ge3$ and $QB=4$ against the original good orientation. No core
vertices have been deleted.

Any two vertices in the union of new-core and selected $B$ can coexist
in a neutral core. Within either group this is established as before.
For a cross pair, insert the strong selected $A$ while avoiding the
other vertex's $D$ slot. Consequently every subset of this union has
nonnegative label-3 deficit, up to $o(1)$. The union contains every
initial $B$. If $h_{\rm new}$ and $h_{\rm sel}$ are the new-core and
selected deficits, then
\[
 h_{\rm new}+h_{\rm sel}\ge h_3-o(1).
\]
The selected $B$ gain is exactly $tz-h_{\rm sel}$. Thus
$-\kappa h_{\rm new}-h_{\rm sel}\le-\kappa h_3+o(1)$.

Write $R(r)$ for the right side of \eqref{eq:rawcore} without its
$-\kappa h_3$, and with $-I_3$ retained. The contact comparison gives
\begin{align}
 C_{AP}\le{}&R(r)+t(2z+d+2q-6r)-2t^2
 +E_{AA}+j+a+e_{12}+e_{13}+e_{23}+I_3\notag\\
 &+\frac{3t}{r}a+(t/r-\kappa)h_3+o(1).
 \label{eq:jointAP}
\end{align}
Here compatibility $AP=1,AQ=2$ is useful in the omitted $CD$ pairs:
$PD=1$ forces $BC=4$, and $PC=1$ forces $BD=4$, by replacing the
three original $AB,CD,PQ$ edges. Hence their total contribution is at
most $2t(q-r)+e_{23}$, with no missing-$CC$ charge. Selected $PP$
losses use $E_{PP}\le I_3$. The $X_1$ half bounds supply $e_{12}+e_{13}$;
the remaining bad-pair allowance is $th_3/r$. These give
\eqref{eq:jointAP} with all contacts outside the matching controlled
by the exact outward caps.

\subsection{The saturated alternative}
Set $H=d-r$ and $\tau=(H-a/r)_+$. If the joint matching reaches $\tau$,
stop an augmentation sequence there. Concavity of
$f(t)=t(2z+d+2q-6r)-2t^2$ gives
\[
 f(\tau)\le f(H)+\frac{a}{r}(3d-2q+2r-2z)_+.
\]
This includes $\tau=0$, when the unchanged core is used. Thus
\eqref{eq:jointAP} yields the global cap inequality
\begin{align}
 C_{AP}\le{}&R(r)+f_A+E_{AA}+j+e_{12}+e_{13}+e_{23}+I_3
       +A_{AP}a+(H/r-\kappa)h_3+o(1),\notag\\
 f_A={}&(d-r)(2z-d+2q-4r),\notag\\
 A_{AP}={}&1+3H/r+\frac1r(3d-2q+2r-2z)_+.
 \label{eq:APcap}
\end{align}
There is no loss from rebuilding a smaller core.

\subsection{The deficient alternative}
Otherwise take a minimum vertex cover $A_0,D_0,P_0$ of the joint
bipartite graph, of total mass $r+t<r+\tau$. Put $k=r-|D_0|$.
Write $\eta=a/r$ and note $0\le k\le r$. If $k>\eta$, the empty
$AD$ rectangle and $t<H-\eta$ would give
\[
 a\ge(d-|A_0|)k\ge(d-t-k)k>(r+\eta-k)k
   =r\eta+(k-\eta)(r-k)\ge a,
\]
a contradiction. Therefore $k\le\eta$.
Thus $A_0\cup P_0$ covers the ordinary compatible $AP$ graph with
size $t+k<H$. If $W$ is the label-2 density between all $A$ and good
$P$, its nonedge rectangle implies
\begin{equation}\label{eq:APcover}
 W+e_{23}+\frac{Z}{r}a\ge rZ-H(d-Z)_+.
\end{equation}
The last allowance accounts for the nonstrong rows. The two-neighbor
lemma gives exactly the $Za/r$ allowance used here.

Let $v$ be the additional $P$ mass covered by a maximum blue matching
on retained $C$ and good $P$, starting from the original $CC$ matching.
Write $U_{PC},U_{PP}$ for its ambient blue densities, and $c_*$ for
missing blue pairs within retained $C$. On $v<r$, $Z\ge r+v$,
\begin{equation}\label{eq:jointPCPP}
 (r-v)\{U_{PC}+2(1-r/Z)U_{PP}-vZ\}\le4Zc_*+o(1).
\end{equation}
To prove this, take a maximum $PP$ matching covering mass $p\le v$,
then a maximum $PC$ matching of size $k$ after deleting those $P$.
The remaining $C$ graph and the matching bound $e(G)\le\nu(G)|V(G)|$
give $p+k\le v+4c_* /(r-v)+o(1)$. Also
$U_{PC}\le rp+k(Z-p)$ and $U_{PP}\le pZ/2+o(1)$.
Substitution gives \eqref{eq:jointPCPP}. All matchings used in this
argument are disjoint on their prescribed vertices.

For a second actual comparison, assign $r$ distinct good $PQ$ pairs
uniformly to the actual $AD$ links, and perform
$AD(1),PQ(3)\to AQ(2),DP(2)$ whenever compatible. Its expected failed
mass is at most $e_{23}/Z$: $PD=1$ would contradict primary minimality
when $AQ\le2$, and the other failures are old cross deficits. Reverting
a failed group raises only its $P,Q$ births from 2 to 3. The half
bounds charge at most $2x+2y+r$ label-2 contacts per failed pair.
The resulting actual matching therefore satisfies
\begin{align}
 C_{AD}-C_r\le{}&r(4y-d)-\frac rZ W+\frac rZ U_{PC}
       +\left(\frac rZ\right)^2(U_{PP}+I_3)\notag\\
 &+a+\frac{2x+2y+3r}{Z}e_{23}+o(1).
 \label{eq:ADcomparison}
\end{align}
In the formal count, core blue gains are $5r^2/2$, the retained
$PC,PD$ losses total $2r^2$, and selected $PP$ losses give $r^2/2$.
The $QA$ loss against all $A$ is $rd$; promotions against $X_1$ cost
at most $4ry$. This explains the main term and the three blue-density
corrections in \eqref{eq:ADcomparison}.

For the exact certificate, further retained domains are
$Z\ge3r/2$, $d\le23Z/20$, and $r\ge7q/8$; their exceptional regions
are excluded in Section~\ref{sec:certificates}. Choose
$0<\alpha\le1/2$ for each ratio box. If $v\ge\alpha r$, stop the
blue augmentation at $\alpha r$ and use the $CP$ comparison \eqref{eq:CPcomparison}, whose actual matching
and edge count are given in Section~\ref{sec:auxiliary-counts}.
If $v\le\alpha r$, combine \eqref{eq:APcover},
\eqref{eq:jointPCPP}, \eqref{eq:ADcomparison}, and
Lemma~\ref{lem:joint-packing-resource}.
Since $r/Z\le2/3$, the blue-density contribution is at most
$\alpha r^2+4c_* /(1-\alpha)$. The cover error obeys
\[
 \frac rZ H(d-Z)_+\le\frac3{20}r(d-r),
\]
since $d/Z\le23/20$. The jointly maximal core gives, on a box with
$r/q\in[\rho_0,\rho_1]$ and $h/q\in[v_0,v_1]$, the coefficients
\[
 K_C=\frac{4}{(1-\alpha)(5-4/\rho_0)},\qquad
 \beta_0=\frac1{1+v_0}.
\]
The small-$v$ count has main polynomial
\[
 f_R+r(4y-d-r)+\frac3{20}r(d-r)+\alpha r^2-K_CK,
 \qquad K=(q^2-r^2)/2,
\]
and coefficients at most $2+\beta_0K_C$ on $a$, $K_C$ on $c$,
$57/8$ on $e_{23}$, and $-5/8$ on $h_3$. To see the cross-deficit
constant explicitly, \eqref{eq:APcover} contributes $r/Z$ and
\eqref{eq:ADcomparison} contributes $(2x+2y+3r)/Z$; adding the
core coefficient $y/z-1$ gives at most $57/8$ on the retained
domains. All other unused terms are nonpositive. These are the
quadratic inequalities used in Section~\ref{sec:certificates}.