\section{Restrictions on labels near the retained vertices}\label{sec:resources}
The count bounds will depend on missing edges both inside and outside
the retained sets. This section relates those quantities and proves
label restrictions that permit later matching replacements without
creating critical edges to uncovered vertices.

\subsection{\texorpdfstring{Edges incident to the $Q$ endpoint of a good pair}{Edges incident to the Q endpoint of a good pair}}\label{sec:global-Q}
The core replacements imply
\begin{equation}\label{eq:Q-no-blue-X2}
 Qv\ne1\qquad\text{for every good $PQ$ and every old }v\in X_2.
\end{equation}
Indeed the neutral core is a union of disjoint alternating paths,
each consisting of four original red label-2 edges and three blue
links, with two $B$ endpoints. Four retained $B$ neighbors of $P$
belong to at least two such paths. Given an old red edge $uv$, choose
one of these paths disjoint from $uv$, and name its endpoints $B,B'$
so that $PB=3$. If $Qu=1$, replace the four red edges of the path,
$uv(2)$, and $PQ(3)$ by its three blue links and
$PB(3),Qu(1),vB'(\le4)$. The primary weight decreases from seven
to at most six. This proves \eqref{eq:Q-no-blue-X2} using the original
matching, with no assumption that $u$ belongs to the retained core.

A good $Q$ also has label four to every originally uncovered vertex.
Re-pair a core edge $BB'(4)$ with $PB=3$; then
$BB'(4),PQ(3)\to PB(3),Qu(\le3)$ would reduce primary weight,
leaving $B'$ uncovered instead of $u$. Including an original top
edge gives the analogous conclusion $QT=4$. These label restrictions
justify the later increases in the matching labels of good $Q$
vertices, including when arbitrarily many vertices remain uncovered.
\subsection{A Tutte--Berge packing stability inequality}
Let $c_*$ be the missing-blue density inside the actual retained $C$
set, and put $K=(q^2-r^2)/2$. The following relation holds when
$d\ge q$:
\begin{equation}\label{eq:TB-core}
 \left(5-\frac{4q}{r}\right)c_*
 \le c-K+\frac{2r}{d}a+\frac{2a^2}{d^2}+o(1)
 \le c-K+\frac{59}{25}a+o(1).
\end{equation}
Its left coefficient is nonnegative on $r\ge4q/5$.

Here is the argument. Extend the retained blue $CC$ matching to a
maximum matching covering mass $R$, without uncovering its initial
vertices. Then $R\ge r$. Match the original $D$ mates of its covered
vertices to $A$ using blue edges. The bipartite matching bound leaves
at most $aM/d+O(1)$ of these $D$ vertices unmatched. Deleting every
$CC$ edge meeting such a vertex's mate removes at most
$2aM/d+O(1)$ endpoints and produces a neutral core. Maximality of
the fixed core therefore gives $R\le r+2a/d+o(1)$.
Let $c_R$ count missing pairs
within the covered set, and let $\delta=q-R$. A Tutte--Berge equality
set $S$, of mass $s$, is fully matched to distinct odd components of
the graph outside $S$; each remaining odd component has one unmatched
vertex. Thus unmatched-to-covered blue edges have density at most
$\delta s+o(1)$: edges to $S$ give this term, while contacts within
components total only $O(M)$. The covered vertices outside $S$ occupy
at least $sM$ distinct components. Concentrating all remaining vertices
in one component minimizes their missing pairs, giving
\[
 c_R\ge s(R-3s/2)-o(1)\ge sR/4-o(1),\qquad s\le R/2.
\]
Since the unmatched set is independent,
\[
 c\ge\frac{q^2-R^2}{2}+c_R-\delta s-o(1)
 \ge\frac{q^2-R^2}{2}+\left(5-\frac{4q}{R}\right)c_R-o(1).
\]
Use $c_*\le c_R$, the bounds on $R$, and $a\le9d^2/50$ to obtain
\eqref{eq:TB-core}. The only standard input is the
Tutte--Berge formula~\cite{Tutte,Berge,Schrijver}; the displayed stability estimate follows from
the component count above.

In particular, at $a=0$, $c=K$, and $r>4q/5$, this forces $c_*=o(1)$.
The same missing-$CC$ allowance cannot both reduce the core size and
make its retained graph substantially incomplete.

\input{dense-tutte}
\input{joint-packing}

\subsection{The internal count on the label-3 class}
Write $Z$ for the mass of good original pairs $P_iQ_i$ and
$\beta=z-Z$ for the remaining label-3 pair mass. Set
$I_3=z^2/2-C_{33}\ge0$, and let $E_{PP}$ count good-$P$ pairs with
label at least 3. Then
\begin{equation}\label{eq:PP-deficit}
 E_{PP}\le I_3+o(1).
\end{equation}
First, a good $Q_i$ has no label-at-most-2 edge to any endpoint $R$
of another original pair $RS(3)$. In the neutral core, the replacement
\[
 BB'(4),P_iQ_i(3),RS(3)
 \longrightarrow P_iB(3),Q_iR(\le2),B'S(\le4)
\]
would reduce primary cost from 8 to at most 7.
Second, if $RP_i\le2$ and $SP_j\le2$ for distinct good indices,
replace the three \emph{original} label-3 matching edges by
\[
 RP_i(\le2),\quad SP_j(\le2),\quad Q_iQ_j(\le4).
\]
Primary cost is at most 6; if it equals 6, the secondary cost decreases
from 3 to 2. This contradicts the choice of $\M_0$.
Each remaining pair therefore has at most $ZM+O(1)$ critical contacts
to the good pairs. Summing the good, mixed, and remaining internal
counts gives
\[
 C_{33}\le Z^2/2-E_{PP}+Z\beta+\beta^2/2+o(1)
          =z^2/2-E_{PP}+o(1),
\]
which proves \eqref{eq:PP-deficit}.