\section{Bounds on the size of the neutral core}\label{sec:core-size}
Let $a=E_{AD}$ and $c=E_{CC}$ be the missing blue-edge densities in
the indicated rectangles, let $m=\min(d,q)$, and let $D=\max(d,q)$.
All densities are normalized by $M^2$, while vertex-set sizes are
normalized by $M$. The inequalities below have uniform $o(1)$ errors
in the finite problem. We suppress these errors in the displayed
asymptotic formulas.
The formulas involving $1/m$ or $1/D$ concern $m>0$; this is
guaranteed on the parameter domain retained in
Section~\ref{sec:base-certificate}.

First find a maximum blue $AD$ matching, of size $tM$ with
$t\ge m-a/D$, by the bipartite matching bound in
Section~\ref{sec:foundations}. Then find a maximum blue $CC$ matching
on the corresponding $C$ endpoints. Its endpoints and their assigned
$AD$ edges form a neutral core. In the dense regime $E\le9m^2/50$,
\eqref{eq:dense-matching-corollary} gives
\[
 r^2\ge t^2-2c\ge(m-a/D)^2-2c.
\]
Indeed $c\le9m^2/50-a\le(9/50)(m-a/D)^2$, so the density
hypothesis of that corollary holds. Consequently
\begin{equation}\label{eq:split-packing}
 \frac mD a+c\ge\frac{m^2-r^2}{2}+\frac{a^2}{2D^2}.
\end{equation}
Missing $AA$ and $AC$ edges have no role in this packing obstruction.
Keeping $a$ and $c$ separate will allow the later count bounds to
use the constraints on each edge class.

In the larger-$CD$ branch, write $d=m$ and $q=d+h$. If
$h\ge3d/10$ and $E\le9d^2/50$, first find a maximum blue matching
in the \emph{entire} $C$ graph. Its covered vertex mass is at least
\[
 \sqrt{q^2-2c}\ge\sqrt{(13d/10)^2-9d^2/25}
 =\sqrt{133/100}\,d>d.
\]
Here \eqref{eq:dense-matching-corollary} is applicable since
$c\le9d^2/50\le9q^2/50$.
Select $dM+O(1)$ endpoints that are paired by this blue matching.
Between their old $D$ mates and $A$, a maximum blue matching misses
at most $(a/d)M+O(1)$ vertices, by the bipartite vertex-cover bound.
Delete every selected $CC$ matching edge that meets one of these
unmatched $D$ mates. At most twice as many endpoints are deleted.
The surviving $AD$ and $CC$ edges give disjoint alternating paths,
and hence an actual neutral core satisfying
\begin{equation}\label{eq:reverse-packing}
 r\ge d-\frac{2a}{d},\qquad
 a\ge\frac{d(d-r)}2.
\end{equation}
Choose a core of maximum size among all joint blue $AD$ and $CC$
packings in the peeled template. Such a core is at least as large as
each of the preceding constructions. Increasing its size preserves
the lower bounds in both
\eqref{eq:split-packing} and \eqref{eq:reverse-packing}. All switching
and counting statements used subsequently apply to this actual core.

In particular, if $r<d$ in this branch, \eqref{eq:reverse-packing}
forces a positive number of missing $AD$ edges, regardless of the
number of missing $CC$ edges.

\subsection{Averaging the reverse construction}
We can also average the choice of the $dM$ endpoints in this
construction. Suppose a blue $CC$ matching covers $LM$ endpoints,
where $L\ge\sqrt{q^2-2c}-o(1)$. Choose $dM/2$ of its matching edges
uniformly. The expected number of missing $AD$ edges on the chosen
$D$ mates is at most $(d/L)aM^2$. Some choice attains this bound.
Its balanced $AD$ matching therefore misses at most $(a/L)M+O(1)$
vertices. Deleting whole $CC$ edges again costs at most twice this
number. Choosing the largest actual core among all constructions gives
\begin{equation}\label{eq:averaged-reverse}
 2a\ge\sqrt{q^2-2c}\,(d-r)-o(1).
\end{equation}

For a ratio interval $v_0\le h/d\le v_1$ with $v_0\ge3/10$, set
\[
 \alpha=\frac1{1+v_0},\qquad \beta=\frac1{9(1+v_0)},\qquad
 K=\frac{d^2-r^2}{2},\quad J=\frac{q(d-r)}2.
\]
The elementary inequality
$\sqrt{1-2t}\ge1-10t/9$ for $0\le t\le9/50$ follows by squaring.
Since $(d-r)/q\le1/[5(1+v_0)]$, the two packing inequalities imply
\begin{equation}\label{eq:bin-packing}
 \alpha a+c\ge K,\qquad a+\beta c\ge J.
\end{equation}
These inequalities apply to the fixed maximum core of
Section~\ref{sec:neutral-core}. A core constructed later on a
restricted set of vertices will have its own size parameter.