\subsection{Derivation of the core comparison}\label{sec:core-derivation}
We derive \eqref{eq:rawcore} by counting the changes on disjoint edge
classes. The retained
sets $A,B,C,D$ each have mass $r$. The new matching uses blue $AD$
and $CC$ edges and label-4 $BB$ edges, whereas the original matching
used $AB(2)$ and $CD(2)$. Every alternating four-red, three-blue path
has a label-4 edge between its two $B$ ends: an edge of label at most
three would replace primary cost four by cost at most two. Thus the
rematching is primary-minimal. Its entire new label-4 class is a
label-4 clique, and its $B$ vertices may be paired arbitrarily.

For each original label-3 edge $PQ$, its two endpoints cannot have
distinct label-3 neighbors $B,B'$ in the retained set: re-pairing the
$B$ vertices would permit $BB'(4),PQ(3)\to BP(3),B'Q(3)$, which
decreases the primary weight. Thus its total label-3 degree into
retained $B$ is at most $rM+O(1)$. If the pair is exceptional, that
degree is at most six. Summing proves $H_3\le rz+o(1)$ and
the exceptional-mass bound $h_3/r+o(1)$ used in
Section~\ref{sec:neutral-core}.

For each retained $D$, write $A$ for its new mate and $B$ for the old
mate of $A$. For every other retained $B'$, the switch
\[
 BB'(4),AD(1)\longrightarrow B'D(\le3),AB(2)
\]
would reduce primary weight. Hence there are at most $rM$ exceptional
$BD$ contacts. The analogous switch including an original top edge
shows $DT=4$ for every old top vertex $T$.

Let $PQ(3)$ be good, with $P$ having at least four retained $B$
neighbors of label three. Switching $PQ$ with a suitably re-paired
$BB'$ shows $QB=4$. Choosing a neighbor $B'$ distinct from the old
mate $B$ of the $A$ paired with a given $D$, the switch
\[
 BB'(4),AD(1),PQ(3)
 \longrightarrow PB'(3),QD(\le3),AB(2)
\]
shows $QD=4$. If $CC'$ is a core edge and $D$ is the old mate of
$C'$, adding $CC'$ to this switch and replacing it by
$QC(\le2),C'D(2)$ proves $QC\ge3$. These arguments use only strict
primary improvements. The secondary order will be used separately
for the internal label-3 deficit.

Put $h_1=2ry-H_1$, where $H_1$ counts contacts of labels at most
three between retained $B$ and old $X_1$. The changes from the old
matching to the core satisfy the following count table:
\[
\begin{array}{c|c}
 \text{Pairs affected}&\text{Upper bound for the change}/M^2\\\hline
 \text{inside the four retained sets}&-r^2+E+o(1)\\
 \text{retained versus omitted }X_2&r(x-2r)+o(1)\\
 \text{retained versus }X_1&-2ry-h_1+e_{12}+o(1)\\
 \text{retained versus }X_3&rz-h_3\\
 \text{retained versus }X_4&0
\end{array}
\]
For the first row, at least $3r^2-E-o(1)$ old blue contacts were
critical; afterwards only $BA,BC$ and $O(M)$ exceptional $BD$
contacts can be critical, giving at most $2r^2+o(1)$. For the second
row, only $B$ promotions can increase the count. Pairwise minimality
between a new $BB(4)$ edge and an omitted label-2 matching edge bounds
its critical cross contacts by two.

For the third row, the old selected $AB,CD$ pairs have at least
$4ry-e_{12}$ blue contacts with $X_1$, since the cross-rectangle
deficits are individually nonnegative. The change is their removal
plus the $H_1$ contacts made critical by $B$. Against $X_3$, only
the newly critical label-3 contacts from $B$ contribute; all other
birth changes are below the old birth three. Against $X_4$ no maximum
endpoint birth changes. Both matchings have no outward critical edges.
Adding these changes to \eqref{eq:original} gives
\[
 C_r\le B-b^2+r(x+z-2y)-3r^2-h_1-h_3
 -\sum_{(i,j)\ne(1,2)}e_{ij}+o(1).
\]

It remains to relate $h_1$ and $h_3$. For a good $PQ$ let $g_P$ be
its retained-$B$ label-3 degree, and let $b_Q,d_P$ count, respectively,
the retained $A,D$ contacts that fail to have label at most two.
The original half bounds and the exclusions $QB=QD=4,QC\ge3$ imply
$b_Q+d_P\le e_P+r-g_P$, where $e_P$ is its share of $e_{23}$.
For $V\in X_1$ and $T\in X_4$, let $a_V,c_T$ count the failures of
$VB\le3$ and $TC\le3$ in the aligned core slots. Then
$\sum_Va_V=h_1$ and $\sum_Tc_T\le e_{24}$.

For original matching edges $UV(1),TT'(4),AB(2),CD(2),PQ(3)$,
the replacement
\[
 UT(\le3),\ T'C(\le3),\ DP(\le2),\ QA(\le2),\ BV(\le3)
\]
has weight at most $8+3\varepsilon$, whereas the old weight is
$8+3\varepsilon+\varepsilon^2$. Thus whenever $UT$ is critical,
the four failures cover the aligned core indices, yielding
$a_V+c_{T'}\ge g_P-e_P$.
Sum first over good pairs. A nongood pair has only $O(1)$ retained
$B$ contacts of label three, so the omitted contribution is $o(1)$
after normalization. Sum next over the $2yw-e_{14}$ critical
$X_1$--$X_4$ edges. The endpoint degree bounds $2w$ and $2y$ give
\[
 (2yw-e_{14})(rz-h_3-e_{23})
 \le2z(wh_1+ye_{24})+o(1).
\]
On the localized domain $z,w>0$, expanding and discarding favorable
products proves
\[
 h_1\ge ry-\frac yz h_3-\frac r{2w}e_{14}
              -\frac yz e_{23}-\frac yw e_{24}-o(1).
\]
Substituting in the count table proves \eqref{eq:rawcore}. The same proof
retains $-I_3$ if the old internal-$X_3$ count is written as
$z^2/2-I_3$.