\section{A matching replacement on alternating paths}\label{sec:neutral-core}
We construct several $M$-edge matchings and bound the smallest of their
critical-edge counts. Throughout the four-label upper-bound proof,
vertex-set sizes are divided by $M$ and edge counts by $M^2$. In
particular, write $C_0=C_\lambda(\M_0)/M^2$, and use analogous
subscripts for the other matchings. A bound $C\le2G+o(1)$ therefore
implies $\gamma(4)\le G$. An expression such as $UV=3$ means
$\lambda(UV)=3$, and $UV\le3$ means $\lambda(UV)\le3$.
All $o(1)$ terms tend to zero uniformly in $N\ge2M$; only $O(M)$
covered vertices enter the auxiliary graphs.

Write $y,x,z,w$ for the normalized numbers of edges of labels
$1,2,3,4$ in $\M_0$, respectively; their sum is one. Thus $|X_1|/M=2y$,
and similarly for the other three classes. We call vertices of $X_4$
\emph{top vertices}. The decomposition in
Section~\ref{sec:foundations} gives label-2 pairs $AB$ and $CD$, of
masses $d,q$, where $d+q=x$. Set
\[
 b=|d-q|/2,\quad m=\min(d,q)=x/2-b,\quad
 B=1-y^2-x^2/4-z^2/2-w^2,\qquad B_0=B-b^2.
\]
We write $h=|d-q|=2b$ when specifying the imbalance between the
two families of label-2 pairs.
Let $E_{AA},E_{AC},E_{AD},E_{CC}$ count missing blue pairs in the
four permitted blue regions, and let $E$ be their sum. Let $C_{ij}$
be the normalized original critical count between $X_i$ and $X_j$
when $i\ne j$, and within $X_i$ when $i=j$. For $i<j$,
define the cross-class deficit $e_{ij}$ as the difference between
the two-critical-edges-per-matching-pair bound and the actual critical
count between $X_i$ and $X_j$. For example, if this count is
$C_{12}M^2$, then $e_{12}=2xy-C_{12}\ge0$.
We also use the abbreviations
\[
 a=E_{AD},\qquad c=E_{CC},\qquad j=E_{AC}.
\]
The original count satisfies
\begin{equation}\label{eq:original}
 C_0\le B-b^2-E-\sum_{i<j}e_{ij}+o(1).
\end{equation}
Here the $X_2$ count is at most $3x^2/4-b^2-E+o(1)$,
the $X_3$ count is at most $z^2/2+o(1)$, and the other internal
counts vanish. Summing these and the cross-class bounds proves
\eqref{eq:original}.

\subsection{The retained alternating paths}
Choose a blue matching from some $A$ vertices to some $D$ vertices,
and a blue perfect matching on the original $C$ mates of those $D$
vertices. These two matchings, together with the original red edges,
form vertex-disjoint alternating paths of the form
\[
 B_1A_1D_1C_1C_2D_2A_2B_2.
\]
The edges $B_1A_1,D_1C_1,C_2D_2,A_2B_2$ are original label-2
edges, and $A_1D_1,C_1C_2,D_2A_2$ are blue. Replace the four
original edges by those three blue edges and $B_1B_2$.
Minimality forces $B_1B_2$ to have label 4, so the primary weight
remains four. We call a union of these paths a \emph{neutral core}.
The four sets of vertices retained in it each have the same mass $r$,
where $0\le r\le\min(d,q)$.

Fix a neutral core with maximum possible $r$ among all such joint
choices of blue $AD$ and $CC$ matchings. Its replacement matching
uses $AD$ and $CC$ edges at label 1 and $BB$ edges at label 4,
and agrees with $\M_0$ elsewhere. Subsequent references to retained
or omitted vertices are relative to this choice, unless a different
core is explicitly constructed. Its normalized critical count is $C_r$.

Let $H_3M^2$ count the label-3 edges from retained $B$ to $X_3$,
and put $h_3=rz-H_3$. We call an original label-3 pair \emph{good}
if it can be oriented $PQ$ with $P$ having at least four label-3
neighbors in retained $B$; the remaining pairs are \emph{exceptional}.
The switching argument below shows that $h_3\ge-o(1)$ and that
the exceptional pair mass is at most $h_3/r+o(1)$ when $r$ is
bounded away from zero. This condition, and other parameter
conditions used below, are established in
Sections~\ref{sec:base-certificate} and~\ref{sec:certificates} before
the corresponding inequalities are used in the final optimization.
For brevity write $f_R=r(x+z-3y)-3r^2$.

For $z,w>0$, the core comparison derived in
Section~\ref{sec:core-derivation} gives
\begin{align}
 C_r\le{}&B-b^2+r(x+z-3y)-3r^2
 -(1-y/z)h_3-e_{13}-e_{34}\notag\\
 &+(r/(2w)-1)e_{14}+(y/z-1)e_{23}
 +(y/w-1)e_{24}+o(1).\label{eq:rawcore}
\end{align}
\input{core-derivation}

In particular, on $2y\le z$, $y\le w$, $r\le2w$,
\begin{equation}\label{eq:core}
 C_r\le B-b^2+r(x+z-3y)-3r^2-h_3/2-e_{13}-e_{34}+o(1).
\end{equation}
If $y\le3z/8$, the coefficient $1/2$ of $h_3$ in
\eqref{eq:core} can be replaced by $5/8$, directly from
$1-y/z\ge5/8$ in \eqref{eq:rawcore}.