\subsection{A stronger bound when the retained graph is dense}
\label{sec:dense-tutte}
In the proof of \eqref{eq:TB-core}, the barrier set $S$ has normalized
size $s\le R/2$. A density hypothesis forces $s$ to be smaller and
therefore strengthens both the internal missing-edge estimate and
the bound on blue edges from retained to omitted vertices.

\begin{lemma}\label{lem:dense-tutte}
Suppose $r\ge7q/8$ and $c\le q^2/8$. Then
\begin{align}
 \frac{7-4q/r}{3}\,c_*
   &\le c-K+\frac{2r}{d}a+\frac{2a^2}{d^2}+o(1),
       \label{eq:dense-tutte-core}\\
 U&\le\frac{2r}{d}a+
 \frac{4(q-r)}{7r-4q}
       \left(c-K+\frac{2r}{d}a+\frac{2a^2}{d^2}\right)+o(1).
       \label{eq:dense-tutte-out}
\end{align}
Here $c_*$ is the missing-blue density inside the retained $C$ set,
$K=(q^2-r^2)/2$, and $U$ is its cross-blue density to omitted $C$.
\end{lemma}
\begin{proof}
Extend the retained blue matching to a maximum one covering mass $R$,
where $r\le R\le r+2a/d+o(1)$, and let $c_R$ be the missing density
inside its covered set. The barrier calculation proving
\eqref{eq:TB-core} gives
\[
 c\ge\frac{q^2-R^2}{2}+
       \left(5-\frac{4q}{R}\right)c_R-o(1).
\]
If $c_R\ge R^2/8$, its right side is at least
\[
 \frac{q^2}{8}+\frac{(q-R)(3q-R)}8-o(1).
\]
Consequently, in the limiting region $R<q$ and $c\le q^2/8$, we
have $c_R<R^2/8$. The concave function $s(R-3s/2)$ is at least
$R^2/8$ throughout $s\in[R/6,R/2]$. Hence the barrier mass satisfies
$s<R/6$, and therefore
\[
 c_R\ge\frac{3R}{4}s-o(1).
\]
Its unmatched-to-covered blue density is at most
$(q-R)s+o(1)\le4(q-R)c_R/(3R)+o(1)$. Thus
\[
 c\ge\frac{q^2-R^2}{2}+
       \frac{7-4q/R}{3}c_R-o(1).
\]
Use $c_*\le c_R$ and
$(R^2-r^2)/2\le2ra/d+2a^2/d^2+o(1)$ to obtain
\eqref{eq:dense-tutte-core}.

To obtain the cross-blue bound, contacts with the newly covered
vertices cost at most $r(R-r)\le2ra/d+o(1)$. The remaining contacts
are at most $4(q-R)c_R/(3R)$; substitute the last inequality and
use the monotonicity of $4(q-R)/(7R-4q)$ in $R$.
This proves \eqref{eq:dense-tutte-out}.

For completeness the boundary and finite errors do not require a
strict hypothesis $c<q^2/8$. Given any sequence of finite instances,
pass to a subsequence on which the bounded normalized parameters
converge. If $R<q$ in the limit, the preceding argument applies.
If $R=q$, the unmatched mass of the full maximum matching tends to zero, so
its unmatched-to-covered blue density is $o(1)$ (the separate
newly-covered contribution $r(R-r)$ to $U$ remains), and the
coefficient in the internal inequality tends to
one. These alternatives prove the displayed inequalities with uniform
$o(1)$ on the compact localized domain. The original finite errors
are $O(M)$ and involve only the $O(M)$ covered vertices, so this
uniformity is independent of $N$.
\end{proof}

The complementary region $c\ge q^2/8$ is easy to exclude separately.
There the original and core bounds give
\[
 C_0\le B_0-q^2/8+o(1),\qquad C_r\le B_0+f_R+o(1).
\]
A rational positive-definite certificate on $7q/8\le r\le q$
proves $\gamma(4)\le0.43321$ in this region. Its checker reconstructs
the weighted quadratic form and subtracts only nonnegative multiples
of $(x+y+z+w)(r-7q/8)$ and $(r-7q/8)(q-r)$.
Thus the stronger resource estimates apply throughout the remaining
region. On a ratio box $h/q\in[v_0,v_1]$, $r/q\in[\rho_0,\rho_1]$,
we may take
\[
 L=\frac{4(1-\rho_0)}{7\rho_0-4},\qquad
 \beta=\frac{2\rho_1}{1+v_0}+\frac{9/25}{(1+v_0)^2}.
\]