\section{Local optimality under the addition of one or two blocks}
\label{sec:lower-local-stability}
We next ask whether the six-part construction can be improved by
adjoining parts of small positive mass. Throughout this section the
labels between its six existing parts remain fixed. The result is
local: neither a change to those labels nor a configuration far from
the specified mass vector is covered by the theorem.

\begin{theorem}\label{thm:lower-local-one-two}
There is a neighborhood of $(a^*,0,0)$ such that adjoining one or two
new blocks to the fixed matrix $L$, with arbitrary labels in
$\{1,2,3,4\}$, and choosing masses in that neighborhood cannot produce
a minimum critical density greater than $\alpha_L$. The six old masses
may vary simultaneously, and all masses sum to one. Zero masses for
one or both new blocks are allowed.
\end{theorem}

We use the notation $T=1-\alpha_L$ and $C_k,\lambda_k$ from
Section~\ref{sec:lower-exact}. To rule out a larger minimum critical
density, it suffices to exhibit a matching with noncritical density
at least $T$. A weighted average of the noncritical densities of
several feasible matchings is a lower bound for their maximum, so
the same conclusion follows from such an average being at least
$T$.

Write the three active flow maps from the table in the proof of
Theorem~\ref{thm:lower-exact} as $x=P_k a$, and put
$H_0=\sum_k\lambda_k C_k$. An interval $LDL^T$ calculation verifies
$H_0\succ0$. Together with $H_0a^*=T\mathbf1$, this gives
\begin{equation}\label{eq:lower-mixed-support}
 a^TH_0a\ge T\quad\text{whenever }\mathbf1^Ta=1.
\end{equation}
Indeed, writing $a=a^*+h$ with $\mathbf1^Th=0$ gives
$a^TH_0a=T+h^TH_0h$.
The flows defining this inequality remain feasible whenever their six
strictly positive basic coordinates remain positive; this holds in
an open neighborhood of $a^*$.

\subsection{One added block}
The \emph{label profile} of a new block $G$ is the tuple consisting
of its six labels to $A,B,C,D,E,F$ and its internal label $L_{GG}$.
For a block $G$ of mass $s$, an extension of flow $k$
chooses a partner $j\in\{A,B,C,D,E,F,G\}$. If $j\ne G$, match the
new mass to $j$ and apply the old flow map to $a-se_j$. If $j=G$,
match $G$ internally and apply the old map to $a$. These are feasible
for all sufficiently small $s$ and old masses near $a^*$. Let
$c_{k,j}$ be the derivative of their noncritical count with respect
to $s$, holding the old masses fixed at $a^*$. Since the old gradients
satisfy
\[
 \sum_k\lambda_k\nabla(a^TC_ka)|_{a=a^*}=2T\mathbf1,
\]
a selected set of partners gives a strictly positive first-order
increase in the weighted noncritical density, after accounting for
the mass removed from the six existing blocks, whenever
\begin{equation}\label{eq:lower-insertion-price}
 \sum_k\lambda_kc_{k,j_k}-2T>0.
\end{equation}
The subtraction of $2T$ accounts for the constraint that the total
mass remains one. The identity for the old gradients shows that
arbitrary simultaneous first-order variations of the six existing
masses have the same contribution.

It suffices to consider \emph{capped} profiles, meaning that
$L_{Gi}\le\max(L_{GG},L_{ii})$. Indeed lowering an uncapped contact
to that maximum weakly improves the construction: the affected
cross-matching type is dominated by its two internal types both
before and after the change, while the ambient edge label decreases.
There are exactly
\[
48+384+1296+4096=5824
\]
capped profiles. The checker supplies explicit partners for each.
For 5801 profiles, the left side of
\eqref{eq:lower-insertion-price} is strictly greater than $19/1000$.
The smallest certified lower endpoint exceeds $0.0190748046715198$.
For the remaining 23 profiles, the weighted first-order expression
vanishes; call these profiles \emph{neutral}. They fall into the
following classes. The first column specifies a part with which the
new mass will be combined in the quadratic identity below; profile
entries are ordered $A,B,C,D,E,F$.
\[
\begin{array}{c|c|c|c}
\text{old role}&(L_{GA},L_{GB},L_{GC},L_{GD},L_{GE},L_{GF})
 &L_{GG}&\text{number}\\\hline
A&(1,2,1,1,1,1)&1,2,3,4&4\\
B&(2,4,3,4,4,1)&4&1\\
C&(1,3,1,2,2,1)&1,2,3,4&4\\
D&(1,4,2,3,3,1)&3,4&2\\
E&(1,4,2,3,4,1)&4&1\\
F&(1,1,1,1,1,k)&\delta, 1\le k\le\delta\le4&10\\
D&(1,4,2,4,3,1)&4&1
\end{array}
\]
For each neutral profile there are explicit partners such that its
extended quadratic in strategy $k$ is exactly
\begin{equation}\label{eq:lower-neutral-merge}
 Q_k^{\rm ext}(a,s)=Q_k(a+se_{\rm role})+\kappa_k s^2,
 \qquad\kappa_k\ge0.
\end{equation}
These are identities of rational quadratic matrices. Thus every
neutral one-block extension is excluded exactly by
\eqref{eq:lower-mixed-support}, not merely to first order.
For a nonneutral profile, positive definiteness on the old mass
coordinates and its positive linear $s$ term imply
$\sum_k\lambda_kQ_k^{\rm ext}(a,s)\ge T$ in a sufficiently small
neighborhood. Cross terms are absorbed by the positive old quadratic
and the linear $s$ margin. This proves the one-block assertion.

\subsection{Two added blocks}
For two neutral profiles of masses $s,t$, extend the same three flows
by their designated partners independently. Their mixed quadratic is
\[
 Q_0(a+se_i+te_j)+\kappa s^2+\kappa't^2+2Kst,
 \qquad Q_0(x)=x^TH_0x,\quad\kappa,\kappa'\ge0.
\]
There are 979 unordered neutral-profile/contact-label cases after
capping the new mutual contact. Exact checks find 453 cases with
$K=0$, 469 with $K>0$, and 57 with $K<0$. The first two classes
are excluded by \eqref{eq:lower-mixed-support}. In every negative
case, at least one active flow admits a strictly positive
first-order increase in noncritical density: match equal portions of the two new blocks
to each other instead of their designated old partners. Its weighted
first-order gain is greater than $19/500$ per unit of this matched
mass, verified by rational intervals. At old masses near $a^*$,
this remains a positive margin. Taking a portion of size
$\min(s,t)$ gains a positive constant times $\min(s,t)$, while
$|2Kst|\le2|K|(s+t)\min(s,t)$ and all quadratic changes of the
response have the same smaller order. This excludes an improvement
for sufficiently small $s+t$.

If at least one new profile is nonneutral, its positive linear mass
term similarly absorbs all mixed terms involving it. A neutral
profile's remaining contribution is controlled by
\eqref{eq:lower-neutral-merge}. Since the profile lists are finite,
a single sufficiently small neighborhood works for all cases.
This completes the proof of Theorem~\ref{thm:lower-local-one-two}.

The exact checks for this section are
\path{verify_atom_stability.py} and
\path{check_neutral_pairs_exact.py}, with data in
\path{atom-certificate.json}, all in the
\texttt{lower-certificate} directory. They reconstruct the selected flow
extensions directly from $L$ and $P_k$, check all quadratic identities,
and use the exact algebraic root intervals for every strict sign.
The theorem allows new labels and simultaneous old-mass changes, but
keeps all old label cells fixed and is local in the added mass.
It does not exclude improvements from larger changes in the masses
or from changes to labels between the six existing parts.