\subsection{Fixed directions with more added blocks and uncovered vertices}
For any fixed finite number of added blocks, a related statement
holds along each direction in which their masses tend to zero at
positive relative rates. We also allow a small excess of vertices
over those covered by the required matching. The threshold for the
added mass may depend on the direction; thus this statement differs
from the neighborhood valid for all one- and two-block extensions
in Theorem~\ref{thm:lower-local-one-two}.

\begin{theorem}\label{thm:lower-directional-surplus}
Keep the old six-by-six label matrix fixed and add a fixed finite collection of
blocks with fixed labels. Measure masses per $2M$, where the required
matching has $M$ edges. Suppose the new masses are
$u\alpha_s+o(u)$, with every $\alpha_s>0$ and $\sum_s\alpha_s=1$,
and the old masses are $a^*+uh+o(u)$. Assume the total mass is at least
one and $\sum_i h_i\ge-1$. For each sufficiently small fixed positive
$u$, the minimum critical-edge count divided by $\binom{2M}{2}$ is
at most $\alpha_L+o_M(1)$ in the associated finite blow-ups.
\end{theorem}

\begin{proof}
First consider total mass one. Use the active flow maps, weights,
and $T=1-\alpha_L$ from Section~\ref{sec:lower-exact}. As in the
preceding section, an increase in noncritical density provides a
matching that prevents an improvement of the lower construction.
The check allowing profiles without the capping restriction enumerates
all $4^7=16384$ new-block profiles. Of these, 16354 have the strictly
positive insertion-price margin $>19/1000$. The remaining thirty
profiles each have a \emph{parent}, namely an existing part $i$
whose incident labels are no larger than the corresponding labels
of the new part, including its internal label. The two label profiles
agree on every matching type used by each of the three active flows.
These are finite comparisons of integer labels. The identities giving the
zero weighted first-order price are checked as linear forms in the
old masses.

If a profile with positive first-order noncritical gain has positive
relative mass, that gain dominates all quadratic errors.
Suppose therefore all profiles are neutral. If a pair of new blocks
with parents $i,j$ has mutual label $k<L_{ij}$, matching equal portions
to each other instead of their designated old partners produces a
strict first-order noncritical gain in at least one active strategy.
The exact checker examines all 81 such profile-pair/label cases and
proves an unweighted gain greater than $9/100$ in an active strategy.
The strategy weight is strictly positive, as are both relative block
masses. This again prevents an improvement for small $u$.

It remains to treat neutral profiles whose mutual labels are all at
least the corresponding parent labels. Aggregate their masses with
their parents, obtaining $s$ with $\sum_i s_i=1$. Let $\beta_i$ be
the fraction of aggregate class $i$ lying in added parts. For small
$u$, all $\beta_i<1/2$ and the old active flows $P_ks$ remain feasible.
Lift each such flow so each added part inherits its parent's matching-label
distribution, using only new--old and old--old matching edges. For a
cross flow $t_{ij}$, route $\beta_it_{ij}$ from new $i$ to old $j$,
$\beta_jt_{ij}$ from old $i$ to new $j$, and
$(1-\beta_i-\beta_j)t_{ij}$ between old endpoints. For an internal
endpoint flow $t_{ii}$, match $\beta_it_{ii}$ new endpoints to the
same mass of old endpoints, and match the remaining
$(1-2\beta_i)t_{ii}$ old endpoints internally. Divide every new quota
among added parts with the same parent in proportion to their masses.
All matching-edge labels
are the same as in the parent flow. Ambient labels can only increase,
so the lifted noncritical density is at least $Q_k(s)$. Thus
\[
 Q_{\max}\ge\sum_k\lambda_kQ_k(s)=s^TH_0s\ge T.
\]
This is exact, and resolves the remaining directions at every order.

Now allow total mass larger than one. The old $B$ column is an
entrywise maximum of each old row:
\[
 L_{iB}\ge L_{ij}\qquad\text{for every old }i,j.
\]
Leave the entire surplus, of mass $u(1+\sum_i h_i)+o(u)$, uncovered
inside old block $B$. This is possible for small $u$ because $a_B^*>0$.
The remaining covered set has total mass exactly one, and the preceding
perfect-matching insertion argument applies to it.

For a strict first-order case, the old--old matching endpoints have
no critical edges to the uncovered $B$ set by the displayed
maximum-column property. Only the $O(u)$ new endpoints and the
$O(u)$ old endpoints now matched to them can contribute outward
critical edges. The uncovered set also has mass $O(u)$; hence this
extra contribution is $O(u^2)$ and cannot cancel the strict linear
improvement in the covered-set count.

In the residual neutral case, the proportional lift has no outward
critical edges at all. Every lifted old endpoint has a parent birth
at most its label to $B$. A new part with parent $i$ inherits the
parent's birth distribution and has label to $B$ at least $L_{iB}$.
Thus the exact covered-set bound remains valid after reinstating the
uncovered vertices. The normalization is unchanged because all masses
were measured per $2M$.
\end{proof}

The certificate \path{lower-certificate/verify_stationarity.py} checks
the 16384 profiles, parent domination, active-edge agreement, and all
81 strict pair responses. Its stationary root lies in the same
rational box as the exact algebraic construction of
Theorem~\ref{thm:lower-exact}. This result is directional: it does not
assert one radius uniform over varying numbers of added blocks or
relative masses tending to zero. It also keeps all labels between
the six existing parts fixed and does not concern configurations a
fixed positive distance away.