\subsection{Late entrances into a compact core range}
\label{sec:late-physical-core}

At the growing-wall entrance just obtained, the other walls are
quadratically small in $\Delta$. A projection preserving the dominant
wall height puts this state near a compact family of exact single-wall
solutions. Continuous dependence on initial data over a finite time
interval then gives a core entrance on the original orbit.

Suppose $q=(p,s,n,b,c)\in\M$, $0<n\leq1/4$, and
$\mathcal P_\Delta(q)$ holds with $\Delta\leq n/8$. Define
\begin{equation}
 \ell=\sqrt{\frac{1-n^2}{p^2+s^2}},\qquad
 \bar q=(\ell p,\ell s,n,0,0).
 \label{eq:single-wall-physical-projection}
\end{equation}
The constraint gives
\[
 0\leq p^2+s^2-(1-n^2)
      =2n(b+c)-(b-c)^2\leq\Delta^2,
 \qquad 15/16\leq p^2+s^2\leq2.
\]
Here nonnegativity follows from $b+c\leq n/128\leq2n$.
Thus $0\leq\ell\leq1$, the projected state has constraint one,
and $1-\ell\leq2\Delta^2$. Since $|p|,|s|\leq2$,
\begin{equation}
 \begin{split}
 |\ell p-p|,\ |\ell s-s|&\leq4\Delta^2,\\
 \dist(\bar q,q)&\leq2\Delta^2/n,\\
 |\lambda_1(\bar q)-\lambda_1(q)|&\leq9\Delta^2.
 \end{split}
 \label{eq:single-wall-projection-errors}
\end{equation}
For the distance bound, each of the four nonzero coordinate errors
is at most $\Delta^2/n$, using $n\leq1/4$ and
$b+c\leq\Delta^2/(2n)$. For the last inequality use the exact identity
\[
 \lambda_1(\bar q)-\lambda_1(q)
       =p^2+s^2-(1-n^2)+2(\ell p-p).
\]

Suppose in addition that $\lambda_1(q)\geq c_0>0$, every Taub
distance is at least $\epsilon$, and
\begin{equation}
 n^2\leq c_0/4,\quad \Delta^2\leq c_0/18,\quad
 n\leq\epsilon^2/32,\quad 2\Delta^2/n\leq\epsilon/2.
 \label{eq:single-wall-reference-smallness}
\end{equation}
Then $\lambda_1(\bar q)\geq c_0/2$ and $\bar p>1/2$.
Order the two stable axes so that the transformed state is
\[
 z=\left(-\frac{\bar p+\sqrt3|\bar s|}{2},
          \frac{\sqrt3\bar p-|\bar s|}{2},0,n,0\right).
\]
Its height $A=1+z_p$ is positive. If its projective coordinates are
$(M,0,v,0)$ and $k^2=(M-1)^2+v^2$, direct substitution gives
\begin{equation}
 (M-1)A=2\bar p-1>0,\qquad
 k^2 A=4A+4\sqrt3|\bar s|.
 \label{eq:ordered-single-wall-identities}
\end{equation}
In particular $k\geq2$. Axis permutations preserve Euclidean
distance. Since $\bar q$ is at least $\epsilon/2$ from all Taub
points and $\dist(z,T_1)^2=2A$, we have $A\geq\epsilon^2/8$.
Also $A\leq2$, and $(3+M^2+v^2)A=6$. These formulas imply
\begin{equation}
 2\leq k<8/\epsilon,\qquad n/2\leq v\leq1/2,
 \qquad (M,0,v,0)=(1+\sqrt{k^2-v^2},0,v,0).
 \label{eq:physical-compact-reference}
\end{equation}
For the upper bound on $v$, use $v=\sqrt3n/A$ and
$n\leq\epsilon^2/32$. For $k$, use $k^2<3+M^2+v^2$.

\begin{lemma}
\label{lem:compact-physical-core}
Fix $K$ and $a>0$. There are $\delta>0$ and $C\geq1$ such that
every constrained physical state within Euclidean distance $e\leq\delta$
of the reference family
\[
 \mathcal Q(k,v)=\operatorname{proj}^{-1}
       (1+\sqrt{k^2-v^2},0,v,0),\qquad
       2\leq k\leq K,\quad a\leq v\leq1/2,
\]
has a first positive upward $V=1$ arrival in this frame, at a time
$T\leq C$. Its endpoint satisfies $A>0$, $M>1$,
\begin{equation}
 |\sqrt{(M-1)^2+1}-k|\leq Ce,
 \qquad N_2\geq n_K:=\frac{\sqrt3}{4+(1+K)^2}.
 \label{eq:compact-physical-core-estimate}
\end{equation}
The arrival is transverse, and its time and state are locally $C^2$
in ordinary physical initial coordinates.
\end{lemma}

\begin{proof}
Each reference solution reaches $V=1$ with $M=1+\sqrt{k^2-1}>1$.
The entrance construction of Sections~\ref{sec:entrance-phase-transport}
and~\ref{sec:entrance-variation} gives a valid first arrival on an
ordinary neighborhood of each reference initial state. Composing
with the smooth projective coordinate maps gives $C^2$ time,
projective endpoint, physical endpoint and radius there.
The reference initial states form a compact set. Cover it by finitely
many smaller balls whose doubled balls remain in these neighborhoods.
Bound the derivatives of these four observables on the larger balls
and apply the mean value estimate along line segments. A common
positive radius and a common constant $C$ then control their values
and their differences from the reference values.

At the reference endpoint $M\geq2$ and
\[
 N_2=\frac{2\sqrt3}{4+(1+\sqrt{k^2-1})^2}\geq2n_K.
\]
Reduce $\delta$ until $C\delta\leq\min(1/2,n_K)$.
The physical wall lower bound and $M>1$ follow. The actual first
arrival, its direction, and positivity of $A$ hold on the same
neighborhoods by their construction. This proves all the assertions
for one and the same event.
\end{proof}

\begin{proposition}
\label{prop:late-small-compact-core}
Fix $\epsilon>0$. There are $N$ and $0<\gamma<1/4$ such that, for
every $B>0$, the following holds. If $q\in\M$ eventually stays at
least $\epsilon$ from all Taub points and
\begin{equation}
 N_i(\phi_tq)N_j(\phi_tq)\longrightarrow0\qquad(i\ne j),
 \label{eq:pair-product-decay}
\end{equation}
then, after every prescribed time $R$, its orbit has a positive-time
upward core entrance in some ordered frame, with
\[
 Y=(r,u,w)\in\mathcal K_{N,\gamma},\qquad
 u,w>0,\quad u+w\leq B.
\]
The physical core state satisfies uniform Taub avoidance at distance
$\epsilon$.
\end{proposition}

\begin{proof}
Choose $N,\gamma,B_0$ from
Proposition~\ref{prop:avoidance-compact-phase}.
Choose the growth constants $\nu_0,c_0$ of
Lemma~\ref{lem:away-wall-growth-gap}, and fix
\[
 0<\nu\leq\min\{\nu_0,c_0/4,\epsilon^2/32\}.
\]
After a sufficiently late time $s_0\geq\max(0,R)$, avoidance holds
and $\mathcal P_\Delta$ holds forever, where $\Delta>0$ may be
chosen arbitrarily small before choosing $s_0$.
Proposition~\ref{prop:late-fixed-wall-entrance} gives a later growing
wall of height $\nu$. Permute it to the first axis.
For sufficiently small $\Delta$, conditions
\eqref{eq:single-wall-reference-smallness} hold. The projection and
ordering above put the actual state within $2\Delta^2/\nu$ of the
compact reference family with $K=8/\epsilon$ and $a=\nu/2$.
Lemma~\ref{lem:compact-physical-core} supplies the actual upward
core, with a radius error at most $C_1\Delta^2$ and $N_2\geq n_K$.

At this core, $V=1$ means $A=\sqrt3N_2$. Consequently
\begin{equation*}
 u+w=\frac{N_1+N_3}{N_2}
       \leq\frac{\Delta^2}{2N_2^2}
       \leq\frac{\Delta^2}{2n_K^2}.
\end{equation*}
Choose $\Delta$ further so that $C_1\Delta^2\leq\gamma/2$ and the
last bound is at most $\min(B,B_0)$. The reference radius is at least
two, so the actual radius is at least $2-\gamma/2$.
The actual endpoint inherits avoidance from the original orbit.
Proposition~\ref{prop:avoidance-compact-phase} now places it in
$\mathcal K_{N,\gamma}$. Both passages have positive elapsed time,
so their final arrival is strictly later than $R$. All frames used
are among the six fixed axis permutations.
\end{proof}