\subsection{Amplitudes and directions at the delayed entrance}
\label{sec:small-section-delayed-core}

The upward arrival has a nonzero radial response. To continue
through the core, its transverse amplitudes must also be small
relative to the radius, and its tangent must lie in the core
cone. Both properties follow from the wall powers accumulated
during the slow decay. The bound $U+W=o(a)$ supplies the
core amplitude condition, while the square roots of the walls absorb
the logarithmic loss in their derivatives.

Keep the endpoint family $Y=(M,U,V,W)$ and its upward arrival
from the preceding subsection. Write
\begin{equation}
 Z=\mathcal P\circ Y=(M_+,U_+,1,W_+),\qquad
 \kappa=r(Z),\qquad \Lambda=1+|\log a|.
 \label{eq:delayed-core-family}
\end{equation}
The labels satisfy
\begin{equation}
 \begin{gathered}
 0\leq\varepsilon\leq1/128,\qquad
 0<\zeta\leq\min(a^P,2\varepsilon a),\qquad
 0<\nu\leq2\varepsilon a,\\
 a\leq\rho\leq2a,\qquad
 t_0=\frac q2\log(1/a),\quad T=\frac Da\log(1/a).
 \end{gathered}
 \label{eq:delayed-core-labels}
\end{equation}
All derivatives are ordinary derivatives in $\rho$, with the
other parameters fixed.

The endpoint estimates give
\begin{equation}
 U\leq20ha^{q-2},\qquad
 V\leq40\zeta a^{-6D-2},\qquad
 W\leq80a^{D/2-1}.
 \label{eq:delayed-core-wall-powers}
\end{equation}
In the last inequality we have used $\varepsilon\leq1$.
The following two consequences have distinct uses.

\begin{lemma}
\label{lem:delayed-core-wall-smallness}
For $q>2$, $D>2$, and $P>6D+2$, each $\delta>0$
admits a sufficiently small scale threshold such that
\begin{equation}
 \Lambda\sqrt{U+W}\leq\delta.
 \label{eq:delayed-core-weighted-root}
\end{equation}
If $q>3$ and $D>4$, then each $\eta>0$ also admits
a sufficiently small threshold such that
\begin{equation}
 U+VW\leq\frac{\eta a}{128}
                    \leq\frac{\eta}{2k},\qquad k=r(Y).
 \label{eq:delayed-core-incoming-budget}
\end{equation}
The thresholds are uniform over~\eqref{eq:delayed-core-labels}.
\end{lemma}

\begin{proof}
Put $b_a=20ha^{q-2}+80a^{D/2-1}$. For $h\geq0$,
squaring the nonnegative right-hand side verifies
\[
 \sqrt{b_a}\leq(20h+1)a^{(q-2)/2}
                         +80a^{D/4-1/2}.
\]
Both exponents are positive. The limit
$a^r(1+|\log a|)\to0$ for $r>0$ therefore gives
$\Lambda\sqrt{b_a}\leq\delta$ for all sufficiently small
$a>0$. Since $U+W\leq b_a$, this proves
\eqref{eq:delayed-core-weighted-root}.

Under the stronger exponent assumptions,
\[
 \frac{b_a}{a}=20ha^{q-3}+80a^{D/2-2}\longrightarrow0.
\]
Thus $b_a\leq\eta a/128$. Use $V<1$ and
$k\leq64/a$ from Theorem~\ref{thm:delayed-entrance-inputs}
to obtain~\eqref{eq:delayed-core-incoming-budget}.
All bounds used to select the thresholds are independent
of the labels.
\end{proof}

\begin{theorem}
\label{thm:delayed-core-admission}
There is $h_*>0$ such that, for each $0<h\leq h_*$,
$q>3$, $D>4$, $P>6D+2$, $0<\eta\leq10^{-4}$, and
$\theta>0$, all sufficiently small $a>0$ have the
following properties uniformly over~\eqref{eq:delayed-core-labels}.

The actual upward arrival $Z$ lies on the incoming boundary
of the closed core region with reference radius $\kappa$,
and
\begin{gather}
 \kappa\geq6,\qquad M_+>1,\qquad
 z(Z)=-\operatorname{arcosh}\kappa,\notag\\
 0<U_+\leq20ha^{q-2},\qquad
 0<W_+\leq80a^{D/2-1},\qquad
 U_++W_+\leq\frac{\eta}{\kappa}.
 \label{eq:delayed-core-admission}
\end{gather}
For every real $d\leq\min(q-2,D/2-1)$,
\begin{equation}
 U_++W_+\leq(20h+80)a^d.
 \label{eq:delayed-core-common-power}
\end{equation}
The curve $Z$ is ordinarily differentiable, $\kappa'<0$,
and its tangent satisfies
\begin{equation}
 |U_+'|\leq\theta\sqrt{U_+}\,|\kappa'|,\qquad
 |W_+'|\leq\theta\sqrt{W_+}\,|\kappa'|.
 \label{eq:delayed-core-narrow-cone}
\end{equation}
\end{theorem}

\begin{proof}
Take $h_*$ small enough for the endpoint and upward arrival
estimates. The endpoint has $k\geq8$, $M>1$, $0<V<1$,
and $0<U,W\leq10^{-4}$.
Lemma~\ref{lem:delayed-core-wall-smallness} supplies
$U+VW\leq\eta/(2k)$.
Lemma~\ref{lem:entrance-core-smallness} now places $Z$ on
the incoming core boundary, with
\[
 \kappa\geq6,\qquad U_+\leq U,\qquad W_+\leq VW,
 \qquad U_++W_+\leq\eta/\kappa.
\]
It also gives the angle in~\eqref{eq:delayed-core-admission}.
The walls are positive by their multiplicative equations.
Combining these inequalities with
\eqref{eq:delayed-core-wall-powers} proves the amplitude
bounds. For $0<a\leq1$ and the stated $d$, both wall
powers are at most $a^d$, proving
\eqref{eq:delayed-core-common-power} without any additional
restriction depending on $d$.

For the tangent estimate, choose the universal wall threshold
and constant $C$ in
Proposition~\ref{prop:entrance-ordinary-direction}.
Theorem~\ref{thm:delayed-entrance-inputs} supplies its direction
hypotheses with constant $B\Lambda$: the $M$ derivative
bound may be enlarged by $\Lambda\geq1$.
The two output wall estimates give
\begin{align*}
 |U_+'|&\leq CB\Lambda k^2\sqrt{UV}\sqrt{U_+},\\
 |W_+'|&\leq CB\Lambda k\sqrt{WV}\sqrt{W_+}.
\end{align*}
Since $V<1$ and $k\geq8$, both square-root factors
at the input are bounded by $\sqrt{U+W}$, and the
second power of $k$ may replace the first in the second
line. The radial speed from
Theorem~\ref{thm:delayed-entrance-phase} gives
\begin{equation*}
 k^2\leq\frac{64^2}{a^2}
          \leq R_0|\kappa'|,\qquad R_0=64^2\cdot36864.
\end{equation*}
Apply~\eqref{eq:delayed-core-weighted-root} with
$\delta=\theta/(R_0CB)$. Substitution proves both
inequalities in~\eqref{eq:delayed-core-narrow-cone}.
Ordinary differentiability follows by composition with
the $C^2$ arrival map; the sign of $\kappa'$
is already part of Theorem~\ref{thm:delayed-entrance-phase}.
\end{proof}

The amplitude and direction thresholds may therefore be chosen
together. Taking $\eta$ to be a core threshold and $\theta=1$
puts the delayed family directly in the domain of the full
core return and its invariant cone.