\subsection{The phase at the delayed core entrance}
\label{sec:small-section-delayed-entrance}

At the end of the long decay interval, the active wall still lies
below the core section. The remaining incoming passage brings it
to $V=1$. Its effect on the radius is controlled by the small
transverse walls. These same walls absorb the logarithmic factor
in their initial derivatives, so the phase remains close to the
original section phase even when the active wall is arbitrarily small.

Write $\mathcal P$ for the first upward arrival of
Proposition~\ref{prop:incoming-passage}, and use
$r(y)=H(M-1,V)$ for the core radius. We first give the
direction estimate needed to apply this passage to the delayed
family.

\begin{lemma}[Weighted incoming directions]
\label{lem:delayed-weighted-entrance}
There are universal constants $A>0$ and
$0<\delta\leq10^{-4}$ with the following property.
Let $Y=(M,U,V,W)$ be an ordinarily differentiable curve at $\rho$,
and write
\[
 Y(\rho)=(m,u,v,w),\qquad k=r(Y(\rho)).
\]
Suppose that $B\geq0$, $\Lambda\geq1$, and
\begin{gather}
 k\geq8,\qquad m\geq1,\qquad
 0<u,w\leq\delta,\qquad 0<v<1,\qquad
 \Lambda(u+w)\leq1,\notag\\
 |M'|\leq Bk^2,\qquad
 |U'|\leq B\Lambda ku,\quad
 |V'|\leq B\Lambda kv,\quad
 |W'|\leq B\Lambda kw.
 \label{eq:delayed-weighted-entrance-input}
\end{gather}
Then $K=r\circ Y$ and $\kappa=r\circ\mathcal P\circ Y$
are ordinarily differentiable at $\rho$, and
\begin{equation}
 |\kappa(\rho)-k|\leq\frac Ak,\qquad
 |\kappa'(\rho)-K'(\rho)|\leq ABk.
 \label{eq:delayed-weighted-entrance-output}
\end{equation}
\end{lemma}

\begin{proof}
Take the wall threshold from
Proposition~\ref{prop:entrance-ordinary-direction}, and apply that
proposition to $Y'(\rho)$ with direction constant $B\Lambda$.
The $M$ component satisfies the enlarged bound because
$\Lambda\geq1$. With
\[
 b=u+w,\qquad J=uv+v^2b^2,
\]
the proposition gives, for a universal $A_0>0$,
\[
 |\kappa-k|\leq\frac{A_0}k(b^2+J),\qquad
 |\kappa'-K'|\leq A_0B\Lambda(b^2+kJ).
\]
Now $0\leq b\leq1$ and $J\leq b+b^2$. Moreover,
\[
 \Lambda b^2\leq1,\qquad \Lambda J\leq2.
\]
Consequently $b^2+J\leq3$ and
$\Lambda(b^2+kJ)\leq1+2k\leq3k$.
Taking $A=3A_0$ proves both estimates. Differentiability follows
from the ordinary local arrival map and the chain rule.
\end{proof}

Return to the positive shear family with
\begin{gather*}
 q>2,\qquad D>2,\qquad P>6D+2,\qquad
 t_0=\frac q2\log(1/a),\quad T=\frac Da\log(1/a),\\
 0\leq\varepsilon\leq1/128,\qquad
 0<\zeta\leq\min(a^P,2\varepsilon a),\qquad
 0<\nu\leq2\varepsilon a,\qquad a\leq\rho\leq2a.
\end{gather*}
Let $Y(\rho)$ be the projective coordinates of
$\phi_{t_0+T}(\Xi_h(\rho,\zeta,\nu))$, and put
\begin{equation}
 k(\rho)=r(Y(\rho)),\qquad
 \kappa(\rho)=r(\mathcal P(Y(\rho))).
 \label{eq:delayed-entrance-radii}
\end{equation}
All parameters other than $\rho$, including the two elapsed
times, remain fixed in derivatives.

\begin{proposition}[Radius at the upward arrival]
\label{prop:delayed-entrance-radius}
There is $h_*>0$ such that, for each
$0<h\leq h_*$, $q>2$ and $D>2$, there is a constant $C>0$
with the following property. For every $P>6D+2$, all sufficiently
small $a>0$ satisfy the conclusions below uniformly over the
stated labels and $\rho\in[a,2a]$.

The first upward arrival from $Y(\rho)$ exists. Its time and state
are $C^2$ on an ordinary neighborhood of this input. The radii
in~\eqref{eq:delayed-entrance-radii} are ordinarily
differentiable and satisfy
\begin{gather}
 \frac1{48a}\leq\kappa\leq\frac{128}a,\qquad
 k\geq\frac1{48a},\notag\\
 |\kappa-k|\leq Ca,\qquad
 |\kappa'-k'|\leq\frac Ca,\qquad
 |k'|\leq\frac{6144}{a^2}.
 \label{eq:delayed-entrance-radius}
\end{gather}
The constant $C$ is chosen before $P$.
\end{proposition}

\begin{proof}
Choose $A,\delta$ from Lemma~\ref{lem:delayed-weighted-entrance}.
Theorem~\ref{thm:delayed-entrance-inputs}, applied with this
$\delta$, gives
\[
 k\geq8,\qquad \frac1{24a}\leq k\leq\frac{64}a,
 \qquad M>1,\qquad 0<U,W\leq\delta,\quad 0<V<1.
\]
Thus Proposition~\ref{prop:incoming-passage} supplies the actual
arrival and its local $C^2$ time and state maps. Throughout this
incoming passage the radius stays within $k/100$ of $k$; in
particular,
\[
 \frac{99}{100}k\leq\kappa\leq\frac{101}{100}k.
\]
This proves the radius bounds in~\eqref{eq:delayed-entrance-radius}.

The same input theorem gives~\eqref{eq:delayed-weighted-entrance-input}
with $\Lambda=1+|\log a|$ and a constant $B=B_{h,q,D}>0$.
The weighted estimate therefore yields
\[
 |\kappa-k|\leq24Aa,\qquad
 |\kappa'-k'|\leq\frac{64AB}a.
\]
Both constants are independent of $P$.

To bound $k'$, use the projected phase $f=2/k$ from
Theorem~\ref{thm:delayed-projection-phase}. Its bounds give
\[
 f\geq a/32,\qquad |f'|\leq3.
\]
Since $k=2/f$ on an ordinary neighborhood of the parameter,
\[
 |k'|=\frac{2|f'|}{f^2}\leq\frac{6144}{a^2}.
\]
Taking $C=24A+64AB+1$ completes the proof after the common
restriction on $a$.
\end{proof}

Define the phase at this upward arrival by
\begin{equation}
 \Phi(\rho)=\frac2{\kappa(\rho)},\qquad
 f_0(\rho)=\Phi_h(\rho,\zeta,\nu).
 \label{eq:delayed-entrance-phase-definition}
\end{equation}
The small radius displacement in the preceding proposition also
controls its reciprocal derivative. The lower radius bounds are
essential here: both reciprocal factors are of order $a$.

\begin{theorem}[Phase and radial speed at the delayed entrance]
\label{thm:delayed-entrance-phase}
For a sufficiently small universal $h_*>0$ and each
$0<h\leq h_*$, $q>2$ and $D>2$, there is $C=C_{h,q,D}>0$
such that, for every $P>6D+2$ and all sufficiently small $a>0$,
the actual arrival above satisfies
\begin{gather}
 |\Phi-f_0|\leq Ca^2,\qquad
 |\Phi'-f_0'|\leq Ca,\notag\\
 \frac1{4\sqrt3}\leq\Phi'\leq\frac2{\sqrt3}+1.
 \label{eq:delayed-entrance-phase}
\end{gather}
These are ordinary derivatives, and the estimates are uniform
over the stated labels and $\rho\in[a,2a]$. In addition,
\begin{equation}
 \frac1{48a}\leq\kappa\leq\frac{128}a,\qquad
 \frac1{36864a^2}\leq-\kappa'\leq\frac{24576}{a^2}.
 \label{eq:delayed-entrance-speed}
\end{equation}
\end{theorem}

\begin{proof}
Write $B$ for the constant in
Proposition~\ref{prop:delayed-entrance-radius}, and assume $a\leq1$.
Set $e=\kappa-k$ and $f=2/k$. The preceding proposition gives
\[
 |e|\leq B,\qquad |e'|\leq B/a,\qquad
 |k'|\leq6144/a^2,\qquad \kappa,k\geq1/(48a).
\]
The reciprocal identities
\[
 \Phi-f=-\frac{2e}{\kappa k},\qquad
 (\Phi-f)'=-\frac{2e'}{\kappa^2}
       +\frac{2e(\kappa+k)k'}{k^2\kappa^2}
\]
then imply
\begin{equation}
 |\Phi-f|\leq2\cdot48^2Ba^2,\qquad
 |\Phi'-f'|\leq A_1a,\qquad
 A_1=2\cdot48^2B+4\cdot6144\cdot48^3B.
 \label{eq:delayed-entrance-reciprocal-error}
\end{equation}
For the second term in the derivative, use
$(\kappa+k)/(k^2\kappa^2)
=1/(k^2\kappa)+1/(k\kappa^2)\leq2(48a)^3$.

Let $K$ be the phase comparison constant in
Theorem~\ref{thm:delayed-projection-phase}. Adding its value and
derivative errors to~\eqref{eq:delayed-entrance-reciprocal-error}
proves the first two bounds in~\eqref{eq:delayed-entrance-phase}
with $C=A_1+K$. Choose $a$ still smaller so that
$Ca\leq1/(4\sqrt3)$. The reference slope from
Proposition~\ref{prop:small-section-reference-slope} is
\[
 \frac1{2\sqrt3}\leq f_0'\leq\frac2{\sqrt3}.
\]
It gives the asserted bounds on $\Phi'$.

Finally, the radius bounds imply
$a/64\leq\Phi\leq96a$, while the slope bounds imply
$1/8\leq\Phi'\leq3$. Differentiating $\kappa=2/\Phi$ gives
\[
 -\kappa'=\frac{2\Phi'}{\Phi^2}.
\]
Substitution proves the two constants in
\eqref{eq:delayed-entrance-speed}.
\end{proof}

Thus the original shear coordinate parametrizes the actual upward
entrance with a positive phase slope and a decreasing radius.
The latter has the scale $a^{-2}$ needed to compare transverse
wall responses with radial motion through the core.