\subsection{Projected directions after the slow decay}
\label{sec:small-section-delayed-directions}

The endpoint of the slow interval must also supply controlled
ordinary derivatives in the projective coordinates. The
denominator $A=1+p$ is of order $a^2$, so its relative
derivative requires more information than a bound on the
full physical tangent. The constraint expresses this
derivative through the shear response and the product $bL$.
The latter remains small because the wall energy absorbs
the time growth of the logarithmic response.

We use the positive-wall family of
Theorem~\ref{thm:small-section-slow-variation} and its notation
$z(t,\rho)=(p,s,n,x,y)$, $b$, $L$, and $L_0$.
Throughout this subsection, a prime denotes the ordinary
derivative in $\rho$, with $h,a,q,D,\zeta,\nu$ and both
times $t_0,T$ fixed. Set
\begin{equation}
 Y(t,\rho)=(M,U,V,W)(t,\rho)
             =\frac{\sqrt3}{A(t,\rho)}(s,n,x,y)(t,\rho).
 \label{eq:delayed-projective-family}
\end{equation}
The positive height, shear, and walls make all four
coordinates positive.

\begin{lemma}
\label{lem:delayed-weighted-energy}
Let $C\geq32$ be the absolute constant in
Theorem~\ref{thm:small-section-slow-variation}. For the same
family and sufficiently small $a$, put
$K=13+20Cq+D$. Then, for $0\leq t\leq T$,
\begin{equation}
 \begin{split}
 b(t)&\leq5a^2,\qquad tb(t)\leq(3+D)a,\\
 (L_0+t)b(t)&\leq Ka,\qquad b(t)L(t)\leq20Ka.
 \end{split}
 \label{eq:delayed-weighted-energy}
\end{equation}
\end{lemma}

\begin{proof}
The regional bounds and the wall rates give
\[
 n(t)\leq a^2e^{-2t},\qquad
 x(t)\leq a^2,\qquad y(t)\leq ae^{-at/2}.
\]
Since $n+x+y\leq3a$, it follows that
\begin{equation}
 b(t)\leq3a^3e^{-2t}+a^4+a^2e^{-at}.
 \label{eq:delayed-energy-envelope}
\end{equation}
For $a\leq1$, this is at most $5a^2$.
The elementary bound $u e^{-u}\leq1$ for $u\geq0$ gives
\[
 t(3a^3e^{-2t})\leq\frac32a^3,\qquad
 t(a^2e^{-at})\leq a.
\]
Also, since $t\leq T=(D/a)\log(1/a)$ and
$a\log(1/a)\leq1$,
\[
 ta^4\leq Da^2\,a\log(1/a)\leq Da.
\]
Multiplying \eqref{eq:delayed-energy-envelope} by $t$
proves the second bound in
\eqref{eq:delayed-weighted-energy}.

By \eqref{eq:slow-variation-actual-bounds},
$L_0\leq2+4Cq|\log a|$. Thus
\[
 L_0b(t)\leq5a^2(2+4Cq|\log a|)
                         \leq(10+20Cq)a.
\]
Adding the estimate for $tb(t)$ gives
$(L_0+t)b(t)\leq Ka$. Finally,
$L(t)\leq2L_0+20t\leq20(L_0+t)$, which proves the last
bound.
\end{proof}

\begin{proposition}
\label{prop:delayed-projective-directions}
For each $q>2$ and $D\geq0$, there is a constant
$C_{q,D}>0$, independent of $h,P$ and the weak-wall labels,
such that the following holds under the thresholds of
Theorem~\ref{thm:small-section-slow-variation}.
Throughout $0\leq t\leq T$, the projective family has
ordinary derivatives and
\begin{align}
 |(\log A)'|+|(\log M)'|
       &\leq\frac{C_{q,D}}a,
       \label{eq:delayed-height-shear-log}\\
 |(\log U)'|+|(\log V)'|+|(\log W)'|
       &\leq C_{q,D}\left(\frac1a+L_0+t\right),
       \label{eq:delayed-wall-logs}\\
 |M'|&\leq\frac{C_{q,D}}{a^2}.
       \label{eq:delayed-shear-direction}
\end{align}
\end{proposition}

\begin{proof}
The ordinary physical derivatives and the positive height
give ordinary derivatives of the quotient in
\eqref{eq:delayed-projective-family}. Their logarithms
satisfy the exact identities
\begin{equation}
 \begin{aligned}
 (\log A)'&=\frac{p'}A,
       &(\log M)'&=\frac{s'}s-\frac{p'}A,\\
 (\log U)'&=\frac{n'}n-\frac{p'}A,
       &(\log V)'&=\frac{x'}x-\frac{p'}A,\\
 (\log W)'&=\frac{y'}y-\frac{p'}A.
 \end{aligned}
 \label{eq:delayed-log-identities}
\end{equation}
They hold on an ordinary neighborhood of each initial
parameter, including the endpoints of $[a,2a]$.

The polynomial bound
\eqref{eq:slow-variation-polynomial-bounds} and
Lemma~\ref{lem:delayed-weighted-energy} imply
\[
 |Q'|\leq4bL\leq80Ka.
\]
Differentiating the constraint yields
\[
 (-p)p'=ss'+\frac12Q'.
\]
Use $-p\geq1/2$, $s\leq3a$, and $|s'|\leq5$ to obtain
$|p'|\leq(30+80K)a$. Since $A\geq a^2/10$ and
$s\geq a/2$,
\begin{equation*}
 \left|\frac{p'}A\right|\leq\frac{300+800K}{a},
       \qquad \left|\frac{s'}s\right|\leq\frac{10}{a}.
\end{equation*}
The first two identities in
\eqref{eq:delayed-log-identities} prove
\eqref{eq:delayed-height-shear-log} with a constant depending
only on $q,D$.

Each of $|n'/n|$, $|x'/x|$, and $|y'/y|$ is at most
$L(t)\leq2L_0+20t$. Thus the sum of the last three
logarithmic derivatives is at most
\[
 6L_0+60t+\frac{900+2400K}{a}.
\]
Enlarging the same constant proves
\eqref{eq:delayed-wall-logs}.
Finally, the regional height and shear bounds imply
$0<M\leq60/a$. The identity $M'=M(\log M)'$ therefore
gives \eqref{eq:delayed-shear-direction}, after one more
enlargement of $C_{q,D}$.
\end{proof}

At the endpoint write
\begin{equation}
 \begin{gathered}
 Y(\rho)=Y(T,\rho)=(M,U,V,W)(\rho),\\
 k(\rho)=\sqrt{(M(\rho)-1)^2+V(\rho)^2},\qquad
 \Lambda=1+|\log a|.
 \end{gathered}
 \label{eq:delayed-entrance-family}
\end{equation}
The parameter $k$ is the radius of the growing
core circle. It converts the preceding bounds in inverse
powers of $a$ into the natural scales of the core orbit.

\begin{theorem}
\label{thm:delayed-entrance-inputs}
There exists $h_*>0$ such that, for every $0<h\leq h_*$,
$q>2$, and $D>2$, there is $B>0$ with the following
property. For every $P>6D+2$ and $0<\delta\leq10^{-4}$,
all sufficiently small $a>0$ give, uniformly over
\eqref{eq:slow-variation-label-domain}, an ordinarily
differentiable endpoint family \eqref{eq:delayed-entrance-family}
such that
\begin{equation}
 \begin{gathered}
 \frac1{24a}\leq k\leq\frac{64}{a},\qquad k\geq8,
       \qquad M>1,\qquad 0<V<1,\\
 0<U,W\leq\delta,\qquad \Lambda(U+W)\leq1.
 \end{gathered}
 \label{eq:delayed-entrance-inputs}
\end{equation}
The ordinary directions obey
\begin{equation}
 |M'|\leq Bk^2,\qquad
 |U'|\leq B\Lambda kU,\qquad
 |V'|\leq B\Lambda kV,\qquad
 |W'|\leq B\Lambda kW.
 \label{eq:delayed-entrance-directions}
\end{equation}
The constant $B$ is chosen before $P$ and $\delta$.
\end{theorem}

\begin{proof}
Choose $h_*$ to meet the thresholds of the preceding
results and keep $a\leq1/256$.
The height and shear bounds give
\begin{equation}
 \frac1{12a}\leq M=\frac{\sqrt3s}{A}\leq\frac{60}{a}.
 \label{eq:delayed-shear-scale}
\end{equation}
In particular $M>1$. The bound $0<V<1$ follows from
Theorem~\ref{thm:small-section-slow-decay} and positivity
of the walls. For the core radius,
\[
 M-1\leq k\leq(M-1)+V<M.
\]
Since $1/(24a)\geq1$, the lower bound in
\eqref{eq:delayed-shear-scale} implies $k\geq1/(24a)$.
This is at least $8$ when $a\leq1/256$; the upper bound
gives $k\leq64/a$.

The endpoint wall powers
\eqref{eq:slow-decay-projective-endpoint} imply
\[
 U+W\leq20ha^{q-2}+80a^{D/2-1}.
\]
Both exponents are positive because $q>2$ and $D>2$.
For every $r>0$,
$a^r(1+|\log a|)\to0$ as $a\downarrow0$.
We may therefore decrease the threshold for $a$ so that
\begin{equation}
 \bigl(20ha^{q-2}+80a^{D/2-1}\bigr)\Lambda\leq\delta.
 \label{eq:delayed-entrance-wall-scale}
\end{equation}
It follows that $\Lambda(U+W)\leq\delta\leq1$ and,
since $\Lambda\geq1$, each of $U,W$ is at most $\delta$.
Their positivity follows from the positive initial walls.

It remains to put the derivatives in the core scale.
Let $N=3+4Cq+D$, where $C$ is the absolute constant in
\eqref{eq:slow-variation-actual-bounds}.
The identities $aT=D|\log a|$ and $a\leq1$ give
\begin{equation*}
 \frac1a+L_0+T\leq\frac{N\Lambda}{a}.
\end{equation*}
Indeed, multiply by $a$ and use
$aL_0\leq2a+4Cqa|\log a|\leq2+4Cq|\log a|$.
Proposition~\ref{prop:delayed-projective-directions}
therefore gives
\[
 |M'|\leq\frac{C_{q,D}}{a^2},\qquad
 |(\log U)'|,|(\log V)'|,|(\log W)'|
                       \leq\frac{C_{q,D}N\Lambda}{a}.
\]
Use $1/a\leq24k$ and multiply the logarithmic estimates
by their respective positive coordinates. Choosing $B$
to dominate $24^2C_{q,D}$ and $24C_{q,D}N$ proves
\eqref{eq:delayed-entrance-directions}.
This choice depends only on the already fixed constants
and $q,D$, so it can be made before $P,\delta$ and all
subsequent scale choices.
\end{proof}

The logarithmic factor in the wall directions depends
only on the radial scale $a$. The endpoint estimates
therefore remain uniform as the two positive initial
weak walls become arbitrarily unequal. The smallness in
\eqref{eq:delayed-entrance-wall-scale} preserves room for
this logarithmic factor when the orbit is followed to
the upward core entrance.