\subsection{Projection after the slow decay}
\label{sec:small-section-delayed-projection}

The physical phase at the end of the longer interval must
now be expressed in the coordinates of the growing core.
The projection divides by a height of order $a^2$.
Its radial response is controlled because the strong wall
and the decaying weak wall have both reached order $a^2$,
and their ordinary derivatives have reached order $a$.
These are precisely the scales needed to reuse the compact
comparison of Section~\ref{sec:small-section-projection}.

We keep the positive-wall family
\eqref{eq:slow-variation-label-domain}, with $q>2$,
$D>2$, and $P>6D+2$. Write its endpoint as
\begin{equation}
 z_T(\rho)=\phi_{t_0+T}\Xi_h(\rho,\zeta,\nu)
                  =(p,s,n,x,y)(\rho),\qquad
 X_T(\rho)=(n,s,x,y)(\rho).
 \label{eq:delayed-projection-state}
\end{equation}
All primes in this subsection denote ordinary derivatives in
$\rho$ with $h,a,q,D,\zeta,\nu$ fixed. In particular,
the two times
\[
 t_0=\frac q2\log\frac1a,\qquad
 T=\frac Da\log\frac1a
\]
are constant along the family.

\begin{proposition}
\label{prop:delayed-projection-endpoint}
There are constants $C\geq32$ and $h_*>0$ such that,
for every $0<h\leq h_*$, $q>2$, $D>2$, and $P>6D+2$,
the following holds for all sufficiently small $a>0$,
uniformly over \eqref{eq:slow-variation-label-domain}.
The endpoint \eqref{eq:delayed-projection-state} belongs
to \eqref{eq:slow-decay-region}, has positive walls and
positive height, and is ordinarily differentiable in
$\rho$. With
\begin{equation}
 B=13+16Cq+40D,
 \label{eq:delayed-projection-response-constant}
\end{equation}
its coordinates satisfy
\begin{equation}
 \begin{gathered}
 0<n\leq a^2,\qquad 0<y\leq a^2,\\
 |s'|\leq B,\qquad |x'|\leq B,\qquad
 |n'|\leq Ba,\qquad |y'|\leq Ba.
 \end{gathered}
 \label{eq:delayed-projection-endpoint-bounds}
\end{equation}
The scale threshold may depend on $h,q,D,P$.
\end{proposition}

\begin{proof}
Use the constant $C$ and the thresholds of
Theorem~\ref{thm:small-section-slow-variation}, decreasing
$h_*$ so that $h_*\leq1$ and
Theorem~\ref{thm:small-section-slow-decay} also applies.
The actual endpoint then has all the asserted geometric
and regularity properties. Its wall bounds give
\begin{equation}
 n\leq ha^qe^{-2T}\leq a^q,
 \qquad y\leq4\varepsilon a^{1+D/2}
                         \leq a^{1+D/2}.
 \label{eq:delayed-projection-wall-powers}
\end{equation}
Both exponents are strictly greater than two.
For any fixed $r>2$, the elementary limit
$a^{r-2}(1+|\log a|)\to0$ as $a\downarrow0$ permits
the common choice
\begin{equation}
 a^q(1+|\log a|)\leq a^2,
 \qquad a^{1+D/2}(1+|\log a|)\leq a^2.
 \label{eq:delayed-projection-log-absorption}
\end{equation}
These inequalities already prove the two wall size bounds.

At the endpoint, let $L$ be the logarithmic wall response
from \eqref{eq:slow-variation-variables}, and put
$K_0=4+8Cq+20D$. Since $aT=D|\log a|$ and $a\leq1$,
\eqref{eq:slow-variation-actual-bounds} implies
\begin{equation*}
 aL\leq K_0(1+|\log a|).
\end{equation*}
Multiplying $|n'|\leq nL$ by $a$ and using
\eqref{eq:delayed-projection-wall-powers} and
\eqref{eq:delayed-projection-log-absorption} gives
$a|n'|\leq K_0a^2$. The same argument applies to $y$.
Thus $|n'|,|y'|\leq K_0a\leq Ba$.

For $x$, the regional bound $x\leq a^2$ gives
\[
 |x'|\leq xL\leq K_0a(1+|\log a|)\leq2K_0,
\]
where $a|\log a|\leq1$ was used in the last step.
Finally $|s'|\leq5$ by the same variation theorem.
The identity $B=2K_0+5$ proves the remaining bounds.
\end{proof}

Let $A=1+p$ at the endpoint, and use the projective
coordinates \eqref{eq:projective-coordinates}. Define
\begin{equation}
 \begin{gathered}
 M=\frac{\sqrt3s}{A},\qquad V=\frac{\sqrt3x}{A},\qquad
 k=\sqrt{(M-1)^2+V^2},\\
 f_{\mathrm p}=\frac2k,\qquad
 g=\frac{\sqrt{s^2+(x-y)^2}}{\sqrt3}.
 \end{gathered}
 \label{eq:delayed-projection-phases}
\end{equation}
Here $g$ is the physical phase already controlled in
Corollary~\ref{cor:small-section-slow-physical-phase}.
The projection error admits the same scaled expression
as in \eqref{eq:projection-scaled-error}.
We use maximum norms and their induced operator norms.

\begin{proposition}
\label{prop:delayed-projection-error}
There is an absolute constant $C_{\mathrm e}\geq1$ such
that, after reducing the scale threshold in
Proposition~\ref{prop:delayed-projection-endpoint}, the
error $e=f_{\mathrm p}-g$ has an ordinary derivative and
\begin{equation}
 |e(\rho)|\leq C_{\mathrm e}a^2,\qquad
 |e'(\rho)|\leq2C_{\mathrm e}Ba
       \quad(a\leq\rho\leq2a).
 \label{eq:delayed-projection-error}
\end{equation}
The constants $C_{\mathrm e}$ and the additional scale
restriction are independent of $h,q,D,P$.
\end{proposition}

\begin{proof}
Use the scaled function $F(u,z)$ from
\eqref{eq:projection-rescaling}, now with the fixed
compact set
\[
 \mathcal K=\{z:\|z\|\leq4,\ |z_1|\geq1/2\}.
\]
The proof of \eqref{eq:projection-scaled-bounds} gives
constants $\delta>0$ and $C_{\mathrm e}\geq1$ on this
set. In particular, the function is smooth near every
$(u,z)$ with $\|u\|\leq\delta$ and $z\in\mathcal K$,
and
\begin{equation}
 \begin{split}
 |F(u,z)|&\leq C_{\mathrm e}\|u\|,\\
 |DF(u,z)(u',z')|
    &\leq C_{\mathrm e}\|u'\|
           +C_{\mathrm e}\|u\|\|z'\|.
 \end{split}
 \label{eq:delayed-projection-compact-bounds}
\end{equation}
The second term retains $\|u\|$ because $F(0,z)$
vanishes on the entire limiting slice.

At the actual endpoint take
\[
 u(\rho)=\left(a,\frac{n(\rho)}a,\frac{y(\rho)}a\right),
 \qquad
 z(\rho)=\left(\frac{s(\rho)}a,\frac{x(\rho)}a\right).
\]
The regional bounds $a/2\leq s\leq3a$ and $x\leq a^2$
place $z$ in $\mathcal K$, while
\eqref{eq:delayed-projection-endpoint-bounds} gives
\begin{equation*}
 \|u\|\leq a,\qquad \|u'\|\leq B,
                         \qquad \|z'\|\leq\frac Ba.
\end{equation*}
Choose $a\leq\delta$. Since $a$ is fixed under
differentiation, \eqref{eq:projection-scaled-error} and
\eqref{eq:delayed-projection-compact-bounds} yield
\[
 \left|\mathcal P(X_T)-g\right|\leq C_{\mathrm e}a^2,
 \qquad
 \left|\frac{\dd}{\dd\rho}
               (\mathcal P(X_T)-g)\right|
     \leq aC_{\mathrm e}\left(B+a\frac Ba\right)
     =2C_{\mathrm e}Ba.
\]

It remains to identify this smooth expression with the
actual projective phase also in its derivative.
At each initial point the chart radicand is positive,
the total time $t_0+T$ is nonnegative, and the endpoint
has $p<0$ and $1+p>0$. The three strict inequalities
persist for nearby chart parameters. The constraint
continues to hold for their actual flow values, so
\eqref{eq:projection-phase-identity} gives
$\mathcal P(X_T)=f_{\mathrm p}$ on an ordinary
neighborhood in those parameters.
The scaled expression is smooth on a neighborhood as
well. Hence this identity transfers both the value and
the ordinary derivative bounds to $e$, including at the
endpoints of $[a,2a]$.
\end{proof}

\begin{theorem}
\label{thm:delayed-projection-phase}
There exists $h_*>0$ such that, for every $0<h\leq h_*$,
$q>2$, and $D>2$, one can choose $K>0$ with the following
property. For every $P>6D+2$ and all sufficiently small
$a>0$, uniformly over \eqref{eq:slow-variation-label-domain},
the actual phase $f_{\mathrm p}$ in
\eqref{eq:delayed-projection-phases} has an ordinary
derivative and satisfies
\begin{equation}
 \begin{split}
 |f_{\mathrm p}(\rho)-\Phi_h(\rho,\zeta,\nu)|
                                  &\leq Ka^2,\\
 \left|f_{\mathrm p}'(\rho)
       -\frac{\dd}{\dd\rho}\Phi_h(\rho,\zeta,\nu)\right|
                                  &\leq Ka,\\
 \frac1{4\sqrt3}\leq f_{\mathrm p}'(\rho)
                                  &\leq\frac2{\sqrt3}+1.
 \end{split}
 \label{eq:delayed-projection-reference-comparison}
\end{equation}
The constant $K$ is chosen before $P$ and the weak-wall
labels; the scale threshold may depend on $h,q,D,P$.
\end{theorem}

\begin{proof}
Choose $h_*$ to meet the preceding propositions and
Corollary~\ref{cor:small-section-slow-physical-phase},
and require $h_*\leq1/16$. Let $K_{\mathrm r}$ be the
constant in that corollary, and put
\[
 K_{\mathrm p}=2C_{\mathrm e}B,\qquad
 K=K_{\mathrm p}+K_{\mathrm r}.
\]
These constants are selected after $h,q,D$ and before
$P$. Since $B\geq1$, Proposition~\ref{prop:delayed-projection-error}
bounds both projection errors by $K_{\mathrm p}a^2$
and $K_{\mathrm p}a$, respectively. The identity
\[
 f_{\mathrm p}-\Phi_h=e+(g-\Phi_h)
\]
and its ordinary derivative prove the first two lines
of \eqref{eq:delayed-projection-reference-comparison}.

For the derivative bounds, compare directly with the
original reference phase. By
Proposition~\ref{prop:small-section-reference-slope},
\[
 \frac1{2\sqrt3}
       \leq\frac{\dd}{\dd\rho}\Phi_h(\rho,\zeta,\nu)
       \leq\frac2{\sqrt3}.
\]
Decrease the common scale threshold so that
$Ka\leq1/(4\sqrt3)\leq1$.
Combining these bounds with the derivative error gives
the last line of
\eqref{eq:delayed-projection-reference-comparison}.
\end{proof}

The extra waiting time has therefore preserved the
reference phase through the subsequent projection.
The projected state still has $V<1$ by
\eqref{eq:slow-decay-actual-height}; its further approach
to the growing core starts from this controlled phase.