\subsection{Core coordinates in the angular region}
\label{sec:angular-core-coordinates}

On the families of the preceding subsection, the second wall
remains comparable to the radial scale throughout the initial
decay. It gives a positive quadratic contribution to the
constraint even when the shear vanishes. We use this
contribution to control the height of the projective chart
and to determine the initial core coordinates.

Keep $\theta$ and $\zeta$ fixed, and write
\begin{equation}
 \begin{split}
 T&=\frac q2\log(1/a),\qquad
 x(\rho)=\phi_T\Xi_h(\theta\rho,\rho,\zeta),\\
 x(\rho)&=(p,s,n,w_2,w_3),\qquad
 X(\rho)=(n,s,w_2,w_3),\qquad M_L=8(L+1).
 \end{split}
 \label{eq:angular-cutoff-state}
\end{equation}
All derivatives below are ordinary derivatives in $\rho$,
with $a,h,q,\theta,\zeta$, and hence $T$, held fixed.
Finite-dimensional norms are maximum norms.

\begin{proposition}
\label{prop:angular-cutoff-state}
There is a constant $C\geq32$ such that, for
$0<h\leq1/(4C)$, $L\geq0$ and $q>2$, the following
holds for all sufficiently small $a>0$.
Uniformly for $|\theta|\leq L$, $0\leq\zeta\leq a^2$
and $a\leq\rho\leq2a$, the state in
\eqref{eq:angular-cutoff-state} exists, has ordinary
derivatives, and satisfies
\begin{equation}
 \begin{aligned}
 p&\leq-15/16,&0<n&\leq ha^q,&\|X\|&\leq M_La\leq1/16,\\
 w_2&\geq a/2,&0\leq w_3&\leq2\zeta,
       &n,w_3&\leq w_2/16,\\
 \|X'\|&\leq2(L+1),&|n'|&\leq\frac{2(L+1)n}{a},
       &|w_3'|&\leq\frac{8C(L+1)w_3}{a}.
 \end{aligned}
 \label{eq:angular-cutoff-bounds}
\end{equation}
Its height $A=1+p$ has an ordinary derivative and obeys
\begin{equation}
 \frac{a^2}{16}\leq A\leq10M_L^2a^2,
 \qquad |A'|\leq64M_L(L+1)a.
 \label{eq:angular-height-bounds}
\end{equation}
\end{proposition}

\begin{proof}
Enlarge the common constant in the orbit, variation and
weak-wall estimates of
Sections~\ref{sec:small-section-nonlinear}--\ref{sec:small-section-projection}.
Set $b=M_La$ and decrease the scale threshold so that
\[
 0<a\leq1/64,\qquad b\leq1/16,\qquad
 CbT\leq1/8,\qquad aT\leq1.
\]
These conditions hold for small $a$, since $a\log(1/a)\to0$.
The initial state satisfies
$\|Z_{\theta,\zeta}(\rho)\|\leq2(L+1)a<b/2$,
and $\zeta\leq a/9$.
The orbit and derivative estimates give
\[
 p\leq-15/16,\qquad 0<n\leq he^{-2T}=ha^q,
 \qquad \|(s,w_2,w_3)\|\leq b,
 \qquad \|X'\|\leq2(L+1).
\]
Here the factor $L+1$ bounds the direction $(\theta,1,0)$.
Also $n\leq a^2\leq a\leq b$, so $\|X\|\leq b$.

By Lemma~\ref{lem:small-section-weak-factors},
$w_2=\rho\mu_2$ and $w_3=\zeta\mu_3$, with
$1/2<\mu_i<2$. Thus $w_2\geq a/2$ and
$0\leq w_3\leq2\zeta\leq2a^2$.
Together with $a\leq1/64$, these inequalities give
$n,w_3\leq w_2/16$.
The strong-wall derivative retains its factor $n$:
$|n'|\leq2(L+1)n\leq2(L+1)n/a$.
The initial third wall is constant along the curve, so
\eqref{eq:projection-weak-direction} gives
\[
 |w_3'|\leq8CT(L+1)w_3
             \leq\frac{8C(L+1)w_3}{a}.
\]
This proves~\eqref{eq:angular-cutoff-bounds}.

For the height, use the defect $d=s^2+Q(n,w_2,w_3)$
of~\eqref{eq:projection-height}. The wall ordering implies
\[
 Q(n,w_2,w_3)=(w_2-n-w_3)^2-4nw_3
       \geq\left(\frac{7w_2}{8}\right)^2
                  -4\left(\frac{w_2}{16}\right)^2
       =\frac{3w_2^2}{4}.
\]
Writing $R=M_La$, we therefore have
\[
 \frac{3a^2}{16}\leq d
       \leq s^2+(n+w_2+w_3)^2\leq10R^2\leq\frac12.
\]
The constraint and $p<0$ give
$A=\mathcal H(X)=d/(1+\sqrt{1-d})$.
Its denominator lies between one and two, yielding the
two bounds on $A$ in~\eqref{eq:angular-height-bounds}.
As in Section~\ref{sec:small-section-projection},
$|Dd(X)v|\leq32\|X\|\|v\|$, and hence
\[
 |A'|=\left|\frac{Dd(X)X'}{2\sqrt{1-d}}\right|
      \leq32R\,2(L+1).
\]
The initial chart radicand is strictly positive and the
negative constraint branch persists on an ordinary
neighborhood. Thus $A=\mathcal H(X)$ is a local identity,
which also establishes the ordinary derivative asserted
at the endpoints of the radial interval.
\end{proof}

Define the actual projective state and its core angle by
\begin{equation}
 Y=(M,U,V,W)=\frac{\sqrt3}{A}(s,n,w_2,w_3),
 \qquad z=\operatorname{arsinh}\frac{1-M}{V}.
 \label{eq:angular-projective-state}
\end{equation}
Since $A>0$ and $w_2>0$, these are smooth coordinate
expressions on an ordinary neighborhood of each state.

\begin{corollary}
\label{cor:angular-projective-bounds}
Under the hypotheses of Proposition~\ref{prop:angular-cutoff-state},
and after decreasing the scale threshold, $Y$ and $z$ have
ordinary derivatives and
\begin{equation}
 \begin{aligned}
 2\leq V,\qquad \frac1{20M_L^2a}&\leq V\leq\frac{32M_L}{a},
       &0<U&\leq32ha^{q-2},\\
 0\leq W&\leq\frac{64\zeta}{a^2},
       &0\leq VW&\leq\frac{2048M_L\zeta}{a^3},\\
 |z|&\leq\operatorname{arsinh}(4M_L).
 \end{aligned}
 \label{eq:angular-projective-bounds}
\end{equation}
\end{corollary}

\begin{proof}
Divide~\eqref{eq:angular-cutoff-bounds} by the height
bounds, using $1\leq\sqrt3\leq2$. In particular,
\[
 \frac{a/2}{10M_L^2a^2}\leq V
       \leq\frac{2M_La}{a^2/16}=\frac{32M_L}{a}.
\]
The lower bound is at least two when
$a\leq1/(40M_L^2)$. The estimates for $U,W$ follow
in the same way, and multiplication gives the one for $VW$.

Since $M_La\leq1/16$, the upper height bound also gives
$A\leq10(M_La)^2\leq M_La$. Therefore
\[
 \left|\frac{1-M}{V}\right|
    =\frac{|A/\sqrt3-s|}{w_2}
    \leq\frac{2M_La}{a/2}=4M_L.
\]
Oddness and monotonicity of $\operatorname{arsinh}$ prove
the angle bound.
\end{proof}

Thus the initial decay places these families at a large
core radius and at an angle bounded in terms of $L$ alone.
The weak product $VW$ retains the factor $\zeta$.
We next compare the reciprocal core radius with the
reference phase, keeping its ordinary radial derivative.